Review Article Computational Methods for Fracture in

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2013 Article ID 849231 38 pageshttpdxdoiorg1011552013849231

Review ArticleComputational Methods for Fracture in Brittle and Quasi-BrittleSolids State-of-the-Art Review and Future Perspectives

Timon Rabczuk

Institute of Structural Mechanics Bauhaus-Universitat Weimar Marienstraszlige 15 99423 Weimar Germany

Correspondence should be addressed to Timon Rabczuk timonrabczukuni-weimarde

Received 1 August 2012 Accepted 3 September 2012

Academic Editors S Li and R Samtaney

Copyright copy 2013 Timon Rabczuk This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An overview of computational methods to model fracture in brittle and quasi-brittle materials is given The overview focuses oncontinuummodels for fracture First numerical difficulties related tomodelling fracture for quasi-brittlematerials will be discussedDifferent techniques to eliminate or circumvent those difficulties will be described subsequently In that context regularizationtechniques such as nonlocal models gradient enhanced models viscous models cohesive zone models and smeared crack modelswill be discussed The main focus of this paper will be on computational methods for discrete fracture (discrete cracks) Elementerosion technques inter-element separation methods the embedded finite element method (EFEM) the extended finite elementmethod (XFEM) meshfree methods (MMs) boundary elements (BEMs) isogeometric analysis and the variational approach tofracture will be reviewed elucidating advantages and drawbacks of each approach As tracking the crack path is of major concernin computational methods that preserve crack path continuity one section will discuss different crack tracking techniques Finallycracking criteria will be reviewed before the paper ends with future research perspectives

1 Introduction

The prediction of material failure is of major importance inengineering and materials Science In the past two decadesa huge effort was made to develop novel efficient and accu-rate computational methods for fracture and an enormousprogress was madeThis paper will give an overview on thoseadvancesThe paper focuses on continuummodels for brittleand quasi-brittle fracture It will not discuss issues related toductile fracture with plastic deformation prior to localizationsee for example the excellent work of the group of Prof Li[1ndash7]

There are three failure modes see Figure 1 Mode-I failureis related to crack opening mode-II failure (sliding) is a pureshear failure mode and mode-III failure (tearing) can beconsidered as out-of-plane shearing In many applicationsmaterials will fail due to a mixed mode failure Fracture inbrittle materials can be modelled by linear elastic fracturemechanics (LEFMs) Brittle materials are characterized bylinear elastic material behavior in the bulk and a smallfracture process zone However LEFM cannot be used whenthe failure process zone is of the order of the size of the

structure The relative size of the fracture process zone (119897)

with respect to the smallest critical dimension of the structure(119863) is important for the choice of the fracture model [8]For 119863119897 gt 100 LEFM is valid while for 5 lt 119863119897 lt 100 a(nonlinear) quasi-brittle fracture approach should be usedinstead see also the contributions by [9 10] Gdoutos [8]recommends the use of nonlocal damage models for119863119897 lt 5The length of the fracture process zone is of the order of thecharacteristic length [11]

119897119888ℎ

=119864119866

119891

1198912119905

or 119897119888ℎ

=119864119866

119891

(1 minus 1205992) 1198912119905

(1)

where 119864 is the modulus of elasticity 120599 is the Poissonrsquos ratio119866119891is the fracture energy and 119891

119905denotes the tensile strength

of the material Carpinteri [12] introduced a nondimensionalbrittleness number 119904

119864= 119866

119891(119891

119905119887) where 119887 is a geometric

measure He tested pre-notched beams under 3-point bend-ingThe distance between the notch and the upper boundaryof the beam is 119887 If the process zone is small compared to119887 the failure is brittle and LEFM is applicable Otherwise

2 ISRN Applied Mathematics

Mode I Mode II Mode III

Figure 1 Different failure modes

in the absence of plasticity we may call the failure quasi-brittle The stress-strain curve for quasi-brittle materials suchas concrete ceramics and rock is characterized by a dropof the stress after the peak stress is reached This softeningcan be explained by damage mechanics due to the initiationgrowth and coalescences of microcracks that reduce theeffective cross-sectional area Figure 2 For more details thereader is referred to numerous articles and books on damagemechanics [13ndash26]

The use of strain-softening material models leads toan ill-posed boundary value problem (BVP) [27ndash31] Thesolution of ill-posed BVPs with computationalmethods leadsto certain difficulties Bazant and Belytschko [32] showedthat the deformations localize in a set of measure zero in1D the localization will occur in a point in 2D in a lineand in 3D in a surface They further showed that the energydissipation approaches zero with increasing refinement of thediscretization yielding physically meaningless results

The difficulties that emerge with the material instabilitiesdue to the ill-posedness of the BVP can be avoided by regular-ization techniques such as higher order continuum modelsgradient based models and polar theories (the most well-known theory is probably the Cosserat continuum) nonlocalmodels viscous models or cohesive zone models Theseregularization techniques introduce a characteristic lengthinto the discretization In other words the local character ofthe deformation is lost and the fracture is ldquosmearedrdquo over acertain domain involving several elements Therefore a veryfine mesh has to be used in order to resolve the crack As thecrack introduces a jump in the displacement field methodsthat smear the crack over a certain region are not capableof representing the correct crack kinematics Moreover thedifferent length scales (of the structure and the characteristiclength) may significantly increase the computational costA lot of effort has been devoted to develop methods thatare capable of capturing the crack and take advantage ofregularization techniques that can be coupled to these strongdiscontinuity approaches Modeling fracture with strong-discontinuity approaches require two key ingredients

