Research ArticleStudy on Thermally Induced Crack Propagation Behavior ofFunctionally Graded Materials Using a ModifiedPeridynamic Model

Fei Wang Yursquoe Ma Yanning Guo and Wei Huang

School of Aeronautics Northwestern Polytechnical University Xirsquoan 710072 China

Correspondence should be addressed to Yursquoe Ma mayuenwpueducn

Received 11 August 2019 Accepted 6 November 2019 Published 25 January 2020

Academic Editor Charles C Sorrell

Copyright copy 2020 Fei Wang et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Peridynamic (PD) theory is used to study the thermally induced cracking behavior of functionally graded materials (FGMs) Amodified thermomechanical peridynamic model is developed e thermal crack propagation of a ceramic slab in quenching iscalculated to validate the modified PD model e results predicted by the modified PD model agree with previously publishednumerical and experimental ones Compared with the original PD model the calculation accuracy of the modified PD model forthermal cracking is improved e thermal cracking in FGMs is also simulated e effects of material shape initial temperatureand ceramic fracture toughness on thermal crack propagation behaviors are studied It can be found that the thermal cracks inFGMs are still in periodical and hierarchical forms e metal materials in FGMs can prevent crack initiation and arrest the longcracks e crack number tends to be increased with the increasing initial temperature while the strengthened ceramic fracturetoughness can decrease it

1 Introduction

Metals and traditional layered composite materials processhigh strength and fracture toughness at room temperatureBut they become incapable for application at high tem-peratures Because of the low fracture toughness of ceramicmaterials its employment in heat shielding structure islimited [1] Functionally graded materials (FGMs) are a newgeneration of advanced composite materials which combinethe advantages of ceramic and metal materials Becausemechanical properties of FGMs continuously change withspatial location delamination which induces traditionalcomposites or jointed dissimilar materials to fail does nothappen in it FGMs can also provide excellent heat shieldingcapability and high composite bonding strength withengineered material gradient and orientation [2] is fea-ture makes FGMs preferably be used in high temperatureenvironments [3] e applications of FGMs were sum-marized in [4ndash6] Since FGMs are increasingly applied inaerospace structures and components study on thermally

induced crack propagation behavior of FGMs is necessaryfor assessing and enhancing structural integrity

Various analytical and computational methods havebeen studied to simulate thermal fracture of FGMs eanalytical methods such as elastic fracture theoreticalmethod singular integral method and homogenous mul-tilayered method (HM method) were mainly employed toderive crack-tip fields and cracking angle e materialproperties of FGMs were usually specified as simple func-tions of spatial locations for example the exponentialfunction [7] or the power-law function [8] e crack-tipthermal stress field and the plastic singular field of FGMswere conducted based on the fracture theory [9]e Fouriertransform can be used to reduce the thermal and mechanicalproblems of FGMs into integral equation systems [10] ethermal stress intensity factor (TSIF) in FGMs with a crackwas obtained by solving integral equation systems [11]Petrova and Schmauder [12 13] employed the integralequation method to evaluate crack deflection angle and TSIFof functionally gradedhomogeneous bimaterials with an

HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 1317965 17 pageshttpsdoiorg10115520201317965

interface crack In the HM method FGMs is divided intoseveral layers with constant properties along the gradeddirection Guo et al [14] used the HMmethod to investigateTSIFs of a piecewise exponential FGM plate with differentproperties e explicit monitoring of crack initiation andpropagation in FGMs is still difficult to be simulated due tothe limitation of analytical techniques

In order to model crack initiation and propagation inFGMs with complex material properties and loading somenumerical methods such as finite element method (FEM)and extend finite element method (XFEM) were oftenapplied Fujimoto and Noda [15] used FEM to calculatethermal fracture of the FGM plate with an edge crack andstudied the effects of thermal condition and ceramicfracture toughness on crack path In [16] the crack paths ofFGM plate with two cracks were studied by FEM eremeshing algorithm is needed in FEM to be implementedat every crack increasement [17] e time-consuming isincreased by this algorithm in numerical simulationMoreover a fracture criterion such as the maximumtangential stress criterion is required to predict crackkinking angle for every step [18] e XFEM was imple-mented to predict thermal crack paths of isotropic andorthotropic FGMs in [19] Ivanov et al [20] used XFEM tocalculate crack propagation of FGM strips with an edgecrack under thermal shock Pathak [21] applied XFEM tostudy crack interaction in FGMs under thermal and me-chanical loads e crack discontinuity modelling wasfacilitated by extrinsically enriched approximationBhardwaj et al [22] used extended isogeometric analysis(XIGA) to simulate cracked FGM plates In XIGA XFEMand isogeometric analysis (IGA) were coupled Nonuni-form rational B-splines (NURBS) was used to define ge-ometry and solution Although XFEM permits crack topropagate through elements without remeshing someextra crack modelling techniques such as crack growthcriterion level set method and enrichment functions arerequired during analysing Furthermore coupling of themesh-free method with FEM is also a choice to calculatecrack propagation as well Pathak [23] performed a methodwhich coupled FEM and extrinsically enriched element-free Galerkin (XEFG) to analyse crack growth in FGMsunder cyclic thermal and mechanical loading environ-ments e enrichment functions were still required tomodel crack discontinuity

To avoid the disadvantages of numerical methods based oncontinuum mechanics a nonlocal mechanical frameworkcalled peridynamics was proposed by Silling [24] e basiccontrol equation of peridynamics is reformulated by inte-gration of forces instead of using divergence of stresses Eachmaterial point in peridynamics is connected to others inside aradius regionwhich is called horizone interactions betweenpoints can be assumed as peridynamic bonds Discontinuitiessuch as cracks can be introduced by eliminating all the bondsthat pass though the crack surface A bond breaks when itsstate variables reach a critical value for failure Damage of apoint is characterized by the broken bond number [25] encrack path can be represented by damages of all the points onthe structure domain e extra fracture criterion which

determines crack growth direction and extension increment isnot required Other discontinuity modelling technique toovercome mathematical inconsistences is not needed in per-idynamics as well Cracks will naturally propagate whenperidynamic bonds break Additionally peridynamics also hasadvantages of the meshless method owing to the feature of anintegrodifferential equation

Peridynamics has been used to study thermal crackpropagation behaviors of brittle solids Kilic and Madenci[26] employed the bond-based peridynamics (BB-PD) topredict crack growth patterns in quenched glass plates esimulation results agreed with the experimental observationsin [27] DrsquoAntuono and Morandini [28] gave a weaklycoupled thermomechanical ordinary state-based peridy-namic (OSB-PD) model and simulated the thermal shockcracking behaviors of a brittle slab and a honeycomb plateOterkus andMadenci [29] calculated the crack initiation andpropagation of fuel pellet using oxygen diffusion andthermomechanical model within the framework of BB-PDZhang and Qiao [30] presented an extended OSB-PD modelto predict damage growth of bimaterial structures Wanget al [31] developed a coupled BB-PD thermomechanicalmodel to study thermal cracking behavior of ceramic nuclearpellet under power cycles ey also developed a BB-PDmodel enriched with shear deformation to investigatethermal shock crack branching instability in brittle solids[32] Furthermore Wang et al [33] presented a dual-ho-rizon BB-PD model to simulate thermal fracture Wang andZhou [34] introduced the shear failure into OSB-PD andcalculated the thermal cracking in rocks Giannakeas et al[35] combined the BB-PD and FEM to calculate thermalcracking in ceramics and studied the effects of hot and coldshock in brittle solids However numerical studies onthermal crack propagation behavior of FGMs using theperidynamic model have not been published

In this paper a modified BB-PD thermomechanicalmodel is developed to calculate thermally induced cracking inFGMs e peridynamic material parameters in discretizedform and the energy weight functions of bond constants aregiven e goals of this work are to correct the calculationaccuracy of thermal crack propagation and to study the effectsof material shape initial temperature and ceramic fracturetoughness on crack path in FGMs under thermal shock

2 Peridynamic Theory

21 Modified Heat Conduction Formulas e differencebetween peridynamics and classical local theory is that theperidynamics takes all the material point interactions withina horizon into accounts However the classical local theoryconsiders the interactions that are only adjacent to a pointe horizon means a neighborhood of point x with a limitedradius δ (see Figure 1)

e heat conduction equation based on peridynamics forhomogenous materials can be derived by the EulerndashLagrangeequation or energy conservation law [36]

ρcv_T(x t) 1113946

Hx

μfq xprime x t( 1113857dVxprime + hs(x t) (1)

2 Advances in Materials Science and Engineering

where ρ and cv are respectively the density and the heatspecific of materials hs (x t) is the heat source function perunit volume at point x μ is defined as 0 or 1 fq (xprime x t) iscalled the thermal response function or the ldquokernelrdquo function(see Figure 1) e form of fq (xprime x t) for bond-basedperidynamics is given as follows [37]

fq xprime x t( 1113857 κτ xprime x t( 1113857

|ξ|n (2)

where κ is the microconductivity of materials and it meansthe thermal conductivity in peridynamics ξ is the distancevector of xprime and x ie ξ xprime minus x τ (xprime x t) is the tem-perature difference of xprime and x at time t

τ xprime x t( 1113857 T xprime t( 1113857 minus T(x t) (3)

e integer n is called the ldquoshape factorrdquo It equals to 0 1or 2 More details of ldquoshape factorrdquo can be found in [38]

e classical thermal potential can be given by

ZCM

(x t) 12nablaT(x t) middot knablaT(x t) (4)

where nabla is the gradient operator and k is the thermalconductivity in classical heat transfer theory e peridy-namic thermal potential ZPD (xprime x t) is a function ofmicrothermal potential z (xprime x t) which is given by

ZPD

(x t) 12

1113946Hx

z xprime x t( 1113857dVxprime (5)

e thermal response function can be related to z (xprime x t)by

fq zz

zτ (6)

Hence the microthermal potential can be derived as

z τ2

2|ξ|n (7)

When a linear temperature field is applied κ can beobtained from the equivalent relation of classical and

peridynamic thermal potential After the integrals aresolved the expression of κ for 2D in the original peridy-namic model are given as

κ xprime x( 1113857 2(4 minus n)k

hπδ4minus n for 2D (8)

where h is the thicknessemicroconductivity is obtained under the assumption

that material points are always within a full horizon esurface corrections are needed for the material points locatedclose to surfaces e surface correction methods weresummarized in [39] In these methods the correction factorsare needed to be derived for revising the full horizon materialparameters Here a modified peridynamic model is devel-oped to correct surface effect without calculating extra cor-rection factors e linear temperature field assumption isapplied as well e equivalent relation of classical andperidynamic thermal potential can be also written in dis-cretized form

k κ4

1113944

l

j1

Vxprime

|ξ|nminus 2 for 2D (9)

where l is the material point number within a horizon usthe expression of κ in the modified peridynamic model canbe derived as

κ 4k

1113936lj1Vxprime |ξ|

nminus 2 for 2D (10)

22 Modified 0ermomechanical Formulas e derivationof the peridynamic equation is contained in [40] e basicmotion equation is given by

ρ(x) eurou(x t) 1113946Hx

μf xprime x t( 1113857dVxprime + b(x t) (11)

where f is the pairwise bond force density function as shownin Figure 2 e value of this function is decided by materialproperties and its deformation through the constitutivemodel u is the acceleration of point x e function b is thebody force density

e pairwise force function f can be derived from themicropotential w(xprime x t)

f xprime x t( 1113857 zw

zη (12)

in which

η u xprime t( 1113857 minus u(x t) (13)

and u (x t) is the displacement of x at time of t Amicropotential that leads to a linear thermomechanicalmaterial model can be expressed as [41]

w 12

cs2|ξ| minus cαΔTs|ξ| (14)

where c is a parameter called the micromodulus and α is thecoefficient of thermal expansion ΔT is temperature

xdVx

dVxprime

δ

fq(xprime x)

Hx horizon of x

xprime

Figure 1 ermal response function

Advances in Materials Science and Engineering 3

difference of a bond ΔT [T (xprime t) +T (x t) minus T0 (xprime t) minus T0(xprime t)]2 T0 is initial temperature field s is the bond strainand it is defined by

s |η + ξ| minus |ξ|

|ξ| (15)

Corresponding to the micropotential given in equation(14) the pairwise force density function can be given as

f xprime x t( 1113857 c(s minus αΔT)M if |ξ|lt δ

0 ortherwise1113896 (16)

where M is a unit vector in the direction of deformed bondIt can be written as

M η + ξ

|η + ξ| (17)

e peridynamic strain energy density can be given by

WPD

(x t) 12

1113946Hx

w xprime x t( 1113857dVxprime (18)

e parameter micromodulus c can be derived bymatching the peridynamic strain energy density and thevalue from the classical elasticity theory For plane stress themicromodulus in the original peridynamic model can bederived as

c 12K

hπδ3 (19)

where K is the bulk modulus and it can be given as

K E

2(1 minus ]) (20)

where E is the elastic modulus and ] is Poissonrsquos ratioSimilarly the micromodulus in the modified peridy-

namic model can be derived as

c 8K

1113936lj1|ξ|Vxprime

(21)

e thermomechanical damage can be introduced by thescalar-valued function μ which is defined by

μ xprime x t( 1113857 1 s minus αΔTlt s0

0 ortherwise1113896 (22)

where s0 is the critical relative elongation for bond failureSilling and Askari [25] related s0 with the fracture energyGICfor plane stress and it can be derived as

s0

4πGIC

9Eδ

1113970

(23)

e local damage at a point can be directly defined asthe ratio of broken bond number to total initial bondnumber

φ(x t) 1 minus1113938

Hxμ xprime x t( 1113857dVxprime

1113938HxdVxprime

(24)

23 Microconductivity and Micromodulus of a BondWhen material point is located closely to surface or thematerial is heterogeneous the material parameters (iemicroconductivity or micromodulus) of the PD modelvary with the point location Hence the parameters of abond are determined by the values on two material pointsof this bond Cheng et al [42] applied the average valuefunction to calculate bond constants of heterogeneousmaterials In this work the energy weight functions areused to evaluate bond constants e microconductivityand micromodulus of a bond can be respectivelyexpressed as

κ Gthxκx + Gthxprimeκxprime

c Gexcx + Gexprimecxprime(25)

where Gth and Ge are the weight parameters of micro-conductivity and micromodulus ey are respectivelydefined as

Gthx 12

h(x)

Gthxprime 12

h xprime( 1113857

Gex 12

rx xprime x( 1113857

gx(x)1113890 1113891

2

+ry xprime x( 1113857

gy(x)1113890 1113891

2

⎛⎝ ⎞⎠

minus 12

Gexprime 12

rx xprime x( 1113857

gx xprime( 11138571113890 1113891

2

+ry xprime x( 1113857

gy xprime( 1113857⎡⎣ ⎤⎦

2

⎛⎝ ⎞⎠

minus 12

(26)

where

xdVx

dVxprime

δ

f(xprime x)

Hx horizon of x

xprime

Figure 2 Pairwise bond force density function

4 Advances in Materials Science and Engineering

h(x) ZCM(x)

ZPD(x)

h xprime( 1113857 ZCM xprime( 1113857

ZPD xprime( 1113857

g(x) gx(x) gy(x)1113960 1113961T

WCM

x (x)

WPDx (x)

WCM

y (x)

WPDy (x)

