Int J Fract (2009) 159:151–165DOI 10.1007/s10704-009-9391-y

ORIGINAL PAPER

Computational modeling of mixed-mode fatigue crackgrowth using extended finite element methods

Yangjian Xu · Huang Yuan

Received: 13 August 2008 / Accepted: 4 August 2009 / Published online: 30 August 2009© Springer Science+Business Media B.V. 2009

Abstract The extended finite element method(XFEM) combined with a cyclic cohesive zone model(CCZM) is discussed and implemented for analysis offatigue crack propagation under mixed-mode loadingconditions. Fatigue damage in elastic-plastic materi-als is described by a damage evolution equation in thecohesive zone model. Both the computational imple-mentation and the CCZM are investigated based onthe modified boundary layer formulation under mixed-mode loading conditions. Computational results con-firm that the maximum principal stress criterion givesaccurate predictions of crack direction in comparisonwith known experiments. Further popular multi-axialfatigue criteria are compared and discussed. Computa-tions show that the Findley criterion agrees with tensilestress dominant failure and deviates from experimentsfor shear failure. Furthermore, the crack propagationrate under mixed mode loading has been investigatedsystematically. It is confirmed that the CCZM can agreewith experiments.

Keywords Mixed-mode fatigue crack propagation ·Extended finite element method · Cyclic cohesive zonemodel · Boundary layer formulation

Y. Xu · H. Yuan (B)Department of Mechanical Engineering,University of Wuppertal, 42097 Wuppertal, Germanye-mail: h.yuan@uni-wuppertal.de

1 Introduction

Since damage mechanics models confront difficultiesin computational convergence as well as in strain local-izations, the cohesive zone model (CZM) has beenpopular in simulating material failure (Scheider et al.2006; Tan et al. 2005; Yuan and Cornec 1990; Yuanet al. 1996), which provides an alternative way todescribe crack propagation. In these works the crackpropagations are only modeled for simple loading his-tory, i.e. fracture under monotonic loading conditions.The idea in monotonic fracture is that the material failsas soon as the critical fracture energy is exceeded, whilefatigue cracks follow accumulative damage. The cohe-sive zone model has been extended to model fatiguein recent years (Nguyen et al. 2001; Siegmund 2004;Yang et al. 2001). Compared to the crack simulationunder simple loading history, the development of thecyclic cohesive zone model (CCZM) is still in the earlystages. Both cyclic damage accumulation and mixed-mode failure are involved in controversial discussions.A cyclic cohesive model for engineering applicationsneeds substantial further understanding and validationwith experiments.

An additional limitation for the cohesive zone modelis related to the finite element method. In most ofthe published works the crack path was pre-assumed(Scheider et al. 2006; Tan et al. 2005; Yuan and Cornec1990; Yuan et al. 1996). The conventional finite ele-ment method does not allow new crack surfaces intoan element. Therefore, investigations of cohesive zone

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152 Y. Xu, H. Yuan

models are mainly limited to the cohesive traction-separation law with zero threshold under mode I load-ing. However, in engineering applications a fatiguecrack initiates under the shear loading condition (StageI) and propagate further under the mode I condition(Stage II) (Socie and Marquis 2000). In this sensefatigue crack propagation is generally mixed-mode.Especially in nonproportional multi-axial fatigue, thecrack direction can vary with crack growth, and in thesecases the crack path cannot be pre-assumed. Addition-ally, the cohesive traction-separation law calculatedfrom atomistic simulations implies a finite thresholdvalue for the cohesive zone (Krull and Yuan 2009). Asdiscussed in Zhai et al. (2004), it is difficult to sim-ulate the curve crack propagation, even impracticablefor 3D mixed-mode crack problems using FEM (Chenet al. 2005).

Recently several promising computational methodsfor considering discontinuity (Oden et al. 1998; Ziet al. 2005), generally referred as extended finite ele-ment methods (XFEM), have been developed based onMelenk and Babuska’s work on the approach of par-tition of unity (Melenk and Babüska 1996). In thesetype of methods, the discontinuity is introduced bymeans of additional degrees of freedom (DOFs) onthose elements where the discontinuity crosses. Themost appealing feature is that the XFEM inherits thefinite element framework and its advantageous proper-ties, such as sparsity and symmetry. Combining withthe XFEM, the cohesive zone model can be natu-rally incorporated to simulate mixed-mode crack pro-pagation and has found wide application (Jirasek andZimmermann 2001; Mergheim et al. 2005; Meschkeand Dumstorff 2007; Wells and Sluys 2001; Simoneet al. 2003; Zi and Belytschko 2003). Using the XFEM,one can consider crack initiation at an arbitrary materialpoint and crack propagation in an arbitrary direction,without adding extra nodes and elements. Additionally,it allows multi-cracks nucleation, growth and coales-cence without remeshing.

Under mixed-mode loading conditions crack prop-agation depends on loading mode mixity and intensity.The loading mode mixity affects crack direction, whilethe intensity determines the crack growth rate. In thepast decades many different fatigue criteria were sug-gested based on experimental observations and frac-ture mechanics considerations (Qian and Fatemi 1996;Richard 1984, 1985; Socie and Marquis 2000). A popu-lar criterion was developed based on maximal principal

stress around the crack tip, which was applied in brittlefracture analysis (Richard 1984, 1985). More experi-mental observations reveal that both fatigue and frac-ture of a cracked specimen share the same crack angle,at least for finite/high cycle fatigue (Richard 1989;Sander and Richard 2006). That is, one may use theconventional fracture criterion to predict fatigue crackdirection. On the other hand, multi-axial fatigue inves-tigation in uncracked specimens generally suggeststhat fatigue failure is related to the shear stress/strainamplitude and the normal stress on the shear plane.A systematic review of different fatigue models canbe found in Karolczuk and Macha (2005), Socie andMarquis (2000). As Findley (1956) model introduced[ f = (τ/2 + kσn)max], the damage indicator formulti-axial fatigue is based on shear stress/strain ampli-tude and normal stress. The index ‘max’ denotes thatthe damage indicator f reaches its maximum, i.e. thecritical plane, which gives the direction of material fail-ure under mixed-mode condition. Obviously, the Find-ley’s model differs from the maximal principal stresscriterion used in fracture mechanics.

