Research Article XFEM for Thermal Crack of Massive …downloads.hindawi.com/journals/mpe/2013/343842.pdfResearch Article XFEM for Thermal Crack of Massive Concrete GuoweiLiu,YuHu,QingbinLi,andZhengZuo

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 343842 9 pageshttpdxdoiorg1011552013343842

Research ArticleXFEM for Thermal Crack of Massive Concrete

Guowei Liu Yu Hu Qingbin Li and Zheng Zuo

State Key Laboratory of Hydroscience and Engineering Tsinghua University Beijing 100084 China

Correspondence should be addressed to Qingbin Li qingbinlimailtsinghuaeducn

Received 28 June 2013 Revised 2 September 2013 Accepted 2 September 2013

Academic Editor Song Cen

Copyright copy 2013 Guowei Liu et al This 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

Thermal cracking of massive concrete structures occurs as a result of stresses caused by hydration in real environment conditionsThe extended finite element method that combines thermal fields and creep is used in this study to analyze the thermal crackingof massive concrete structures The temperature field is accurately simulated through an equivalent equation of heat conductionthat considers the effect of a cooling pipe system The time-dependent creep behavior of massive concrete is determined by theviscoelastic constitutive model with Prony series Based on the degree of hydration we consider the main properties related tocracking evolving with time Numerical simulations of a real massive concrete structure are conducted Results show that thedeveloped method is efficient for numerical calculations of thermal cracks on massive concrete Further analyses indicate thata cooling system and appropriate heat preservation measures can efficiently prevent the occurrence of thermal cracks

1 Introduction

In massive hardening concrete the thermal gradients underthe heat of hydration during hardening can cause thermalcracking The thermal cracking of concrete structures is aserious concern in construction Some cracks with wideopenings can particularly result in durability problems

To address this concern various numerical models havebeen proposed to investigate the nature of thermal cracksfor massive concrete by means of finite element calculationsde Borst and van den Boogaard [1] studied deformation andcracking in early-age concrete modeled by finite elementmethod (FEM) Xiang et al [2] conducted thermoelasticanalysis by examining time-dependent evolutions of heatrelease caused by hydration and material properties Mazarsand Bournazel [3] Waller et al [4] and Lackner and Mang[5] performed thermal-elastic (or plastic) analyses of massiveconcrete without considering creep strain de Schutter [6]simulated crack initiation and propagation by employing asimple stress-based cracking criterion and a smeared crackmodel based on the heat of hydration in hardening massiveconcrete elements Benboudjema and Torrenti [7] developeda numerical model to predict early-age cracking of massiveconcrete through Kelvin-Voigt elements and an elastic dam-age model that considered autogenous and thermal shrink-age Briffaut et al [8 9] examined the early-age cracking

of massive concrete structures caused by thermal restrainedshrinkage by using the thermal active restrained shrinkagering test and FE methods Buffo-Lacarriere et al [10] pro-posed FE modeling to predict thermal cracking through theevolution of the damage variable law which considered creepand damage behavior based on nonlinear viscoelasticity

Massive concrete has been widely used in civil engineer-ing such as in the construction of dams buildings andbridges Traditional finite element methods have been widelyapplied in the thermal analysis of massive concrete Howeverconventional FEM is formulated with continuous mediaso additional remeshing is necessary to accurately predictirregular crack propagation [11] The extended finite elementmethod (XFEM) [12ndash15] enhances conventional FEM capa-bilities by excluding the mesh requirement to conform todiscontinuities XFEM has been widely used in numerousfields with discontinuous problems particularly in fracturemechanics because XFEM is an excellent method of address-ing discrete crack propagation in various types of materialsXFEM was applied to thermal problems in [16] and to shearband problems with thermal effects in [17] It was applied byDuflot for the first time in steady-state thermoelastic fracturemechanics [18] Zamani and Eslami [19 20] investigatedthermoelastic fractures by implementing XFEM for dynamicfractures and by improving accuracy through high-ordercrack tip enrichments In these papers the authors employed

2 Mathematical Problems in Engineering

XFEM to characterize both the displacement discontinuityand the temperature discontinuity In addition to XFEM theembedded finite element method (EFEM) [21ndash23] meshfreemethods (MMs) [24ndash27] and boundary elements (BEMs)[28ndash31] are also efficient computationalmethods based on thecontinuum-mechanics for discrete cracks

An alternative pathway to simulate crack initiation andpropagation offers methods based on the discrete-mechanicssuch as the particle simulation method [32ndash37] and theparticle element method [38 39] The methods have beenrecently developed to discretize the continuum into particlesso that crack generation at a contact between two particles isa natural process Zhao et al [40] simulatedmagma intrusionproblems using solid particles for the surrounding rock andfluid particles for the intruded magma respectively Theparticle simulation method can efficiently simulate hydraulicfracturing because the fluid particles are allowed to movethrough the cracks between the solid particles In additionthe particle simulation method has also been used to solve abroad range of scientific and engineering problems [41ndash43]

In accordance with the literature review this paperaims to develop a numerical method based on XFEM toanalyze thermal cracks in massive concrete structures Apractical thermal model that considers an artificial water-cooling system is adopted to simulate the temperature fieldTime-dependent viscoelastic behavior is implemented forcreep As input parameters for XFEM that combines thetemperature field and the viscoelastic constitutive model themain properties develop over time when thermal cracking iscalculatedThe reliability of the developed method is verifiedby a case study on a lift of an arch dam which underwenta cold wave during its construction Subsequently furtheranalysis on thermal cracking is conducted

2 XFEM for Thermal Cracking ofMassive Concrete

21 Briefing on XFEM Compared with the classical FEMXFEM provides significant benefits in modeling crack prop-agation The crack geometry in XFEM does not need tobe aligned with the element edges XFEM is based on thepartition of unity concept and introduces additional degreesof freedom which are tied to the nodes of the elementsintersected by the crack [12 13]

XFEM displacement approximation around the crackuℎ(x) is enriched with the step function to model thecrack surface and with asymptotic displacement fields tomodel crack tips The displacement approximation takes thefollowing form [13]

uℎ (x) = sum119899isin119873119899

119873119899 (x)u119894 + sum119895isin119873cr

119873119895 (x) ℎ (x) a119895

+ sum119896isin119873tip

119873119896(

4

sum119897=1

120595119897 (x) b119896119897)

(1)

where the first term is the classical FE approximation 119873119899is the FE shape function and u119894 is the vector of the regular

degrees of nodal freedom The other terms are the enrichedterms where 119873cr and 119873tip stand for the set of nodes thathave crack faces and crack tips in their support domainrespectively a119895 and b119896119897 denote the vectors of additionaldegrees of nodal freedom for modeling crack faces and cracktips respectively

Step function ℎ(x) simulates the displacement at the cracklocation with the help of the signed distance function 120601(x) tothe crack which is also called the normal level set [13 15]