⎡⎣ ⎤⎦

T

g xprime( 1113857 gx xprime( 1113857 gy xprime( 11138571113960 1113961T

WCM

x xprime( 1113857

WPDx xprime( 1113857

WCM

y xprime( 1113857

WPDy xprime( 1113857

⎡⎣ ⎤⎦

T

r xprime minus xxprime minus x

11138681113868111386811138681113868111386811138681113868

rx ry1113960 1113961T

(27)

e variables Z and W are thermal potential and strainenergy density of a material point e superscripts CM andPD mean continuum mechanics and peridynamics evalues of Z andW can be calculated by equations (5) and (18)with the linear field assumption r is the direction vector ofpoint x and xprime

3 Numerical Implementations

e time differential term in heat conduction equation isreplaced by forward difference form e adaptive dynamicrelaxation (ADR) technique is chosen to calculate the qua-sistatic displacements because it is reliable and has a simplealgorithm [41 43] e Gauss midpoint quadrature scheme isapplied to calculate the integrations in heat conduction andmotion equations e volume correction is also consideredin both calculations [36] e peridynamic numerical algo-rithm for thermally induced cracking is shown in Figure 3

In Figure 3 t tn and tc are the calculating time of heatconduction the total calculating time and the time step forcracking respectively int means that the value type is in-teger For example when tc 1 s the cracking procedure isimplemented for every second

4 Validation

41 Ceramic Slab in Quenching To validate the modifiedthermomechanical peridynamic model the Al2O3 ceramicslab in the quenching test is simulated and thermal crackpaths predicted by the modified peridynamic model arecompared with the previously published numerical andexperimental results Due to the symmetry of the ceramicslab only lower left part is calculated In Figure 4 the lengthand the width are 25mm and 5mm respectively ematerial properties of Al2O3 ceramics are shown in Table 1[33] e vertical displacement and velocity are fixed on topsurface e horizontal displacement and velocity are fixedon the right surface Moreover the adiabatic boundaryconditions are assumed on top and right surfaces econvective heat transfer conditions are applied on bottomand left surfaces [33]

hs(x t) hc

Tinfin minus T(x t)( 1113857

Δ (28)

where hc is the convective heat transfer coefficient Tinfin isthe environment temperature and Δ is the cross-sectionalarea

e convective heat transfer coefficient hc under dif-ferent initial temperatures is listed in Table 2 e specifiedvalue of hc is dissimilar due to the differences of numericalmethods and mesh densities e time step for heat con-duction Δt 1e minus 5 s and the time step for crackingtc 2e minus 3 s are specified Plane stress is assumed

Begin

Input model parameters and structure discretization

Generate horizon and initial crack

Setup time parameters and initial conditions

Loop in thermal time and apply thermal boundary conditions

Calculate temperature field

End

t = tn

Yes

NoEnergy balance

Loop in all material points and calculate thermomechanical bond constants according to μ to calculate fi and update μ

Calculate displacement fields

Loop in all material points and calculate thermal bond constants according to μ to calculate fi

Yes

No

ttc = int

Yes

No

Loop in ADR and setup mechanical boundary conditions

Figure 3 Peridynamic numerical algorithm for thermal cracking

25mm

5mm

Figure 4 Al2O3 ceramic slab in quenching

Advances in Materials Science and Engineering 5

According to the results of Sicsic et al [46] the thermalcrack initiation is related to thermal shock and geometry aswell as the characteristic length of materials In [47] thecharacteristic length of Al2O3 ceramics is specified as92e minus 5m Silling [48] suggested that the horizon size inperidynamics is equivalent to the material characteristiclength Hence the point spacing Δx is selected as 25e minus 5mand m 3015 e horizon size can be obtained byδ mΔxasymp76e minus 5m which is less than the characteristiclength of Al2O3 ceramics

42 Results and Comparisons e thermally induced crackpropagation in the ceramic slab at an initial temperature of500degC is calculatede results predicted by the modified PDmodel in present work are compared with the publishednumerical ones (see Figure 5) Because the selected time stepand hc value are different the calculation times of crackinitiating and crack propagating to the same length have aslight difference

In Figure 5 the thermally induced cracks are initiated onthe fixed free surfaces (bottom and left surfaces) of theceramic slab where the thermal load is applied (seeFigure 5(a)) e initiated directions of the cracks areperpendicular to the edges of the slab and the thermal cracksare basically parallel to each other e crack lengths aresimilar and initiated locations are equally spaced Somecracks are arrested over calculation time and some aredriven to propagate toward the inner of the slab (seeFigure 5(b) and Figure 5(c)) Hence the cracks can beclassified into different levels according to their lengthsDuring the crack propagating the oscillations and slightturning of crack paths can be found But the cracks are stillspaced equally and the crack lengths still keep in hierarchicalform (see Figure 5(d))

In the results predicted by the modified PD model thesurface damages on the skin of the slab are occurred (seeFigure 6) e damages are located at the middle parts of thebottom and left surfaces Crack initiations on these parts aredelayed but this effect on crack propagation result can beneglected Furthermore the damages also can be found atthe left-lower corner e reason is that the material issoftened by the surface effect of the PD model

e thermal crack paths at different initial temperaturesare calculatede crack paths predicted by the modified PD

model are compared with the experimental ones from [41]as shown in Figure 7 e periodical and hierarchical formsof the thermal cracks can be captured by the modified PDmodel

It is assumed that a is the depth of crack tip to the slabedge L2 is the width of the ceramic slab which is equal to5mm Dimensionless crack depth a aL2 e dimen-sionless depth can be used to classify the thermal cracks intothree hierarchical levels When agt 06 the crack is defined aslong crack If 03lt alt 06 the thermal crack is defined asmedium-sized crack When alt 03 the crack is called asshort cracks e short medium-sized long and total cracknumbers are counted from the modified PD simulationresults and the experimental tests (see Figure 8) emaximum error number of short cracks predicted by themodified PDmodel is 3 when T0 400degC while it is 1 for thepredicted medium-sized cracks when T0 400degC For thelong cracks it is also 1 at T0 300degC and for total cracknumber it is 2 at T0 300degC and T0 400degC So it can befound that the thermal crack numbers reproduced by themodified PD model are in good agreement with the ex-perimental results

e thermal crack paths at T0 600degC are also calculatedand they are shown in Figure 9(b) e previous publishedexperimental and numerical results are shown in Figure 9 aswell Compared with the original PD model result the skindamages on bottom and left surfaces predicted by themodified PD model are thinner Hence the surface cor-rection of the modified PD model is better than that of theoriginal one e periodical and hierarchical forms of thethermal cracks in the modified PD model result are moreprecise (see Figures 9(b) and 9(c))

e crack numbers at T0 600degC are given in Figure 10e differences of short medium-sized long and total cracknumbers between the modified PD model results and theexperimental ones are respectively 1 0 0 and 1 e cracknumber differences for the original PD model are 19 4 1and 13 e crack numbers predicted by the modified PDmodel agree with the experimental results Compared withthe results of the original PD model the calculations of themodified model are relatively more satisfying e cracknumber differences calculated by FEM are 2 1 0 and 2 Forthe nonlocal approach they are 15 1 1 and 17 e shortcracks are initiated more than that observed from theexperiment

Table 1 Material properties of Al2O3 ceramics

E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) GIC (Jm2)

370 13 3980 31 880 75e minus 6 243

Table 2 Convective heat transfer coefficient hc under different initial temperatures (Wm2middotdegC)

T0 300degC 350degC 400degC 500degC 600degCPresent work 65000 85000 82000 65000 45000Jiang et al [44] 54500 100000 86000 57000 45000Li et al [45] 65000 90000 82000 70000 60000

6 Advances in Materials Science and Engineering

5 Thermal Crack PropagationBehavior in FGMs

51 Sizes andMaterial Properties of FGMSlab e FGM slabis composed by Al metal coating Al2O3 ceramics (seeFigure 11) e geometries and the boundary conditions ofthe FGM slab are as the same as those in Figure 4

e FGM slab has a continuously graded compositionprofile of metal and ceramicse gradient direction is alongthe vertical coordinate e material properties of the FGMslab are assumed as [15]

k k0 1 minus P23

y1113872 1113873k0 + P23y ka1113960 1113961

P13y k0 + 1 minus P13

y1113872 1113873 1 minus P23y1113872 1113873k0 + P23

y ka1113960 1113961

k0 kc 1 +3 km minus kc( 1113857Vm

3kc + km minus kc( 1113857Vc

1113890 1113891

ρ ρ0 1 minus Py1113872 1113873 + ρaPy

ρ0 ρmVm + ρcVc

cv c0ρ0 1 minus Py1113872 1113873 + caρaPy

ρ0 1 minus Py1113872 1113873 + ρaPy

c0 cmρmVm + ccρcVc

ρmVm + ρcVc

] ]0

]0 ]mVm + ]cVc

E E0 1 minus Py1113872 1113873

1 + Py 5 + ]0( 1113857 37 minus 8]0( 11138578 1 + ]0( 1113857 23 + 8]0( 1113857

E0 Ec

Ec + Em minus Ec( 1113857V23m

Ec + Em minus Ec( 1113857 V23m minus Vm( 1113857

1113890 1113891

α α0

α0 1 minus ]c( 1113857αmEmVm + 1 minus ]m( 1113857αcEcVc

1 minus ]c( 1113857EmVm + 1 minus ]m( 1113857EcVc

KIC KICminus mVm + KICminus cVc

(29)

where k ρ and cv are respectively the thermal conductivitythe density and the specific heat of FGMs v E α and KICare Poissonrsquos ratio the elastic modulus the linear thermalexpansion coefficient and the fracture toughness e re-lation of critical energy release rate GIC and fracturetoughness KIC for plane stress are given by

(a)

(b)

(c)

(d)

Figure 5ermally induced crack initiation and propagation (left side results predicted by themodified PDmodel in present work right sidenumerical results in [49] (T0 500degC)) (a) t 0002 s and t 00031 s (b) t 0006 s and t 00075 s (c) t 005 s and t 00501 s (d) t 09 sand t 11552 s

Figure 6 Surface damage on slab skin (result of the modified PDmodel in present work t 0002 s)

Advances in Materials Science and Engineering 7

(a)

(b)

(c)

(d)

Figure 7 ermal crack paths at different initial temperatures (left side results predicted by the modified PD model in present work rightside experimental results in [44]) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC

Modified PDExperiment

0

5

10

15

20

25

Shor

t cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(a)

Modified PDExperiment

0

1

2

3

4

5

6

7

8

Med

ium

-siz

ed cr

ack

num

ber

350degC 400degC 500degC300degCInitial temperature

(b)

Figure 8 Continued

8 Advances in Materials Science and Engineering

GIC K2

ICE

(30)

e subscripts m c and a indicate Al metal Al2O3ceramics and air e material properties of Al2O3 are givenin Table 1 e properties of Al and air are shown in Table 3

Variable Py is the porosity of FGMs along the verticaldirection e volume fractions of metal and ceramics (ieVm and Vc) are defined as

Vm y

b1113874 1113875

My

Vc 1 minus Vm

(31)

e porosity is expressed as

Py Py

y

b1113874 1113875

ny

1 minusy

b1113874 1113875

zy

1113876 1113877 (32)

where

ny + zy1113872 1113873zy

1 minus ny ny + zy1113872 11138731113960 1113961geAy ge 0 (33)

in which Ay ny and zy are the control parameters of FGMporosity Here Ay 0 is specified e values of ny and zy areselected as 1 e width of FGM slab b 5mm My is thematerial shape parameter e volume fraction Vm varieswith vertical coordinate y and My It is shown in Figure 12

52 0e Effect of Material Shape of FGMs e temperaturefields and thermal crack paths of FGM slab in quenching withdifferentmaterial shape parameters are calculatedematerialshape parameter My is specified as shown in Figure 12 eenvironment and initial temperatures are 20degC and 500degC econvective heat transfer coefficient hc is given in Table 2 e

temperature fields and the thermally induced crack paths of theFGM slab at t 1 s are shown in Figures 13 and 14

In Figure 13 the material shape parameter has a slighteffect on temperature field in this case Due to the in-creasing of the material shape parameter the volumefraction of ceramics is enhanced Hence temperature onthe lower side of the FGM slab is decreased with the in-creasing material shape parameter e temperature jumpsare more obvious as a result of thermal crack initiation andpropagation

In Figure 14 there are no thermal crack initiated whenthe material shape parameter My is selected as 001 and 02(Figures 14(a) and 14(b)) While My 1 (Figure 14(c))three cracks are initiated on the right-lower side of theFGM slab which are located close to the fixed boundary AtMy 5 (Figure 14(d)) many cracks are observed on thelower side of the FGM slab e thermal cracks are still inperiodical and hierarchical forms But the cracks arearrested at the horizonal center line and no thermal crackis initiated at the left-upper side AtMy 70 (Figure 14(e))the cracks are not only initiated on the lower side of theFGM slab but also produced on the left side And theirlengths are obviously greater than those when My 5

In Figure 15 the total crack number is increased whenMy is increasing With the increasing My the fracturetoughness of FGMs is softened In Table 4 the crack lengthis increased when My becomes larger e existence ofmetal makes the fracture toughness of FGMs strength-ened It can prevent cracks from initiating and arrestthem

530e Effect of Initial Temperature e temperature fieldsand thermal crack paths with different initial temperaturesare calculated e initial temperature T0 is respectivelyselected as 300degC 350degC 400degC 500degC and 600degC

Modified PDExperiment

0

1

2

3

4

5

6

7

8

9

Long

crac

k nu

mbe

r

350degC 400degC 500degC300degCInitial temperature

(c)

Modified PDExperiment

0

5

10

15

20

25

30

35

Tota

l cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(d)

Figure 8ermal crack numbers at different initial temperatures (a) short crack numbers (b) medium-sized crack numbers (c) long cracknumbers (d) total crack numbers

Advances in Materials Science and Engineering 9

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

where ρ and cv are respectively the density and the heatspecific of materials hs (x t) is the heat source function perunit volume at point x μ is defined as 0 or 1 fq (xprime x t) iscalled the thermal response function or the ldquokernelrdquo function(see Figure 1) e form of fq (xprime x t) for bond-basedperidynamics is given as follows [37]

fq xprime x t( 1113857 κτ xprime x t( 1113857

|ξ|n (2)

where κ is the microconductivity of materials and it meansthe thermal conductivity in peridynamics ξ is the distancevector of xprime and x ie ξ xprime minus x τ (xprime x t) is the tem-perature difference of xprime and x at time t

τ xprime x t( 1113857 T xprime t( 1113857 minus T(x t) (3)

e integer n is called the ldquoshape factorrdquo It equals to 0 1or 2 More details of ldquoshape factorrdquo can be found in [38]

e classical thermal potential can be given by

ZCM

(x t) 12nablaT(x t) middot knablaT(x t) (4)

where nabla is the gradient operator and k is the thermalconductivity in classical heat transfer theory e peridy-namic thermal potential ZPD (xprime x t) is a function ofmicrothermal potential z (xprime x t) which is given by

ZPD

(x t) 12

1113946Hx

z xprime x t( 1113857dVxprime (5)

e thermal response function can be related to z (xprime x t)by

fq zz

zτ (6)

Hence the microthermal potential can be derived as

z τ2

2|ξ|n (7)

When a linear temperature field is applied κ can beobtained from the equivalent relation of classical and

peridynamic thermal potential After the integrals aresolved the expression of κ for 2D in the original peridy-namic model are given as