In the present study, a modified boundary layermodel is used to investigate the correlation betweenexternal loading mixity and crack propagation basedon the XFEM in combination with the CCZM. Crackextensions under both simple and cyclic mixed-modeloading are considered. Different fracture criteria areevaluated. The feasibility of predicting fatigue crackpropagation behavior under mixed-mode loading basedon the CCZM is validated by known experiments.

2 Review of the extended finite element method(XFEM)

2.1 The discontinuity in finite elements

Comprehensive overviews of discontinuity methodsin conjunction with partition of unity property havebeen given by numerous publications (De Borst 2006;Belytschko et al. 1988; Jirasek and Zimmermann 2001;Mergheim et al. 2005). The XFEM, based on theapproach of partition of unity, is an efficient way toreduce mesh dependency when it is used to analyzecrack growth. The key lies in that the discontinuityfield space can be directly approximated by introduc-ing the enriched degrees of freedom and the enrichedbasis function ψ j . By combining the function ψ j with

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Using extended finite element methods 153

the standard basis function φi , a good, approximateproperty can be inherited from the standard basis func-tion, and the specific discontinuous property is fromthe enriched function ψ j as well.

Note that the approximation is only locally enrichedin the region of interest such as the damage zone.According to the approach of partition of unity, if thestandard basis function φi fulfills the partition of unity(i.e.

∑ni=1 φi = 1), the enriched field approximation

can be expressed as following:

u =n∑

i=1

φi

⎛

⎝ai +m∑

j=1

ψ j bi j

⎞

⎠ , (1)

where n is the number of nodes associated with thestandard basis functionφi , andψ j is the enriched func-tion with m terms. ai is the vector of the conventionaldegrees of freedom on node i , and bi j is the vector ofthe enhanced degrees of freedom associated with nodei . Since DOFs are enriched on the existing nodes, mod-ification of topology can be avoided.

In the finite element notation, the element shapefunction N fulfils the partition of unity. If the disconti-nuity is contained in the finite element and the enrichedbasis is taken as Heaviside function Hd , the approxi-mation of displacement field can be written as

u = N(a + Hd b) = Na︸︷︷︸standard

+Hd Nb︸ ︷︷ ︸enriched

, (2)

where a and b represent the standard and enriched nodaldegrees of freedom, respectively. The Heaviside stepfunction Hd is defined as

Hd =

0 if x∈ −,

1 if x∈ +.(3)

The whole FEM domain consists of two parts: + and−, with the discontinuity d , as shown in Fig. 1.In the formulation, only the nodes associated withthe discontinuity d have to be enriched. That is, theextra enriched term in Eq. (2) will be considered forthese nodes. Otherwise, the standard shape function isretrieved. By the enriched degrees of freedom, the dis-continuity is introduced directly. Effectively, the stan-dard DOFs, a are used to represent the continuum part,while the enriched DOFs b, stand for a displacementjump across the border d , expressed as

[u] = Nb |d . (4)

In a nonlinear analysis of the finite elements with thediscontinuity, the nodal displacements are computediteratively from the global equilibrium equations.

ΩΩ

+

-

dΓ

m

n t

tΓ

uΓ

Fig. 1 An FEM domain crossed by the discontinuity d andrestrained by the applied boundary conditions. d does not existbefore the material failure starts

2.2 Governing equations

We consider a domain, , with a discontinuity, d ,as shown in Fig. 1. Without taking body forces intoaccount, the equilibrium equations can be expressed asfollowing,

∇ · σ = 0 in , (5)

where ∇ is the gradient operator and σ is the Cauchystress tensor. The essential and natural boundary con-ditions are, respectively provided as

σ · m = t on d , σ · n = t on t ,

u = u on u . (6)

Herein m is the inward unit vector perpendicular to thediscontinuity d and n is the outward normal vectorof the boundary. Correspondingly, t is the inner trac-tion in d and t is the prescribed traction vector at theboundary t . u is the prescribed displacement at theboundary u .

Equation (5) can be converted into the governingequation of the XFEM by using the Galerkin formula-tion. If the discontinuity d is not traction-free, due tothe cohesive zone for example, we may introduce anadditional function to connect + and − as

t = T[u] = T(Nb)|d . (7)

The function T describes behavior of the discontinuityd . Generally speaking, d is determined during com-putation and contains nonlinear features so that all ofthe finite element equations are to be solved incremen-tally. Using the Galerkin method and considering thenonlinear behavior of the discontinuity curve, the lin-earized governing equations of the XFEM (Wells andSluys 2001) can be obtained as

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154 Y. Xu, H. Yuan

[Kaa Kab

Kba Kbb

] [ab

]

=[

fa,ext

fb,ext

]

−[

fa,int

fb,int

]

(8)

with

fa,int =∫

BT σd,

fb,int =∫

+BT σd +

∫

d

NT td,

fa,ext =∫

t

NT td,

fb,ext =∫

+t

NT td,

and

Kaa =∫

BT DBd,

Kab =∫

+BT DBd,

Kba = KTab =

∫

+BT DBd,

Kbb =∫

+BT DBd +

∫

d

NT TNd.