ℎ (x) = sign (120601 (x)) (2)

The four following branch functions which define thediscontinuity near the crack tip enrich the nodes with a dis-tance to the front inferior to a prescribed enrichment radius

120595 (x)

= [radic119903 sin 120579

2radic119903 cos 120579

2radic119903 sin 120579

2cos 120579

2radic119903 cos 120579

2cos 120579]

(3)

where r and 120579 are polar coordinates in the local crack-tipcoordinate system

Duflot [18] applied the same procedure to temperatureenrichment The step function ℎ(x) can account for thetemperature jump The leading term of the asymptotic fieldfor the temperature of an adiabatic crack can be written as

119879 = minus119870119879

120582radic2119903

120587sin(120579

2) (4)

where coefficient119870119879 provides the strength of flux singularityat the crack tipThe temperature field is discretized similar tothe displacement field but with only the second branch func-tion (4) which is the only discontinuous branch functionTheapproximation of the temperature field 119879ℎ(x) is then writtenas

119879ℎ(x) = sum

119899isin119873

119873119899 (x) 119879119894 + sum119895=119873cr

119873119895 (x) ℎ (x) 119888119895


1198731198961205952 (x) 119889119896(5)

As in standard FEM performing numerical integrationover the element domain is necessary to compute the elementstiffness matrix The subdomain method [44] in which thecrack is one of the subdomain boundaries to carry out thenumerical integration is adopted for the elements that are cutby the crack

22 Equivalent Equation of Heat Conduction An isotropichomogenous domain Ω bounded by boundary Γ is consid-ered The thermal properties of this domain are independentof temperature so the equation of thermal conduction is

q = minus120582nabla119879

minusnablaq + 120588119888 = (6)

Mathematical Problems in Engineering 3

where q is the heat flux 120582 is the thermal conductivity119879 is thetemperature 120588 is the density 119888 is the specific heat and isthe heat source

For massive concrete structures the hydration heat ofconcrete which varies with age is the main heat sourceTemperature control is one of the critical problems in theconstruction of massive concrete and embedding water-cooling pipes in massive concrete is one of the key methodsto address this problem Zhu [45] suggested that heat removalthrough a cooling system can be regarded as a negative heatsource Therefore the heat source consists of the followingtwo parts in consideration of the cooling pipe system

119876 = 1205881198881205851015840(119905 +

Δ119905

2) + 120588119888 [119879 (119905) minus 119879119908] 120593

1015840(119905 +

Δ119905

2) (7)

In the equation the first part is the heat source caused byhydration and the second part is the ldquocool sourcerdquo causedby the water cooling system 120585(119905) is the adiabatic temperaturerise of concrete at time 119905 Δ119905 is the increment step 119879119908 isthe temperature of the cooling water inlet and 120593(119905) is thecooling function illustrated by Yang et al [46] Yang et al [46]emphasized that although the temperature gradient near thecooling pipes cannot be obtained uniformly dispersing heatthroughout the solution domain is feasible

For the adiabatic temperature rise of concrete Zhu [47]proposed a compound exponential formula that is convenientfor mathematical operation and is in accordance with experi-mental resultsThehydrationmodel can bewritten as follows

120585 (119905) =

119899

sum119894=1

120585119894 (1 minus 119890minus119898119894119905) (8)

where 120585119894 and 119898119894 are the parameters that can be obtainedthrough the optimization method or the trial method whosedetails can be found in the literature [47]

23 Thermal Boundary Conditions When the concrete is incontact with air we assume that the heat flux on the concretesurface is proportional to the difference between concretesurface temperature 119879surf and air temperature 119879air Thereforethe thermal boundary conditions at the concrete surface canbe written as

minus120582120597119879

120597119899= ℎ119879 (119879surf minus 119879air) (9)

where ℎ119879 is the surface heat convection coefficient and 119899

represents the normal direction of outer surfaces

24 Viscoelastic Creep Model Early-age concrete exhibitshigh creep which reduces the stresses significantly morethan any other influencing parameter as well as assistingin mitigating thermal cracking by reducing thermal stressBazant [48] briefly summarized the method of calculatingconcrete creep

The mechanism of concrete creep is not entirely clearbut the viscoelastic constitutive model composed of springand dashpots can describe its performance In this studycreep is included in time-dependent viscoelastic behavior

The constitutive relationship of viscoelastic materials can berepresented in an integral form as follows [49]

120590119894119895 (119905) = 2int119905

0

119866(119905 minus 1199051015840)120597120576

dev119894119895

1205971199051015840d1199051015840

+ 3int119905

0

119870(119905 minus 1199051015840)120597120576119896119896 (119905

1015840)

1205971199051015840d1199051015840

(10)

where 119866 and 119870 are the shear and bulk moduli respectivelywhich are functions of reduced time 119905 Superscript dotsdenote differentiation with respect to time 1199051015840 and 120576dev

119894119895and

120576119896119896 are the mechanical deviatory and volumetric strainsrespectively The bulk (K) and shear (G) moduli can bedefined individually through Prony series representationwhich is expressed as

119866 (119905) = 1198660 [1 minus

119899119866

sum119894=1

119892119894 (1 minus 119890minus119905120591119866

119894 )]

119870 (119905) = 1198700 [1 minus

119899119870

sum119894=1

119896119894 (1 minus 119890minus119905120591119870

119894 )]

(11)

where 119892119894 120591119866

119894 119896119894 and 120591119870

119894are material constants Variable

numbers of Prony series parameters end in 119899119870 and 119899119866 1198660 =119866(0) and 1198700 = 119870(0) are the instantaneous relaxation moduliobtained by the instantaneous Youngrsquos moduli as follows

1198660 =1198640

2 sdot (1 + 120592)

1198700 =1198640

3 sdot (1 minus 2120592)

(12)

where 120592 is the Poissonrsquos ratioTo obtain the relaxation modulus the normalized shear

and bulk moduli can be defined as functions of the creepcoefficient which is expressed as follows [50]

119866 (119905)

1198660=

1

(1 + 120593 (119905 1199050))

119870 (119905)

1198700=

1

(1 + 120593 (119905 1199050))

(13)

where 120593(119905 1199050) is the creep coefficient which can be calculatedas follows [51]

120593 (119905 1199050) = 120593(119905 1199050)stan sdot 120573 (1199050) 120573 (ℎ0) 120573 (119891119888119896) sdot 120573119879120573119891120573120572 (14)

with

120593(119905 1199050)stan = 37419(1 minus (11520 minus 119905

11520)21528

)

0263

(15)

where 120573(1199050) 120573(ℎ0) 120573(119891119888119896) 120573119879 120573119891 and 120573120572 represent a seriesof correction coefficients that were described in detail in theliterature [51] Through nonlinear least-squares fit the Pronyseries parameters in (11) are calculated


