Research Article XFEM-Based Analysis for Crack Growth ...downloads.hindawi.com/journals/mpe/2013/820871.pdf · XFEM-Based Analysis for Crack Growth Characteristics of Diffusion Bonded

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 820871 8 pageshttpdxdoiorg1011552013820871

Research ArticleXFEM-Based Analysis for Crack Growth Characteristics ofDiffusion Bonded Laminates of Titanium Alloy with LocalizedNonwelded Zone

Yang Liu1 Yongcun Zhang1 Shutian Liu1 Shan Xiao1

Xiangming Wang2 and Yanpeng Sun2

1 State Key Laboratory of Structural Analysis for Industrial Equipment Department of Engineering MechanicsDalian University of Technology Dalian 116024 China

2 Shenyang Aircraft Design Research Institute Shenyang 110035 China

Correspondence should be addressed to Shutian Liu stliudluteducn

Received 24 June 2013 Accepted 7 August 2013

Academic Editor Song Cen

Copyright copy 2013 Yang Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Titanium-alloy laminates fabricated by sheet materials using diffusion bonding process have drawnmore andmore attention in therecent years Proper placement of nonwelded zones on the diffusion bonding (DB) interfacewithin titanium-alloy laminates as crackarrest zones can improve damage tolerance To achieve the optimal damage tolerance via designing non-welded zones it is necessaryto study fatigue crack growth characteristics for this type of laminates by adjusting all the relevant parameters such as geometricalsizes locations and the number of the nonwelded zones which is highly time consuming Therefore it is essential to develop areliable and quick method to analyze the fatigue crack growth characteristics for titanium-alloy laminates with non-welded zonesIn this paper the extended finite elementmethod (XFEM)whichwas employed to simulate the fatigue crack growth process and theapplicability of this method to capture fatigue crack growth characteristics of titanium-alloy laminates with localized non-weldedzones was also studied The numerical results were compared with the experiment data and the agreement on numerical andexperimental results illustrated that the specific crack growth characteristics can be captured by using XFEM thereby verifying theapplicability of XFEM in the analysis of fatigue crack growth of the laminates with non-welded zonesThe influence of non-weldedzones on the fatigue crack growth was then discussed

1 Introduction

Titanium alloys have over the years proven themselves to betechnically superior and cost effective materials for a widerange of applications spanning the industries of aerospacemarine and even commercial products [1] This is becauseof their excellent combination of mechanical properties suchas high specific strength immunity to corrosion in seawater environment good erosion resistance in environmentsspanning a range of aggressiveness and importantly theiracceptable mechanical properties at elevated temperaturescoupled with an intrinsic capability to withstand and safelyfunction at elevated temperatures The noticeable attractiveproperties of titanium alloys have facilitated their selectionand enhanced their use in aviation and a spectrum of otherperformance-critical applications [2]

However there is a disadvantage that the residual lifeis relatively short once the damage appears in the structurewhich is made of common titanium alloy (Damage isinevitable actually) Thus the common titanium alloys areineffective in damage tolerance design [3] and developingthe technology that is able to improve the damage tolerancebecomes an important issue At present some damagetolerance type titanium alloys such as Ti-6Al-4VELI Ti-6-22-22S have been applied in advanced airplane whichgreatly improved the service life and combat effectiveness ofairplanes

Damage-tolerance-design strategy gets more and moreattention in modern aircraft design It takes the intrinsicdiscrete damage large area manufacturing flaws or severeaccidental damage into consideration and ensures that

2 Mathematical Problems in Engineering

the remaining structure will withstand reasonable loadswithout failure or excessive structural deformation until thedamage is detected In order to improve the applicability oftitanium alloys to damage-tolerance-design one traditionalmethod is through microstructural control and anothermethod is through the use of submacro laminated structures[4] Some experimental investigations demonstrate that theDB interfaces in the titanium alloy-laminates which aremadeby diffusion bonding process have significant inhibitory effecton the growth of the fatigue crack [5 6] Something moreinteresting is that the diffusion bonded laminates of TC4titanium alloy may have longer fatigue life if the localizednonwelded zone exists in the DB interfaces This indicatesthat the defects may achieve significant inhibition of crackgrowth when the defects lie in the proper position It will be agood approach to increase the structure fatigue life by reason-ably arranging the defects in DB interfaces To implement thereasonable arrangement of defects the influence of the defecteffects on the fatigue crack growth characteristics should beknown clearly At present the crack growth characteristicsmostly are conducted through experiments and hence it isnecessary to develop the rapid numerical analytical methodfor studying the mechanism of the crack growth in thetitanium alloy-laminates

The recently proposed XFEM can describe the disconti-nuity and singularity by introducing the enrichment functionto the shape function of the conventional finite elementmethod (CFEM) and it exhibits a unique advantage inthe analysis of discontinuous problem [7ndash12] Crack is atypical discontinuity and XFEM will be an effective methodin researching fatigue crack growth characteristics in thelaminates of titanium alloys

In this paper the XFEM method based on the ABAQUSplatform is employed for analyzing crack growth characteris-tics in the diffusion bonded laminates of TC4 titanium alloysand the results obtained are compared with the experimentalresults given in [6] to verify that XFEM is able to get thetrue crack growth characteristic and an effective method forcrack growth analysis in the laminates of titanium alloys Inaddition through the comparison between the crack growthcharacteristics in the laminates with and without nonweldedzones this paper reveals the mechanism that is how thenonwelded zone increases the fatigue life And the results willgive the technical support in the damage tolerance design oftitanium-alloy laminates