In the linearized equation system (8), a and bdenote the increment of nodal displacements in anincremental step. In addition, D is the material stiff-ness matrix; T is the cohesive stiffness matrix and B isthe strain matrix.

In the present work the crack tip is located at theend of the cohesive zone so that the stress at the cracktip becomes non-singular. One gets the same numeri-cal accuracy by the XFEM without using special shapefunctions.

3 Cyclic cohesive zone models for fatigue crackgrowth

3.1 The cohesive law for fracture

Through experiments it has been proven that the crackpropagation is dominated by the mode I loading case inboth brittle and ductile material for cracked specimens(Richard 1985, 1989). Based on this consideration, thefailure is dominated by the normal traction, Tn (Fig. 2).

-0.2

0

0.2

0.4

0.6

0.8

1

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Tn/

σm

ax,0

Monotonic Loading

Cyclic Loading (one cycle)

Degradation

δ n

Fig. 2 The cyclic cohesive zone model under pure normal sep-aration. Under the simple loading condition the cohesive lawis identical with the model of Xu and Needleman (1994), asκ0 = 0 in (11). Under unloading and reloading the tractiondecreases/increases linearly with separation. The material dam-age is characterized by diminishing tensile strength Tn togetherwith decreasing material stiffness

Therefore, the cohesive strength is considered to bean exponential function of the separation as suggestedby Xu and Needleman (1994), while the shear tractionTt is assumed to be linearly related to the tangentialseparation. Under monotonic loading conditions, therelationship between tractions and separations in thecohesive law is written as

Tn = φn

δ0exp

(

−κn

δ0− κt

2

δ20

)

, (9)

Tt = Gtκt

δ0, (10)

where

κn = κn + κ0 (11)

denotes the effective normal separation. κn is the nor-mal separation from the FEM computation. κ0 standsfor a model parameter to consider the threshold valueof the cohesive traction. For κ0 = 0 the model recursto the original version of the model of Xu and Needle-man (1994). Gt is the shear stiffness and κt is the tan-gential separation. The characteristic cohesive length isdenoted by δ0. The fracture energy, φn , resulting fromnormal separation for failure is calculated as

φn = eσmaxδ0, (12)

wherein the Euler’s number e = exp(1), and σmax isthe cohesive normal strength of material. In the crack

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Using extended finite element methods 155

simulation under simple monotonic loading conditions,the parameter σmax is a model parameter to fit experi-mental curves, together with δ0 (Scheider et al. 2006).Under cyclic loading conditions the material may failat a significantly lower loading level than the mate-rial strength. This implies that σmax decreases with thecyclic loading history.

3.2 The cyclic fatigue damage evolution

A CCZM should be able to describe both fractureunder monotonic loading and fatigue failure, i.e. dueto accumulative damage (Nguyen et al. 2001; Roe andSiegmund 2003; Siegmund 2004). The accumulativedamage will be described by an additional damageparameter, Dc, which is determined by a damage evo-lution equation for the cyclic loading process, as sug-gested by Roe and Siegmund (2003) and Siegmund(2004)

Dc = | κn |δ

(Tn

σmax− σ f

σmax,0

)

H(κ − δ0) (13)

with Dc ≥ 0 and κ = ∫ | κn | dt . In the aboveexpression H represents the Heaviside function whichprescribes that the damage accumulation starts oncethe accumulated material separation κ is greater thanthe characteristic length δ0. The cohesive strength isaffected by Dc, which can be written as

σmax = (1 − Dc)σmax,0

with σmax,0 as the initial cohesive strength. Obviously,this consideration is a direct extension of continuumdamage mechanics (Lemaitre 1996). Additionally, σ f

is the cohesive zone endurance limit, and δ is the accu-mulated cohesive length which scales the increment ofmaterial separation κn .

In this investigation, the unloading and reloadingare assumed to proceed along a current stiffness, kn =σmax,0(1 − dc)e/δ0, which is equal to the slope of thecurrent cohesive curve at zero separation. This assump-tion will lead to the presence of a residual separation.The material damage is presumed to be accumulatingduring the whole process of unloading and reloading,except for the case of compression, i.e. κn < 0. As forthe potential compressing case, from a viewpoint ofnumerical treatment, the material penetration at the dis-continuity is reduced by enhancing the material com-pression stiffness.

4 Implementation of the XFEM into ABAQUS

The XFEM (8) combined with the cyclic cohesivemodel (9)–(13) has been implemented within ABA-QUS via the user element subroutine (UEL) (ABAQUS2006). The subroutine UEL allows the user to define anew element in which all variables, e.g. strains, stressesand residual forces as well as the element stiffnessmatrix, have to be passed in and updated for a newiteration. In the present work, the degrees of freedomat enriched nodes are covered by some additional vari-ables to include displacement discontinuity within anelement. In the present UEL subroutine the main fol-lowing functions are achieved: (1) updating strains andstresses; (2) generating an element stiffness matrix; (3)determining tractions and separations in accordancewith cohesive law; (4) defining crack initiation andpropagation; (5) creating data for post-processing. Forsimplicity only two kinds of discontinuity patterns areimplemented, as shown in Fig. 3. That is, a cohe-sive zone can go through two opposite element edges(Fig. 3a) or through two adjacent edges of one element(Fig. 3b). All other crack paths can be represented bythese two patterns.

The integration scheme is crucial in the implemen-tation of the XFEM. Here, an intersected element isdivided into 8 triangular zones and integrated basedon 24 integration points, together with two integrationpoints for the discontinuity curve, as shown in Fig. 3.Note that the partition of the element is merely forintegration based on iso-parametric interpolations overall the triangles, while no additional node is needed.Additionally, to reduce computational efforts, only thepotential damage zone is covered with user-defined ele-ments, otherwise the conventional isotropic elementsare adopted. Connections between two zones are real-ized without special algorithms.