(1) a method that is capable of capturing the crackkinematics and

(2) a cracking criterion that determines the orientationand the length of the crack

Computational methods that describe the topology of thecrack as continuous surface require also methods to describethe crack surface and track the crack path The paper isstructured as follows In the next section popular regulariza-tion techniques are summarized including gradient enhancedmodels non-local models viscous models cohesive zonemodels and smeared crack models Section 3 gives anoverview of the state-of-the-art computational methods for(discrete) fracture of brittle and quasi-brittle solids Section 4is devoted to crack tracking procedures as it is a majorchallenge for computational methods preserving crack pathcontinuity Section 5 discusses cracking criterion that deter-mine the transition from the continuum to the discontinuumThe article finishes with future research directions and someconcluding remarks

2 Regularization

21 Gradient-Enhanced Models Gradient-enhanced modelsor briefly called gradient models are typically describedby differential equations that contain higher order spatialderivativesThe coefficientsmultiplying the terms of differentorder have different physical dimensions It is possible todeduce the characteristic length from their ratio Gradientmodels can be incorporated into plasticity format [33 34]damage format [22 35] or in a combination of both [15] Sincethey require higher order spatial derivatives higher orderelements need to be used Gradient models are sometimesreferred to asweakly non-localmodels see Bazant and Jirasek[14] though some recent models [22] are also classified asstrongly non-local

22 Nonlocal Models Strongly non-local models are modelsof the integral type see Figure 3 In non-local formulationsthe stress at a given material point does not only depend onthat local strain but also on the strain of the neighboringpoints This effect is obtained by averaging certain internalvariables in a given neighborhood Figure 3 The size ofthis neighborhood is a material parameter that dependsalso on the geometry of the structure of interest In otherwords changing the dimensions of the problem requiresrecalibration of the parameters of the non-local model Theycan be obtained by an inverse analysis Bazant and Jirasek [14]

ISRN Applied Mathematics 3

Real damaged material Undamaged equivalent continuum

120601(119863)

120590 120576120590eff 120576eff

Figure 2 The equivalence principe of continuum damage mechanics

1

Volume equivalence

119877 119877119903

119871119889120572119899119897

119883119877

119903

119878

Figure 3 Principle of nonlocal constitutive models with the typical bell-shaped domain of influence

give an excellent overview on non-localmodels of the integraltypes and physical motivations (see also Bazant [36]) as wellas suggestions for calibrating material parameters

23 Viscous Models The introduction of a viscosity canalso restore the well-posedness of the BVP or initial BVP(IBVP) It can be regarded as introducing higher order timederivatives similar to the gradient models Considering thedimensions of the viscosity 120578 (kg(ms)) the dimensions of theYoungrsquosmodulus119864 (kg(ms2)) and the dimension of themassdensity 984858(kgm3) there is indeed a length scale 119897

119888ℎassociated

with the viscosity given by [14]

119897119888ℎ

=120578

radic119864984858 (2)

where 119888 = radic119864984858 is the longitudinal propagation velocityin 1D However the well-posedness of the IBVP is onlyguaranteed during the time span of the viscosity (119905

0=

119897119888ℎ119888) For visco-plastic models Etse and Willam [37] have

shown that after discretization hyperbolicity of the linearizedmomentum equation can be lost if a critical time step isexceeded

The introduction of a viscosity can sometimes be physi-cally motivated The strain rate effect and the correspondingdynamic strength increase for example can be captured byviscous damage models [38]

24 Cohesive Zone Models Cohesive zone models (CZMs)also restore the well-posedness of the (I)BVP In contrast tothe models described before CZMs can be combined withcomputational methods that maintain the local character ofthe crack In cohesive cracks a traction-separation modelis applied across the crack surface that links the cohesivetraction transmitted by the discontinuity surface to thedisplacement jump characterized by the separation vectorCZMs go back to the 60rsquos and were originally developedfrom Dugdale [39] and Barenblatt [40] They were appliedin metal plasticity to take into account friction along neigh-boring grains Hillerborg et al [41] extended this concept tomodel crack growth in concrete They called their approach119891119894119888119905119894119905119894119900119906119904 119888119903119886119888119896 119898119900119889119890119897 The main difference between thefictitious crackmodel of Hillerborg and the CZM byDugdale[39] and Barenblatt [40] is that crack initiation and propaga-tion is not restricted along a predetermined path but crackscan initiate anywhere in the structure

The principal idea of cohesive cracks is shown in Figure 4The cohesive model is used in the so-called process zonesometimes also called cohesive zoneThe complex stress statearound the crack tip is lumped into a single surface Thefirst approach by Hillerborg et al [41] was limited to mode-Ifracture Meanwhile many models have been developed thatare able to handle mixed mode fracture and other complexphenomena including irreversible deformations stress tri-axiality and rate dependence [42ndash47] Some CZMs include


Zone of microcracking

Cohesive zone

Macrocrack

Cohesive tractionCohesive crack

120596119898119886119909

Figure 4 Principle of cohesive crack models

fine scale effects obtained from micromechanical modelsor atomistic simulations [48ndash51] Note that the methodsdeveloped in [52ndash54] are different to conventional cohesivezone models as the interphase zone is a finite size zone andtherefore enables the simulation of complex mixed-modefracture and thermomechanical fracture They also allow thecombination of the Cauchy-Born rule with cohesive strength