2 XFEM-Based Crack Growth Analysis

The enrichment function in XFEM can accurately describethe crack or material interfacersquos discontinuity and singularitythat exit inside the element So the mesh in XFEM canbe independent of the structurersquos geometrical or physicalinterface and that avoids the limit that there must be veryfine grids near crack tip or material interface when usingtraditional finite element method and it is more attractivethat there needs no remeshing in analyzing the crack growthproblem in XFEM XFEM has all the advantages in CFEM

Crack front

Crack surface

120595

120593

120595 = 0

nabla

nabla

120593 = 0

Figure 1 Level set method for interface described

For this reason the form of element stiffness matrix in XFEMis inlinewith that inCFEM and the implementation ofXFEMcan make full use of the CFEM program that already exists

21 Description of Discontinuous Interface The level setmethod is a numerical technique for tracking the evolutionof interfaces In this method the interface is represented asthe zero level set of a function 120593(119909 119905) This function is onedimension higher than the dimension of the interface Ingeneral amoving interface in 119899 dimensional (2Dor 3D) spacecan be formulated as

Γ (119905) = 119909 isin 119877119899 120593 (119909 119905) = 0

120593 (119909 119905) = plusmnmin119909119903isin119903(119905)

1003817100381710038171003817119909 minus 1199091199031003817100381710038171003817

(1)

The level set function usually uses signal distance func-tion to be constructed Where 119909119903 are the points located onthe crack interface and 119903(119905) represent the whole crack surfaceThe whole domain is cut into two parts 120593 gt 0 and 120593 lt0 by the zero level set function 120593 = 0 Another level setfunction 120595(119909 119905) is introduced to describe the crack tip bythe intersection set of the two level set functions shown inFigure 1

Consider120595 (119909 119905) = (119909 minus 119909

lowast) sdot ( 119899 times

997888119905 ) (2)

where 997888119905 is the unit tangent vector and 997888119899 is the unit vectornormal to the crack surface 119909lowast is the crack tip coordinates

The description of crack growth can be got from theevolution of the level set and the details refer to the literature[14]

22 XFEM Displacement Mode The displacement field 119906ℎ inXFEM consists of two parts

119906ℎ= sum

119868

119873119868 (119909) 119906119868 + 120601 (119909) (3)

where 119873119868(119909) is the conventional shape function and 119906119868 isthe degree of freedom (DOF) of standard nodes 120601(119909) is

Mathematical Problems in Engineering 3

the enrichment function In the crack tip element 119906ℎ can berepresented [7]

119906ℎ= sum

119868

119873119868 (119909) 119906119868 +sum

119868

119873119868 (119909)Φ (119909) 119902119868 (4)

where Φ(119909) always takes linear combination of the followingfunction base

Φ (119909) = radic119903 [sin 1205792 sin 120579

2sin 120579 cos 120579

2 cos 120579

2sin 120579] (5)

where 119903 and 120579 are the polar coordinates in the crack tiplocal coordinates and the enrichment function 120601(119909) uses thefollowing form in the elements crossed by crack [15]

120601 (119909) = 119873119868119867(120593 (119909)) (6)

where 120593(119909) is the level set function and119867(120593(119909)) is the jumpfunction

Consider

119867(120593 (119909)) = 1 120593 (119909) ge 0

minus1 120593 (119909) lt 0(7)

The displacement mode in the crossed elements is

119906ℎ= sum

119868

119873119868 (119909) 119906119868 +sum

119868

119873119868 (119909)119867 (120593 (119909)) 119902119868 (8)

The junction between the enriched elements and thecommon elements will produce mixed elements and theliterature [16ndash19] discussed that particularly

23 XFEM Discrete Control Equation The weak form ofXFEM discrete control equation can be described as followsin linear elasticity problems [12]

intΩℎ120590 120575120576

ℎdΩ = intΩℎ

b 120575uℎdΩ + int120597Ωℎ119905

t sdot 120575uℎdΓ

forall120575uℎ isin Uℎ0 (9)

where t is the surface force and b is the body force uℎ and120575uℎ are the trial function and test function in XFEM Takingthe uℎ and 120575uℎ into (9) and using the arbitrariness of nodevariation we can get the discrete linear system of equations

Kd = f (10)

The form of equilibrium equation previously mentionedis inline with the CFEM K is the global stiffness matrixf is the load vector and d is the displacement vector Kand f consist of each element follows the node number Thecontribution of each element is expressed as follow

k119890119894119895 =[[[[[

[

k119906119906119894119895 k119906119886119894119895 k119906119887119894119895

k119886119906119894119895 k119886119886119894119895 k119886119887119894119895

k119887119906119894119895 k119887119886119894119895 k119887119887119894119895

]]]]]

]

f119890119894 = f119906119894 f119886119894 f1198871119894 f1198872119894 f1198873119894 f1198874119894 119879

(11)

The details of k and f are as follows

k119903119904119894119895 = intΩ119890(B119903119894 )119879DB119904119895dΩ (119903 119904 = 119906 119886 119887)

f119906119894 = int120597Ωℎ119905cap120597Ω119890

119873119894tdΓ + intΩ119890119873119894bdΩ

f119886119894 = int120597Ωℎ119905cap120597Ω119890

119873119894119867tdΓ + intΩ119890119873119894119867bdΩ

f119887120572119894 = int120597Ωℎ119905cap120597Ω119890

119873119894Φ120572tdΓ + intΩℎ119873119894Φ120572bdΩ (120572 = 1ndash4)