In the XFEM programme, another two built-inABAQUS subroutines are adopted. Using subroutineUEXTERNALDB, one can exchange data betweenexternal databases and user subroutines at the requiredsteps. Based on the subroutine, initial user-definedparameters can be passed into subroutine UEL, and theresults obtained from the user-defined elements can bewritten to the output file. The other important built-in subroutine is URDFIL by which the results of thegeneral elements from ABAQUS (e.g. element typeCPE4) can be explored. With these two subroutines,the user-defined elements and ABAQUS elements can

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156 Y. Xu, H. Yuan

(a) (b)

Fig. 3 Two types of intersected elements implemented in ABA-QUS. The middle line with two dots denotes the discontinuity inthe XFEM-element. a The discontinuity divides a quadrilateralinto two sub-quadrilaterals. b The discontinuity separates an ele-ment into one triangle and another pentagon. In both cases the

enhanced elements are divided into 8 sub-triangles for integra-tion. Each sub-triangle possesses three integration points (opencircles). The discontinuity curve is integrated using two integra-tion points (solid circles)

be coupled to calculate and export results. Since thepost-processor of ABAQUS does not support the user-defined elements, the output database (ODB) has to bemodified by the user in order to visualize the results.In the present work, a Python script is used to pro-duce an ODB file which is compatible with the ABA-QUS/Viewer.

5 Computational models

The XFEM and CCZM discussed previously have beenapplied to the investigations of three-point and four-point bending cases. The results are in good agree-ment with the experimental observations (Xu and Yuan2009a,b). In order to understand further the fracture andfatigue mechanisms, the modified boundary layer for-mulation is investigated. In fracture analysis, the mod-ified boundary layer formulation is popular to identifythe relation between loading mixity and crack tip field(Anderson 1995; Larsson and Carlsson 1973; Yuan2002). In this case both loading mixity and intensitycan be characterized by K I and K I I in the remotefield. In the present work the modified boundary layerformulation is used to predict crack angles and crackgrowth rate under the given mixed-mode loading. Fora small amount of crack growth the loading mixity of

inner crack will not be significantly affected so that onemay correlate crack propagation with external load-ing parameters K I and K I I . The modified boundaryformulation method states that the desired load casewith respect to different combination of K I and K I I

fields can be directly applied on the external boundary.The displacement-controlled loads can be obtained bysuperposing mode I and mode II displacement fieldsfrom fracture mechanics, as

• the displacements from mode I:

uI x = K I

G(1 + ν′)

√r

2πcos

θ

2

×[

1 − ν′ + (1 + ν′) sin2 θ

2

]

,

uI y = K I

G(1 + ν′)

√r

2πsin

θ

2(14)

×[

2 − (1 + ν′) cos2 θ

2

]

;

• the displacements from mode II:

uI I x = K I I

G(1 + ν′)

√r

2πsin

θ

2

×[

2 + (1 + ν′) cos2 θ

2

]

,

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Using extended finite element methods 157

uI I y = K I I

G(1 + ν′)

√r

2πcos

θ

2

[(−1 + ν′) (15)

+(1 + ν′) sin2 θ

2

]

.

Wherein, G is shear modulus and ν′ = ν/(1 − ν) withν as Poisson’s ratio. The applied displacements, ux anduy , are obtained by superposing uI with uI I like so:

ux = uI x + uI I x ,

uy = uI y + uI I y(16)

at the remote boundary with the radius of r0 =√x2

0 + y20 and the polar angle θ = tan−1 (y0/x0). The

initial crack tip is located at the origin of the geome-try (r = 0). The cyclic loading is imposed by varyingthe stress intensity factors, K I and K I I . For simplic-ity we only consider pulsating load with the loadingratio R = Kmin/Kmax = 0 so that K = Kmax. Therange of stress intensity factor K is correlated withthe incremental energy release rate G as

G = (1 − ν2)K 2

E. (17)

The global finite element mesh and the boundary con-ditions are plotted in Fig. 4. To reduce the artificialeffect of K fields on the crack, crack propagation isconfined to a small region, and its area is negligible incomparison to the whole specimen (less than 1%). Theradius of the crack extension region, l, is smaller than5%r0, where r0 represents the radius of the boundarylayer model. As shown in Fig. 4b, the mesh is highlyrefined in the crack extension region.

Computations were performed under the plane straincondition in ABAQUS, in which the CPE4-type ele-ment was employed, together with the user-definedelements discussed in the previous sections. To savecomputational time, only those elements near the crackpath were defined as the user-defined elements.

The material properties of the specimen are normal-ized by yield stress σ0 which is not needed in elasticmaterials. Accordingly, elastic modulus E = 300σ0,and Poisson’s ratio ν = 0.3. For the cohesive model,the cohesive strength and length are taken as σmax,0 =6.7σ0 and δ0 = 0.0153 mm, respectively. In this inves-tigation, in accordance with a suggestion from Roe andSiegmund (2004), the parameters σ f and δ are set to0.25σmax,0 and 4δ0, respectively.

6 Results and discussion

6.1 Crack kinking angles

It is a common postulate that a crack in brittle mate-rials propagates toward mode I, i.e. in the directionfollowing K I I = 0. Among known fracture criteria,the maximum tangential stress (MTS) criterion and themaximum energy release rate (MERR) criterion aremost popular, and have been widely applied in frac-ture mechanics analysis (Anderson 1995). Since thephilosophy of fracture mechanics differs from that offatigue assessment, a criterion for a fatigue crack couldbe distinctive. Under multi-axial loading conditions,for instance, the shear stress amplitude is responsiblefor fatigue failure and the fatigue crack follows the so-called critical plane of the damage indicator (Socie andMarquis 2000).