CZMs can be classified into initially rigid models (some-times also called extrinsic models) Figure 5(d) and initiallyelastic models Figure 5(e) While the initially rigid modelsare characterized by a monotonic decrease in the cohesivetraction initially elastic models exhibit initially a positiveslope Figure 5(e) As the cohesive traction increases ininitially elastic models the stresses in the neighboring areaswill increase as well leading to spurious cracking Cohesivecracking will extend to a finite zone with infinitely closecracks with infinitely small crack openings whichmeans thatlocalized cracking is in fact forbidden [55]

Initially rigid models cause numerical difficulties whenelastic unloading occurs at an early stage such that thestiffness for the unloading case tends to infinity It is believedthat initially rigid models are better suited particularly inthe context of dynamic fracture A good overview and adiscussion on initially elastic and initially rigid models aregiven for example in Papoulia et al [56] and referencestherein

An important material parameter of the CZM is thefracture energy 119866

119891 In the one-dimensional case of pure

mode I-failure the fracture energy 119866119891= int

119908max

0

119905(119908)119889119908 is theenergy that is dissipated during crack opening It is related tothe area of the cohesive crack 119905 is the cohesive traction 119908 is

Cohesive traction

Crack opening

(a)

Cohesive traction

Crack opening

(b)

Cohesive traction

Crack opening

(c)

Cohesive traction

Crack opening

(d)

Cohesive traction

Crack opening

(e)

Cohesive traction

Crack opening

(f)

Figure 5 Typical one-dimensional cohesive models (a) The modelby Dugdale1 for fracture across grain boundaries in metal (b) amodified Dugdale model (c) linear decaying cohesive model (d)exponential decaying model (e) model with positive initial stiffnessfor brittle fracture and (f) non-admissible cohesive model

the crack opening and 119908max is the crack opening at which nocohesive forces are transmitted see also Figure 4

25 Smeared Crack Models Themain idea of smeared crackmodels is to spread the energy release along the width of thelocalization band usually within a single element so that itbe objective This is achieved by calibrating the width of theband such that the dissipated energy is the correct one Thisintroduces a characteristic length into the discretization thatdepends on the size of the elements

A milestone in smeared crack models is the crack bandmodel by Bazant and Oh [57] It is based on the observationthat the process zone localizes in a single element the so-called crack band Therefore Bazant and Oh [57] modifiedthe constitutive model after localization so that the correctenergy is dissipated Since the width of the numericallyresolved fracture process zone depends on the element sizeand tends to zero if the mesh is refined the crack bandmodel cannot be considered as real localization limiter It onlypartially regularizes the problem in the sense that the globalcharacteristics do not exhibit spurious mesh dependenceWhen the element size exceeds the size of the process zonethe stress-strain curve has to be adjusted horizontally as


120590

120576pre 120576post

120576

(a)

120590

120576pre 120576post

120576

(b)

120590

120576

(c)

Figure 6 Adjustment of the stress-strain curve in the crack bandmodel (a) master curve and decomposition of the strains intoprelocalizaton and post-localization parts (b) horizontal scaling foran element larger than the physical process zone and (c) snap backfor too large elements reproduced from Bazant and Jirasek [14]

shown in Figure 6(b) This makes the stress-strain curvein the post-localization region steeper and the appropriatescaling ratio achieves the dissipation energy remains thesameThe crack bandmodel was successfully used for mode-I fracture The extension to mixed mode failure and complexthree-dimensional stress states is difficult Besides exact scal-ing leads to a system of non-linear equations that can only besolved iteratively Moreover when the elements are too large(larger than the physical process zone) the adjusted post-localization diagramwill develop a snap back see Figure 6(c)In turn when the elements aremuch smaller than the process

zone the adjusted post-localization diagram becomes lesssteep and the band of cracking elements converges to a setof measure zero Thus the limit corresponds to the solutionof a cohesive crack model [58]

The smeared crack models can be further classified intorotating and fixed crack models In the fixed crack modelthe orientation of the crack is not allowed to change Fixedcrack models can be again classified into single fixed crackmodels and multiple fixed crack models In the latter casemore than one crack initiates in an element Often a secondcrack is only allowed to be initiated perpendicular to the firstcrack (this should avoid the initiation of too many cracks)see Figure 7(b)

The orientation of the crack is typically chosen tobe perpendicular to the direction of the principal tensilestrain or principal tensile stress Since the direction of theprincipal tensile strain changes during loading fixed crackmodels result in too stiff system responses Fixing the crackorientation leads to stress locking meaning stresses aretransmitted even over wide open cracks mainly caused dueto shear stresses generated by a rotation of the principalstrain axes after the crack initiation Therefore rotating crackmodels have been introduced where the crack is rotatedwith the principal strain axis see Figure 7(a) HoweverJirasek and Zimmermann [59] have shown that even rotatingcrack models suffer from stress locking caused by the poorkinematic representation of the discontinuous displacementfield around the macroscopic crack Unless the directionof the macroscopic crack that is represented by a bandof cracked elements is parallel to the element sides thedirections of the maximum principal strain determined fromthe finite element interpolation at individual material pointsdeviate from the normal to the macroscopic crack Thelateral principal stress has a nonzero projection on the cracknormal which generates cohesive forces acting across thecrack even at very late stages of the cracking process whenthe crack should be completely stress-free Therefore Jirasekand Zimmermann [60] proposed to combine rotating crackmodels with scalar damage models Since in scalar damagemodels all stress components tend to zero when the crackopens widely no spurious stress transfer occurs any more