(12)

where B119886119894 B119906119894 and B

119887119894 are the derivative matrixes of the shape

function

B119906119894 = [[

119873119894119909 0

0 119873119894119910

119873119894119910 119873119894119909

]

]

B119886119894 =[[

[

(119873119894119867)119909 0

0 (119873119894119867)119910

(119873119894119867)119910 (119873119894119867)119909

]]

]

B119887120572119894 =[[

[

(119873119894Φ120572)119909 0

0 (119873119894Φ120572)119910

(119873119894Φ120572)119910 (119873119894Φ120572)119909

]]

]

(120572 = 1 minus 4)

B119887119894 = [B1198871119894 B1198872119894 B1198873119894 B1198874119894 ]

(13)

The derivation of the discrete control equation refers tothe literature [20]

24 Fatigue Crack Growth Rule and Description of CrackGrowth We get the three energy release rate componentsat crack tip by virtual crack closure technique (VCCT) andcalculate the equivalent energy release rate through (14)which Wu and Reuter [21] suggested (In the equation 1198861 =1198862 = 1198863 = 1)When themax equation energy release rate119866MEis bigger than the threshold of fatigue crack growth the crackbegins to grow When the crack growth rule is satisfied thecrack growth length can be calculated by the Paris formula(15) and the crack grows normal to the direction of themaximum circumferential stress

119866119864

119866119862

= (119866119868

119866119868119862

)

1198861

+ (119866119868119868

119866119868119868119862

)

1198862

+ (119866119868119868119868

119866119868119868119868119862

)

1198863

(14)

119889119886

119889119873= 119862(Δ119866)

119899 (15)

The XFEM in ABAQUS lacks the description of cracktip inside element therefore the crack tip has to grow tothe boundary of element So there are some differences incalculating the crack growth length from the traditionalmethod It calculates the needed cycles that the crack growsgiven length (the length of crossed element) which instead ofcalculating the length crack grows given cycles 119873 Where 119873gets from the needed cycles119873119895 that each element at the cracktip is crossed by and get min(119873119895) as the119873 in current step


L

T

W

A

A

A

A

Initial crack locationInitialcrack

locationΦnminusw

L2

T1

T2

Φhole

Figure 2 Model of the floor plan

Table 1 Mechanical properties of TC4 [13]

Elasticmodulus

Ultimatestrength

Fracturetoughness

Poissonrsquosratio

119864 (GPa) 120590119887 (MPa) 119870IC (MParadicm) ]

110 913 783 034

3 The Specimen Properties

Crack growth characteristics of diffusion bonded laminatesof titanium alloys with localized nonwelded zone can beobtained by the XFEM based on the ABAQUS platform Inorder to compare with the experimental data given in [6] thespecimen shown in Figure 2 is investigated as an exampleThelength 119871 is 180mm and the width119882 is 40mm and there isa hole with Φhole = 6mm in the center The overall thickness119879 is 8mm which consists of three layers of titanium-alloyplate and their thicknesses are 3mm 2mm and 3mm fromthe upper layer to lower layer respectively A single-edgesquare notchwith side length 05mmand thickness 02mm islocated at the edge of the hole on the plate surface In order toreveal the mechanism of how the nonwelded zone increasesthe fatigue life two cases are taken into consideration Incase one there are circular non-welded zones with diameterΦ119899minus119908 = 15mm in the twoDB interfaces as shown in Figure 2and there is no non-welded zone in the DB interfaces in theother case

The boundary condition of two ends is clamp and it is inkeepingwith the tensile experiment just like it was consideredin [6] The tension-tension cyclic loading uses the stress rate119877 = 0093 and the stress peak 120590max = 276MPa The Parisconstants are 119862 = 525 times 10minus8 and 119899 = 285 and the formulaconstants and material properties are from the literature [13]The material properties are in Table 1 as follows

4 Results and Discussion

41 Comparisonwith Experiment Thewhole process of crackgrowth in the specimen is simulated by XFEM The crackgrowth path is described in Figure 3 and the relation betweencrack growth rate and surface crack length along the directionof the aperture is shown in Figure 4 The experimental resultof the relation between crack growth rate and surface cracklength in [6] is also shown in Figure 5

Figure 4 shows that the results obtained by XFEM agreewell with the experimental data Combined with the detailcrack growth path described in Figure 3 the whole growthprocess of surface crack can be divided into three distinctstages In the first stage the crack growth rate increases asthe fracture crack length gets longer When the crack frontalong the thickness direction reaches the non-welded zonethe first stage ends and then the second stage begins Thecrack growth rate decreases obviously and this indicatesthat the non-welded zone can inhibit the crack growthsignificantly The third stage will start when the crack frontbypasses the boundary of the nonwelded zone meanwhilethe crack growth rate increases rapidly and the residualstrength of the structure decreases sharply These evidenceshows that the numerical simulation by XFEM can capturethe characteristics of the crack growth

The experimental data shown in Figure 6 illustrates thatthe times of fatigue cycle in the stage of crack initiationhave huge discrepancy for different specimens In order tocompare the fatigue life in the crack growth stage with theresults forecasted by XFEM the crack initiation is ignoredand the relationship between crack length and the cycles isconsidered from a given surface crack length (the length 1198860 =10mm in this paper)The comparison between the predictedresults and experiments is shown in Figure 7 Where Δ119873 =119873 minus1198730 and1198730 represents the cycle number when the cracklength reaches 1198860 From Figure 7 we can see that there arethe same tendencies for three specimens and the results byXFEM are accord with the experimental data These indicateagain that the simulation by XFEM is very validity