The developed XFEM method that combines the equiv-alent equation of heat conduction and the viscoelastic creepmodel as previously described can be used in the numericalcalculation of the thermal cracking of massive concreteTherefore considering hydration and creep we can simulatea few discrete thermal cracks on amassive concrete structure

3 Fracture Criteria and Evolution ofMaterial Properties

The temperature gradient in massive concrete leads to tensilestress which in turn causes crack initiation and evolution Inmodeling thermal cracking evolution an appropriate law oncrack initiation and damage evolution should be definedThemechanical properties of concrete are essential to predict thedevelopment of thermal stress in massive concrete elements

31 Criterion for Initiation As a possible first approximationthe maximum principal stress criterion can be applied tosimulate the onset of thermal cracks This criterion can bedefined as follows

119891 = ⟨120590max⟩

1205900max (16)

where 119891 is the maximum principle stress ratio and 120590max and1205900max denote the maximum principle stress and maximumallowable principle stress (tensile strength) respectively Thesymbol ⟨⟩ represents the so-called Macaulay brackets whichenable ⟨120590max⟩ to have an alternative value of 0 (when 120590max lt0) and 120590max (when 120590max ge 0) The crack initiates when 119891 = 1

32 Criterion for Evolution In this study the analysis of crackgrowth is based on the approach of linear elastic fracturemechanics and is conducted with the framework of theextended-FEM method The power law criterion proposedby Wu and Reuter [52] can be extended to 3D problems asfollows [53]

(119866I119866I119888

)

120572

+ (119866II119866II119888

)

120573

+ (119866III119866III119888

)

120594

= 1 (17)

where 120572 120573 and 120594 are parameters with values of 1 to 2respectively 119866I 119866II and 119866III denote the strain energy releaserate parameters of mode I mode II and mode III compo-nents respectively 119866I119888 119866II119888 and 119866III119888 are the critical valuesof the strain energy release rate (the fracture toughness) forthe three fracture modes The law forms an energy envelopesurface and if the energy release rate exceeds this surface thecrack propagates

33 Evolution of Material Properties During the construc-tion period in which concrete changes from an almost liquidstate to a solid state most of the mechanical propertiesrapidly vary in relation to the age of concrete Amongthese properties modulus of elasticity tensile strength andfracture toughness are the key parameters used in analyzingthermal cracking

Degree of hydration is a fundamental parameter thatdescribes concrete behavior during hardening de Schutterand Taerwe [54 55] proposed a degree-of-hydration-baseddescription of these material properties as follows

119864 (120574)

119864 (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198860

1205900max (120574)

1205900max (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198870

119866119888

119866119888 (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198880

(18)

where 119864(120574) 1205900max(120574) and 119866119888(120574) are the Youngrsquos modulustensile strength and critical values of strain energy releaserate at degree of reaction 120574 respectively and 119864(120574 = 1)1205900max(120574 = 1) and 119866119888(120574 = 1) correspond to these values when120574 = 1 1205740 a0 b0 and c0 are parameters that depend on theconcrete composition

A practical estimation of the degree of hydration ateach moment 119905 can be expressed as the ratio between theaccumulated heat of hydration at time 119905 and that at the endof reaction which can be calculated through the adiabatictemperature rise

120574 (119905) =int119905

0120585 (120591) 119889120591

int119905end

0120585 (120591) 119889 (120591)

(19)

where 120574(119905) is the degree of hydration at time t and 119905end is thetime at end of hydration reaction

The coefficient of thermal expansion of concrete isanother key parameter for thermal cracking analysis Itsmeasured values tend to be stable within two days [56] soa constant for it is adopted in this paper

4 Numerical Examples

41 Introduction A 286m high parabolic double-curvaturearch dam under construction is located in Southwest Chinaat an altitude of 610m above sea level (level of crest) Thedam consists of 31monoliths each containing a series of liftsThe 10th lift of the 13th monolith was poured on September5 2009 and underwent a cold wave before the next liftwas poured This lift was selected for the thermal crackinganalysis The 9th lift was also used in our calculation as thelower boundary of the 10th lift

Figure 1 illustrates the finite element mesh of the twolifts with cast corners (they were used to facilitate theconstruction and increase the rigidity of the bottom of thedam) at downstream each lift (15m high) contains four-layer elements Each element in Figure 1 has an edge lengthof about 0984m and a thickness of 0375m The numericalmodel includes a total of 15488 elements in which 5544elements (the 10th lift) are used in the XFEMmesh

42 Boundary Conditions During construction the 10th liftof themonolith had been standing for 86 days before the next


The river direction

The 10th liftThe 9th lift

63

m

Cast corners

The positions of the elements in Figure 5

Figure 1 Element mesh of the two lifts (9th and 10th)

0 10 20 30 40 50 60 70 80 90 100 110 120 130Time (day)

Air

tem

pera

ture

(∘C)

35

30

25

20

15

10

5

0

Figure 2 Development of air temperature

lift was poured The increasing upstream and downstreamsurface of the poured concrete slabs and the top surface ofthe latest slab were assumed to be directly exposed to theenvironment where heat transfer by convention occurredBecause the other lifts under the 9th lift and in the othermonoliths were not considered the boundary condition ofthe lower surface of the 9th lift was assumed to be adiabaticwith fixed vertical displacement

The calculations correspond to a period of 128 days (theduration between starting the 9th lift and starting the 11thlift) and the 10th lift started to be poured 42 days after the9th lift was poured The daily mean air temperature curve atthe dam site as shown in Figure 2 was used in this analysis

43 Properties of Materials According to [47] the hydrationmodel in the engineering case analyzed in this paper may bewritten as follows

120585 (119905) = 164 (1 minus 119890minus0516119905

) + 109 (1 minus 119890minus0052119905

) (20)

The equation is in accordance with the experimental dataEach concrete slab contains one-layer staggered heteroge-neous iron cooling pipes in the middle for artificial coolingrelated property values are illustrated in [46 57]

Equations (18) to (19) show that the Youngrsquos modulus themaximum allowable principle stress and the critical values of

Table 1 Main properties of concrete

Property Values NoteDensity 120588 26630 kgm3 ConstantThermal conductivity 120582 1849 kJmsdotday ∘C ConstantSpecific heat 119888 086 kJkg ∘C ConstantHeat convectioncoefficient ℎ119879

12000 kJm2sdotday ∘C Constant

1198860 140 Constant1198870 062 Constant1198880 088 Constant1205740

025 ConstantElastic modulus 119864 (28 d) 40GPa Time-dependentPoissonrsquos ratio 120592 0167 ConstantCoefficient of thermalexpansion 120572 7 times 10minus6∘C Constant

Allowable maximumprinciple 1205900max (28 d)

198MPa Time-dependent

Critical energy releaserate for model I 119866I119888(28 d)

4792Nm Time-dependent

0 100 200 300 400 500 600 700 800 900 1000Time (day)