In the present section we are computing fatiguecrack growth based on various criteria with verificationof experimental data from literature. Richard (1985)and Sander and Richard (2006) carried out a number oftests for the mixed-mode crack extension under the sim-ple or cyclic loading cases. Most tests were based on theCompact Tension Shear (CTS)-specimen which wasput forward first by Richard (1984, 1985, 1989). Withthis specimen the mode-mixity in fracture mechanicscan be effectively and conveniently calculated, makingit easy to evaluate the crack extension by the fracturemechanics method. Different loading mode mixitiescan be approached by varying the loading angle (β) ofthe specimen in 15-steps, from β = 0 to 90. PureMode I (β = 0), pure Mode II (β = 90) or mixed-mode(β = 15, . . . , 75) situations can be generatedby the use of just a uniaxial testing system. The detailscan be obtained from Richard (1984, 1985, 1989).

6.1.1 Maximum tangential stress criterion

The MTS criterion is one of the most popular fracturemodels for material strength. It was proposed by Erdo-gan and Sih (1963) as

σθθ (θ)|θ=θ0 =[

K I

(

3 cosθ

2+ cos

3θ

2

)

−3K I I

(

sinθ

2+ sin

3θ

2

)]

max(18)

for the angular function of the circumferential stresscomponent σθθ . According to this criterion, the crack

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158 Y. Xu, H. Yuan

Fig. 4 The finite elementmesh for the modifiedboundary layer formulation(MBLF). The remoteboundary (r = r0) is loadedby the given displacementsaccording to the plane strainK -fields in fracturemechanics. a The wholemesh. b The refined mesharound initial crack tip

will advance along the radial direction θ = θ0, in whichthe circumferential stress σθθ becomes maximum, i.e.∂σθθ/∂θ = 0. It follows

tan

(θ0

2

)

= 1

4

(K I

K I I

)

± 1

4

√(

K I

K I I

)2

+ 8. (19)

In this criterion, K I and K I I are loading parameters andare directly used to determine the cracking orientation.This criterion has been verified by many experimentalresults.

6.1.2 Nonlocal maximum principal stress criterion

A nonlocal treatment becomes significant if materialfailure is related to a certain material volume, but notaffected by the stress state at a single mathematicalpoint. From discussions on nonlocal damage mechan-ics we learn that the local damage assumption maylead to a paradox in computations. A nonlocal crite-rion accounts for a weighted averaging of the damageover a certain area, which can reduce effects of thelocal stress distribution. That is, the nonlocal criterionmay diminish the influence of the finite element size.For this reason, the maximum principal stress criterionbased on the averaging method is becoming popular inthe computational mechanics community (Mergheimet al. 2005; De Borst 2006). In the present work we usethe nonlocal maximum principal stress criterion to pre-dict mixed-mode crack growth, as suggested by Wellsand Sluys (2001).

The nonlocal stress is computed as a weighted aver-age of the stress over ngp Gaussian points around the

cohesive zone tip and within a prescribed radius. Aweighted Gaussian function (Mergheim et al. 2005) isdefined as

w(r) = w(r)∑ngp

i=1 wi Aiwith

w(r) = 1

ρ√

2πexp

( −r2

[2ρ]2

)

. (20)

The nonlocal stress is calculated by the followingexpression,

σ =ngp∑

i=1

σiwi Ai . (21)

In the above equations r is the distance of the Gauss-ian point to the cohesive zone tip and ρ denotes theradius of the nonlocal zone. The nonlocal stress tensorresults from the sum of the local stresses at the Gauss-ian points i , the weight function wi and the associatedarea Ai . In the 2-dimensional case the orientation ofmaximum principal stress can be calculated from thenonlocal stresses as

tan(2θ0) = −2τxy

σx − σy. (22)

In the XFEM computations the maximum principalstress criterion includes two aspects: propagation ofthe cohesive zone, and direction of the cohesive zonepropagation. During computations the maximum prin-cipal stress ahead of the cohesive zone tip is calculatedand checked against the fatigue limit of the material.Once it exceeds the fatigue limit, the enhanced degreesof freedom in this element are activated. That is, the

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Using extended finite element methods 159

cohesive zone propagates into this element. As regardsthe direction of crack extension, the criterion states thatthe crack will grow in the perpendicular direction to thatof nonlocal maximum principal stress.

The nonlocal maximum principal stress differs fromthe local maximum principal stress just due to averag-ing over a certain region. In XFEM the NMPS is easierto realize in comparison with the MTS criterion.

In Fig. 5 the crack paths of two mixed-mode load-ing cases are plotted, and they are predicted based onthe NMPS criterion. The loading ratios, K I I /(K I +K I I ), for these two cases are 0.22 and 0.47, respec-tively, which correspond to the loading orientations ofthe CTS-specimen (Richard 1984, 1985), β = 30 andβ = 60. Whereas K I I /(K I + K I I ) = 0.22 showsa straight crack path, the higher mode II, K I I /(K I +K I I ) = 0.47, causes varying kinking angles after crackinitiation. With crack growth the angle θ increases fromθ0 to θ1 and finally reaches a steady state.

Figure 6a summarizes the steady crack angles asa function of the loading ratio. The computations arebased on three different criteria: MTS, NMPS andMERR. The experimental results were taken from theworks of Richard (1984, 1985, 1989). All criteria fallin a small scatter band in the whole loading ratio andagree with experimental results (Richard 1984, 1985,1989). The predictions based on the MTS and MERRcriteria come from the analytical solution (Anderson1995). The results of the nonlocal maximum principalstress criterion (NMPS) were acquired from XFEM sothat the crack path can be characterized in detail.