κ xprime x( 1113857 2(4 minus n)k

hπδ4minus n for 2D (8)

where h is the thicknessemicroconductivity is obtained under the assumption

that material points are always within a full horizon esurface corrections are needed for the material points locatedclose to surfaces e surface correction methods weresummarized in [39] In these methods the correction factorsare needed to be derived for revising the full horizon materialparameters Here a modified peridynamic model is devel-oped to correct surface effect without calculating extra cor-rection factors e linear temperature field assumption isapplied as well e equivalent relation of classical andperidynamic thermal potential can be also written in dis-cretized form

k κ4

1113944

l

j1

Vxprime

|ξ|nminus 2 for 2D (9)

where l is the material point number within a horizon usthe expression of κ in the modified peridynamic model canbe derived as

κ 4k

1113936lj1Vxprime |ξ|

nminus 2 for 2D (10)

22 Modified 0ermomechanical Formulas e derivationof the peridynamic equation is contained in [40] e basicmotion equation is given by

ρ(x) eurou(x t) 1113946Hx

μf xprime x t( 1113857dVxprime + b(x t) (11)

where f is the pairwise bond force density function as shownin Figure 2 e value of this function is decided by materialproperties and its deformation through the constitutivemodel u is the acceleration of point x e function b is thebody force density

e pairwise force function f can be derived from themicropotential w(xprime x t)

f xprime x t( 1113857 zw

zη (12)

in which

η u xprime t( 1113857 minus u(x t) (13)

and u (x t) is the displacement of x at time of t Amicropotential that leads to a linear thermomechanicalmaterial model can be expressed as [41]

w 12

cs2|ξ| minus cαΔTs|ξ| (14)

where c is a parameter called the micromodulus and α is thecoefficient of thermal expansion ΔT is temperature

xdVx

dVxprime

δ

fq(xprime x)

Hx horizon of x

xprime

Figure 1 ermal response function

Advances in Materials Science and Engineering 3

difference of a bond ΔT [T (xprime t) +T (x t) minus T0 (xprime t) minus T0(xprime t)]2 T0 is initial temperature field s is the bond strainand it is defined by

s |η + ξ| minus |ξ|

|ξ| (15)

Corresponding to the micropotential given in equation(14) the pairwise force density function can be given as

f xprime x t( 1113857 c(s minus αΔT)M if |ξ|lt δ

0 ortherwise1113896 (16)

where M is a unit vector in the direction of deformed bondIt can be written as

M η + ξ

|η + ξ| (17)

e peridynamic strain energy density can be given by

WPD

(x t) 12

1113946Hx

w xprime x t( 1113857dVxprime (18)

e parameter micromodulus c can be derived bymatching the peridynamic strain energy density and thevalue from the classical elasticity theory For plane stress themicromodulus in the original peridynamic model can bederived as

c 12K

hπδ3 (19)

where K is the bulk modulus and it can be given as

K E

2(1 minus ]) (20)

where E is the elastic modulus and ] is Poissonrsquos ratioSimilarly the micromodulus in the modified peridy-

namic model can be derived as

c 8K

1113936lj1|ξ|Vxprime

(21)

e thermomechanical damage can be introduced by thescalar-valued function μ which is defined by

μ xprime x t( 1113857 1 s minus αΔTlt s0

0 ortherwise1113896 (22)

where s0 is the critical relative elongation for bond failureSilling and Askari [25] related s0 with the fracture energyGICfor plane stress and it can be derived as

s0

4πGIC

9Eδ

1113970

(23)

e local damage at a point can be directly defined asthe ratio of broken bond number to total initial bondnumber

φ(x t) 1 minus1113938

Hxμ xprime x t( 1113857dVxprime

1113938HxdVxprime

(24)

23 Microconductivity and Micromodulus of a BondWhen material point is located closely to surface or thematerial is heterogeneous the material parameters (iemicroconductivity or micromodulus) of the PD modelvary with the point location Hence the parameters of abond are determined by the values on two material pointsof this bond Cheng et al [42] applied the average valuefunction to calculate bond constants of heterogeneousmaterials In this work the energy weight functions areused to evaluate bond constants e microconductivityand micromodulus of a bond can be respectivelyexpressed as

κ Gthxκx + Gthxprimeκxprime

c Gexcx + Gexprimecxprime(25)

where Gth and Ge are the weight parameters of micro-conductivity and micromodulus ey are respectivelydefined as

Gthx 12

h(x)

Gthxprime 12

h xprime( 1113857

Gex 12

rx xprime x( 1113857

gx(x)1113890 1113891

2

+ry xprime x( 1113857

gy(x)1113890 1113891

2

⎛⎝ ⎞⎠

minus 12

Gexprime 12

rx xprime x( 1113857

gx xprime( 11138571113890 1113891

2

+ry xprime x( 1113857

gy xprime( 1113857⎡⎣ ⎤⎦

2

⎛⎝ ⎞⎠

minus 12

(26)

where

xdVx

dVxprime

δ

f(xprime x)

Hx horizon of x

xprime

Figure 2 Pairwise bond force density function

4 Advances in Materials Science and Engineering

h(x) ZCM(x)

ZPD(x)

h xprime( 1113857 ZCM xprime( 1113857

ZPD xprime( 1113857

g(x) gx(x) gy(x)1113960 1113961T

WCM

x (x)

WPDx (x)

WCM

y (x)

WPDy (x)

⎡⎣ ⎤⎦

T

g xprime( 1113857 gx xprime( 1113857 gy xprime( 11138571113960 1113961T

WCM

x xprime( 1113857

WPDx xprime( 1113857

WCM

y xprime( 1113857

WPDy xprime( 1113857

⎡⎣ ⎤⎦

T

r xprime minus xxprime minus x

11138681113868111386811138681113868111386811138681113868

rx ry1113960 1113961T

(27)

e variables Z and W are thermal potential and strainenergy density of a material point e superscripts CM andPD mean continuum mechanics and peridynamics evalues of Z andW can be calculated by equations (5) and (18)with the linear field assumption r is the direction vector ofpoint x and xprime

3 Numerical Implementations

e time differential term in heat conduction equation isreplaced by forward difference form e adaptive dynamicrelaxation (ADR) technique is chosen to calculate the qua-sistatic displacements because it is reliable and has a simplealgorithm [41 43] e Gauss midpoint quadrature scheme isapplied to calculate the integrations in heat conduction andmotion equations e volume correction is also consideredin both calculations [36] e peridynamic numerical algo-rithm for thermally induced cracking is shown in Figure 3

In Figure 3 t tn and tc are the calculating time of heatconduction the total calculating time and the time step forcracking respectively int means that the value type is in-teger For example when tc 1 s the cracking procedure isimplemented for every second

4 Validation

41 Ceramic Slab in Quenching To validate the modifiedthermomechanical peridynamic model the Al2O3 ceramicslab in the quenching test is simulated and thermal crackpaths predicted by the modified peridynamic model arecompared with the previously published numerical andexperimental results Due to the symmetry of the ceramicslab only lower left part is calculated In Figure 4 the lengthand the width are 25mm and 5mm respectively ematerial properties of Al2O3 ceramics are shown in Table 1[33] e vertical displacement and velocity are fixed on topsurface e horizontal displacement and velocity are fixedon the right surface Moreover the adiabatic boundaryconditions are assumed on top and right surfaces econvective heat transfer conditions are applied on bottomand left surfaces [33]

hs(x t) hc

Tinfin minus T(x t)( 1113857

Δ (28)

where hc is the convective heat transfer coefficient Tinfin isthe environment temperature and Δ is the cross-sectionalarea

e convective heat transfer coefficient hc under dif-ferent initial temperatures is listed in Table 2 e specifiedvalue of hc is dissimilar due to the differences of numericalmethods and mesh densities e time step for heat con-duction Δt 1e minus 5 s and the time step for crackingtc 2e minus 3 s are specified Plane stress is assumed

Begin

Input model parameters and structure discretization

Generate horizon and initial crack

Setup time parameters and initial conditions

Loop in thermal time and apply thermal boundary conditions

Calculate temperature field

End

t = tn

Yes

NoEnergy balance

Loop in all material points and calculate thermomechanical bond constants according to μ to calculate fi and update μ

Calculate displacement fields

Loop in all material points and calculate thermal bond constants according to μ to calculate fi

Yes

No

ttc = int

Yes

No

Loop in ADR and setup mechanical boundary conditions

Figure 3 Peridynamic numerical algorithm for thermal cracking

25mm

5mm

Figure 4 Al2O3 ceramic slab in quenching

Advances in Materials Science and Engineering 5

According to the results of Sicsic et al [46] the thermalcrack initiation is related to thermal shock and geometry aswell as the characteristic length of materials In [47] thecharacteristic length of Al2O3 ceramics is specified as92e minus 5m Silling [48] suggested that the horizon size inperidynamics is equivalent to the material characteristiclength Hence the point spacing Δx is selected as 25e minus 5mand m 3015 e horizon size can be obtained byδ mΔxasymp76e minus 5m which is less than the characteristiclength of Al2O3 ceramics

42 Results and Comparisons e thermally induced crackpropagation in the ceramic slab at an initial temperature of500degC is calculatede results predicted by the modified PDmodel in present work are compared with the publishednumerical ones (see Figure 5) Because the selected time stepand hc value are different the calculation times of crackinitiating and crack propagating to the same length have aslight difference

In Figure 5 the thermally induced cracks are initiated onthe fixed free surfaces (bottom and left surfaces) of theceramic slab where the thermal load is applied (seeFigure 5(a)) e initiated directions of the cracks areperpendicular to the edges of the slab and the thermal cracksare basically parallel to each other e crack lengths aresimilar and initiated locations are equally spaced Somecracks are arrested over calculation time and some aredriven to propagate toward the inner of the slab (seeFigure 5(b) and Figure 5(c)) Hence the cracks can beclassified into different levels according to their lengthsDuring the crack propagating the oscillations and slightturning of crack paths can be found But the cracks are stillspaced equally and the crack lengths still keep in hierarchicalform (see Figure 5(d))

In the results predicted by the modified PD model thesurface damages on the skin of the slab are occurred (seeFigure 6) e damages are located at the middle parts of thebottom and left surfaces Crack initiations on these parts aredelayed but this effect on crack propagation result can beneglected Furthermore the damages also can be found atthe left-lower corner e reason is that the material issoftened by the surface effect of the PD model

e thermal crack paths at different initial temperaturesare calculatede crack paths predicted by the modified PD

model are compared with the experimental ones from [41]as shown in Figure 7 e periodical and hierarchical formsof the thermal cracks can be captured by the modified PDmodel

It is assumed that a is the depth of crack tip to the slabedge L2 is the width of the ceramic slab which is equal to5mm Dimensionless crack depth a aL2 e dimen-sionless depth can be used to classify the thermal cracks intothree hierarchical levels When agt 06 the crack is defined aslong crack If 03lt alt 06 the thermal crack is defined asmedium-sized crack When alt 03 the crack is called asshort cracks e short medium-sized long and total cracknumbers are counted from the modified PD simulationresults and the experimental tests (see Figure 8) emaximum error number of short cracks predicted by themodified PDmodel is 3 when T0 400degC while it is 1 for thepredicted medium-sized cracks when T0 400degC For thelong cracks it is also 1 at T0 300degC and for total cracknumber it is 2 at T0 300degC and T0 400degC So it can befound that the thermal crack numbers reproduced by themodified PD model are in good agreement with the ex-perimental results

e thermal crack paths at T0 600degC are also calculatedand they are shown in Figure 9(b) e previous publishedexperimental and numerical results are shown in Figure 9 aswell Compared with the original PD model result the skindamages on bottom and left surfaces predicted by themodified PD model are thinner Hence the surface cor-rection of the modified PD model is better than that of theoriginal one e periodical and hierarchical forms of thethermal cracks in the modified PD model result are moreprecise (see Figures 9(b) and 9(c))

e crack numbers at T0 600degC are given in Figure 10e differences of short medium-sized long and total cracknumbers between the modified PD model results and theexperimental ones are respectively 1 0 0 and 1 e cracknumber differences for the original PD model are 19 4 1and 13 e crack numbers predicted by the modified PDmodel agree with the experimental results Compared withthe results of the original PD model the calculations of themodified model are relatively more satisfying e cracknumber differences calculated by FEM are 2 1 0 and 2 Forthe nonlocal approach they are 15 1 1 and 17 e shortcracks are initiated more than that observed from theexperiment

Table 1 Material properties of Al2O3 ceramics

E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) GIC (Jm2)

370 13 3980 31 880 75e minus 6 243

Table 2 Convective heat transfer coefficient hc under different initial temperatures (Wm2middotdegC)

T0 300degC 350degC 400degC 500degC 600degCPresent work 65000 85000 82000 65000 45000Jiang et al [44] 54500 100000 86000 57000 45000Li et al [45] 65000 90000 82000 70000 60000

6 Advances in Materials Science and Engineering

5 Thermal Crack PropagationBehavior in FGMs

51 Sizes andMaterial Properties of FGMSlab e FGM slabis composed by Al metal coating Al2O3 ceramics (seeFigure 11) e geometries and the boundary conditions ofthe FGM slab are as the same as those in Figure 4

e FGM slab has a continuously graded compositionprofile of metal and ceramicse gradient direction is alongthe vertical coordinate e material properties of the FGMslab are assumed as [15]

k k0 1 minus P23

y1113872 1113873k0 + P23y ka1113960 1113961

P13y k0 + 1 minus P13

y1113872 1113873 1 minus P23y1113872 1113873k0 + P23

y ka1113960 1113961

k0 kc 1 +3 km minus kc( 1113857Vm

3kc + km minus kc( 1113857Vc

1113890 1113891

ρ ρ0 1 minus Py1113872 1113873 + ρaPy

ρ0 ρmVm + ρcVc

cv c0ρ0 1 minus Py1113872 1113873 + caρaPy

ρ0 1 minus Py1113872 1113873 + ρaPy

c0 cmρmVm + ccρcVc

ρmVm + ρcVc

] ]0

]0 ]mVm + ]cVc

E E0 1 minus Py1113872 1113873

1 + Py 5 + ]0( 1113857 37 minus 8]0( 11138578 1 + ]0( 1113857 23 + 8]0( 1113857

E0 Ec

Ec + Em minus Ec( 1113857V23m

Ec + Em minus Ec( 1113857 V23m minus Vm( 1113857

1113890 1113891

α α0

α0 1 minus ]c( 1113857αmEmVm + 1 minus ]m( 1113857αcEcVc

1 minus ]c( 1113857EmVm + 1 minus ]m( 1113857EcVc

KIC KICminus mVm + KICminus cVc

(29)

where k ρ and cv are respectively the thermal conductivitythe density and the specific heat of FGMs v E α and KICare Poissonrsquos ratio the elastic modulus the linear thermalexpansion coefficient and the fracture toughness e re-lation of critical energy release rate GIC and fracturetoughness KIC for plane stress are given by

(a)

(b)

(c)

(d)

Figure 5ermally induced crack initiation and propagation (left side results predicted by themodified PDmodel in present work right sidenumerical results in [49] (T0 500degC)) (a) t 0002 s and t 00031 s (b) t 0006 s and t 00075 s (c) t 005 s and t 00501 s (d) t 09 sand t 11552 s