Some models combine the rotating and fixed smearedcrack approach [61] Up to a certain crack opening the crackis allowed to rotate Then the crack orientation is fixedOther interesting contributions can be found for example in[17 61ndash65]

Smeared crack models in finite element analysis oftenexhibit a so-called mesh alignment sensitivity or mesh ori-entation bias see Figure 8 meaning the orientation of thesmeared crack depends on the orientation of the discretiza-tion When smeared crack models are used in the contextof meshfree methods with isotropic (circular) support theldquomeshrdquo orientation bias can be alleviated

3 Computational Methods for Fracture

31 Element Erosion The easiest way to deal with discretefracture is the element erosion algorithm [66ndash69] Element


119910

2

119909

Φ119900

(1)Φ119903

1

(at crack initiation)

119909 119910 global system

1 2 principal strain system

(a)

1Crack system principal strain system

(b)

Figure 7 Principle of (a) rotating crack models (b) fixed crackmodels

erosion algorithms do not require any representation of thecrackrsquos topology Most approaches do not really delete theelements but set the stresses in the ldquodeletedrdquo elements to zeroApproaches that indeed remove elements often replace thedeleted elementswith rigidmasses In order to account for thecorrect energy dissipation in the post-localization domainconcepts from smeared crack models are applied Song etal [70] compared different computational methods for cer-tain benchmark problems in dynamic fracture They reportextreme mesh sensitivity for element erosion algorithmsand conclude they are not well suited for dynamic fractureHowever due to their simplicity they are incorporated incommercial software packages such as LS DYNA

For brittle fracture Pandolfi and Ortiz [71] derived aninteresting eigenerosion approach Their idea is based onthe eigenfracture approach by Schmidt et al [72] wherethe eigendeformation is restricted in a binary sense Theelement can be either intact or fractured In the first case the

(a)

(b)

Figure 8 Mesh orientation bias for smeared crack models

element behavior is elastic Otherwise the element is erodedThey showed that in contrast to many other element erosionapproaches [66 67 73 74] the eigenerosion approach con-verges to Griffith fracture with increasing mesh refinement

32 Interelement-Separation Methods Standard finite ele-ments have difficulties to capture the crack kinematics sincethey use continuous trial functions that are not particularlywell adapted for solutions with discontinuous displacementfields One of the first models capable of modelling crackswithin the FEM are the so-called interelement-separationmethods [46 47 75ndash80] Interelement seperation methodsweremostly applied to dynamic fracturewith cohesive cracksIn the approaches by Xu and Needlemen [75 76] cohesivesurfaces were introduced at the beginning of the simulationIn contrast Camacho and Ortiz [46] adaptively introducedcohesive surfaces at element edges when a certain crackingcriteria is met In interelement-separation methods cracksare only allowed to develop along existing interelement edgesThis endows the method with comparative simplicity butcan result in an overestimate of the fracture energy whenthe actual crack paths are not coincident with element edgesThe results depend not only on the mesh size (and formof the chosen element) but also on the mesh bias thatcan be compensated only by remeshing that is particularlycumbersome in 3D It has been noted that the solutionssometimes depend on mesh refinement see Falk et al [81]


This sensitivity has been mollified by adding randomness tothe strength as in Zhou and Molinairi [80] and Espinosaet al [82] For dynamic problems interelement-separationmethods are often used with rigid cohesive zone modelsSpecial care has to be taken in order to guarantee timecontinuity Time continuity requires temporal convergencewhen the time step tends to zero Violating time continuitymight lead to unstable or oscillatory responses and the resultsseverely depend on the chosen time step [56 83]

Many interesting problems have been studied byinterelement-separation methods [47 75 79 82ndash84] Theyare most popular for dynamic fracture and problemsinvolving numerous cracks with complex fracture patterns

Special crack tip elements for linear fracture problemswere developed in the early 70rsquos [85ndash88] They account forthe crack tip singularity and were applied to stationarycracks Recently such approaches were incorporated in thesmoothed finite element method by Liu et al [89 90]These approacheswere extended to anisotropicmaterials [91]interface cracks [92] and plastic fracture mechanics [93] In[94 95] these approaches were applied to crack propagationproblems in two dimensions Adaptive remeshing techniqueswere employed As the crack is aligned to the crack bound-ary these methods can also be classified as interelement-separation methods One drawback is that only the radic119903-term is accounted for in the crack tip element but not thediscontinuity behind the crack tip