The predictions from MTS and MERR assume astraight crack, while the XFEM computational analysisshows a curved crack path. In Fig. 6b the crack initiationangles θ0 calculated from both XFEM computation andanalytical solution are summarized, as well as the sta-bilized crack angles θ1 from the XFEM computation,and the experimental results. After a crack initiates withkinking, the stress field around the crack tip changes.It follows that the crack no longer propagates in theinitial direction. The computations show that the crackinitiation angles are generally significantly smaller thanthe experimental data, while the stabilized angles agreewith experimental observation. Additionally, as shownin Fig. 5b, the transient phase of crack initiation inmixed-mode cracking is too small to be measured inexperiments. However, it can be observed from ournumerical XFEM calculation. To validate that the initialkinking is not induced by numerical errors, the analyt-

ical solutions are further derived from the mixed-modeanalytical crack stress fields and maximum principalstress criterion. The calculations indicate that the ana-lytical solutions are in good agreement with the XFEMresults, as shown in Fig. 6b.

6.1.3 The modified Findley’s multi-axial fatiguecriterion

Multi-axial fatigue criteria such as Findley (1956),Fatemi-Socie and Marquis (2000) and McDiarmid(1994), etc. have been proposed for predicting fatiguefailure. Combined with the critical plane concept, themulti-axial fatigue criteria try to predict crack prop-agation direction (Karolczuk and Macha 2005; Socieand Marquis 2000). Some experimental observationsin multi-axial fatigue seem, however, not to agree withfracture mechanics predictions.

Development of multi-axial fatigue life assessmentrelies mainly on extensive experimental observations(Karolczuk and Macha 2005; Socie and Marquis 2000).Life models are trying to give a prediction of com-ponent durability in service, without detailed infor-mation about crack initiation and crack propagation.Crack growth is caused by cyclic loading and will notbe described explicitly by fracture parameters, but bythe macro-stresses and strains. Therefore, fatigue dif-fers from fracture significantly. Based on experimentalobservations Sines (Socie and Marquis 2000) predictedthat the multi-axial fatigue failure is related to the shearstress amplitude and the total normal stress, but is inde-pendent of total shear stress and less dependent on thenormal stress amplitude. It turns out that the fatiguedamage indicator is a function of the shear stress ampli-tude, τ , and the normal stress at the shear plane, σn .Using the critical plane concept one can predict the fail-ure direction (Socie and Marquis 2000). Such consid-erations have been further extended to both high cyclefatigue and low cycle fatigue. This concept seems not tobe compatible with the mixed-mode fracture mechan-ics. In this section we are implementing the Findley’smodel to re-analyze crack propagation under mixed-mode loading conditions.

Findley took the idea of Sines (Socie and Marquis2000) and assumed that the material failure is charac-terized by the following parameter (Findley 1956),

f =(

τ

2+ βσn

)

max. (23)

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160 Y. Xu, H. Yuan

Fig. 5 Crack pathspredicted by the NMPScriterion: aK I /(K I + K I I ) = 0.22corresponds to the loadingorientation β = 30; bK I /(K I + K I I ) = 0.47corresponds to β = 60.The crack angle changesfrom the initiation angle θ0to the steady-state angle θ1

OA

B

OA

B

Xθ0

θ1

(a) (b)

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1KII/ (KI+KII)

θ (Ο

)

Experimental Value

MPS initial comp. solu.

MPS steady comp. solu.MPS analy. solu.

0

10

20

30

40

50

60

70

80

0 0.2 0.4 0.6 0.8 1KII/ (KI+KII)

θ (Ο

)

Experimental valueMPS steady comp. solu.MTS analy. solu.MERR analy. solu.

(a) (b)

Fig. 6 Crack propagation angles under different mixed-modeloading conditions, predicted by different criteria: MTS, NMPSand MERR. Experimental data are taken from the works ofRichard (1984, 1985). a The stabilized angle as a function of

the loading mixity. b Comparisons between the initiation angleθ0 and the steady-state angle θ1 based on analytical and compu-tational analyses using nonlocal MPS criterion

In Findley’s multi-axial fatigue model the normal stressis introduced in order to consider the effects of the meanstress, primarily for smooth specimens. Usually β issignificantly smaller than 0.5 which implies that fatigueis dominated by the shear stress amplitude. The cri-terion is effective for a proportional combination oftension and torsion loading configurations (Socie andMarquis 2000). The damage occurs at the plane withthe biggest value of f , the critical plane. According toFindley’s model (Socie and Marquis 2000) the criticalplane depends on the direction of the shear stress ampli-tude, the mean normal stress and the coefficient β. Forductile materials β is small and the position of the crit-ical plane approaches the maximum shear stress. Forbrittle materials β is large and the critical plane turnsperpendicular to the maximum principal stress.

To use the Findley’s model for cracked specimens,one has to reformulate the definition of the damageindicator (23). To consider normal stress and shearstress simultaneously, we re-write Eq. (23) into

f = (aσn + bτ) max. (24)

Both a and b are defined as material coefficients so thatthe effects of both shear stress amplitude and normalstress can be adjusted simultaneously. The expressionidentifies a critical plane for fatigue crack initiation andpropagation depending on both alternated shear stressand maximum normal stress at that plane. The crackwill propagate along the critical plane on which theresultant value of these two terms in Eq. (24) is max-imal and surpasses the failure limit f . Observing thisexpression, one finds that this fatigue criterion tends

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Using extended finite element methods 161

-60

-40

-20

0

20

40

60

80

0 0.2 0.4 0.6 0.8 1KII/ (KI+KII)

θ (Ο

)

Experimenta=0.0,b=1.0a=0.3,b=0.7a=0.5,b=0.5a=0.7,b=0.3a=1.0,b=0.0

Criteria: (aσ n+b∆τ )max= f

Fig. 7 Crack angle as a function of the loading mixity predictedby the modified Findley’s model (24)

to approach the maximum principal stress criterion infracture mechanics as b goes to zero.