Figure 6 Surface damage on slab skin (result of the modified PDmodel in present work t 0002 s)

Advances in Materials Science and Engineering 7

(a)

(b)

(c)

(d)

Figure 7 ermal crack paths at different initial temperatures (left side results predicted by the modified PD model in present work rightside experimental results in [44]) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC

Modified PDExperiment

0

5

10

15

20

25

Shor

t cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(a)

Modified PDExperiment

0

1

2

3

4

5

6

7

8

Med

ium

-siz

ed cr

ack

num

ber

350degC 400degC 500degC300degCInitial temperature

(b)

Figure 8 Continued

8 Advances in Materials Science and Engineering

GIC K2

ICE

(30)

e subscripts m c and a indicate Al metal Al2O3ceramics and air e material properties of Al2O3 are givenin Table 1 e properties of Al and air are shown in Table 3

Variable Py is the porosity of FGMs along the verticaldirection e volume fractions of metal and ceramics (ieVm and Vc) are defined as

Vm y

b1113874 1113875

My

Vc 1 minus Vm

(31)

e porosity is expressed as

Py Py

y

b1113874 1113875

ny

1 minusy

b1113874 1113875

zy

1113876 1113877 (32)

where

ny + zy1113872 1113873zy

1 minus ny ny + zy1113872 11138731113960 1113961geAy ge 0 (33)

in which Ay ny and zy are the control parameters of FGMporosity Here Ay 0 is specified e values of ny and zy areselected as 1 e width of FGM slab b 5mm My is thematerial shape parameter e volume fraction Vm varieswith vertical coordinate y and My It is shown in Figure 12

52 0e Effect of Material Shape of FGMs e temperaturefields and thermal crack paths of FGM slab in quenching withdifferentmaterial shape parameters are calculatedematerialshape parameter My is specified as shown in Figure 12 eenvironment and initial temperatures are 20degC and 500degC econvective heat transfer coefficient hc is given in Table 2 e

temperature fields and the thermally induced crack paths of theFGM slab at t 1 s are shown in Figures 13 and 14

In Figure 13 the material shape parameter has a slighteffect on temperature field in this case Due to the in-creasing of the material shape parameter the volumefraction of ceramics is enhanced Hence temperature onthe lower side of the FGM slab is decreased with the in-creasing material shape parameter e temperature jumpsare more obvious as a result of thermal crack initiation andpropagation

In Figure 14 there are no thermal crack initiated whenthe material shape parameter My is selected as 001 and 02(Figures 14(a) and 14(b)) While My 1 (Figure 14(c))three cracks are initiated on the right-lower side of theFGM slab which are located close to the fixed boundary AtMy 5 (Figure 14(d)) many cracks are observed on thelower side of the FGM slab e thermal cracks are still inperiodical and hierarchical forms But the cracks arearrested at the horizonal center line and no thermal crackis initiated at the left-upper side AtMy 70 (Figure 14(e))the cracks are not only initiated on the lower side of theFGM slab but also produced on the left side And theirlengths are obviously greater than those when My 5

In Figure 15 the total crack number is increased whenMy is increasing With the increasing My the fracturetoughness of FGMs is softened In Table 4 the crack lengthis increased when My becomes larger e existence ofmetal makes the fracture toughness of FGMs strength-ened It can prevent cracks from initiating and arrestthem

530e Effect of Initial Temperature e temperature fieldsand thermal crack paths with different initial temperaturesare calculated e initial temperature T0 is respectivelyselected as 300degC 350degC 400degC 500degC and 600degC

Modified PDExperiment

0

1

2

3

4

5

6

7

8

9

Long

crac

k nu

mbe

r

350degC 400degC 500degC300degCInitial temperature

(c)

Modified PDExperiment

0

5

10

15

20

25

30

35

Tota

l cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(d)

Figure 8ermal crack numbers at different initial temperatures (a) short crack numbers (b) medium-sized crack numbers (c) long cracknumbers (d) total crack numbers

Advances in Materials Science and Engineering 9

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

difference of a bond ΔT [T (xprime t) +T (x t) minus T0 (xprime t) minus T0(xprime t)]2 T0 is initial temperature field s is the bond strainand it is defined by

s |η + ξ| minus |ξ|

|ξ| (15)

Corresponding to the micropotential given in equation(14) the pairwise force density function can be given as

f xprime x t( 1113857 c(s minus αΔT)M if |ξ|lt δ

0 ortherwise1113896 (16)

where M is a unit vector in the direction of deformed bondIt can be written as

M η + ξ

|η + ξ| (17)

e peridynamic strain energy density can be given by

WPD

(x t) 12

1113946Hx

w xprime x t( 1113857dVxprime (18)

e parameter micromodulus c can be derived bymatching the peridynamic strain energy density and thevalue from the classical elasticity theory For plane stress themicromodulus in the original peridynamic model can bederived as

c 12K

hπδ3 (19)

where K is the bulk modulus and it can be given as

K E

2(1 minus ]) (20)

where E is the elastic modulus and ] is Poissonrsquos ratioSimilarly the micromodulus in the modified peridy-

namic model can be derived as

c 8K

1113936lj1|ξ|Vxprime

(21)

e thermomechanical damage can be introduced by thescalar-valued function μ which is defined by

μ xprime x t( 1113857 1 s minus αΔTlt s0

0 ortherwise1113896 (22)

where s0 is the critical relative elongation for bond failureSilling and Askari [25] related s0 with the fracture energyGICfor plane stress and it can be derived as

s0

4πGIC

9Eδ

1113970

(23)

e local damage at a point can be directly defined asthe ratio of broken bond number to total initial bondnumber

φ(x t) 1 minus1113938

Hxμ xprime x t( 1113857dVxprime

1113938HxdVxprime

(24)

23 Microconductivity and Micromodulus of a BondWhen material point is located closely to surface or thematerial is heterogeneous the material parameters (iemicroconductivity or micromodulus) of the PD modelvary with the point location Hence the parameters of abond are determined by the values on two material pointsof this bond Cheng et al [42] applied the average valuefunction to calculate bond constants of heterogeneousmaterials In this work the energy weight functions areused to evaluate bond constants e microconductivityand micromodulus of a bond can be respectivelyexpressed as

κ Gthxκx + Gthxprimeκxprime

c Gexcx + Gexprimecxprime(25)

where Gth and Ge are the weight parameters of micro-conductivity and micromodulus ey are respectivelydefined as

Gthx 12

h(x)

Gthxprime 12

h xprime( 1113857

Gex 12

rx xprime x( 1113857

gx(x)1113890 1113891

2

+ry xprime x( 1113857

gy(x)1113890 1113891

2

⎛⎝ ⎞⎠

minus 12

Gexprime 12

rx xprime x( 1113857

gx xprime( 11138571113890 1113891

2

+ry xprime x( 1113857

gy xprime( 1113857⎡⎣ ⎤⎦

2

⎛⎝ ⎞⎠

minus 12

(26)

where

xdVx

dVxprime

δ

f(xprime x)

Hx horizon of x

xprime

Figure 2 Pairwise bond force density function

4 Advances in Materials Science and Engineering

h(x) ZCM(x)

ZPD(x)

h xprime( 1113857 ZCM xprime( 1113857

ZPD xprime( 1113857

g(x) gx(x) gy(x)1113960 1113961T

WCM

x (x)

WPDx (x)

WCM

y (x)

WPDy (x)

⎡⎣ ⎤⎦

T

g xprime( 1113857 gx xprime( 1113857 gy xprime( 11138571113960 1113961T

WCM

x xprime( 1113857

WPDx xprime( 1113857

WCM

y xprime( 1113857

WPDy xprime( 1113857

⎡⎣ ⎤⎦

T

r xprime minus xxprime minus x

11138681113868111386811138681113868111386811138681113868

rx ry1113960 1113961T

(27)

e variables Z and W are thermal potential and strainenergy density of a material point e superscripts CM andPD mean continuum mechanics and peridynamics evalues of Z andW can be calculated by equations (5) and (18)with the linear field assumption r is the direction vector ofpoint x and xprime

3 Numerical Implementations

e time differential term in heat conduction equation isreplaced by forward difference form e adaptive dynamicrelaxation (ADR) technique is chosen to calculate the qua-sistatic displacements because it is reliable and has a simplealgorithm [41 43] e Gauss midpoint quadrature scheme isapplied to calculate the integrations in heat conduction andmotion equations e volume correction is also consideredin both calculations [36] e peridynamic numerical algo-rithm for thermally induced cracking is shown in Figure 3

In Figure 3 t tn and tc are the calculating time of heatconduction the total calculating time and the time step forcracking respectively int means that the value type is in-teger For example when tc 1 s the cracking procedure isimplemented for every second

4 Validation

41 Ceramic Slab in Quenching To validate the modifiedthermomechanical peridynamic model the Al2O3 ceramicslab in the quenching test is simulated and thermal crackpaths predicted by the modified peridynamic model arecompared with the previously published numerical andexperimental results Due to the symmetry of the ceramicslab only lower left part is calculated In Figure 4 the lengthand the width are 25mm and 5mm respectively ematerial properties of Al2O3 ceramics are shown in Table 1[33] e vertical displacement and velocity are fixed on topsurface e horizontal displacement and velocity are fixedon the right surface Moreover the adiabatic boundaryconditions are assumed on top and right surfaces econvective heat transfer conditions are applied on bottomand left surfaces [33]

hs(x t) hc

Tinfin minus T(x t)( 1113857

Δ (28)

where hc is the convective heat transfer coefficient Tinfin isthe environment temperature and Δ is the cross-sectionalarea

e convective heat transfer coefficient hc under dif-ferent initial temperatures is listed in Table 2 e specifiedvalue of hc is dissimilar due to the differences of numericalmethods and mesh densities e time step for heat con-duction Δt 1e minus 5 s and the time step for crackingtc 2e minus 3 s are specified Plane stress is assumed

Begin

Input model parameters and structure discretization

Generate horizon and initial crack

Setup time parameters and initial conditions

Loop in thermal time and apply thermal boundary conditions

Calculate temperature field

End

t = tn

Yes

NoEnergy balance

Loop in all material points and calculate thermomechanical bond constants according to μ to calculate fi and update μ

Calculate displacement fields

Loop in all material points and calculate thermal bond constants according to μ to calculate fi

Yes

No

ttc = int

Yes

No

Loop in ADR and setup mechanical boundary conditions

Figure 3 Peridynamic numerical algorithm for thermal cracking

25mm

5mm

Figure 4 Al2O3 ceramic slab in quenching

Advances in Materials Science and Engineering 5

According to the results of Sicsic et al [46] the thermalcrack initiation is related to thermal shock and geometry aswell as the characteristic length of materials In [47] thecharacteristic length of Al2O3 ceramics is specified as92e minus 5m Silling [48] suggested that the horizon size inperidynamics is equivalent to the material characteristiclength Hence the point spacing Δx is selected as 25e minus 5mand m 3015 e horizon size can be obtained byδ mΔxasymp76e minus 5m which is less than the characteristiclength of Al2O3 ceramics

42 Results and Comparisons e thermally induced crackpropagation in the ceramic slab at an initial temperature of500degC is calculatede results predicted by the modified PDmodel in present work are compared with the publishednumerical ones (see Figure 5) Because the selected time stepand hc value are different the calculation times of crackinitiating and crack propagating to the same length have aslight difference

In Figure 5 the thermally induced cracks are initiated onthe fixed free surfaces (bottom and left surfaces) of theceramic slab where the thermal load is applied (seeFigure 5(a)) e initiated directions of the cracks areperpendicular to the edges of the slab and the thermal cracksare basically parallel to each other e crack lengths aresimilar and initiated locations are equally spaced Somecracks are arrested over calculation time and some aredriven to propagate toward the inner of the slab (seeFigure 5(b) and Figure 5(c)) Hence the cracks can beclassified into different levels according to their lengthsDuring the crack propagating the oscillations and slightturning of crack paths can be found But the cracks are stillspaced equally and the crack lengths still keep in hierarchicalform (see Figure 5(d))

In the results predicted by the modified PD model thesurface damages on the skin of the slab are occurred (seeFigure 6) e damages are located at the middle parts of thebottom and left surfaces Crack initiations on these parts aredelayed but this effect on crack propagation result can beneglected Furthermore the damages also can be found atthe left-lower corner e reason is that the material issoftened by the surface effect of the PD model

e thermal crack paths at different initial temperaturesare calculatede crack paths predicted by the modified PD

model are compared with the experimental ones from [41]as shown in Figure 7 e periodical and hierarchical formsof the thermal cracks can be captured by the modified PDmodel

It is assumed that a is the depth of crack tip to the slabedge L2 is the width of the ceramic slab which is equal to5mm Dimensionless crack depth a aL2 e dimen-sionless depth can be used to classify the thermal cracks intothree hierarchical levels When agt 06 the crack is defined aslong crack If 03lt alt 06 the thermal crack is defined asmedium-sized crack When alt 03 the crack is called asshort cracks e short medium-sized long and total cracknumbers are counted from the modified PD simulationresults and the experimental tests (see Figure 8) emaximum error number of short cracks predicted by themodified PDmodel is 3 when T0 400degC while it is 1 for thepredicted medium-sized cracks when T0 400degC For thelong cracks it is also 1 at T0 300degC and for total cracknumber it is 2 at T0 300degC and T0 400degC So it can befound that the thermal crack numbers reproduced by themodified PD model are in good agreement with the ex-perimental results

e thermal crack paths at T0 600degC are also calculatedand they are shown in Figure 9(b) e previous publishedexperimental and numerical results are shown in Figure 9 aswell Compared with the original PD model result the skindamages on bottom and left surfaces predicted by themodified PD model are thinner Hence the surface cor-rection of the modified PD model is better than that of theoriginal one e periodical and hierarchical forms of thethermal cracks in the modified PD model result are moreprecise (see Figures 9(b) and 9(c))

e crack numbers at T0 600degC are given in Figure 10e differences of short medium-sized long and total cracknumbers between the modified PD model results and theexperimental ones are respectively 1 0 0 and 1 e cracknumber differences for the original PD model are 19 4 1and 13 e crack numbers predicted by the modified PDmodel agree with the experimental results Compared withthe results of the original PD model the calculations of themodified model are relatively more satisfying e cracknumber differences calculated by FEM are 2 1 0 and 2 Forthe nonlocal approach they are 15 1 1 and 17 e shortcracks are initiated more than that observed from theexperiment

Table 1 Material properties of Al2O3 ceramics

E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) GIC (Jm2)

370 13 3980 31 880 75e minus 6 243

Table 2 Convective heat transfer coefficient hc under different initial temperatures (Wm2middotdegC)

T0 300degC 350degC 400degC 500degC 600degCPresent work 65000 85000 82000 65000 45000Jiang et al [44] 54500 100000 86000 57000 45000Li et al [45] 65000 90000 82000 70000 60000