33 Embedded Elements (EFEM) In 1987 Ortiz et al [77]modified the approximation of the strain field to captureweakdiscontinuities in finite elements to improve the resolutionof shear bands Therefore they enriched the strain field toobtain the kinematic relations shown in Figure 9(a) Basedon this idea Belytschko et al [96] allowed two parallel weakdiscontinuity lines in a single element Figure 9(b) so thatthe element was able to contain a band of localized strainDvorkin et al [97] were the first who developed a methodthat was able to deal successfully with strong discontinuitiesin finite elements the class of embedded elements (EFEM)The name comes from the fact that the localization zoneis embedded in a single element see Figure 9(c) This waycrack growth can be modeled without remeshing This classof methods are much more flexible than schemes that allowdiscontinuities only at element interfaces and it eliminatesthe need for continuous remeshing Several different versionsof the EFEMwere developed All techniques had in commonthat the approximation of the displacement fieldwas enrichedwith additional parameters to capture the jump in thedisplacement field in a single element These parametersin the following called enrichment were inherent to thecracked element and the jump in the displacement fielddepends only on this enrichmentThe differentmethodswerederived from various principles starting from the extendedprinciple of virtual work over Hellinger Reissner to Hu-Washizu variational principle using an enhanced assumedstrain (EAS) or B-bar format One major advantage of theEFEM (compared to XFEM discussed in the next section)is that the additional unknowns could be condensed on the

(a)

(b)

119878

Ω+0Ωminus0

(c)

Figure 9 Element with (a) one weak discontinuity (b) two weakdiscontinuities and (c) one strong discontinuity

element level so that discontinuities could be captured withvery small changes of the existing code

The first version of the EFEM is often called staticaloptimal symmetric (SOS) since traction continuity is fulfilledbut it is not possible to capture the correct crack kinematicsThe correct relative rigid body motion of the element isnot guaranteed and it has been shown by Jirasek [98] thatSOS formulations lead to stress locking The kinematicaloptimal symmetric (KOS) version by Lotfi and Shing [99]ensures the correct crack kinematics but violates the tractioncontinuity condition Consequently the criteria for the onsetof localization written in terms of stresses in the bulk are


(a)

(b)

Figure 10 (a) Piecewise constant crack opening in embeddedelements and (b) linear crack opening for XFEM

no longer equivalent to the same criteria written in termsof the tractions on the discontinuity area for example forthe Rankine criterion the normal traction at the onset oflocalization should be equal to the tensile strength andthe shear traction should be zero This cannot be properlyreproduced by the KOS formulation The kinematical andstatical optimal nonsymmetric (KSON) version of embeddedelements (see Dvorkin et al [97] Klisinski et al [100])guarantees traction continuity and the appropriate crackkinematics but leads to a non-symmetric stiffnessmatrix withall its disadvantages with respect to solving the linearizedsystem of equations

The approximation of the displacement field in the EFEMis given by [101]

uℎ (X) = sum119868isinS

119873119868(X) u

119869+M

(119890)

119904(X)

u(119890)119868

(119883)

(3)

where u119869are the nodal parameters of the usual approxi-

mation u is the enrichment and the superimposed ldquo(119890)rdquo

indicates the enrichment on element levelThe functionM(119890)

119904

is given by

M(119890)

119904(X) =

0 forall (119890) notin 119878

119867(119890)

119904minus 120588(119890) forall (119890) isin 119878

120588(119890) =

119873+

119890

sum119868=1

119873+

119868(X)

(4)

where 119867119904is the step function acting on the crack line 119878 and

119873+

119890is the number of nodes of element (119890) that belong to the

domain Ω+0 see Figure 9(c) The discretization of the strain

field can be written as

120598ℎ

(X) = sum119868isinS

(nabla0119873119868(X) otimes u

119868)119878

minus (nabla120588(119890)

otimes

u(119890)119868

(119883)

)119878

+120578(119890)119904

119896(

u(119890)119868

(119883)

otimes n)

119878

(5)

where the superimposed 119878 denotes the symmetric part of atensor 120578(119890)

119904119896 is a regularized Dirac delta function and 120578(119890)

119904is

a collocation function defined as

120578(119890)

119904=

1 forallX isin 119878119896

119890

0 forallX notin 119878119896119890

(6)

with the thickness 119896 of the localization band Considering theequilibrium equation in elastostatics and with the trial andtest function of the structure of (3) the discrete equations inmatrix form can be written as

[K(119890) K

(119890)

K(119890) K

(119890)

]u(119890)u(119890)119868

=

Fext119868

0 (7)

with

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879lowastCB 119889Ω

0

K(119890)

= intΩ0


0

(8)

whereC is the elasticity tensor B is the B operator containingthe derivatives of the shape functions and B is the B operatorof the enrichment that depends on the embedded elementformulation (SOS and KOS or KSON) For the KSON formu-lation B

lowast= B that will result in a non-symmetric stiffness

matrix For the SOS and KOS formulation Blowast

= B Theelemental enrichment u(119890)

119868 can be condensed on the element

levelu(119890)119868

= minus[K(119890)

]minus1

K(119890)u(119890) (9)


that leads with the expression for u(119890)119868

to the system ofequations

Ku = f (10)

with

K = K

minus KKminus1K (11)

The crack in embedded elements is modelled by a set ofcrack segments with piecewise constant crack opening seeFigure 10 though recent versions of the EFEM can handlelinear crack opening [102] The EFEM has been applied tomany interesting problems [103ndash106] including piezoelectricmaterials [107] geomaterials [108] coupled formulations[109] finite strains [110] crack branching [111] and dynamicfracture [112] to name a few

Oliver et al [113] compared embedded elements withthe extended finite element method (that will be discussedin the next section) (XFEM) for several selected examplesand showed that embedded elements can be as accurate asXFEM However examples in LEFM that could be comparedto analytical solutions are missing