Figure 7 shows crack angle as a function of theloading mixity computed using the XFEM with var-ious values for a and b. The results confirm that thecrack angle is sensitive to both parameters a and b. As bincreases the effect of shear stress rises. That means thecrack angle will kink even under mode I loading con-ditions, which seems not to coincide with the knownexperimental observations in a pre-cracked specimen(Richard 1989). On the other side, effects of a and bon the predicted crack angle are nonlinear. For b < 0.5and a > 0.5 the crack angle does not deviate from theknown experimental results significantly.

6.2 Fatigue crack propagation rates

Siegmund (2004) has used the cyclic cohesive modelfor fatigue crack growth in various specimens and loadcases. Their extensive investigations indicate the fea-sibility of CCZM for life prediction, and the computa-tions reproduce Paris’ law in cracked specimens basedon the incremental energy release rate G. However,their discussions are limited to some special cases dueto the limitations of FEM mentioned in the Sect. 1.Based on the current proposed method, the compu-tational prediction can be extended to more generalmixed-mode cases. In this section, the fatigue crackpropagation rate will be further investigated via themodified boundary layer formulation.

Firstly, the pure mode I loading is considered, illus-trating the detailed procedure of fatigue life predictionand investigating the validity in terms of applying theXFEM combined with CCZM in fatigue life assess-ment. In Fig. 8a, the approximately linear relation-ship between the crack extension and loading cyclenumber in the double-logarithmic coordinate systemis depicted. The slopes of these curves (i.e. crack prop-agation rates) depend on K I . As K I increases thecrack growth rate rises. On the basis of these data, thecrack propagation rates can be expressed as a functionof G/φn according to Expression (17). As shown inFig. 8b, it can be seen that d(a/δ0)/d N is linearlydependent on G/φn in double-logarithmic coordi-nates, which can be curve-fitted in Paris’ law

d(a/δ0)

d N= C

(G

φn

)m

(25)

with C = 67.35 and m = 2.16. Paris’ law onlydescribes stationary fatigue crack growth, whereas theCCZM also can predict fatigue crack growth in low andhigh loading amplitudes. By introducing the threshold,σ f in the damage evolution Eq. (13), one may shiftthe fatigue limit in the life curve. That is, the fatiguedamage predicted in the XFEM approaches a thresholdbelow which the material degradation will not occur.On the other hand, the normalized energy release raterange possesses an upper limit which is characterizedby the general cohesive law without cyclic damage.

The traction degradation processes on three materialpoints ahead of the initial crack tip under K I loadingconditions are recorded in Fig. 9, which shows dif-ferent damage processes in a cracked specimen. Allthree points are located on the crack path along theinitial crack direction, as shown in Fig. 4. Point A isat the initial crack tip. Point C is 10 elements awayfrom Point A, i.e. approximately, 0.3 l (l denotes thelength of the refined region). B is located in the middleof A and C. Since A is located at the initial crack tipwhere the stress concentration occurs initially and thestress level is highest, the cohesive zone goes throughA earlier than the other points. Once the criterion ofcohesive zone growth has been satisfied, the discon-tinuity will immediately be introduced into these ele-ments ahead of the crack tip. Thereafter, with the cyclicdamage accumulation and the cohesive zone propaga-tion, the loading capacity of the material point beginsto sink and the material degrades gradually. As shownin Fig. 9, Point A experiences the highest peak traction,

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162 Y. Xu, H. Yuan

-1.3

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-1.5 -1.4 -1.3 -1.2 -1.1 -1

log(∆G/fn)

log(

d(a

/d0)

/dN

)

KII=0

2. 61

0 . 5367)/( ⎛

⎝ ⎛⎝∆=

n

G

dN

ad

φδ

0

5

10

15

20

25

30

35

0 50 100 150 200 250N (cycles)

∆a/

d0

KI=2

KI=2.25

KI=2.5

KI=2.75

∆ K I=2.00

∆ K I=2.25

∆ K I=2.50

∆ K I=2.75

KII =0

(a) (b)

Fig. 8 Correlation between crack growth and mode I loading: a variation of crack growth with loading amplitude K I ; b dependenceof crack growth rate on G

-0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100Cycles (N)

T n/ σ

max

,0

-0.2

0

0.2

0.4

0.6

0.8

1

Dc

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.17 0.27 0.37 0.47 0.57δ n / δ 0

Tn/ σ

max

,0

point A

point B

point CTn at A

Tn at B Tn at C

Dc at B Dc at C Dc at A

(a) (b)

Fig. 9 Evolution of material damage predicted by XFEM: a the predicted traction-separation responses for three points A, B, C alongthe crack path; b the relationships between traction and damage evolution of the three points

Tn/σmax,0 ≈ 0.75, which is reached in the first cycle,and subsequently it decreases gradually with loadingcycles, till complete failure.

The material separation processes at B and C aredistinctly different from that at A. Both Fig. 9a, b indi-cate that the material damage process at B is similar tothat at C. Their traction first experiences the hardeningprocess, then drops as they form the envelope curves.In both cases the peaks of traction are nearly the same,Tn/σmax,0 ≈ 0.5, and the shapes of the envelope curves

are also similar, except that the process of Point C ispostponed. The similarity of the traction losing processfor Points B and C indicates that the damage processtends to be steady after a small crack growth phase.These results are in good agreement with experimentalobservation, and the steady state propagation of fatiguecrack indicates the features of Paris’ law.