6 Advances in Materials Science and Engineering

5 Thermal Crack PropagationBehavior in FGMs

51 Sizes andMaterial Properties of FGMSlab e FGM slabis composed by Al metal coating Al2O3 ceramics (seeFigure 11) e geometries and the boundary conditions ofthe FGM slab are as the same as those in Figure 4

e FGM slab has a continuously graded compositionprofile of metal and ceramicse gradient direction is alongthe vertical coordinate e material properties of the FGMslab are assumed as [15]

k k0 1 minus P23

y1113872 1113873k0 + P23y ka1113960 1113961

P13y k0 + 1 minus P13

y1113872 1113873 1 minus P23y1113872 1113873k0 + P23

y ka1113960 1113961

k0 kc 1 +3 km minus kc( 1113857Vm

3kc + km minus kc( 1113857Vc

1113890 1113891

ρ ρ0 1 minus Py1113872 1113873 + ρaPy

ρ0 ρmVm + ρcVc

cv c0ρ0 1 minus Py1113872 1113873 + caρaPy

ρ0 1 minus Py1113872 1113873 + ρaPy

c0 cmρmVm + ccρcVc

ρmVm + ρcVc

] ]0

]0 ]mVm + ]cVc

E E0 1 minus Py1113872 1113873

1 + Py 5 + ]0( 1113857 37 minus 8]0( 11138578 1 + ]0( 1113857 23 + 8]0( 1113857

E0 Ec

Ec + Em minus Ec( 1113857V23m

Ec + Em minus Ec( 1113857 V23m minus Vm( 1113857

1113890 1113891

α α0

α0 1 minus ]c( 1113857αmEmVm + 1 minus ]m( 1113857αcEcVc

1 minus ]c( 1113857EmVm + 1 minus ]m( 1113857EcVc

KIC KICminus mVm + KICminus cVc

(29)

where k ρ and cv are respectively the thermal conductivitythe density and the specific heat of FGMs v E α and KICare Poissonrsquos ratio the elastic modulus the linear thermalexpansion coefficient and the fracture toughness e re-lation of critical energy release rate GIC and fracturetoughness KIC for plane stress are given by

(a)

(b)

(c)

(d)

Figure 5ermally induced crack initiation and propagation (left side results predicted by themodified PDmodel in present work right sidenumerical results in [49] (T0 500degC)) (a) t 0002 s and t 00031 s (b) t 0006 s and t 00075 s (c) t 005 s and t 00501 s (d) t 09 sand t 11552 s

Figure 6 Surface damage on slab skin (result of the modified PDmodel in present work t 0002 s)

Advances in Materials Science and Engineering 7

(a)

(b)

(c)

(d)

Figure 7 ermal crack paths at different initial temperatures (left side results predicted by the modified PD model in present work rightside experimental results in [44]) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC

Modified PDExperiment

0

5

10

15

20

25

Shor

t cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(a)

Modified PDExperiment

0

1

2

3

4

5

6

7

8

Med

ium

-siz

ed cr

ack

num

ber

350degC 400degC 500degC300degCInitial temperature

(b)

Figure 8 Continued

8 Advances in Materials Science and Engineering

GIC K2

ICE

(30)

e subscripts m c and a indicate Al metal Al2O3ceramics and air e material properties of Al2O3 are givenin Table 1 e properties of Al and air are shown in Table 3

Variable Py is the porosity of FGMs along the verticaldirection e volume fractions of metal and ceramics (ieVm and Vc) are defined as

Vm y

b1113874 1113875

My

Vc 1 minus Vm

(31)

e porosity is expressed as

Py Py

y

b1113874 1113875

ny

1 minusy

b1113874 1113875

zy

1113876 1113877 (32)

where

ny + zy1113872 1113873zy

1 minus ny ny + zy1113872 11138731113960 1113961geAy ge 0 (33)

in which Ay ny and zy are the control parameters of FGMporosity Here Ay 0 is specified e values of ny and zy areselected as 1 e width of FGM slab b 5mm My is thematerial shape parameter e volume fraction Vm varieswith vertical coordinate y and My It is shown in Figure 12

52 0e Effect of Material Shape of FGMs e temperaturefields and thermal crack paths of FGM slab in quenching withdifferentmaterial shape parameters are calculatedematerialshape parameter My is specified as shown in Figure 12 eenvironment and initial temperatures are 20degC and 500degC econvective heat transfer coefficient hc is given in Table 2 e

temperature fields and the thermally induced crack paths of theFGM slab at t 1 s are shown in Figures 13 and 14

In Figure 13 the material shape parameter has a slighteffect on temperature field in this case Due to the in-creasing of the material shape parameter the volumefraction of ceramics is enhanced Hence temperature onthe lower side of the FGM slab is decreased with the in-creasing material shape parameter e temperature jumpsare more obvious as a result of thermal crack initiation andpropagation

In Figure 14 there are no thermal crack initiated whenthe material shape parameter My is selected as 001 and 02(Figures 14(a) and 14(b)) While My 1 (Figure 14(c))three cracks are initiated on the right-lower side of theFGM slab which are located close to the fixed boundary AtMy 5 (Figure 14(d)) many cracks are observed on thelower side of the FGM slab e thermal cracks are still inperiodical and hierarchical forms But the cracks arearrested at the horizonal center line and no thermal crackis initiated at the left-upper side AtMy 70 (Figure 14(e))the cracks are not only initiated on the lower side of theFGM slab but also produced on the left side And theirlengths are obviously greater than those when My 5

In Figure 15 the total crack number is increased whenMy is increasing With the increasing My the fracturetoughness of FGMs is softened In Table 4 the crack lengthis increased when My becomes larger e existence ofmetal makes the fracture toughness of FGMs strength-ened It can prevent cracks from initiating and arrestthem

530e Effect of Initial Temperature e temperature fieldsand thermal crack paths with different initial temperaturesare calculated e initial temperature T0 is respectivelyselected as 300degC 350degC 400degC 500degC and 600degC

Modified PDExperiment

0

1

2

3

4

5

6

7

8

9

Long

crac

k nu

mbe

r

350degC 400degC 500degC300degCInitial temperature

(c)

Modified PDExperiment

0

5

10

15

20

25

30

35

Tota

l cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(d)

Figure 8ermal crack numbers at different initial temperatures (a) short crack numbers (b) medium-sized crack numbers (c) long cracknumbers (d) total crack numbers

Advances in Materials Science and Engineering 9

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

h(x) ZCM(x)

ZPD(x)

h xprime( 1113857 ZCM xprime( 1113857

ZPD xprime( 1113857

g(x) gx(x) gy(x)1113960 1113961T

WCM

x (x)

WPDx (x)

WCM

y (x)

WPDy (x)

⎡⎣ ⎤⎦

T

g xprime( 1113857 gx xprime( 1113857 gy xprime( 11138571113960 1113961T

WCM

x xprime( 1113857

WPDx xprime( 1113857

WCM

y xprime( 1113857

WPDy xprime( 1113857

⎡⎣ ⎤⎦

T

r xprime minus xxprime minus x

11138681113868111386811138681113868111386811138681113868

rx ry1113960 1113961T

(27)

e variables Z and W are thermal potential and strainenergy density of a material point e superscripts CM andPD mean continuum mechanics and peridynamics evalues of Z andW can be calculated by equations (5) and (18)with the linear field assumption r is the direction vector ofpoint x and xprime

3 Numerical Implementations

e time differential term in heat conduction equation isreplaced by forward difference form e adaptive dynamicrelaxation (ADR) technique is chosen to calculate the qua-sistatic displacements because it is reliable and has a simplealgorithm [41 43] e Gauss midpoint quadrature scheme isapplied to calculate the integrations in heat conduction andmotion equations e volume correction is also consideredin both calculations [36] e peridynamic numerical algo-rithm for thermally induced cracking is shown in Figure 3

In Figure 3 t tn and tc are the calculating time of heatconduction the total calculating time and the time step forcracking respectively int means that the value type is in-teger For example when tc 1 s the cracking procedure isimplemented for every second

4 Validation

41 Ceramic Slab in Quenching To validate the modifiedthermomechanical peridynamic model the Al2O3 ceramicslab in the quenching test is simulated and thermal crackpaths predicted by the modified peridynamic model arecompared with the previously published numerical andexperimental results Due to the symmetry of the ceramicslab only lower left part is calculated In Figure 4 the lengthand the width are 25mm and 5mm respectively ematerial properties of Al2O3 ceramics are shown in Table 1[33] e vertical displacement and velocity are fixed on topsurface e horizontal displacement and velocity are fixedon the right surface Moreover the adiabatic boundaryconditions are assumed on top and right surfaces econvective heat transfer conditions are applied on bottomand left surfaces [33]

hs(x t) hc

Tinfin minus T(x t)( 1113857

Δ (28)

where hc is the convective heat transfer coefficient Tinfin isthe environment temperature and Δ is the cross-sectionalarea

e convective heat transfer coefficient hc under dif-ferent initial temperatures is listed in Table 2 e specifiedvalue of hc is dissimilar due to the differences of numericalmethods and mesh densities e time step for heat con-duction Δt 1e minus 5 s and the time step for crackingtc 2e minus 3 s are specified Plane stress is assumed

Begin

Input model parameters and structure discretization

Generate horizon and initial crack

Setup time parameters and initial conditions

Loop in thermal time and apply thermal boundary conditions

Calculate temperature field

End

t = tn

Yes

NoEnergy balance

Loop in all material points and calculate thermomechanical bond constants according to μ to calculate fi and update μ

Calculate displacement fields

Loop in all material points and calculate thermal bond constants according to μ to calculate fi

Yes

No

ttc = int

Yes

No

Loop in ADR and setup mechanical boundary conditions

Figure 3 Peridynamic numerical algorithm for thermal cracking

25mm

5mm

Figure 4 Al2O3 ceramic slab in quenching

Advances in Materials Science and Engineering 5

According to the results of Sicsic et al [46] the thermalcrack initiation is related to thermal shock and geometry aswell as the characteristic length of materials In [47] thecharacteristic length of Al2O3 ceramics is specified as92e minus 5m Silling [48] suggested that the horizon size inperidynamics is equivalent to the material characteristiclength Hence the point spacing Δx is selected as 25e minus 5mand m 3015 e horizon size can be obtained byδ mΔxasymp76e minus 5m which is less than the characteristiclength of Al2O3 ceramics

42 Results and Comparisons e thermally induced crackpropagation in the ceramic slab at an initial temperature of500degC is calculatede results predicted by the modified PDmodel in present work are compared with the publishednumerical ones (see Figure 5) Because the selected time stepand hc value are different the calculation times of crackinitiating and crack propagating to the same length have aslight difference

In Figure 5 the thermally induced cracks are initiated onthe fixed free surfaces (bottom and left surfaces) of theceramic slab where the thermal load is applied (seeFigure 5(a)) e initiated directions of the cracks areperpendicular to the edges of the slab and the thermal cracksare basically parallel to each other e crack lengths aresimilar and initiated locations are equally spaced Somecracks are arrested over calculation time and some aredriven to propagate toward the inner of the slab (seeFigure 5(b) and Figure 5(c)) Hence the cracks can beclassified into different levels according to their lengthsDuring the crack propagating the oscillations and slightturning of crack paths can be found But the cracks are stillspaced equally and the crack lengths still keep in hierarchicalform (see Figure 5(d))

In the results predicted by the modified PD model thesurface damages on the skin of the slab are occurred (seeFigure 6) e damages are located at the middle parts of thebottom and left surfaces Crack initiations on these parts aredelayed but this effect on crack propagation result can beneglected Furthermore the damages also can be found atthe left-lower corner e reason is that the material issoftened by the surface effect of the PD model

e thermal crack paths at different initial temperaturesare calculatede crack paths predicted by the modified PD

model are compared with the experimental ones from [41]as shown in Figure 7 e periodical and hierarchical formsof the thermal cracks can be captured by the modified PDmodel

It is assumed that a is the depth of crack tip to the slabedge L2 is the width of the ceramic slab which is equal to5mm Dimensionless crack depth a aL2 e dimen-sionless depth can be used to classify the thermal cracks intothree hierarchical levels When agt 06 the crack is defined aslong crack If 03lt alt 06 the thermal crack is defined asmedium-sized crack When alt 03 the crack is called asshort cracks e short medium-sized long and total cracknumbers are counted from the modified PD simulationresults and the experimental tests (see Figure 8) emaximum error number of short cracks predicted by themodified PDmodel is 3 when T0 400degC while it is 1 for thepredicted medium-sized cracks when T0 400degC For thelong cracks it is also 1 at T0 300degC and for total cracknumber it is 2 at T0 300degC and T0 400degC So it can befound that the thermal crack numbers reproduced by themodified PD model are in good agreement with the ex-perimental results

e thermal crack paths at T0 600degC are also calculatedand they are shown in Figure 9(b) e previous publishedexperimental and numerical results are shown in Figure 9 aswell Compared with the original PD model result the skindamages on bottom and left surfaces predicted by themodified PD model are thinner Hence the surface cor-rection of the modified PD model is better than that of theoriginal one e periodical and hierarchical forms of thethermal cracks in the modified PD model result are moreprecise (see Figures 9(b) and 9(c))

e crack numbers at T0 600degC are given in Figure 10e differences of short medium-sized long and total cracknumbers between the modified PD model results and theexperimental ones are respectively 1 0 0 and 1 e cracknumber differences for the original PD model are 19 4 1and 13 e crack numbers predicted by the modified PDmodel agree with the experimental results Compared withthe results of the original PD model the calculations of themodified model are relatively more satisfying e cracknumber differences calculated by FEM are 2 1 0 and 2 Forthe nonlocal approach they are 15 1 1 and 17 e shortcracks are initiated more than that observed from theexperiment

Table 1 Material properties of Al2O3 ceramics

E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) GIC (Jm2)

370 13 3980 31 880 75e minus 6 243

Table 2 Convective heat transfer coefficient hc under different initial temperatures (Wm2middotdegC)

T0 300degC 350degC 400degC 500degC 600degCPresent work 65000 85000 82000 65000 45000Jiang et al [44] 54500 100000 86000 57000 45000Li et al [45] 65000 90000 82000 70000 60000

6 Advances in Materials Science and Engineering

5 Thermal Crack PropagationBehavior in FGMs

51 Sizes andMaterial Properties of FGMSlab e FGM slabis composed by Al metal coating Al2O3 ceramics (seeFigure 11) e geometries and the boundary conditions ofthe FGM slab are as the same as those in Figure 4

e FGM slab has a continuously graded compositionprofile of metal and ceramicse gradient direction is alongthe vertical coordinate e material properties of the FGMslab are assumed as [15]

k k0 1 minus P23

y1113872 1113873k0 + P23y ka1113960 1113961

P13y k0 + 1 minus P13

y1113872 1113873 1 minus P23y1113872 1113873k0 + P23

y ka1113960 1113961

k0 kc 1 +3 km minus kc( 1113857Vm

3kc + km minus kc( 1113857Vc

1113890 1113891

ρ ρ0 1 minus Py1113872 1113873 + ρaPy

ρ0 ρmVm + ρcVc

cv c0ρ0 1 minus Py1113872 1113873 + caρaPy

ρ0 1 minus Py1113872 1113873 + ρaPy

c0 cmρmVm + ccρcVc

ρmVm + ρcVc

] ]0

]0 ]mVm + ]cVc

E E0 1 minus Py1113872 1113873

1 + Py 5 + ]0( 1113857 37 minus 8]0( 11138578 1 + ]0( 1113857 23 + 8]0( 1113857

E0 Ec

Ec + Em minus Ec( 1113857V23m

Ec + Em minus Ec( 1113857 V23m minus Vm( 1113857

1113890 1113891

α α0

α0 1 minus ]c( 1113857αmEmVm + 1 minus ]m( 1113857αcEcVc

1 minus ]c( 1113857EmVm + 1 minus ]m( 1113857EcVc

KIC KICminus mVm + KICminus cVc

(29)

where k ρ and cv are respectively the thermal conductivitythe density and the specific heat of FGMs v E α and KICare Poissonrsquos ratio the elastic modulus the linear thermalexpansion coefficient and the fracture toughness e re-lation of critical energy release rate GIC and fracturetoughness KIC for plane stress are given by