Cree

p co

effici

ent

16

14

12

10

08

06

04

00

02

Figure 3 Development of the creep coefficient

strain energy release rate can be written as time-dependentfunctions (119864(119905) 1205900max(119905) and 119866119888(119905)) when (20) is used Inthe simplest case the fracture process can be isotropic andparameters 119866I119888(119905) = 119866II119888(119905) = 119866III119888(119905) and 120572 = 120573 = 120594 = 1 Themain properties of concrete are presented in Table 1 Figure 3shows the inclusion of the creep coefficient in viscoelasticmaterial behavior during the cracking analysis for the 10thlift

44 Results and Discussions Accurate temperature fields arerequired to calculate thermal cracking Figure 4 compares thetemperature histories of the concrete in the calculated resultsand the actual measured temperature from thermometersburied at middle height of the lift Overall the calculatedresults obtained from the methods used in this study agreewith themeasured data A sudden temperature drop occurred


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

24

26

28

30

22

20

18

16

14

12

10

Tem

pera

ture

(∘C)

Measured (upstream)Measured (middle)

Measured (downstream)Computed

Figure 4 Development of concrete temperature

in the position of the buried thermometers five days afterthe cold wave and an inflection point was observed in thecomputed curve two days in advanceThese conditions can berelated to the positions of the thermometers and the thermalconductivity

Both elastic and viscoelastic models were applied in thestress analysis Four elements of different heights at thecenter of the slab (see Figure 1) were selected with the time-dependent maximum principal stress at the centroid of theseelements shown in Figure 5 Overall creep can reduce thestress levels by approximately 20 At the beginning thebottom element bears low tensile stress levels because ofthe effects from the lower lift The three other elementsbear compressive stresses because of the restraint of thermaldilatation and the rise to their peaks seven days after thelift casting After the temperature decreases in the coreas a result of artificial water cooling compressive stressesdecrease gradually and then tensile stress occurs Affectedby the ambient temperature the stresses fluctuate over timeand increase suddenly as a result of the cold wave The stressof the surface element that contacts with air has the largestvaried amplitude and exceeds the tensile strength eventually

The cold wave results in a thermal gradient (betweenthe core and the skin of the lift) which causes tensile stressthat is particularly serious on the surface The cracks occurat discrete locations where the maximum principle stressexceeds the allowable values and propagates based on thecriterion The cracks simulated by XFEM and observed onthe slab are presented in Figure 6 This figure illustrates thatthe numerical results are generally consistent with real crackoccurrence Most of the main cracks are initiated at the edgesof the lift where high-temperature gradients occur and thiscondition favors crack initiation The evolution paths of thesimulated cracks are almost close to the lines because of theisotropic assumption for the material and the crack evolutionlaw The discrepancy between the simulated and real crackdirections in the middle of the lift surface is a result of

A (without considering creep)A (considering creep)B (without considering creep)B (considering creep)C (without considering creep)C (considering creep)D (without considering creep)D (considering creep)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

350030002500200015001000

5000

minus3500minus3000minus2500minus2000minus1500minus1000minus500

Max

prin

cipl

e stre

ss (K

pa)

AB

CD

Figure 5 Development of stresses of different heights

excluding the other lifts (except the 9th and 10th) and thefoundation in the calculation In addition crack crossingcannot be simulated because of the limitations of XFEMThenumerical simulation indicates that ambient temperature isamong the main reasons for the thermal cracking of massiveconcrete structures

The casting temperature obviously affects the crackingrisk if no cooling system is used [58] The concrete pouringtemperature in the preceding analysis is 67∘C and then wegradually increase the casting temperature to the ambientone (303∘C) The maximum principle stresses with differentpouring temperatures are presented in Figure 7 The resultindicates that an increase in casting temperature does notexert significant effects on the stresses during the early stagebecause the embedded water-cooling system decrease theconcrete temperature effectively A negligible discrepancy isobserved between these stresses at a later stage

Heat preservation measures can be adopted during theintermission between the pouring of two lifts to reducethe thermal gradient of concrete In this study we useddifferent heat convection coefficients to simulate differentlevels of thermal insulation effectsThe stress evolutionwith aseries of heat convection coefficients is illustrated in Figure 8The stress levels decrease significantly with a decreasein the coefficients particularly from 400 kJmsdotday ∘C to200 kJmsdotday ∘C In contrast to the case during the early stagea small convection coefficient assumes a critical functionwhen the cold wave arrives The results demonstrate thatappropriate heat insulation measures can effectively reducethe risk of thermal cracking


(a) (b)

Figure 6 Comparison between simulated and actual thermal cracks (a) simulated via XFEM and (b) actual crack occurrence

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

5000

minus3000minus2500minus2000minus1500minus1000minus500

Max

prin

cipl

e stre

ss (K

pa)

30∘C

25∘C

20∘C

15∘C

10∘C

Figure 7 Development of stresses with different pouring tempera-tures

5 Conclusions

This study presents an effective XFEM method to simulatethe thermal cracking of massive concrete The temperaturefield in the slab can be obtained by the equivalent equationof heat conduction which includes the hydration heat ofconcrete and the effect of cooling pipe systems The time-dependent viscoelastic behavior presented by the Prony seriesfor concrete is included in the prediction of stresses todescribe the stress relaxation that results from creep TheYoungrsquos modulus tensile strength and critical values ofstrain energy release rate which evolve over time are thekey parameters that affect the likelihood of thermal cracking

0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

500


Max

prin

cipl

e stre

ss (K

pa)

1000 (kJm2middot

800 (kJm2middot

600 (kJm2middot

400 (kJm2middot

200 (kJm2middot

day ∘C)day ∘C)day ∘C)

day ∘C)day ∘C)

Figure 8 Development of stresses with different heat convectioncoefficients

in massive concrete Thermal cracking of massive concretecan be described through a combination of XFEM with thetemperature field and the viscoelastic creep model

Compared with existing numerical methods to simulatethermal cracks on massive concrete the developed XFEMmethod in this study has advantages in efficiently simulatingmultiple discrete thermal crack propagationswithout presup-posing crack path and remeshingTherefore this method canbe conveniently applied to analyze thermal cracking for mas-sive concrete structures during the design and constructionphases


The precision of the developed method was verified byan engineering application The simulated and actual resultswere in agreement as shown in two comparisons the com-parison between the calculated time-dependent temperaturecurves and the measured ones and the comparison betweenthe numerical simulated cracks and the actual observed crackoccurrence If the structure is embedded with a coolingsystem and the ambient temperature is appropriate thecasting temperature for concrete exerts a limited effect onthermal cracking risk that is different for the case withouta cooling system However heat preservation measures canexert a significant effect on thermal crack control for massiveconcrete particularly when a cold wave occurs These mea-sures deserve attention during the construction of massiveconcrete structures