42 Internal Crack Growth The surface crack growth isdiscussed in the above section Actually the internal growthof crack gets more worthy of attention It is very difficultto observe the internal crack growth law by means ofexperiment Fortunately the numerical simulation is a goodway to research the internal crack growth There are anothertwo performance indexes that are employed to analyze thelaw of the internal crack growth One is the crack lengthalong direction of the thickness and the other is the projectedcrack area that is two-dimensional area measurement of thefracture by projecting its shape onto the plane perpendicularto the length direction

Figure 8 shows the relationship between the crack growthin depth direction and the fracture crack growth rates and


Stage I

n = 739 n = 1544 n = 2244

Stage II

n = 2260 n = 3806 n = 7961

Stage III

n = 15086 n = 16346 n = 17112

Initial crack location Nonwelded zone

Figure 3 The path of crack growth

0

2

4

6

8

10

12

14

0 04 08 12 16 2

The model without nonwelded zonesThe model with nonwelded zones

Fatigue cycles (104)

Surfa

ce cr

ack

leng

tha

(mm

)

Figure 4 Surface crack length varies with fatigue cycles (forecastresults)

Figure 9 shows the relationship between facture cycle timesand the crack growth in depth direction When there is nonon-welded zone in DB interface the crack growth rate in

0

2

4

6

8

10

12

14

0 2 4 6 8 10

XFEM forecast

I II III

Result in the literature [6]

dadN

(10minus4

mm

cyc

le)

Surface crack length a (mm)

Figure 5 Crack growth rate varies with surface crack length

depth direction increases rapidly and then is maintainedin a high rate so the crack length in depth direction alsoincreases rapidly as shown in Figures 8 and 9 By contrastthe crack growth rate decreases sharply when entering


50000 100000 150000 200000 2500000

2

4

6

8

10

12

14

16

18

Specimen III

Specimen IISpecimen I

a(m

m)

N (cycles)

Figure 6 The relationship between crack length and cycles in theexperimental results [6]

0

3

6

9

12

15

18

0 1 2 3 4 5Fatigue cycles (104)

Specimen I in [6]Specimen II in [6]

Specimen III in [6]XFEM forecast

Surfa

ce cr

ack

leng

tha

(mm

)

Figure 7 Crack growth length needed for the fatigue cycles

the second stage of crack growth and the fatigue cyclesincrease from 5000 times to 20000 times and this indicatesthat there are three-fold increased fatigue life These linesof evidence illustrate that the non-welded zone plays animportant role in preventing the crack growth

Actually the internal crack growth happens along thedirection of radius and depth at the same timeThe projectedcrack area as a new parameter is employed to describe theinternal crack growth Similar with definition of the cracklength growth rate the projected crack area growth rate120597119878120597119873 is defined where 119878 is the projected crack area Figures10 and 11 give the relationship curves between the crack areaand the cycle number and the relationship between crack areagrowth rate and the crack area These lines of evidence alsoconfirm that the non-welded zone plays an important role inpreventing the crack growth

0

05

1

15

2

25

3

0 1 2 3 4 5 6 7 8


Thickness direction crack length a (mm)

Crac

k gr

owth

ratedadN

(10minus3

mm

cyc

le)

Figure 8 Fatigue crack growth rate varies with the crack length indepth direction

0

1

2

3

4

5

6

0 04 08 12 16 2 24



Crac

k le

ngth

a(m

m)

Figure 9 Crack length in depth direction varies with fatigue cycles

43 Damage Tolerance Analysis Damage tolerance is a prop-erty of a structure which relates to its ability to sustaindefects safely until the structure is repaired In ensuring thecontinued safe operation of the damage tolerant structureinspection schedules are devised Thus the objective ofdamage tolerance design is to maximize the crack growthlife so that more inspection periods can be undergone or tomaximize the crack length with the same residual life so thatit is easier to be detected

The residual life under the different surface crack lengthis shown in Figure 12 Assuming the inspection interval isabout 5000 cycles the initial notched crack length is longenough to be detected There are about 12000 cyclersquos timesof the model without nonwelded zone and it can experience


0

15

30

45

60

75

90

0 04 08 12 16 2



Crac

k pr

ojec

tive a

reaS

(mm

2)

Figure 10 Projected crack area changes as fatigue cycles


Crack projective area S (mm2)

Crac

k gr

owth

ratedSdN

(10minus3

mm

2c

ycle

)

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70

Figure 11 Fatigue crack growth rate changes as projected crack area

two full inspection periods By contrast there are about20500 cyclersquos times of the model with non-welded zoneand it can experience four full inspection periods whichgreatly improve the structure security If the damage tolerancedesigns demand that there is at least one inspection periodresidual life left before detection the limiting crack lengthshould be 4mm in the structure without non-welded zoneand be 9mm with non-welded zone as shown in Figure 12The non-welded zone makes the limiting length increase onetime

5 Conclusions

This paper analyzes the crack that grows in the diffusionbonded laminates of TC4 titanium alloy using XFEM based



0

05

1

15

2

25

0 5 10 15 20

Resid

ual l

ifeN

(104

cycle

s)