Methods that model the crack as continuous surface andtherefore contain a ldquophysicalrdquo crack tip are expected to bemore accurate However the requirement to represent thecrack surface is a drawback of those methods in particularfor complex crack patterns (eg in three-dimensions or forbranching cracks) EFEM in principle does not require crackpath continuity [114 115] However some authors report thatthe elements then become sensitive to the orientation of thediscontinuity surface with respect to the nodes and in someunfavorable situations the response of the element ceases tobe unique [116]

34 Extended Finite ElementMethod (XFEM) A very flexiblemethod that can handle linear and nonlinear crack openingsis the extended finite element method (XFEM) developedby Belytschko et al [11 117 118] This method employs thelocal partition of unity concept [119ndash121] and introducesadditional nodal parameters for the elements cut by the crackHence the additional unknowns cannot be condensed onthe element level as in EFEM The displacement disconti-nuity depends only on the additional nodal parameters Theprinciple idea of XFEM is shown in Figure 11 simplifiedfor a crack in one dimension Though XFEM has origi-nally been developed for fracture it has been extended tonumerous applications including two-phase flow [122 123]fluid-structure interaction [124 125] biomechanics [126]inverse problems [127 128] multifield problems [129ndash132]among others XFEM has been incorporated meanwhile intocommercial software (XFEM) and has become one of themost popular methods for fracture The generalized finiteelement method (GFEM) [120 121 133ndash139] is similar toXFEM

Thebasic idea of XFEM is to decompose the displacementfield into a continuous part ucont and a discontinuous partudisc

uℎ (X) = ucont (X) + udisc (X) (12)

The continuous part is the standard FE interpolation andadditional information is introduced into the FE interpo-lation through the local partition of unity approach byadding an enrichment (udisc) Therefore additional degreesof freedom are introduced for the discretization of the test-and trial functions Those additional degrees of freedom aremultiplied with the so-called enriched shape functions Theyare the product of the ldquostandardrdquo element shape functionsand enrichment functions that contain information about thesolution Note that the solution does not need to be knownexactly For example a crack results in a discontinuous dis-placement field Therefore a discontinuous function is usedas enrichment function In linear elastic fracture mechanics(LEFM) the analytical near crack tip solution is known aswell and can be included in the XFEM approximation Theapproximation of the displacement field for 119899

119888cracks with119898

119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1

sum119868isinS119888

119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)

where S is the set of nodes in the entire discretization S119905is

the set of nodes around the crack tip andS119888is the set of nodes

associated to elements completely cut by the crack 120595(119873)119868

isthe enrichment function of crack 119873 120601(119872)

119870119868is the enrichment

function for the crack tip119872 a119868and b

119870119868are additional degrees

of freedom that needs to be solved forFor elements completely cut by the crack the jump func-

tion is often chosen to cause a discontinuous displacementfield

120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)

(X) = sign [n sdot (X minus X(119873))]min (X minus X(119873))⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

X119873isinΓ(119873)119888

(15)

where n denotes the normal vector to the crack surface and119891(119873)(X) is a level set function see Figure 12 The zero-isobar119891 = 0 describes the position of the crack surface Often ashifting is employed see the second term on the RHS of (14)and Figure 11 and ensures that the enriched shape function119873119868(x)120595(119873)

119868vanishes in the neighboring element The shifting

also maintains the interpolatory character of the XFEMapproximation or in other words the nodal parameters u

119868

remain the physical displacementsIn LEFM the crack tip enrichment is choosen according

to the analytical solution

120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))

Figure 11 Principle of XFEM for strong discontinuities in 1D

Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903

Figure 12 Representation of the crack surface by level sets 119891 and 119892

where 119903 and 120579 are explained in Figure 12 Early approachesused a so-called topological enrichment Figure 13(a) How-ever a topological enrichment leads to a deterioration ofthe convergence rate as the domain with the accurate tipenrichment shrinks when the mesh is refined Therefore itis important to use a geometrical enrichment Figures 13(b)and 13(c) Note that a geometrical enrichment can drasticallyincrease the condition number of the system matrix andrequires the use of pre-conditioners [140] or other techniques[121 141] For non-linear materials and cohesive cracks atip enrichment should be used that avoids the crack tipsingularity

120601(119872)

119868= 119903

119904 sin(120579

2) (17)

with 119904 ge 1 The sin-term ensures the discontinuity inthe displacement field around the crack tip while 119903119904 guar-antees that the crack closes at the crack tip Note that itis standard to omit the crack tip enrichment for cohesivecracks Omitting the crack tip enrichment tremendously

simplifies the integration (only polynomial terms need to beintegrated) the representation of the crack surface and theentire formulation In that case the crack closes at the elementedge see Figure 17 though more advanced techniques havebeen developed to close the crack inside an element [142ndash144](without tip enrichment)

XFEM has been mainly applied to problems involvingcrack growth of a single crack or a few cracks There areonly a few contributions dealing with fracture problems thatinvolve the growth of numerous (several hundreds of) cracks[143 144] Methodologies have been developed in XFEMto account for complex crack patterns such as branchingcracks and crack coalescence (within a single element) [145146] However the formulation becomes cumbersome withincreasing number of cracks and crack branches Moreoverthosemethods suffer (in practice) from the absence of reliablecrack branching criteria