Acknowledgments

This study was financially supported by National BasicResearch Program of China (2013CB035902) ResearchProject of State Key Laboratory of Hydroscience and Engi-neering of Tsinghua University (2011-KY-4) and the NationalNatural Science Foundation of China (51279087)

References

[1] R de Borst and A H van den Boogaard ldquoFinite-elementmodeling of deformation and cracking in early-age concreterdquoJournal of EngineeringMechanics vol 120 no 12 pp 2519ndash25341994

[2] Y Xiang Z Zhang S He and G Dong ldquoThermal-mechanicalanalysis of a newly cast concrete wall of a subway structurerdquoTunnelling andUnderground Space Technology vol 20 no 5 pp442ndash451 2005

[3] J Mazars and J P Bournazel ldquoModelling of damage processesdue to volumic variations for maturing and matured concreterdquoin Proceedings of the International RILEM Conference on Con-crete From Material to Structure pp 43ndash54 1998

[4] V Waller L drsquoAloıa F Cussigh and S Lecrux ldquoUsing thematurity method in concrete cracking control at early agesrdquoCement and Concrete Composites vol 26 no 5 pp 589ndash5992004

[5] R Lackner and H A Mang ldquoChemoplastic material model forthe simulation of early-age cracking from the constitutive lawto numerical analyses of massive concrete structuresrdquo Cementand Concrete Composites vol 26 no 5 pp 551ndash562 2004

[6] G de Schutter ldquoFinite element simulation of thermal crackingin massive hardening concrete elements using degree of hydra-tion basedmaterial lawsrdquo Computers and Structures vol 80 no27ndash30 pp 2035ndash2042 2002

[7] F Benboudjema and J M Torrenti ldquoEarly-age behaviourof concrete nuclear containmentsrdquo Nuclear Engineering andDesign vol 238 no 10 pp 2495ndash2506 2008

[8] M Briffaut F Benboudjema J M Torrenti and G NahasldquoA thermal active restrained shrinkage ring test to study theearly age concrete behaviour ofmassive structuresrdquoCement andConcrete Research vol 41 no 1 pp 56ndash63 2011

[9] M Briffaut F Benboudjema J M Torrenti and G NahasldquoNumerical analysis of the thermal active restrained shrinkagering test to study the early age behavior of massive concrete

structuresrdquo Engineering Structures vol 33 no 4 pp 1390ndash14012011

[10] L Buffo-Lacarriere A Sellier A Turatsinze and G EscadeillasldquoFinite element modelling of hardening concrete applicationto the prediction of early age cracking for massive reinforcedstructuresrdquo Materials and Structures vol 44 no 10 pp 1821ndash1835 2011

[11] Y Abdelaziz and A Hamouine ldquoA survey of the extended finiteelementrdquo Computers and Structures vol 86 no 11-12 pp 1141ndash1151 2008

[12] T Belytschko and T Black ldquoElastic crack growth in finiteelements with minimal remeshingrdquo International Journal forNumerical Methods in Engineering vol 45 no 5 pp 601ndash6201999

[13] NMoes J Dolbow andT Belytschko ldquoA finite elementmethodfor crack growth without remeshingrdquo International Journal forNumerical Methods in Engineering vol 46 no 1 pp 131ndash1501999

[14] N Sukumar D L Chopp N Moes and T Belytschko ldquoMod-eling holes and inclusions by level sets in the extended finite-element methodrdquo Computer Methods in Applied Mechanics andEngineering vol 190 no 46-47 pp 6183ndash6200 2001

[15] T-P Fries and T Belytschko ldquoThe extendedgeneralized finiteelement method an overview of the method and its applica-tionsrdquo International Journal for Numerical Methods in Engineer-ing vol 84 no 3 pp 253ndash304 2010

[16] R Merle and J Dolbow ldquoSolving thermal and phase changeproblems with the eXtended finite element methodrdquo Compu-tational Mechanics vol 28 no 5 pp 339ndash350 2002

[17] P M A Areias and T Belytschko ldquoTwo-scale method for shearbands thermal effects and variable bandwidthrdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 6 pp658ndash696 2007

[18] M Duflot ldquoThe extended finite element method in thermoe-lastic fracture mechanicsrdquo International Journal for NumericalMethods in Engineering vol 74 no 5 pp 827ndash847 2008

[19] A Zamani and M R Eslami ldquoImplementation of the extendedfinite element method for dynamic thermoelastic fractureinitiationrdquo International Journal of Solids and Structures vol 47no 10 pp 1392ndash1404 2010

[20] A Zamani R Gracie and M R Eslami ldquoHigher order tipenrichment of eXtended Finite Element Method in thermoe-lasticityrdquo Computational Mechanics vol 46 no 6 pp 851ndash8662010

[21] E N Dvorkin A M Cuitino and G Gioia ldquoFinite elementswith displacement interpolated embedded localization linesinsensitive to mesh size and distortionsrdquo International Journalfor Numerical Methods in Engineering vol 30 no 3 pp 541ndash564 1990

[22] C Linder and F Armero ldquoFinite elements with embeddedstrong discontinuities for the modeling of failure in solidsrdquoInternational Journal for Numerical Methods in Engineering vol72 no 12 pp 1391ndash1433 2007

[23] C Linder and F Armero ldquoFinite elements with embeddedbranchingrdquo Finite Elements in Analysis and Design vol 45 no4 pp 280ndash293 2009

[24] T Belytschko Y Y Lu and L Gu ldquoElement-free Galerkinmethodsrdquo International Journal for Numerical Methods in Engi-neering vol 37 no 2 pp 229ndash256 1994

[25] W K Liu S Jun and Y F Zhang ldquoReproducing kernelparticle methodsrdquo International Journal for Numerical Methodsin Fluids vol 20 no 8-9 pp 1081ndash1106 1995


[26] S N Atluri The Meshless Local Petrov-Galerkin (MLPG)Method Tech Science Press Encino Calif USA 2002

[27] S Bordas T Rabczuk and G Zi ldquoThree-dimensional crackinitiation propagation branching and junction in non-linearmaterials by an extended meshfree method without asymptoticenrichmentrdquo Engineering Fracture Mechanics vol 75 no 5 pp943ndash960 2008

[28] M H Aliabadi ldquoBoundary element formulations in fracturemechanicsrdquo Applied Mechanics Reviews vol 50 no 2 pp 83ndash96 1997

[29] E Pan and F G Yuan ldquoBoundary element analysis of three-dimensional cracks in anisotropic solidsrdquo International Journalfor Numerical Methods in Engineering vol 48 no 2 pp 211ndash2372000

[30] G K Sfantos and M H Aliabadi ldquoMulti-scale boundaryelement modelling of material degradation and fracturerdquo Com-puter Methods in Applied Mechanics and Engineering vol 196no 7 pp 1310ndash1329 2007