Figure 12 Structure of residual life changingwith crack length alongdiameter direction

on the ABAQUS platform and studies the crack growth lawunder the TC4 laminates From the comparison between thepredicted results and experiments the XFEM can grasp thetrue crack growth law in the titanium-alloy laminates and theXFEM is proved to be an efficientmethod to analyze the crackgrowth in the titanium-alloy laminates In addition throughthe analysis of the damage tolerance the nonwelded zone setin the DB interface can significantly increase the fatigue lifeand improve the damage tolerance

Acknowledgments

This research is supported by the National Basic ResearchProgram (973Program) of China (Grant no 2011CB610304)National Natural Science Foundation of China (11172052) andChina Aviation Industry Research Project (CXY2011DG34)

References

[1] ASM Metals Handbook vol 2 ASM International MaterialsPark Ohio USA 10th edition 1990

[2] J K Wessel The Handbook of Advanced Materials WileyInterscience 2004

[3] J-K Zhang Z-N Li Z-P Qiu et al ldquoFeasibility study ondamage tolerance design of titanium alloysrdquoActaAeronautica etAstronautica Sinica vol 30 no 4 pp 193ndash197 2009 (Chinese)

[4] D O Cox and A S Tetelman ldquoImproved fracture toughnessof a titanium alloy laminate by controlled diffusion bondingrdquoJournal of Adhesion vol 5 no 4 pp 279ndash300 1973

[5] Z Keyin X U Lai and L Zhaorong ldquoFatigue and fracturecharacteristics of titanium alloy boards bonded by diffusionrdquoJournal of Mechanical Strength vol 20 no 2 pp 112ndash115 1998(Chinese)

[6] X-F He Y-M Liu W-T Liu Y-P Sun and X-M WangldquoCrack growth characteristic for diffusion bonded laminates ofTC4 titanium alloy with localized no-welded areardquo Journal ofAeronautical Materials vol 31 no 5 pp 77ndash81 2011 (Chinese)


[7] 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

[8] J Shi D Chopp J Lua N Sukumar and T Belytschko ldquoAbaqusimplementation of extended finite elementmethod using a levelset representation for three-dimensional fatigue crack growthand life predictionsrdquoEngineering FractureMechanics vol 77 no14 pp 2840ndash2863 2010

[9] E Giner N Sukumar J E Tarancon and F J FuenmayorldquoAn Abaqus implementation of the extended finite elementmethodrdquo Engineering FractureMechanics vol 76 no 3 pp 347ndash368 2009

[10] M Stolarska D L Chopp N Mos and T Belytschko ldquoMod-elling crack growth by level sets in the extended finite elementmethodrdquo International Journal for Numerical Methods in Engi-neering vol 51 no 8 pp 943ndash960 2001

[11] S Loehnert D S Mueller-Hoeppe and P Wriggers ldquo3Dcorrected XFEM approach and extension to finite deformationtheoryrdquo International Journal for Numerical Methods in Engi-neering vol 86 no 4-5 pp 431ndash452 2011

[12] N Sukumar and J-H Prevost ldquoModeling quasi-static crackgrowth with the extended finite element methodmdashpart Icomputer implementationrdquo International Journal of Solids andStructures vol 40 no 26 pp 7513ndash7537 2003

[13] China aeronautical materials handbook editorial board ChinaAeronautical Materials Handbook Standards Press of ChinaBeijing China 2001 Chinese

[14] S Osher and J A Sethian ldquoFronts propagating with curvature-dependent speed algorithms based onHamilton-Jacobi formu-lationsrdquo Journal of Computational Physics vol 79 no 1 pp 12ndash49 1988

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

[16] 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

[17] T Belytschko C Parimi N Moes N Sukumar and S UsuildquoStructured extended finite element methods for solids definedby implicit surfacesrdquo International Journal for Numerical Meth-ods in Engineering vol 56 no 4 pp 609ndash635 2003

[18] N Moes M Cloirec P Cartraud and J-F Remacle ldquoA com-putational approach to handle complex microstructure geome-triesrdquoComputerMethods in AppliedMechanics and Engineeringvol 192 no 28ndash30 pp 3163ndash3177 2003

[19] T-P Fries ldquoA corrected XFEM approximation without prob-lems in blending elementsrdquo International Journal for NumericalMethods in Engineering vol 75 no 5 pp 503ndash532 2008


[21] E M Wu and R C Reuter Jr Crack Extension in FiberglassReinforced Plastics vol 275 University of Illinois 1965

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the remaining structure will withstand reasonable loadswithout failure or excessive structural deformation until thedamage is detected In order to improve the applicability oftitanium alloys to damage-tolerance-design one traditionalmethod is through microstructural control and anothermethod is through the use of submacro laminated structures[4] Some experimental investigations demonstrate that theDB interfaces in the titanium alloy-laminates which aremadeby diffusion bonding process have significant inhibitory effecton the growth of the fatigue crack [5 6] Something moreinteresting is that the diffusion bonded laminates of TC4titanium alloy may have longer fatigue life if the localizednonwelded zone exists in the DB interfaces This indicatesthat the defects may achieve significant inhibition of crackgrowth when the defects lie in the proper position It will be agood approach to increase the structure fatigue life by reason-ably arranging the defects in DB interfaces To implement thereasonable arrangement of defects the influence of the defecteffects on the fatigue crack growth characteristics should beknown clearly At present the crack growth characteristicsmostly are conducted through experiments and hence it isnecessary to develop the rapid numerical analytical methodfor studying the mechanism of the crack growth in thetitanium alloy-laminates