(a)

(b)

(c)

(d)

Figure 5ermally induced crack initiation and propagation (left side results predicted by themodified PDmodel in present work right sidenumerical results in [49] (T0 500degC)) (a) t 0002 s and t 00031 s (b) t 0006 s and t 00075 s (c) t 005 s and t 00501 s (d) t 09 sand t 11552 s

Figure 6 Surface damage on slab skin (result of the modified PDmodel in present work t 0002 s)

Advances in Materials Science and Engineering 7

(a)

(b)

(c)

(d)

Figure 7 ermal crack paths at different initial temperatures (left side results predicted by the modified PD model in present work rightside experimental results in [44]) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC

Modified PDExperiment

0

5

10

15

20

25

Shor

t cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(a)

Modified PDExperiment

0

1

2

3

4

5

6

7

8

Med

ium

-siz

ed cr

ack

num

ber

350degC 400degC 500degC300degCInitial temperature

(b)

Figure 8 Continued

8 Advances in Materials Science and Engineering

GIC K2

ICE

(30)

e subscripts m c and a indicate Al metal Al2O3ceramics and air e material properties of Al2O3 are givenin Table 1 e properties of Al and air are shown in Table 3

Variable Py is the porosity of FGMs along the verticaldirection e volume fractions of metal and ceramics (ieVm and Vc) are defined as

Vm y

b1113874 1113875

My

Vc 1 minus Vm

(31)

e porosity is expressed as

Py Py

y

b1113874 1113875

ny

1 minusy

b1113874 1113875

zy

1113876 1113877 (32)

where

ny + zy1113872 1113873zy

1 minus ny ny + zy1113872 11138731113960 1113961geAy ge 0 (33)

in which Ay ny and zy are the control parameters of FGMporosity Here Ay 0 is specified e values of ny and zy areselected as 1 e width of FGM slab b 5mm My is thematerial shape parameter e volume fraction Vm varieswith vertical coordinate y and My It is shown in Figure 12

52 0e Effect of Material Shape of FGMs e temperaturefields and thermal crack paths of FGM slab in quenching withdifferentmaterial shape parameters are calculatedematerialshape parameter My is specified as shown in Figure 12 eenvironment and initial temperatures are 20degC and 500degC econvective heat transfer coefficient hc is given in Table 2 e

temperature fields and the thermally induced crack paths of theFGM slab at t 1 s are shown in Figures 13 and 14

In Figure 13 the material shape parameter has a slighteffect on temperature field in this case Due to the in-creasing of the material shape parameter the volumefraction of ceramics is enhanced Hence temperature onthe lower side of the FGM slab is decreased with the in-creasing material shape parameter e temperature jumpsare more obvious as a result of thermal crack initiation andpropagation

In Figure 14 there are no thermal crack initiated whenthe material shape parameter My is selected as 001 and 02(Figures 14(a) and 14(b)) While My 1 (Figure 14(c))three cracks are initiated on the right-lower side of theFGM slab which are located close to the fixed boundary AtMy 5 (Figure 14(d)) many cracks are observed on thelower side of the FGM slab e thermal cracks are still inperiodical and hierarchical forms But the cracks arearrested at the horizonal center line and no thermal crackis initiated at the left-upper side AtMy 70 (Figure 14(e))the cracks are not only initiated on the lower side of theFGM slab but also produced on the left side And theirlengths are obviously greater than those when My 5

In Figure 15 the total crack number is increased whenMy is increasing With the increasing My the fracturetoughness of FGMs is softened In Table 4 the crack lengthis increased when My becomes larger e existence ofmetal makes the fracture toughness of FGMs strength-ened It can prevent cracks from initiating and arrestthem

530e Effect of Initial Temperature e temperature fieldsand thermal crack paths with different initial temperaturesare calculated e initial temperature T0 is respectivelyselected as 300degC 350degC 400degC 500degC and 600degC

Modified PDExperiment

0

1

2

3

4

5

6

7

8

9

Long

crac

k nu

mbe

r

350degC 400degC 500degC300degCInitial temperature

(c)

Modified PDExperiment

0

5

10

15

20

25

30

35

Tota

l cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(d)

Figure 8ermal crack numbers at different initial temperatures (a) short crack numbers (b) medium-sized crack numbers (c) long cracknumbers (d) total crack numbers

Advances in Materials Science and Engineering 9

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

According to the results of Sicsic et al [46] the thermalcrack initiation is related to thermal shock and geometry aswell as the characteristic length of materials In [47] thecharacteristic length of Al2O3 ceramics is specified as92e minus 5m Silling [48] suggested that the horizon size inperidynamics is equivalent to the material characteristiclength Hence the point spacing Δx is selected as 25e minus 5mand m 3015 e horizon size can be obtained byδ mΔxasymp76e minus 5m which is less than the characteristiclength of Al2O3 ceramics

42 Results and Comparisons e thermally induced crackpropagation in the ceramic slab at an initial temperature of500degC is calculatede results predicted by the modified PDmodel in present work are compared with the publishednumerical ones (see Figure 5) Because the selected time stepand hc value are different the calculation times of crackinitiating and crack propagating to the same length have aslight difference

In Figure 5 the thermally induced cracks are initiated onthe fixed free surfaces (bottom and left surfaces) of theceramic slab where the thermal load is applied (seeFigure 5(a)) e initiated directions of the cracks areperpendicular to the edges of the slab and the thermal cracksare basically parallel to each other e crack lengths aresimilar and initiated locations are equally spaced Somecracks are arrested over calculation time and some aredriven to propagate toward the inner of the slab (seeFigure 5(b) and Figure 5(c)) Hence the cracks can beclassified into different levels according to their lengthsDuring the crack propagating the oscillations and slightturning of crack paths can be found But the cracks are stillspaced equally and the crack lengths still keep in hierarchicalform (see Figure 5(d))

In the results predicted by the modified PD model thesurface damages on the skin of the slab are occurred (seeFigure 6) e damages are located at the middle parts of thebottom and left surfaces Crack initiations on these parts aredelayed but this effect on crack propagation result can beneglected Furthermore the damages also can be found atthe left-lower corner e reason is that the material issoftened by the surface effect of the PD model

e thermal crack paths at different initial temperaturesare calculatede crack paths predicted by the modified PD

model are compared with the experimental ones from [41]as shown in Figure 7 e periodical and hierarchical formsof the thermal cracks can be captured by the modified PDmodel

It is assumed that a is the depth of crack tip to the slabedge L2 is the width of the ceramic slab which is equal to5mm Dimensionless crack depth a aL2 e dimen-sionless depth can be used to classify the thermal cracks intothree hierarchical levels When agt 06 the crack is defined aslong crack If 03lt alt 06 the thermal crack is defined asmedium-sized crack When alt 03 the crack is called asshort cracks e short medium-sized long and total cracknumbers are counted from the modified PD simulationresults and the experimental tests (see Figure 8) emaximum error number of short cracks predicted by themodified PDmodel is 3 when T0 400degC while it is 1 for thepredicted medium-sized cracks when T0 400degC For thelong cracks it is also 1 at T0 300degC and for total cracknumber it is 2 at T0 300degC and T0 400degC So it can befound that the thermal crack numbers reproduced by themodified PD model are in good agreement with the ex-perimental results

e thermal crack paths at T0 600degC are also calculatedand they are shown in Figure 9(b) e previous publishedexperimental and numerical results are shown in Figure 9 aswell Compared with the original PD model result the skindamages on bottom and left surfaces predicted by themodified PD model are thinner Hence the surface cor-rection of the modified PD model is better than that of theoriginal one e periodical and hierarchical forms of thethermal cracks in the modified PD model result are moreprecise (see Figures 9(b) and 9(c))

e crack numbers at T0 600degC are given in Figure 10e differences of short medium-sized long and total cracknumbers between the modified PD model results and theexperimental ones are respectively 1 0 0 and 1 e cracknumber differences for the original PD model are 19 4 1and 13 e crack numbers predicted by the modified PDmodel agree with the experimental results Compared withthe results of the original PD model the calculations of themodified model are relatively more satisfying e cracknumber differences calculated by FEM are 2 1 0 and 2 Forthe nonlocal approach they are 15 1 1 and 17 e shortcracks are initiated more than that observed from theexperiment

Table 1 Material properties of Al2O3 ceramics

E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) GIC (Jm2)

370 13 3980 31 880 75e minus 6 243

Table 2 Convective heat transfer coefficient hc under different initial temperatures (Wm2middotdegC)

T0 300degC 350degC 400degC 500degC 600degCPresent work 65000 85000 82000 65000 45000Jiang et al [44] 54500 100000 86000 57000 45000Li et al [45] 65000 90000 82000 70000 60000

6 Advances in Materials Science and Engineering

5 Thermal Crack PropagationBehavior in FGMs

51 Sizes andMaterial Properties of FGMSlab e FGM slabis composed by Al metal coating Al2O3 ceramics (seeFigure 11) e geometries and the boundary conditions ofthe FGM slab are as the same as those in Figure 4

e FGM slab has a continuously graded compositionprofile of metal and ceramicse gradient direction is alongthe vertical coordinate e material properties of the FGMslab are assumed as [15]

k k0 1 minus P23

y1113872 1113873k0 + P23y ka1113960 1113961

P13y k0 + 1 minus P13

y1113872 1113873 1 minus P23y1113872 1113873k0 + P23

y ka1113960 1113961

k0 kc 1 +3 km minus kc( 1113857Vm

3kc + km minus kc( 1113857Vc

1113890 1113891

ρ ρ0 1 minus Py1113872 1113873 + ρaPy

ρ0 ρmVm + ρcVc

cv c0ρ0 1 minus Py1113872 1113873 + caρaPy

ρ0 1 minus Py1113872 1113873 + ρaPy

c0 cmρmVm + ccρcVc

ρmVm + ρcVc

] ]0

]0 ]mVm + ]cVc

E E0 1 minus Py1113872 1113873

1 + Py 5 + ]0( 1113857 37 minus 8]0( 11138578 1 + ]0( 1113857 23 + 8]0( 1113857

E0 Ec

Ec + Em minus Ec( 1113857V23m

Ec + Em minus Ec( 1113857 V23m minus Vm( 1113857

1113890 1113891

α α0

α0 1 minus ]c( 1113857αmEmVm + 1 minus ]m( 1113857αcEcVc

1 minus ]c( 1113857EmVm + 1 minus ]m( 1113857EcVc

KIC KICminus mVm + KICminus cVc

(29)

where k ρ and cv are respectively the thermal conductivitythe density and the specific heat of FGMs v E α and KICare Poissonrsquos ratio the elastic modulus the linear thermalexpansion coefficient and the fracture toughness e re-lation of critical energy release rate GIC and fracturetoughness KIC for plane stress are given by

(a)

(b)

(c)

(d)

Figure 5ermally induced crack initiation and propagation (left side results predicted by themodified PDmodel in present work right sidenumerical results in [49] (T0 500degC)) (a) t 0002 s and t 00031 s (b) t 0006 s and t 00075 s (c) t 005 s and t 00501 s (d) t 09 sand t 11552 s

Figure 6 Surface damage on slab skin (result of the modified PDmodel in present work t 0002 s)

Advances in Materials Science and Engineering 7

(a)

(b)

(c)

(d)

Figure 7 ermal crack paths at different initial temperatures (left side results predicted by the modified PD model in present work rightside experimental results in [44]) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC

Modified PDExperiment

0

5

10

15

20

25

Shor

t cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(a)

Modified PDExperiment

0

1

2

3

4

5

6

7

8

Med

ium

-siz

ed cr

ack

num

ber

350degC 400degC 500degC300degCInitial temperature

(b)

Figure 8 Continued

8 Advances in Materials Science and Engineering

GIC K2

ICE

(30)

e subscripts m c and a indicate Al metal Al2O3ceramics and air e material properties of Al2O3 are givenin Table 1 e properties of Al and air are shown in Table 3

Variable Py is the porosity of FGMs along the verticaldirection e volume fractions of metal and ceramics (ieVm and Vc) are defined as

Vm y

b1113874 1113875

My

Vc 1 minus Vm

(31)

e porosity is expressed as

Py Py

y

b1113874 1113875

ny

1 minusy

b1113874 1113875

zy

1113876 1113877 (32)

where

ny + zy1113872 1113873zy

1 minus ny ny + zy1113872 11138731113960 1113961geAy ge 0 (33)

in which Ay ny and zy are the control parameters of FGMporosity Here Ay 0 is specified e values of ny and zy areselected as 1 e width of FGM slab b 5mm My is thematerial shape parameter e volume fraction Vm varieswith vertical coordinate y and My It is shown in Figure 12

52 0e Effect of Material Shape of FGMs e temperaturefields and thermal crack paths of FGM slab in quenching withdifferentmaterial shape parameters are calculatedematerialshape parameter My is specified as shown in Figure 12 eenvironment and initial temperatures are 20degC and 500degC econvective heat transfer coefficient hc is given in Table 2 e

temperature fields and the thermally induced crack paths of theFGM slab at t 1 s are shown in Figures 13 and 14

In Figure 13 the material shape parameter has a slighteffect on temperature field in this case Due to the in-creasing of the material shape parameter the volumefraction of ceramics is enhanced Hence temperature onthe lower side of the FGM slab is decreased with the in-creasing material shape parameter e temperature jumpsare more obvious as a result of thermal crack initiation andpropagation

In Figure 14 there are no thermal crack initiated whenthe material shape parameter My is selected as 001 and 02(Figures 14(a) and 14(b)) While My 1 (Figure 14(c))three cracks are initiated on the right-lower side of theFGM slab which are located close to the fixed boundary AtMy 5 (Figure 14(d)) many cracks are observed on thelower side of the FGM slab e thermal cracks are still inperiodical and hierarchical forms But the cracks arearrested at the horizonal center line and no thermal crackis initiated at the left-upper side AtMy 70 (Figure 14(e))the cracks are not only initiated on the lower side of theFGM slab but also produced on the left side And theirlengths are obviously greater than those when My 5

In Figure 15 the total crack number is increased whenMy is increasing With the increasing My the fracturetoughness of FGMs is softened In Table 4 the crack lengthis increased when My becomes larger e existence ofmetal makes the fracture toughness of FGMs strength-ened It can prevent cracks from initiating and arrestthem

530e Effect of Initial Temperature e temperature fieldsand thermal crack paths with different initial temperaturesare calculated e initial temperature T0 is respectivelyselected as 300degC 350degC 400degC 500degC and 600degC

Modified PDExperiment

0

1

2

3

4

5

6

7

8

9

Long

crac

k nu

mbe

r

350degC 400degC 500degC300degCInitial temperature

(c)

Modified PDExperiment

0

5

10

15

20

25

30

35

Tota

l cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(d)

Figure 8ermal crack numbers at different initial temperatures (a) short crack numbers (b) medium-sized crack numbers (c) long cracknumbers (d) total crack numbers