[31] R N Simpson S P A Bordas J Trevelyan and T Rabczuk ldquoAtwo-dimensional Isogeometric boundary element method forelastostatic analysisrdquo Computer Methods in Applied Mechanicsand Engineering vol 209ndash212 pp 87ndash100 2012

[32] C Zhao B E Hobbs and A Ord Fundamentals of Computa-tional Geoscience Numerical Methods and Algorithms SpringerBerlin Germany 2009

[33] C Zhao B E Hobbs A Ord P Hornby S Peng and L LiuldquoParticle simulation of spontaneous crack generation problemsin large-scale quasi-static systemsrdquo International Journal forNumericalMethods in Engineering vol 69 no 11 pp 2302ndash23292007

[34] C Zhao B E Hobbs A Ord S Peng and L Liu ldquoAn upscaletheory of particle simulation for two-dimensional quasi-staticproblemsrdquo International Journal for Numerical Methods inEngineering vol 72 no 4 pp 397ndash421 2007

[35] C Zhao T Nishiyama and A Murakami ldquoNumerical mod-elling of spontaneous crack generation in brittle materials usingthe particle simulationmethodrdquo Engineering Computations vol23 no 5 pp 566ndash584 2006

[36] C Zhao B E Hobbs A Ord P A Robert P Hornby andS Peng ldquoPhenomenological modelling of crack generation inbrittle crustal rocks using the particle simulation methodrdquoJournal of Structural Geology vol 29 no 6 pp 1034ndash1048 2007

[37] C Zhao B E Hobbs A Ord and S Peng ldquoCritical contactstiffness concept and simulation of crack generation in particlemodels of large length-scalesrdquo Computers and Geotechnics vol36 no 1-2 pp 81ndash92 2009

[38] E O Onate S R Idelsohn F del Pin and R Aubry ldquoTheparticle finite element method an overviewrdquo InternationalJournal of Computational Methods vol 1 no 2 pp 267ndash3072004

[39] S R Idelsohn E Onate F del Pin and N Calvo ldquoFluid-structure interaction using the particle finite element methodrdquoComputer Methods in Applied Mechanics and Engineering vol195 no 17-18 pp 2100ndash2123 2006

[40] C Zhao B E Hobbs A Ord and S Peng ldquoParticle simulationof spontaneous crack generation associated with the laccolithictype of magma intrusion processesrdquo International Journal forNumerical Methods in Engineering vol 75 no 10 pp 1172ndash11932008

[41] C Zhao S Peng L Liu B E Hobbs and A Ord ldquoEffectiveloading algorithm associated with explicit dynamic relaxation

method for simulating static problemsrdquo Journal of Central SouthUniversity of Technology vol 16 no 1 pp 125ndash130 2009

[42] C Zhao ldquoComputational simulation of frictional drill-bitmovement in cemented granular materialsrdquo Finite Elements inAnalysis and Design vol 47 no 8 pp 877ndash885 2011

[43] C Zhao S Peng L Liu B E Hobbs and A Ord ldquoCompu-tational modeling of free-surface slurry flow problems usingparticle simulationmethodrdquo Journal of Central SouthUniversityvol 20 no 6 pp 1653ndash1660 2013

[44] T Belytschko N Moes S Usui and C Parimi ldquoArbitrarydiscontinuities in finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 4 pp 993ndash10132001

[45] B-F Zhu ldquoEquivalent equation of heat conduction in massconcrete considering the effect of pipe coolingrdquo Journal ofHydraulic Engineering vol 3 pp 28ndash34 1991 (Chinese)

[46] J Yang YHu Z Zuo F Jin andQ Li ldquoThermal analysis ofmassconcrete embedded with double-layer staggered heterogeneouscooling water pipesrdquo Applied Thermal Engineering vol 35 no1 pp 145ndash156 2012

[47] B-F Zhu ldquoCompound exponential formula for variation ofthermal and mechanical properties with age of concreterdquoJournal of Hydraulic Engineering vol 42 no 1 pp 1ndash7 2011(Chinese)

[48] Z P Bazant ldquoPrediction of concrete creep and shrinkage pastpresent and futurerdquo Nuclear Engineering and Design vol 203no 1 pp 27ndash38 2001

[49] T Belytschko W K Liu and B Moran Nonlinear FiniteElements for Continua and Structures John Wiley amp SonsChichester UK 2000

[50] R Malm and H Sundquist ldquoTime-dependent analyses of seg-mentally constructed balanced cantilever bridgesrdquo EngineeringStructures vol 32 no 4 pp 1038ndash1045 2010

[51] L Guo Y-M Zhu M-D Zhu and X-Y Zhang ldquoThe formuladesign of creep coefficient for high performance concreterdquoJournal of Sichuan University vol 42 no 2 pp 75ndash81 2010(Chinese)

[52] E M Wu and R C Reuter Jr ldquoCrack extension in fiberglassreinforced plasticsrdquo Tech Rep 275 University of Illinois 1965

[53] V K Goyal E R Johnson and C G Davila ldquoIrreversibleconstitutive law for modeling the delamination process usinginterfacial surface discontinuitiesrdquo Composite Structures vol65 no 3-4 pp 289ndash305 2004

[54] G de Schutter and L Taerwe ldquoDegree of hydration-baseddescription of mechanical properties of early age concreterdquoMaterials and Structures vol 29 no 6 pp 335ndash344 1996

[55] G de Schutter and L Taerwe ldquoFracture energy of concrete atearly agesrdquoMaterials and Structures vol 30 no 196 pp 67ndash711997

[56] D Cusson and T Hoogeveen ldquoAn experimental approachfor the analysis of early-age behaviour of high-performanceconcrete structures under restrained shrinkagerdquo Cement andConcrete Research vol 37 no 2 pp 200ndash209 2007

[57] Z Zuo Y Hu Y Duan and J Yang ldquoSimulation of thetemperature field inmass concrete with double layers of coolingpipes during constructionrdquo Journal of Tsinghua University vol52 no 2 pp 186ndash228 2012 (Chinese)

[58] S Wu D Huang F-B Lin H Zhao and P Wang ldquoEstimationof cracking risk of concrete at early age based on thermal stressanalysisrdquo Journal of Thermal Analysis and Calorimetry vol 105no 1 pp 171ndash186 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


XFEM to characterize both the displacement discontinuityand the temperature discontinuity In addition to XFEM theembedded finite element method (EFEM) [21ndash23] meshfreemethods (MMs) [24ndash27] and boundary elements (BEMs)[28ndash31] are also efficient computationalmethods based on thecontinuum-mechanics for discrete cracks

An alternative pathway to simulate crack initiation andpropagation offers methods based on the discrete-mechanicssuch as the particle simulation method [32ndash37] and theparticle element method [38 39] The methods have beenrecently developed to discretize the continuum into particlesso that crack generation at a contact between two particles isa natural process Zhao et al [40] simulatedmagma intrusionproblems using solid particles for the surrounding rock andfluid particles for the intruded magma respectively Theparticle simulation method can efficiently simulate hydraulicfracturing because the fluid particles are allowed to movethrough the cracks between the solid particles In additionthe particle simulation method has also been used to solve abroad range of scientific and engineering problems [41ndash43]