The recently proposed XFEM can describe the disconti-nuity and singularity by introducing the enrichment functionto the shape function of the conventional finite elementmethod (CFEM) and it exhibits a unique advantage inthe analysis of discontinuous problem [7ndash12] Crack is atypical discontinuity and XFEM will be an effective methodin researching fatigue crack growth characteristics in thelaminates of titanium alloys

In this paper the XFEM method based on the ABAQUSplatform is employed for analyzing crack growth characteris-tics in the diffusion bonded laminates of TC4 titanium alloysand the results obtained are compared with the experimentalresults given in [6] to verify that XFEM is able to get thetrue crack growth characteristic and an effective method forcrack growth analysis in the laminates of titanium alloys Inaddition through the comparison between the crack growthcharacteristics in the laminates with and without nonweldedzones this paper reveals the mechanism that is how thenonwelded zone increases the fatigue life And the results willgive the technical support in the damage tolerance design oftitanium-alloy laminates

2 XFEM-Based Crack Growth Analysis

The enrichment function in XFEM can accurately describethe crack or material interfacersquos discontinuity and singularitythat exit inside the element So the mesh in XFEM canbe independent of the structurersquos geometrical or physicalinterface and that avoids the limit that there must be veryfine grids near crack tip or material interface when usingtraditional finite element method and it is more attractivethat there needs no remeshing in analyzing the crack growthproblem in XFEM XFEM has all the advantages in CFEM

Crack front

Crack surface

120595

120593

120595 = 0

nabla

nabla

120593 = 0

Figure 1 Level set method for interface described

For this reason the form of element stiffness matrix in XFEMis inlinewith that inCFEM and the implementation ofXFEMcan make full use of the CFEM program that already exists

21 Description of Discontinuous Interface The level setmethod is a numerical technique for tracking the evolutionof interfaces In this method the interface is represented asthe zero level set of a function 120593(119909 119905) This function is onedimension higher than the dimension of the interface Ingeneral amoving interface in 119899 dimensional (2Dor 3D) spacecan be formulated as

Γ (119905) = 119909 isin 119877119899 120593 (119909 119905) = 0

120593 (119909 119905) = plusmnmin119909119903isin119903(119905)

1003817100381710038171003817119909 minus 1199091199031003817100381710038171003817

(1)

The level set function usually uses signal distance func-tion to be constructed Where 119909119903 are the points located onthe crack interface and 119903(119905) represent the whole crack surfaceThe whole domain is cut into two parts 120593 gt 0 and 120593 lt0 by the zero level set function 120593 = 0 Another level setfunction 120595(119909 119905) is introduced to describe the crack tip bythe intersection set of the two level set functions shown inFigure 1

Consider120595 (119909 119905) = (119909 minus 119909

lowast) sdot ( 119899 times

997888119905 ) (2)

where 997888119905 is the unit tangent vector and 997888119899 is the unit vectornormal to the crack surface 119909lowast is the crack tip coordinates

The description of crack growth can be got from theevolution of the level set and the details refer to the literature[14]

22 XFEM Displacement Mode The displacement field 119906ℎ inXFEM consists of two parts

119906ℎ= sum

119868

119873119868 (119909) 119906119868 + 120601 (119909) (3)

where 119873119868(119909) is the conventional shape function and 119906119868 isthe degree of freedom (DOF) of standard nodes 120601(119909) is



119906ℎ= sum

119868

119873119868 (119909) 119906119868 +sum

119868

119873119868 (119909)Φ (119909) 119902119868 (4)


Φ (119909) = radic119903 [sin 1205792 sin 120579

2sin 120579 cos 120579

2 cos 120579

2sin 120579] (5)


120601 (119909) = 119873119868119867(120593 (119909)) (6)


Consider

119867(120593 (119909)) = 1 120593 (119909) ge 0

minus1 120593 (119909) lt 0(7)


119906ℎ= sum

119868

119873119868 (119909) 119906119868 +sum

119868

119873119868 (119909)119867 (120593 (119909)) 119902119868 (8)



intΩℎ120590 120575120576

ℎdΩ = intΩℎ

b 120575uℎdΩ + int120597Ωℎ119905




Kd = f (10)


k119890119894119895 =[[[[[

[

k119906119906119894119895 k119906119886119894119895 k119906119887119894119895

k119886119906119894119895 k119886119886119894119895 k119886119887119894119895

k119887119906119894119895 k119887119886119894119895 k119887119887119894119895

]]]]]

]

f119890119894 = f119906119894 f119886119894 f1198871119894 f1198872119894 f1198873119894 f1198874119894 119879

(11)


k119903119904119894119895 = intΩ119890(B119903119894 )119879DB119904119895dΩ (119903 119904 = 119906 119886 119887)

f119906119894 = int120597Ωℎ119905cap120597Ω119890

119873119894tdΓ + intΩ119890119873119894bdΩ

f119886119894 = int120597Ωℎ119905cap120597Ω119890

119873119894119867tdΓ + intΩ119890119873119894119867bdΩ

f119887120572119894 = int120597Ωℎ119905cap120597Ω119890


(12)

where B119886119894 B119906119894 and B


function

B119906119894 = [[

119873119894119909 0

0 119873119894119910

119873119894119910 119873119894119909

]

]