341 Representing the Crack Surface in XFEM XFEM is amethod that ensures crack path continuity Enforcing crackpath continuity is especially challenging in three dimensionsThe crack path can be represented explicitly usually withpiecewise straightplanar crack segments [147 148] or withother techniques such as level sets [149 150] that trace thecrack path in a more elegant way In many publicationsXFEM is closely related to level sets though any othertechnique can be used to describe the crackrsquos topology

Level sets are particularly attractive when a crack tipenrichment is used as geometry quantities can be calculatedthrough the level sets For example the distance of a materialpoint to the crack tip is 119903 = radic1198912 + 1198922 and 120579 = arctan(119892119891)see Figure 12 119892 is a level set orthogonal to 119891 (ie nabla119891sdotnabla119892 = 0)and is needed to uniquely identify the position of a material


Crack

(a)

Crack

(b)

Crack

(c)

Figure 13 (a) Topological enrichment versus (b) and (c) geometri-cal enrichment

point with respect to the crack surface Often the level-setfunction is discretized

119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)

where commonly the shape functions119873119868(X) of the mechan-

ical variables are used Therefore the enrichment can beexpressed in terms of nodal values of both level set functionsPrabel et al [151] used a (finer) finite difference schemefor the discretization of the level set function Howeverthe implementation effort is higher when different meshes

Crack

Level set representationof the crack

Figure 14 Curved crack and its representation with level setfunctions when the level set is discretized with linear triangles

are employed to discretize the level set and the mechanicalvariables On the other hand errors in the crackrsquos geometrywill be introduced for low-order finite elements and curvedcracks Figure 14 As in most applications a straight cracksegment is attached in front of an existing crack front alow-order finite elements should be sufficient for most crackpropagation problems

342 Integration in XFEM Numerical integration in XFEMrequires special attention in particular around the cracktip when employing non-polynomial enrichment functionsLaborde et al [152] reported that several thousand Gauss-points are required to adequately reduce integration error ina ldquobrute-forcerdquo approach For step-enriched (or Heaviside-enriched) elements a simple sub-triangulation is sufficientVentura [153] proposed an integration scheme for piece-wise straight cracks that avoid a sub-triangulation Anotheralternative is to increase the number of integration points[151] For singularities Ventura et al [154] Gracie et al [155]Bordas et al [156] Cheng and Fries [157] proposed efficientintegration schemes One of the most efficient schemes isthe ldquoalmost polar integrationrdquo by Nagarajan and Mukherjee[158] The almost polar integration is based on mapping arectangle onto a triangle and avoids integration of the cracktip singularity It was exploited for the first time in the contextof XFEMby Laborde et al [152] and simultaneously by Bechetet al [159]

343 Blending Elements Another major concern in XFEMare the so-called blending elements Figure 15 Blendingelements are the elements adjacent to the enriched elementsIt was shown by Chessa et al [160] Stazi et al [161] andFries [162] that those blending elements result in the deteri-oration of the convergence rate in XFEM When only a stepenrichment is employed a simple shifting can avoid problemsdue to blending However any non-constant enrichmentfunction causes difficulties Numerous approaches have beendeveloped to eliminate drawbacks associated to blending


Ωblnd

ΩenrΩstd

Figure 15 XFEM discretization with enriched domainΩenr blend-

ing domain Ωblnd and the total (standard) domainΩstd

[154 160 163] A very simple and efficient approach wasdeveloped by Fries [162] The idea is based on the modifi-cation of the enrichment function Therefore he enrichedthe nodes in the blending elements He guaranteed that theenrichment in the enriched elements remains unchanged butvanishes in the ldquostandardrdquo elements

344 Application to Fracture Applying XFEM to linearproblems in elastostatics the final discrete equations for asingle crack are obtained by substituting the test and trialfunctions into the weak form of the equilibrium equationyielding

[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)

with the stiffness matrix

K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)

where the superscripts 119906 119886 and 119887 indicate the ldquostandardrdquoldquostep-enrichedrdquo and ldquotip-enrichedrdquo part of the discretizationd119869= u

119869a119869b119870119869119879 is the vector with the nodal parameters

fext119868

= f119906119868f119886119868f119887119870119868119879

is the external force vector with f119887119870119868

=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)

with body forces b and applied tractions t The stiffnessmatrix is

K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)

where B119894119868 119894 = 119906 119886 119887 is the B operator defined by

B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)

For cohesive cracks without tip enrichment the terms asso-ciated to the tip enrichment are omitted facilitating theimplementation of the method

Although the XFEM approximation is capable of repre-senting crack geometries that are independent of elementboundaries it relies on the interaction between the meshand the crack geometry to determine the sets of enrichednodes [164] This leads to particular crack configurationsthat cannot be accurately be represented by (13) Such casesare shown in Figure 16 As the crack size approaches thelocal nodal spacing the set S

119888of nodes for the Heaviside

or step enrichment is empty Figure 16(b) Moreover node1 for the cracking case in Figure 16(a) or nodes 1 to 4 forthe cracking case in Figure 16(b) respectively contain twobranch enrichments Thus the standard approximation getsdifficulties with this crack configuration since the discon-tinuous function radic119903 sin(1205792) extends too far This problemalways arises when one or more nodal supports containthe entire crack geometry Usually this kind of problemarises whenever a crack nucleates Similar difficulties occurfor approximations without any crack tip enrichment seeFigure 16(d) In order to close the crack within a singleelement the set S