In accordance with the literature review this paperaims to develop a numerical method based on XFEM toanalyze thermal cracks in massive concrete structures Apractical thermal model that considers an artificial water-cooling system is adopted to simulate the temperature fieldTime-dependent viscoelastic behavior is implemented forcreep As input parameters for XFEM that combines thetemperature field and the viscoelastic constitutive model themain properties develop over time when thermal cracking iscalculatedThe reliability of the developed method is verifiedby a case study on a lift of an arch dam which underwenta cold wave during its construction Subsequently furtheranalysis on thermal cracking is conducted

2 XFEM for Thermal Cracking ofMassive Concrete

21 Briefing on XFEM Compared with the classical FEMXFEM provides significant benefits in modeling crack prop-agation The crack geometry in XFEM does not need tobe aligned with the element edges XFEM is based on thepartition of unity concept and introduces additional degreesof freedom which are tied to the nodes of the elementsintersected by the crack [12 13]

XFEM displacement approximation around the crackuℎ(x) is enriched with the step function to model thecrack surface and with asymptotic displacement fields tomodel crack tips The displacement approximation takes thefollowing form [13]

uℎ (x) = sum119899isin119873119899

119873119899 (x)u119894 + sum119895isin119873cr

119873119895 (x) ℎ (x) a119895


119873119896(

4

sum119897=1

120595119897 (x) b119896119897)

(1)

where the first term is the classical FE approximation 119873119899is the FE shape function and u119894 is the vector of the regular

degrees of nodal freedom The other terms are the enrichedterms where 119873cr and 119873tip stand for the set of nodes thathave crack faces and crack tips in their support domainrespectively a119895 and b119896119897 denote the vectors of additionaldegrees of nodal freedom for modeling crack faces and cracktips respectively

Step function ℎ(x) simulates the displacement at the cracklocation with the help of the signed distance function 120601(x) tothe crack which is also called the normal level set [13 15]

ℎ (x) = sign (120601 (x)) (2)

The four following branch functions which define thediscontinuity near the crack tip enrich the nodes with a dis-tance to the front inferior to a prescribed enrichment radius

120595 (x)

= [radic119903 sin 120579

2radic119903 cos 120579

2radic119903 sin 120579

2cos 120579

2radic119903 cos 120579

2cos 120579]

(3)

where r and 120579 are polar coordinates in the local crack-tipcoordinate system

Duflot [18] applied the same procedure to temperatureenrichment The step function ℎ(x) can account for thetemperature jump The leading term of the asymptotic fieldfor the temperature of an adiabatic crack can be written as

119879 = minus119870119879

120582radic2119903

120587sin(120579

2) (4)

where coefficient119870119879 provides the strength of flux singularityat the crack tipThe temperature field is discretized similar tothe displacement field but with only the second branch func-tion (4) which is the only discontinuous branch functionTheapproximation of the temperature field 119879ℎ(x) is then writtenas

119879ℎ(x) = sum

119899isin119873

119873119899 (x) 119879119894 + sum119895=119873cr

119873119895 (x) ℎ (x) 119888119895


1198731198961205952 (x) 119889119896(5)

As in standard FEM performing numerical integrationover the element domain is necessary to compute the elementstiffness matrix The subdomain method [44] in which thecrack is one of the subdomain boundaries to carry out thenumerical integration is adopted for the elements that are cutby the crack

22 Equivalent Equation of Heat Conduction An isotropichomogenous domain Ω bounded by boundary Γ is consid-ered The thermal properties of this domain are independentof temperature so the equation of thermal conduction is

q = minus120582nabla119879

minusnablaq + 120588119888 = (6)




119876 = 1205881198881205851015840(119905 +

Δ119905

2) + 120588119888 [119879 (119905) minus 119879119908] 120593

1015840(119905 +

Δ119905

2) (7)



120585 (119905) =

119899

sum119894=1

120585119894 (1 minus 119890minus119898119894119905) (8)



minus120582120597119879

120597119899= ℎ119879 (119879surf minus 119879air) (9)






120590119894119895 (119905) = 2int119905

0

119866(119905 minus 1199051015840)120597120576

dev119894119895

1205971199051015840d1199051015840

+ 3int119905

0

119870(119905 minus 1199051015840)120597120576119896119896 (119905

1015840)

1205971199051015840d1199051015840

(10)


119894119895and


119866 (119905) = 1198660 [1 minus

119899119866

sum119894=1

119892119894 (1 minus 119890minus119905120591119866

119894 )]

119870 (119905) = 1198700 [1 minus

119899119870

sum119894=1

119896119894 (1 minus 119890minus119905120591119870

119894 )]

(11)

where 119892119894 120591119866

119894 119896119894 and 120591119870



1198660 =1198640

2 sdot (1 + 120592)

1198700 =1198640

3 sdot (1 minus 2120592)

(12)



119866 (119905)

1198660=

1

(1 + 120593 (119905 1199050))

119870 (119905)

1198700=

1

(1 + 120593 (119905 1199050))

(13)


120593 (119905 1199050) = 120593(119905 1199050)stan sdot 120573 (1199050) 120573 (ℎ0) 120573 (119891119888119896) sdot 120573119879120573119891120573120572 (14)

with

120593(119905 1199050)stan = 37419(1 minus (11520 minus 119905

11520)21528

)

0263

(15)







119891 = ⟨120590max⟩

1205900max (16)



(119866I119866I119888

)

120572

+ (119866II119866II119888

)

120573

+ (119866III119866III119888

)

120594

= 1 (17)




119864 (120574)

119864 (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198860

1205900max (120574)

1205900max (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198870

119866119888

119866119888 (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198880

(18)



120574 (119905) =int119905

0120585 (120591) 119889120591

int119905end

0120585 (120591) 119889 (120591)

(19)








The river direction


63

m

Cast corners



0 10 20 30 40 50 60 70 80 90 100 110 120 130Time (day)

Air

tem

pera

ture

(∘C)

35

30

25

20

15

10

5

0





120585 (119905) = 164 (1 minus 119890minus0516119905

) + 109 (1 minus 119890minus0052119905

) (20)












0 100 200 300 400 500 600 700 800 900 1000Time (day)

Cree

p co

effici

ent

16

14

12

10

08

06

04

00

02





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

24

26

28

30

22

20

18

16

14

12

10

Tem

pera

ture

(∘C)








0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

350030002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

AB

CD






(a) (b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

30∘C

25∘C

20∘C

15∘C

10∘C


5 Conclusions


0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

500


Max

prin

cipl

e stre

ss (K

pa)