Advances in Materials Science and Engineering 9

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

5 Thermal Crack PropagationBehavior in FGMs

51 Sizes andMaterial Properties of FGMSlab e FGM slabis composed by Al metal coating Al2O3 ceramics (seeFigure 11) e geometries and the boundary conditions ofthe FGM slab are as the same as those in Figure 4

e FGM slab has a continuously graded compositionprofile of metal and ceramicse gradient direction is alongthe vertical coordinate e material properties of the FGMslab are assumed as [15]

k k0 1 minus P23

y1113872 1113873k0 + P23y ka1113960 1113961

P13y k0 + 1 minus P13

y1113872 1113873 1 minus P23y1113872 1113873k0 + P23

y ka1113960 1113961

k0 kc 1 +3 km minus kc( 1113857Vm

3kc + km minus kc( 1113857Vc

1113890 1113891

ρ ρ0 1 minus Py1113872 1113873 + ρaPy

ρ0 ρmVm + ρcVc

cv c0ρ0 1 minus Py1113872 1113873 + caρaPy

ρ0 1 minus Py1113872 1113873 + ρaPy

c0 cmρmVm + ccρcVc

ρmVm + ρcVc

] ]0

]0 ]mVm + ]cVc

E E0 1 minus Py1113872 1113873

1 + Py 5 + ]0( 1113857 37 minus 8]0( 11138578 1 + ]0( 1113857 23 + 8]0( 1113857

E0 Ec

Ec + Em minus Ec( 1113857V23m

Ec + Em minus Ec( 1113857 V23m minus Vm( 1113857

1113890 1113891

α α0

α0 1 minus ]c( 1113857αmEmVm + 1 minus ]m( 1113857αcEcVc

1 minus ]c( 1113857EmVm + 1 minus ]m( 1113857EcVc

KIC KICminus mVm + KICminus cVc

(29)

where k ρ and cv are respectively the thermal conductivitythe density and the specific heat of FGMs v E α and KICare Poissonrsquos ratio the elastic modulus the linear thermalexpansion coefficient and the fracture toughness e re-lation of critical energy release rate GIC and fracturetoughness KIC for plane stress are given by

(a)

(b)

(c)

(d)

Figure 5ermally induced crack initiation and propagation (left side results predicted by themodified PDmodel in present work right sidenumerical results in [49] (T0 500degC)) (a) t 0002 s and t 00031 s (b) t 0006 s and t 00075 s (c) t 005 s and t 00501 s (d) t 09 sand t 11552 s

Figure 6 Surface damage on slab skin (result of the modified PDmodel in present work t 0002 s)

Advances in Materials Science and Engineering 7

(a)

(b)

(c)

(d)

Figure 7 ermal crack paths at different initial temperatures (left side results predicted by the modified PD model in present work rightside experimental results in [44]) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC

Modified PDExperiment

0

5

10

15

20

25

Shor

t cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(a)

Modified PDExperiment

0

1

2

3

4

5

6

7

8

Med

ium

-siz

ed cr

ack

num

ber

350degC 400degC 500degC300degCInitial temperature

(b)

Figure 8 Continued

8 Advances in Materials Science and Engineering

GIC K2

ICE

(30)

e subscripts m c and a indicate Al metal Al2O3ceramics and air e material properties of Al2O3 are givenin Table 1 e properties of Al and air are shown in Table 3

Variable Py is the porosity of FGMs along the verticaldirection e volume fractions of metal and ceramics (ieVm and Vc) are defined as

Vm y

b1113874 1113875

My

Vc 1 minus Vm

(31)

e porosity is expressed as

Py Py

y

b1113874 1113875

ny

1 minusy

b1113874 1113875

zy

1113876 1113877 (32)

where

ny + zy1113872 1113873zy

1 minus ny ny + zy1113872 11138731113960 1113961geAy ge 0 (33)

in which Ay ny and zy are the control parameters of FGMporosity Here Ay 0 is specified e values of ny and zy areselected as 1 e width of FGM slab b 5mm My is thematerial shape parameter e volume fraction Vm varieswith vertical coordinate y and My It is shown in Figure 12

52 0e Effect of Material Shape of FGMs e temperaturefields and thermal crack paths of FGM slab in quenching withdifferentmaterial shape parameters are calculatedematerialshape parameter My is specified as shown in Figure 12 eenvironment and initial temperatures are 20degC and 500degC econvective heat transfer coefficient hc is given in Table 2 e

temperature fields and the thermally induced crack paths of theFGM slab at t 1 s are shown in Figures 13 and 14

In Figure 13 the material shape parameter has a slighteffect on temperature field in this case Due to the in-creasing of the material shape parameter the volumefraction of ceramics is enhanced Hence temperature onthe lower side of the FGM slab is decreased with the in-creasing material shape parameter e temperature jumpsare more obvious as a result of thermal crack initiation andpropagation

In Figure 14 there are no thermal crack initiated whenthe material shape parameter My is selected as 001 and 02(Figures 14(a) and 14(b)) While My 1 (Figure 14(c))three cracks are initiated on the right-lower side of theFGM slab which are located close to the fixed boundary AtMy 5 (Figure 14(d)) many cracks are observed on thelower side of the FGM slab e thermal cracks are still inperiodical and hierarchical forms But the cracks arearrested at the horizonal center line and no thermal crackis initiated at the left-upper side AtMy 70 (Figure 14(e))the cracks are not only initiated on the lower side of theFGM slab but also produced on the left side And theirlengths are obviously greater than those when My 5

In Figure 15 the total crack number is increased whenMy is increasing With the increasing My the fracturetoughness of FGMs is softened In Table 4 the crack lengthis increased when My becomes larger e existence ofmetal makes the fracture toughness of FGMs strength-ened It can prevent cracks from initiating and arrestthem

530e Effect of Initial Temperature e temperature fieldsand thermal crack paths with different initial temperaturesare calculated e initial temperature T0 is respectivelyselected as 300degC 350degC 400degC 500degC and 600degC

Modified PDExperiment

0

1

2

3

4

5

6

7

8

9

Long

crac

k nu

mbe

r

350degC 400degC 500degC300degCInitial temperature

(c)

Modified PDExperiment

0

5

10

15

20

25

30

35

Tota

l cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(d)

Figure 8ermal crack numbers at different initial temperatures (a) short crack numbers (b) medium-sized crack numbers (c) long cracknumbers (d) total crack numbers

Advances in Materials Science and Engineering 9

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

(a)

(b)

(c)

(d)

Figure 7 ermal crack paths at different initial temperatures (left side results predicted by the modified PD model in present work rightside experimental results in [44]) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC

Modified PDExperiment

0

5

10

15

20

25

Shor

t cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(a)

Modified PDExperiment

0

1

2

3

4

5

6

7

8

Med

ium

-siz

ed cr

ack

num

ber

350degC 400degC 500degC300degCInitial temperature

(b)

Figure 8 Continued

8 Advances in Materials Science and Engineering

GIC K2

ICE

(30)

e subscripts m c and a indicate Al metal Al2O3ceramics and air e material properties of Al2O3 are givenin Table 1 e properties of Al and air are shown in Table 3

Variable Py is the porosity of FGMs along the verticaldirection e volume fractions of metal and ceramics (ieVm and Vc) are defined as

Vm y

b1113874 1113875

My

Vc 1 minus Vm

(31)

e porosity is expressed as

Py Py

y

b1113874 1113875

ny

1 minusy

b1113874 1113875

zy

1113876 1113877 (32)

where

ny + zy1113872 1113873zy

1 minus ny ny + zy1113872 11138731113960 1113961geAy ge 0 (33)

in which Ay ny and zy are the control parameters of FGMporosity Here Ay 0 is specified e values of ny and zy areselected as 1 e width of FGM slab b 5mm My is thematerial shape parameter e volume fraction Vm varieswith vertical coordinate y and My It is shown in Figure 12

52 0e Effect of Material Shape of FGMs e temperaturefields and thermal crack paths of FGM slab in quenching withdifferentmaterial shape parameters are calculatedematerialshape parameter My is specified as shown in Figure 12 eenvironment and initial temperatures are 20degC and 500degC econvective heat transfer coefficient hc is given in Table 2 e

temperature fields and the thermally induced crack paths of theFGM slab at t 1 s are shown in Figures 13 and 14

In Figure 13 the material shape parameter has a slighteffect on temperature field in this case Due to the in-creasing of the material shape parameter the volumefraction of ceramics is enhanced Hence temperature onthe lower side of the FGM slab is decreased with the in-creasing material shape parameter e temperature jumpsare more obvious as a result of thermal crack initiation andpropagation

In Figure 14 there are no thermal crack initiated whenthe material shape parameter My is selected as 001 and 02(Figures 14(a) and 14(b)) While My 1 (Figure 14(c))three cracks are initiated on the right-lower side of theFGM slab which are located close to the fixed boundary AtMy 5 (Figure 14(d)) many cracks are observed on thelower side of the FGM slab e thermal cracks are still inperiodical and hierarchical forms But the cracks arearrested at the horizonal center line and no thermal crackis initiated at the left-upper side AtMy 70 (Figure 14(e))the cracks are not only initiated on the lower side of theFGM slab but also produced on the left side And theirlengths are obviously greater than those when My 5

In Figure 15 the total crack number is increased whenMy is increasing With the increasing My the fracturetoughness of FGMs is softened In Table 4 the crack lengthis increased when My becomes larger e existence ofmetal makes the fracture toughness of FGMs strength-ened It can prevent cracks from initiating and arrestthem

530e Effect of Initial Temperature e temperature fieldsand thermal crack paths with different initial temperaturesare calculated e initial temperature T0 is respectivelyselected as 300degC 350degC 400degC 500degC and 600degC

Modified PDExperiment

0

1

2

3

4

5

6

7

8

9

Long

crac

k nu

mbe

r

350degC 400degC 500degC300degCInitial temperature

(c)

Modified PDExperiment

0

5

10

15

20

25

30

35

Tota

l cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(d)

Figure 8ermal crack numbers at different initial temperatures (a) short crack numbers (b) medium-sized crack numbers (c) long cracknumbers (d) total crack numbers

Advances in Materials Science and Engineering 9

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

GIC K2

ICE

(30)

e subscripts m c and a indicate Al metal Al2O3ceramics and air e material properties of Al2O3 are givenin Table 1 e properties of Al and air are shown in Table 3

Variable Py is the porosity of FGMs along the verticaldirection e volume fractions of metal and ceramics (ieVm and Vc) are defined as

Vm y

b1113874 1113875

My

Vc 1 minus Vm

(31)

e porosity is expressed as

Py Py

y

b1113874 1113875

ny

1 minusy

b1113874 1113875

zy

1113876 1113877 (32)

where

ny + zy1113872 1113873zy

1 minus ny ny + zy1113872 11138731113960 1113961geAy ge 0 (33)

in which Ay ny and zy are the control parameters of FGMporosity Here Ay 0 is specified e values of ny and zy areselected as 1 e width of FGM slab b 5mm My is thematerial shape parameter e volume fraction Vm varieswith vertical coordinate y and My It is shown in Figure 12

52 0e Effect of Material Shape of FGMs e temperaturefields and thermal crack paths of FGM slab in quenching withdifferentmaterial shape parameters are calculatedematerialshape parameter My is specified as shown in Figure 12 eenvironment and initial temperatures are 20degC and 500degC econvective heat transfer coefficient hc is given in Table 2 e

temperature fields and the thermally induced crack paths of theFGM slab at t 1 s are shown in Figures 13 and 14

In Figure 13 the material shape parameter has a slighteffect on temperature field in this case Due to the in-creasing of the material shape parameter the volumefraction of ceramics is enhanced Hence temperature onthe lower side of the FGM slab is decreased with the in-creasing material shape parameter e temperature jumpsare more obvious as a result of thermal crack initiation andpropagation

In Figure 14 there are no thermal crack initiated whenthe material shape parameter My is selected as 001 and 02(Figures 14(a) and 14(b)) While My 1 (Figure 14(c))three cracks are initiated on the right-lower side of theFGM slab which are located close to the fixed boundary AtMy 5 (Figure 14(d)) many cracks are observed on thelower side of the FGM slab e thermal cracks are still inperiodical and hierarchical forms But the cracks arearrested at the horizonal center line and no thermal crackis initiated at the left-upper side AtMy 70 (Figure 14(e))the cracks are not only initiated on the lower side of theFGM slab but also produced on the left side And theirlengths are obviously greater than those when My 5

In Figure 15 the total crack number is increased whenMy is increasing With the increasing My the fracturetoughness of FGMs is softened In Table 4 the crack lengthis increased when My becomes larger e existence ofmetal makes the fracture toughness of FGMs strength-ened It can prevent cracks from initiating and arrestthem

530e Effect of Initial Temperature e temperature fieldsand thermal crack paths with different initial temperaturesare calculated e initial temperature T0 is respectivelyselected as 300degC 350degC 400degC 500degC and 600degC

Modified PDExperiment

0

1

2

3

4

5

6

7

8

9

Long

crac

k nu

mbe

r

350degC 400degC 500degC300degCInitial temperature

(c)

Modified PDExperiment

0

5

10

15

20

25

30

35

Tota

l cra

ck n

umbe

r

350degC 400degC 500degC300degCInitial temperature

(d)

Figure 8ermal crack numbers at different initial temperatures (a) short crack numbers (b) medium-sized crack numbers (c) long cracknumbers (d) total crack numbers

Advances in Materials Science and Engineering 9

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

(a) (b)

(c) (d)

(e)

Figure 9 Crack paths predicted by different methods (T0 600degC) (a) experimental result [44] (b) result of the modified PD model inpresent work (c) result of the original PD model [31] (d) result of FEM [50] (e) result of nonlocal approach [49]

Experiment

Modified peridynamics

Original peridynamics

Finite element method

Nonlocal approach

0

10

20

30

40

50

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 10 Crack number predicted by different methods (T0 600degC)

25mm

5mm

Al

Al2O3

FGMs

Tinfin

Tinfin

uy = 0 vy = 0

ux = 0vx = 0

T0

x

y

Figure 11 FGM slab in quenching

10 Advances in Materials Science and Engineering

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

Table 3 Material properties of Al and air

Materials E (GPa) υ ρ (kgm3) k (J(mmiddotdegC)) cv (J(kgmiddotdegC)) α (1degC) KIC (MPamiddotm12)

Al 71 13 2700 181 880 85e minus 6 60Air mdash mdash 1204 002625 1005 mdash mdash

My = 001

My = 02

My = 1

My = 5

My = 70

0

02

04

06

08

1

Vol

ume f

ract

ion

()

02 04 06 08 10yb

Figure 12 Volume fraction of metal

4080120160200240280320360400

(a)

4080120160200240280320360400

(b)

4080120160200240280320360400

(c)

4080120160200240280320360400

(d)

Figure 13 Continued

Advances in Materials Science and Engineering 11

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

4080120160200240280320360400

(e)

Figure 13 Temperature fields with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

(a) (b)

(c) (d)

(e)

Figure 14 ermal crack paths with different material shapes (t 10 s) (a) My 001 (b) My 02 (c) My 1 (d) My 5 (e) My 70

My = 001My = 002My = 1

My = 5My = 70

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 15 ermal crack numbers with different material shapes (t 10 s)

12 Advances in Materials Science and Engineering

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

Environment temperature is 20degC andMy 5e predictedresults at t 1 s are shown in Figures 16 and 17