B119886119894 =[[

[

(119873119894119867)119909 0

0 (119873119894119867)119910

(119873119894119867)119910 (119873119894119867)119909

]]

]

B119887120572119894 =[[

[

(119873119894Φ120572)119909 0

0 (119873119894Φ120572)119910

(119873119894Φ120572)119910 (119873119894Φ120572)119909

]]

]

(120572 = 1 minus 4)

B119887119894 = [B1198871119894 B1198872119894 B1198873119894 B1198874119894 ]

(13)



119866119864

119866119862

= (119866119868

119866119868119862

)

1198861

+ (119866119868119868

119866119868119868119862

)

1198862

+ (119866119868119868119868

119866119868119868119868119862

)

1198863

(14)

119889119886

119889119873= 119862(Δ119866)

119899 (15)



L

T

W

A

A

A

A


locationΦnminusw

L2

T1

T2

Φhole



Elasticmodulus

Ultimatestrength

Fracturetoughness

Poissonrsquosratio


110 913 783 034











Stage I

n = 739 n = 1544 n = 2244

Stage II

n = 2260 n = 3806 n = 7961

Stage III

n = 15086 n = 16346 n = 17112



0

2

4

6

8

10

12

14

0 04 08 12 16 2



Surfa

ce cr

ack

leng

tha

(mm

)



0

2

4

6

8

10

12

14

0 2 4 6 8 10

XFEM forecast

I II III


dadN

(10minus4

mm

cyc

le)





50000 100000 150000 200000 2500000

2

4

6

8

10

12

14

16

18

Specimen III


a(m

m)

N (cycles)


0

3

6

9

12

15

18




Surfa

ce cr

ack

leng

tha

(mm

)




0

05

1

15

2

25

3

0 1 2 3 4 5 6 7 8



Crac

k gr

owth

ratedadN

(10minus3

mm

cyc

le)


0

1

2

3

4

5

6

0 04 08 12 16 2 24



Crac

k le

ngth

a(m

m)





0

15

30

45

60

75

90

0 04 08 12 16 2



Crac

k pr

ojec

tive a

reaS

(mm

2)




Crac

k gr

owth

ratedSdN

(10minus3

mm

2c

ycle

)

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70



5 Conclusions




0

05

1

15

2

25

0 5 10 15 20

Resid

ual l

ifeN

(104

cycle

s)



Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119906ℎ= sum

119868

119873119868 (119909) 119906119868 +sum

119868

119873119868 (119909)Φ (119909) 119902119868 (4)


Φ (119909) = radic119903 [sin 1205792 sin 120579

2sin 120579 cos 120579

2 cos 120579

2sin 120579] (5)


120601 (119909) = 119873119868119867(120593 (119909)) (6)


Consider

119867(120593 (119909)) = 1 120593 (119909) ge 0

minus1 120593 (119909) lt 0(7)


119906ℎ= sum

119868

119873119868 (119909) 119906119868 +sum

119868

119873119868 (119909)119867 (120593 (119909)) 119902119868 (8)



intΩℎ120590 120575120576

ℎdΩ = intΩℎ

b 120575uℎdΩ + int120597Ωℎ119905




Kd = f (10)


k119890119894119895 =[[[[[

[

k119906119906119894119895 k119906119886119894119895 k119906119887119894119895

k119886119906119894119895 k119886119886119894119895 k119886119887119894119895

k119887119906119894119895 k119887119886119894119895 k119887119887119894119895

]]]]]

]

f119890119894 = f119906119894 f119886119894 f1198871119894 f1198872119894 f1198873119894 f1198874119894 119879

(11)


k119903119904119894119895 = intΩ119890(B119903119894 )119879DB119904119895dΩ (119903 119904 = 119906 119886 119887)

f119906119894 = int120597Ωℎ119905cap120597Ω119890

119873119894tdΓ + intΩ119890119873119894bdΩ

f119886119894 = int120597Ωℎ119905cap120597Ω119890

119873119894119867tdΓ + intΩ119890119873119894119867bdΩ

f119887120572119894 = int120597Ωℎ119905cap120597Ω119890


(12)

where B119886119894 B119906119894 and B


function

B119906119894 = [[

119873119894119909 0

0 119873119894119910

119873119894119910 119873119894119909

]

]

B119886119894 =[[

[

(119873119894119867)119909 0

0 (119873119894119867)119910

(119873119894119867)119910 (119873119894119867)119909

]]

]

B119887120572119894 =[[

[

(119873119894Φ120572)119909 0

0 (119873119894Φ120572)119910

(119873119894Φ120572)119910 (119873119894Φ120572)119909

]]

]

(120572 = 1 minus 4)

B119887119894 = [B1198871119894 B1198872119894 B1198873119894 B1198874119894 ]

(13)



119866119864

119866119862

= (119866119868

119866119868119862

)

1198861

+ (119866119868119868

119866119868119868119862

)

1198862

+ (119866119868119868119868

119866119868119868119868119862

)

1198863

(14)

119889119886

119889119873= 119862(Δ119866)

119899 (15)



L

T

W

A

A

A

A


locationΦnminusw

L2

T1

T2

Φhole



Elasticmodulus

Ultimatestrength

Fracturetoughness

Poissonrsquosratio


110 913 783 034











Stage I

n = 739 n = 1544 n = 2244

Stage II

n = 2260 n = 3806 n = 7961

Stage III

n = 15086 n = 16346 n = 17112



0

2

4

6

8

10

12

14

0 04 08 12 16 2



Surfa

ce cr

ack

leng

tha

(mm

)