1000 (kJm2middot

800 (kJm2middot

600 (kJm2middot

400 (kJm2middot

200 (kJm2middot


day ∘C)day ∘C)






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119876 = 1205881198881205851015840(119905 +

Δ119905

2) + 120588119888 [119879 (119905) minus 119879119908] 120593

1015840(119905 +

Δ119905

2) (7)



120585 (119905) =

119899

sum119894=1

120585119894 (1 minus 119890minus119898119894119905) (8)



minus120582120597119879

120597119899= ℎ119879 (119879surf minus 119879air) (9)






120590119894119895 (119905) = 2int119905

0

119866(119905 minus 1199051015840)120597120576

dev119894119895

1205971199051015840d1199051015840

+ 3int119905

0

119870(119905 minus 1199051015840)120597120576119896119896 (119905

1015840)

1205971199051015840d1199051015840

(10)


119894119895and


119866 (119905) = 1198660 [1 minus

119899119866

sum119894=1

119892119894 (1 minus 119890minus119905120591119866

119894 )]

119870 (119905) = 1198700 [1 minus

119899119870

sum119894=1

119896119894 (1 minus 119890minus119905120591119870

119894 )]

(11)

where 119892119894 120591119866

119894 119896119894 and 120591119870



1198660 =1198640

2 sdot (1 + 120592)

1198700 =1198640

3 sdot (1 minus 2120592)

(12)



119866 (119905)

1198660=

1

(1 + 120593 (119905 1199050))

119870 (119905)

1198700=

1

(1 + 120593 (119905 1199050))

(13)


120593 (119905 1199050) = 120593(119905 1199050)stan sdot 120573 (1199050) 120573 (ℎ0) 120573 (119891119888119896) sdot 120573119879120573119891120573120572 (14)

with

120593(119905 1199050)stan = 37419(1 minus (11520 minus 119905

11520)21528

)

0263

(15)







119891 = ⟨120590max⟩

1205900max (16)



(119866I119866I119888

)

120572

+ (119866II119866II119888

)

120573

+ (119866III119866III119888

)

120594

= 1 (17)




119864 (120574)

119864 (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198860

1205900max (120574)

1205900max (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198870

119866119888

119866119888 (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198880

(18)



120574 (119905) =int119905

0120585 (120591) 119889120591

int119905end

0120585 (120591) 119889 (120591)

(19)








The river direction


63

m

Cast corners



0 10 20 30 40 50 60 70 80 90 100 110 120 130Time (day)

Air

tem

pera

ture

(∘C)

35

30

25

20

15

10

5

0





120585 (119905) = 164 (1 minus 119890minus0516119905

) + 109 (1 minus 119890minus0052119905

) (20)












0 100 200 300 400 500 600 700 800 900 1000Time (day)

Cree

p co

effici

ent

16

14

12

10

08

06

04

00

02





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

24

26

28

30

22

20

18

16

14

12

10

Tem

pera

ture

(∘C)








0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

350030002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

AB

CD






(a) (b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

30∘C

25∘C

20∘C

15∘C

10∘C


5 Conclusions


0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

500


Max

prin

cipl

e stre

ss (K

pa)

1000 (kJm2middot

800 (kJm2middot

600 (kJm2middot

400 (kJm2middot

200 (kJm2middot


day ∘C)day ∘C)






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119891 = ⟨120590max⟩

1205900max (16)



(119866I119866I119888

)

120572

+ (119866II119866II119888

)

120573

+ (119866III119866III119888

)

120594

= 1 (17)




119864 (120574)

119864 (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198860

1205900max (120574)

1205900max (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198870

119866119888

119866119888 (120574 = 1)= (

120574 minus 1205740

1 minus 1205740)

1198880

(18)



120574 (119905) =int119905

0120585 (120591) 119889120591

int119905end

0120585 (120591) 119889 (120591)

(19)








The river direction


63

m

Cast corners



0 10 20 30 40 50 60 70 80 90 100 110 120 130Time (day)

Air

tem

pera

ture

(∘C)

35

30

25

20

15

10

5

0





120585 (119905) = 164 (1 minus 119890minus0516119905

) + 109 (1 minus 119890minus0052119905

) (20)












0 100 200 300 400 500 600 700 800 900 1000Time (day)

Cree

p co

effici

ent

16

14

12

10

08

06

04

00

02





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

24

26

28

30

22

20

18

16

14

12

10

Tem

pera

ture

(∘C)








0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

350030002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

AB

CD






(a) (b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

30∘C

25∘C

20∘C

15∘C

10∘C


5 Conclusions


0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

500


Max

prin

cipl

e stre

ss (K

pa)

1000 (kJm2middot

800 (kJm2middot

600 (kJm2middot

400 (kJm2middot

200 (kJm2middot


day ∘C)day ∘C)






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










The river direction


63

m

Cast corners



0 10 20 30 40 50 60 70 80 90 100 110 120 130Time (day)

Air

tem

pera

ture

(∘C)

35

30

25

20

15

10

5

0





120585 (119905) = 164 (1 minus 119890minus0516119905

) + 109 (1 minus 119890minus0052119905

) (20)












0 100 200 300 400 500 600 700 800 900 1000Time (day)

Cree

p co

effici

ent

16

14

12

10

08

06

04

00

02





0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

24

26

28

30

22

20

18

16

14

12

10

Tem

pera

ture

(∘C)








0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

350030002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

AB

CD






(a) (b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

30∘C

25∘C

20∘C

15∘C

10∘C


5 Conclusions


0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

500


Max

prin

cipl

e stre

ss (K

pa)

1000 (kJm2middot

800 (kJm2middot

600 (kJm2middot

400 (kJm2middot

200 (kJm2middot


day ∘C)day ∘C)






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

24

26

28

30

22

20

18

16

14

12

10

Tem

pera

ture

(∘C)








0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

350030002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

AB

CD






(a) (b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

30∘C

25∘C

20∘C

15∘C

10∘C


5 Conclusions


0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

500


Max

prin

cipl

e stre

ss (K

pa)

1000 (kJm2middot

800 (kJm2middot

600 (kJm2middot

400 (kJm2middot

200 (kJm2middot


day ∘C)day ∘C)






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

5000


Max

prin

cipl

e stre

ss (K

pa)

30∘C

25∘C

20∘C

15∘C

10∘C


5 Conclusions


0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85Time (day)

30002500200015001000

500


Max

prin

cipl

e stre

ss (K

pa)

1000 (kJm2middot

800 (kJm2middot

600 (kJm2middot

400 (kJm2middot

200 (kJm2middot


day ∘C)day ∘C)






Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Acknowledgments


References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article XFEM for Thermal Crack of Massive …downloads.hindawi.com/journals/mpe/2013/343842.pdfResearch Article XFEM for Thermal Crack of Massive Concrete GuoweiLiu,YuHu,QingbinLi,andZhengZuo