Figure 9 additionally indicates that the damage indi-cator Dc grows over-proportionally due to the interac-tion of traction and separation, which is mathematically

123

Using extended finite element methods 163

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

-1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9log (∆G/fn)

log(

d(a/

d0)/

dN)

KII=0KI=0KI/KII=1KI/KII=2KI/KII=0.5

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

KII/ (KI+KII)

Scal

ed s

θθ|m

ax

0

0.2

0.4

0.6

0.8

1

1.2

Scal

ed c

rack

gro

wth

rate

MTS Analy. Solu.

Crack growth rate

σ θθ|max

KI =0

KI /KII =0.5 KI /KII =1 KI /KII =2

KII =0

(a) (b)

Fig. 10 Computational correlations under mixed-mode fatigueconditions: a the fatigue crack growth rate versus the appliedenergy release rate range; b the scaled σθθ |max obtained from

crack tip field according to the MTS criterion versus the scaledcrack growth rates at log(G/φn) = −1.25

described in Eq. (13). Furthermore, comparing the vari-ations of Dc over Points A and C, it can be found thatwhen the material point A is just completely damaged,i.e. Dc = 1, Dc on C is just setting out, meaning thatthe cohesive zone is just now embedded in this element.This result signifies that the length of the cohesive zonein this investigation is the distance between A and C.

Based on experimental observations (Richard 1984,1989; Sander and Richard 2006), the mixed-modefatigue crack in brittle materials or high cycle fatigueis searching in the direction of mode I and the crackpropagation is dominated by the mode I stress field.In this sense, understanding of the mode I crack isessential for understanding of mixed-mode cracks. Toexamine the validity of the proposed method for mixed-mode fatigue, various combinations of K I and K I I

have been considered. In Fig. 10a, fatigue crack growthrate, d(a/δ0)/d N , is plotted with respect to the nor-malized energy release rate range (G/φn), under var-ious mixed-mode loadings. For the mixed-mode case,G = G I + G I I is introduced.

We find a linear correlation between log[d(a/δ0)/

d N ] and log(G/φn) for all cases analyzed with aunique exponent m of Paris’ law (25), regardless ofmode-mixity. That is, even for the mixed-mode fatiguecracking there exists a Paris’ law as, d(a/δ0)/d N =C(G/φn)m , in which m is independent of mode-mix-ity, but the parameter C varies with K I I /(K I + K I I ).

The computations show that the crack growth rate interms of pure K I I is evidently higher than that of pureK I . The maximum rate is reached under mixed-modeloading. This is rather unexpected, because one wouldusually assume the maximum rate under the mode Iloading case, due to normal stress failure assumed inthe model.

To clarify this variation, the crack growth rate fora given loading intensity, log(G/φn) = −1.25, isplotted as a function of loading mixity in Fig. 10b.For comparison, the maximum circumferential stressσθθ |max in accordance with the MTS criterion is cal-culated and plotted in Fig. 10b. Note both curves arescaled by their maximum values and show similar vari-ations with K I I /(K I + K I I ). If one accepts the MTSassumption for the mixed-mode, the highest circumfer-ential stress under the given loading intensity appearsat around K I I /(K I + K I I ) ≈ 0.5, and so does thecracking velocity. From the curves one may concludethat the maximum crack propagation rate has to occurunder a mixed-mode condition.

7 Conclusions

In the present paper the formulation of the XFEMand the CCZM have been discussed and applied forthe analysis of mixed-mode fatigue cracking. Both

123

164 Y. Xu, H. Yuan

implementation and verification of CCZM in combina-tion with XFEM in the frame of ABAQUS have beenreported. Application of XFEM allows displacementdiscontinuities within finite elements so that the curvedcrack propagation can be simulated with less effort.The combination of XFEM and CCZM possesses anappealing ability to predict the fatigue crack curvilin-ear extension under mixed-mode loading.

Based on the modified boundary layer formulation, adetailed investigation has been conducted regarding thecrack propagation direction under mixed mode load-ing. Compared with known experimental results, thenonlocal stress based maximum principal stress cri-terion of the XFEM accurately predicts crack prop-agation behavior in brittle materials. Moreover, thecomputation shows that the initial crack kinking anglediffers from the stabilized crack direction. The tran-sient region, however, is very small. These computa-tions agree with known experimental data.

Additional computations based on a modified Find-ley’s multi-axial fatigue criterion reveal that Findley’smodel approaches experimental observations only withvery small shear stress contribution. This implies thatthis kind of criterion is only suitable for consideringshear stress contribution, for fatigue crack propagationin ductile materials.

In fatigue analysis the crack propagation rate isanother important issue. In this context the fatigue crackrate prediction under mixed mode loading in the frame-work of XFEM combined with CCZM has been inves-tigated as well. The numerical simulation shows thatthe computational results match the known form ofParis’ law under mixed mode condition. Additionally,simulations verify the proposed method is able to sim-ulate the crack initiation and steady propagation pro-cess for very low cycle problems. The computationalresults in fatigue simulation coincide with the experi-mental records in higher loading levels. For finite cyclefatigue the evolution of damage accumulation pre-dicts over-proportional increment of damage indicator.This implies that the damage evolution equation needsfurther modifications.

Although the results obtained so far are impor-tant and many experimental phenomena can besimulated properly, the main difficulties for theapplication of the CCZM are in the formulation ofthe cyclic damage evolution equation as well as in thedescription of the shear stress failure. Especially in duc-tile materials the shear stress becomes significant in

fatigue and cannot be neglected. Additionally, the highcycle fatigue problem needs a robust method beyondcycle-by-cycle computations. For engineering applica-tions more detailed experimental and computationalverifications are necessary.

Acknowledgments The present work is financed by theGerman Science Foundation (DFG) under the contract numberYU119/5-1.

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