In Figure 16 the frequency of temperature jump isenhanced when initial temperature is increasing It can befound from Figures 17 and 18 that the numbers of short andtotal cracks tend to be increased due to the increasing initial

temperaturee long crack is not emerged in the FGM slaband the maximum crack length is about 05L2 In addition atthe left-lower side of the FGM slab thermal cracks areemerged when initial temperature increases Associated withthe results in Figure 14 the location of thermal crack ini-tiated is also affected by the constrained conditions

240

200

160

120

80

40

(a)

2802402001601208040

(b)

280320

2402001601208040

(c)

280320360400

2402001601208040

(d)

280320360400440480

2402001601208040

(e)

Figure 16 Temperature fields with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

Table 4 Average crack lengths with different material shapes (t 10 s)

My 001 02 1 5 70a (aL2) 0 0 0186 0336 0449

Advances in Materials Science and Engineering 13

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

e average crack lengths with different initial tem-peratures are given in Table 5 e average crack length issmaller when initial temperature is higher owing to theincreasing short crack number

54 0e Effect of Ceramic Fracture Toughness e envi-ronment and the initial temperatures are 20degC and 500degCrespectively andMy 5 e temperature fields and thermal

crack paths with a different ceramic fracture toughness arecalculated e ceramic fracture toughness KIC-c is specifiedas 2 3 4 and 5MPamiddotm12 e results at t 1 s are given inFigures 19 and 20

e fracture toughness has no effect on heat con-duction e differences of temperature fields are causedby crack propagation (see Figure 19) Hence the distincttemperature jumps are produced by a different ceramic

(a) (b)

(c) (d)

(e)

Figure 17ermal crack paths with different initial temperatures (t 10 s) (a) T0 300degC (b) T0 350degC (c) T0 400degC (d) T0 500degC(e) T0 600degC

T0 = 300degCT0 = 350degCT0 = 400degC

T0 = 500degCT0 = 600degC

0

5

10

15

20

25

30

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 18 ermal crack numbers with different initial temperatures (t 10 s)

Table 5 Average crack lengths with different initial temperatures (t 10 s)

T0 (degC) 300 350 400 500 600a (aL2) 0411 0382 0345 0336 0301

14 Advances in Materials Science and Engineering

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

fracture toughness which determines the thermal crackpatterns shown in Figure 20

In Figures 20 and 21 the numbers of medium-sizedand total cracks are decreased with the increasing

ceramic fracture toughness e cracks initiated at theleft-lower side are reduced In Table 6 the averagecrack length is increased as a result of short crackarresting

280320360400

2402001601208040

(a)

280320360400

2402001601208040

(b)

280320360400

2402001601208040

(c)

280320360400

2402001601208040

(d)

Figure 19 Temperature fields with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12(c) KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

(a) (b)

(c) (d)

Figure 20 ermal crack paths with a different ceramic fracture toughness (t 10 s) (a) KIC-c 2MPamiddotm12 (b) KIC-c 3MPamiddotm12 (c)KIC-c 4MPamiddotm12 (d) KIC-c 5MPamiddotm12

Advances in Materials Science and Engineering 15

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

6 Conclusions

(1) A modified bond-based peridynamic thermo-mechanical model is developed to calculate thermalcrack propagation e crack numbers and paths ofceramic slab in quenching predicted by the modifiedPD model agree with the previous numerical andexperimental ones Compared with the originalmodel the calculation accuracy of the modifiedmodel for thermal cracking is improved

(2) e thermal cracks in FGMs are still in periodicaland hierarchical forms emetal materials in FGMscan prevent crack initiation and arrest long crackse crack number and length are decreased whenmaterial shape parameter is increasing

(3) e numbers of short and total cracks tend to beincreased with the increasing initial temperaturewhile the strengthened ceramic fracture toughnesscan decrease the crack numbers

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors are grateful for the financial support from theNational Natural Science Foundation of China (Grant no91860128 and Grant no 11572250)

References

[1] V N Burlayenko H Altenbach T Sadowski andS D Dimitrova ldquoComputational simulations of thermalshock cracking by the virtual crack closure technique infunctionally graded platerdquo Computational Materials Sciencevol 116 pp 11ndash21 2016

[2] A Gupta andM Talha ldquoRecent development in modeling andanalysis of functionally graded materials and structuresrdquoProgress in Aerospace Sciences vol 79 pp 1ndash14 2015

[3] F Bobaru ldquoDesigning optimal volume fractions for func-tionally graded materials with temperature-dependent ma-terial propertiesrdquo Journal of Applied Mechanics vol 74 no 5pp 861ndash874 2007

[4] M Kashtalyan and M Menshykova ldquoEffect of a functionallygraded interlayer on three-dimensional elastic deformation ofcoated plates subjected to transverse loadingrdquo CompositeStructures vol 89 no 2 pp 167ndash176 2009

[5] H-T ai and S-E Kim ldquoA review of theories for themodeling and analysis of functionally graded plates andshellsrdquo Composite Structures vol 128 pp 70ndash86 2015

[6] G Li S Guo J Zhang Y Li and L Han ldquoTransient heatconduction analysis of functionally graded materials by amultiple reciprocity boundary face methodrdquo EngineeringAnalysis with Boundary Elements vol 60 pp 81ndash88 2015

[7] G-Y Huang Y-S Wang and S-W Yu ldquoA new model offunctionally graded coatings with a crack under thermalloadingrdquo Journal of 0ermal Stresses vol 27 no 6pp 491ndash512 2004

[8] L Guo and N Noda ldquoInvestigation methods for thermalshock crack problems of functionally graded materials-part Ianalytical methodrdquo Journal of 0ermal Stresses vol 37 no 3pp 292ndash324 2014

[9] N Noda and Z-H Jin ldquoStress singularity of nonhomoge-neous body with crack in thermal stress fieldsrdquo JSME In-ternational Journal Ser A Mechanics and MaterialEngineering vol 38 no 3 pp 364ndash369 1995

[10] N Noda and J Zhi-He ldquoermal stress intensity factors for acrack in a strip of a functionally gradient materialrdquo Inter-national Journal of Solids and Structures vol 30 no 8pp 1039ndash1056 1993

[11] N Noda and B L Wang ldquoTransient thermoelastic responsesof functionally graded materials containing collinear cracksrdquoEngineering Fracture Mechanics vol 69 no 14ndash16pp 1791ndash1809 2002

[12] V Petrova and S Schmauder ldquoCrack-interface crack inter-actions in functionally gradedhomogeneous compositebimaterials subjected to a heat fluxrdquo Mechanics of CompositeMaterials vol 47 no 1 pp 125ndash136 2011

[13] V Petrova and S Schmauder ldquoMathematical modelling andthermal stress intensity factors evaluation for an interfacecrack in the presence of a system of cracks in functionallygradedhomogeneous bimaterialsrdquo Computational MaterialsScience vol 52 no 1 pp 171ndash177 2012

[14] L Guo N Noda and L Wu ldquoermal fracture model for afunctionally graded plate with a crack normal to the surfacesand arbitrary thermomechanical propertiesrdquo Composite Sci-ence and Technology vol 68 no 3-4 pp 1034ndash1041 2008

KICndashc = 2KICndashc = 3

KICndashc = 4KICndashc = 5

0

5

10

15

20

25

30

35

Crac

k nu

mbe

r

Medium-sized Long TotalShortCrack levels

Figure 21 ermal crack numbers with a different ceramicfracture toughness (t 10 s)

Table 6 Average crack lengths with a different ceramic fracturetoughness (t 10 s)

KIC-c (MPamiddotm12) 2 3 4 5a (aL2) 0303 0336 0421 0430

16 Advances in Materials Science and Engineering

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17

[15] T Fujimoto and N Noda ldquoCrack propagation in a functionallygraded plate under thermal shockrdquo Archive of Applied Me-chanics (Ingenieur Archiv) vol 70 no 6 pp 377ndash386 2000

[16] N Noda M Ishihara and N Yamamoto ldquoTwo-crackpropagation paths in a functionally graded material platesubjected to thermal loadingsrdquo Journal of 0ermal Stressesvol 27 no 6 pp 457ndash469 2004

[17] J-H Kim and G H Paulino ldquoSimulation of crack propagationin functionally graded materials under mixed-mode and non-proportional loadingrdquo International Journal of Mechanics andMaterials in Design vol 1 no 1 pp 63ndash94 2004

[18] A Oral J Lambros and G Anlas ldquoCrack initiation infunctionally graded materials under mixed mode loadingexperiments and simulationsrdquo Journal of Applied Mechanicsvol 75 no 5 Article ID 051110 2008

[19] S S Hosseini H Bayesteh and S Mohammadi ldquoermo-mechanical XFEM crack propagation analysis of functionallygraded materialsrdquo Materials Science and Engineering Avol 561 pp 285ndash302 2013

[20] I V Ivanov T Sadowski and D Pietras ldquoCrack propagationin functionally graded strip under thermal shockrdquo 0e Eu-ropean Physical Journal Special Topics vol 222 no 7pp 1587ndash1595 2013

[21] H Pathak ldquoCrack interaction study in functionally gradedmaterials (FGMs) using XFEM under thermal andmechanicalloading environmentrdquo Mechanics of Advanced Materials andStructures pp 1ndash24 2018

[22] G Bhardwaj I V Singh B K Mishra and T Q Bui ldquoNu-merical simulation of functionally graded cracked plates usingNURBS based XIGA under different loads and boundaryconditionsrdquo Composite Structures vol 126 pp 347ndash359 2015

[23] H Pathak ldquoree-dimensional quasi-static fatigue crackgrowth analysis in functionally graded materials (FGMs)using coupled FE-XEFG approachrdquo 0eoretical and AppliedFracture Mechanics vol 92 pp 59ndash75 2017

[24] S A Silling ldquoReformulation of elasticity theory for discon-tinuities and long-range forcesrdquo Journal of the Mechanics andPhysics of Solids vol 48 no 1 pp 175ndash209 2000

[25] S A Silling and E Askari ldquoA meshfree method based on theperidynamic model of solid mechanicsrdquo Computers andStructures vol 83 no 17-18 pp 1526ndash1535 2005

[26] B Kilic and E Madenci ldquoPrediction of crack paths in aquenched glass plate by using peridynamic theoryrdquo Inter-national Journal of Fracture vol 156 no 2 pp 165ndash177 2009

[27] A Yuse and M Sano ldquoTransition between crack patterns inquenched glass platerdquoNature vol 362 no 6418 pp 329ndash3311993

[28] P DrsquoAntuono andMMorandini ldquoermal shock response viaweakly coupled peridynamic thermo-mechanicsrdquo Interna-tional Journal of Solids and Structures vol 129 pp 74ndash89 2017

[29] S Oterkus and E Madenci ldquoPeridynamic modeling of fuelpellet crackingrdquo Engineering Fracture Mechanics vol 176pp 23ndash37 2017

[30] H Zhang and P Qiao ldquoAn extended state-based peridynamicmodel for damage growth prediction of bimaterial structuresunder thermomechanical loadingrdquo Engineering FractureMechanics vol 189 pp 81ndash97 2018

[31] Y Wang X Zhou and M Kou ldquoPeridynamic investigationon thermal fracturing behavior of ceramic nuclear fuel pelletsunder power cyclesrdquo Ceramics International vol 44 no 10pp 11512ndash11542 2018

[32] Y Wang X Zhou and M Kou ldquoNumerical studies onthermal shock crack branching instability in brittle solidsrdquoEngineering Fracture Mechanics vol 204 pp 157ndash184 2018

[33] Y Wang X Zhou and M Kou ldquoAn improved coupledthermo-mechanic bond-based peridynamic model forcracking behaviors in brittle solids subjected to thermalshocksrdquo European Journal of MechanicsmdashASolids vol 73pp 282ndash305 2019

[34] Y-T Wang and X-P Zhou ldquoPeridynamic simulation ofthermal failure behaviors in rocks subjected to heating fromboreholesrdquo International Journal of Rock Mechanics andMining Sciences vol 117 pp 31ndash48 2019

[35] I N Giannakeas T K Papathanasiou and H Bahai ldquoSim-ulation of thermal shock cracking in ceramics using bond-based peridynamics and FEMrdquo Journal of the European Ce-ramic Society vol 38 no 8 pp 3037ndash3048 2018

[36] F Bobaru and M Duangpanya ldquoA peridynamic formulationfor transient heat conduction in bodies with evolving dis-continuitiesrdquo Journal of Computational Physics vol 231no 7 pp 2764ndash2785 2012

[37] A Agwai A peridynamic approach for coupled fields PhDthesis University of Arizona Tucson AZ USA 2011

[38] Z Chen and F Bobaru ldquoSelecting the kernel in a peridynamicformulation a study for transient heat diffusionrdquo ComputerPhysics Communications vol 197 pp 51ndash60 2015

[39] Q V Le and F Bobaru ldquoSurface corrections for peridynamicmodels in elasticity and fracturerdquo Computational Mechanicsvol 61 no 4 pp 499ndash518 2018

[40] E Madenci and E Oterkus ldquoPeridynamic theory and itsapplicationsrdquo Springer New York NY USA 2014

[41] B Kilic Peridynamic theory for progressive failure predictionin homogeneous and heterogeneous materials PhD thesisUniversity of Arizona Tucson AZ USA 2008

[42] Z Cheng G Zhang Y Wang and F Bobaru ldquoA peridynamicmodel for dynamic fracture in functionally graded materialsrdquoComposite Structures vol 133 pp 529ndash546 2015

[43] B Kilic and E Madenci ldquoAn adaptive dynamic relaxationmethod for quasi-static simulations suing the peridynamictheoryrdquo 0eoretical and Applied Fracture Mechanics vol 53no 3 pp 194ndash204 2010

[44] C P Jiang X F Wu J Li et al ldquoA study of the mechanism offormation and numerical simulations of crack patterns inceramics subjected to thermal shockrdquoActaMaterialia vol 60no 11 pp 4540ndash4550 2012

[45] J Li F Song andC Jiang ldquoAnon-local approach to crack processmodeling in ceramic materials subjected to thermal shockrdquoEngineering Fracture Mechanics vol 133 pp 85ndash98 2015

[46] P Sicsic J-J Marigo and C Maurini ldquoInitiation of a periodicarray of cracks in the thermal shock problem a gradientdamage modellingrdquo Journal of the Mechanics and Physics ofSolids vol 63 pp 256ndash284 2014

[47] D Chu X Li and Z Liu ldquoStudy the dynamic crack path inbrittle material under thermal shock loading by phase fieldmodellingrdquo International Journal of Fracture vol 208 no 1-2pp 115ndash130 2017

[48] S A Silling ldquoOrigin and effect of nonlocality in a compositerdquoJournal of Mechanics of Materials and Structures vol 9 no 2pp 245ndash258 2014

[49] J Li F Song and C Jiang ldquoDirection numerical simulationson crack formation in ceramic materials under thermal shockby using a non-local fracture modelrdquo Journal of the EuropeanCeramic Society vol 33 no 13-14 pp 2677ndash2687 2013

[50] S B Tang H Zhang C A Tang and H Y Liu ldquoNumericalmodel for the cracking behavior of heterogeneous brittlesolids subject to thermal shockrdquo International Journal ofSolids and Structures vol 80 pp 520ndash531 2016

Advances in Materials Science and Engineering 17