0

2

4

6

8

10

12

14

0 2 4 6 8 10

XFEM forecast

I II III


dadN

(10minus4

mm

cyc

le)





50000 100000 150000 200000 2500000

2

4

6

8

10

12

14

16

18

Specimen III


a(m

m)

N (cycles)


0

3

6

9

12

15

18




Surfa

ce cr

ack

leng

tha

(mm

)




0

05

1

15

2

25

3

0 1 2 3 4 5 6 7 8



Crac

k gr

owth

ratedadN

(10minus3

mm

cyc

le)


0

1

2

3

4

5

6

0 04 08 12 16 2 24



Crac

k le

ngth

a(m

m)





0

15

30

45

60

75

90

0 04 08 12 16 2



Crac

k pr

ojec

tive a

reaS

(mm

2)




Crac

k gr

owth

ratedSdN

(10minus3

mm

2c

ycle

)

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70



5 Conclusions




0

05

1

15

2

25

0 5 10 15 20

Resid

ual l

ifeN

(104

cycle

s)



Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










L

T

W

A

A

A

A


locationΦnminusw

L2

T1

T2

Φhole



Elasticmodulus

Ultimatestrength

Fracturetoughness

Poissonrsquosratio


110 913 783 034











Stage I

n = 739 n = 1544 n = 2244

Stage II

n = 2260 n = 3806 n = 7961

Stage III

n = 15086 n = 16346 n = 17112



0

2

4

6

8

10

12

14

0 04 08 12 16 2



Surfa

ce cr

ack

leng

tha

(mm

)



0

2

4

6

8

10

12

14

0 2 4 6 8 10

XFEM forecast

I II III


dadN

(10minus4

mm

cyc

le)





50000 100000 150000 200000 2500000

2

4

6

8

10

12

14

16

18

Specimen III


a(m

m)

N (cycles)


0

3

6

9

12

15

18




Surfa

ce cr

ack

leng

tha

(mm

)




0

05

1

15

2

25

3

0 1 2 3 4 5 6 7 8



Crac

k gr

owth

ratedadN

(10minus3

mm

cyc

le)


0

1

2

3

4

5

6

0 04 08 12 16 2 24



Crac

k le

ngth

a(m

m)





0

15

30

45

60

75

90

0 04 08 12 16 2



Crac

k pr

ojec

tive a

reaS

(mm

2)




Crac

k gr

owth

ratedSdN

(10minus3

mm

2c

ycle

)

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70



5 Conclusions




0

05

1

15

2

25

0 5 10 15 20

Resid

ual l

ifeN

(104

cycle

s)



Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Stage I

n = 739 n = 1544 n = 2244

Stage II

n = 2260 n = 3806 n = 7961

Stage III

n = 15086 n = 16346 n = 17112



0

2

4

6

8

10

12

14

0 04 08 12 16 2



Surfa

ce cr

ack

leng

tha

(mm

)



0

2

4

6

8

10

12

14

0 2 4 6 8 10

XFEM forecast

I II III


dadN

(10minus4

mm

cyc

le)





50000 100000 150000 200000 2500000

2

4

6

8

10

12

14

16

18

Specimen III


a(m

m)

N (cycles)


0

3

6

9

12

15

18




Surfa

ce cr

ack

leng

tha

(mm

)




0

05

1

15

2

25

3

0 1 2 3 4 5 6 7 8



Crac

k gr

owth

ratedadN

(10minus3

mm

cyc

le)


0

1

2

3

4

5

6

0 04 08 12 16 2 24



Crac

k le

ngth

a(m

m)





0

15

30

45

60

75

90

0 04 08 12 16 2



Crac

k pr

ojec

tive a

reaS

(mm

2)




Crac

k gr

owth

ratedSdN

(10minus3

mm

2c

ycle

)

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70



5 Conclusions




0

05

1

15

2

25

0 5 10 15 20

Resid

ual l

ifeN

(104

cycle

s)



Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










50000 100000 150000 200000 2500000

2

4

6

8

10

12

14

16

18

Specimen III


a(m

m)

N (cycles)


0

3

6

9

12

15

18




Surfa

ce cr

ack

leng

tha

(mm

)




0

05

1

15

2

25

3

0 1 2 3 4 5 6 7 8



Crac

k gr

owth

ratedadN

(10minus3

mm

cyc

le)


0

1

2

3

4

5

6

0 04 08 12 16 2 24



Crac

k le

ngth

a(m

m)





0

15

30

45

60

75

90

0 04 08 12 16 2



Crac

k pr

ojec

tive a

reaS

(mm

2)




Crac

k gr

owth

ratedSdN

(10minus3

mm

2c

ycle

)

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70



5 Conclusions




0

05

1

15

2

25

0 5 10 15 20

Resid

ual l

ifeN

(104

cycle

s)



Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

15

30

45

60

75

90

0 04 08 12 16 2



Crac

k pr

ojec

tive a

reaS

(mm

2)




Crac

k gr

owth

ratedSdN

(10minus3

mm

2c

ycle

)

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70



5 Conclusions




0

05

1

15

2

25

0 5 10 15 20

Resid

ual l

ifeN

(104

cycle

s)



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-Based Analysis for Crack Growth ...downloads.hindawi.com/journals/mpe/2013/820871.pdf · XFEM-Based Analysis for Crack Growth Characteristics of Diffusion Bonded