119888is empty as well One solution is to refine

the mesh locally such that the characteristic element size fallsbelow that of the crack An admissible crack configuration isshown in Figure 17

More details on XFEM and its applications can be foundin the excellent review papers by Karihaloo and Xiao [165]Fries and Belytschko [166] Belytschko et al [167] or the bookfromMohammadi [168]


Crack

1

(a)

Crack

1 2

34

(b)

Crack

(c) effective crack length (d)

Figure 16 (a) and (b) Crack length that approach the local element size cannot be accurately represented by the standard XFEMapproximation Dots denote single enriched nodes and squares denote double (in our case the node will contain the enrichment of two-crack tips) enriched nodes (c) the dashed line shows the effective crack length (d) even if no crack tip enrichment is used in order to closethe crack within a single element no nodes have to be enriched with a step function

35 Phantom Node Method An alternative to the standardXFEM for strong discontinuities was proposed by A Hansboand P Hansbo [169] They do not model the crack withadditional degrees of freedom but by overlapping elementssee Figure 18 In 1D it was shown by Song et al [170] thatthe crack kinematics of the A Hansbo and P Hansbo [169]version of XFEM can be derived from ldquostandardrdquo XFEMHowever the Hansbo version of XFEM has some advantagesover standard XFEM

(i) It avoids the ldquomixedrdquo terms K119906119886 and K119886119906 (orM119906119886 andM119886119906 in dynamics) (19) facilitating the implementa-tion

(ii) It leads to a better conditioned system matrix

(iii) Standard mass lumping schemes such as the row-sum technique can be used Note that efficient masslumping schemes have meanwhile been proposed forXFEM [171 172] (and XEFG [173])

(iv) Based on the full interpolation basis of the over-lapping elements the Hansbo-XFEM can straight-forwardly integrate enhanced techniques that aremore difficult to integrate in an enriched XFEMformulation [174]

(v) The implementation of the Hansbo XFEM is simplerIt has already been implemented in the commercialsoftware package ABAQUS In ABAQUS it has tobe used in combination with cohesive zone modelsThe ABAQUS implementation of the Hansbo-XFEMcan handle numerous cracks (crack propagation) in


Crack

Step-enriched nodes

Figure 17 Admissible crack representation

statics In contrast XFEM with tip enrichment hasalso been incorporated inABAQUSHowever a crackpropagation algorithm for the tip-enriched XFEMis yet not available in ABAQUS A plugin-in (Mor-feo) that can handle three-dimensional crack growthin LEFM was recently developed by the companyCenaero

On the downside we notice

(i) It is difficult to apply the Hansbo XFEM to otherproblems besides cracks

(ii) It is also difficult to incorporate a tip enrichmentfor the Hansbo-XFEM A crack tip element has beendeveloped for problems in 2D [175] However theextension of such a model to 3D is cumbersome

TheHansbo version of XFEMwas subsequently developed ina static setting by Mergheim et al [176] Areias et al [148]Mergheim and Steinmann [177] and in dynamics by Song etal [170]who called their approach phantomnodemethodThephantom nodemethod is well suited for non-linear problemswith cohesive cracks and problems with numerous cracksin particular when the method needs to be implementedin an existing finite element code It is less accurate forproblems in LEFM due to the absence of a tip enrichmentSubsequently the basic idea of the phantom node method isbriefly summarized

Consider a body that is cracked as shown in Figure 18and the corresponding finite element discretizationThere areelements cut by the crack To have a set of full interpolationbases the part of the cracked elements which belongs in thereal domainΩ

0is extended to the phantom domainΩpThen

the displacement in the real domain Ω0can be interpolated

by using the degrees of freedom for the nodes in the phantomdomain Ωp The nodes are called the phantom nodes and

marked by empty circles in Figure 18 The approximation ofthe displacement field is then given by [170]

uℎ (X 119905) = sum

119868isinW+0WminusP

u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum

119869isinWminus0W+P

u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)

where 119891(X) is the signed distance measured from the crackW+

0Wminus

0W+

P andWminus

P are nodes belonging toΩ+

0Ωminus

0Ω+P and

ΩminusP respectively 119867(119909) is the Heaviside function As can beseen from Figure 18 cracked elements have both real nodesand phantom nodes The jump in the displacement field isrealized by simply integrating only over the area from the sideof the real nodes up to the crack that isΩ+

0andΩminus

0 We note

that there are certain similarities to the visibilitymethod usedoriginally in meshfree methods [178ndash181]

36 Cohesive Segment Method and Cracking Node MethodRemmers et al [182 183] proposed the cohesive segmentmethod that has similarities to XFEM The crack is repre-sented by a set of overlapping cohesive segments that crossthree entire elements In contrast to XFEM no crack pathcontinuity is required in the Cohesive Segment Method Asimilar approachwas developed by Song andBelytschko [184]in theCrackingNodeMethod In theCrackingNodeMethodthe crack is required to directly pass through the node of anelement Also the cracking node method does not requirecrack path continuity

37 Meshfree Methods Thanks to the absence of a meshmeshfree methods (MMs) offer another alternative to modelfracture They are particularly well suited for dynamicfracture and large deformations Moreover adaptive h-refinement can be easily implemented in a meshfree context[185ndash187] Though all methods discussed so far are capableof capturing discrete cracks a certain refinement around thecrack front is still needed for accuracy and efficiency reasons[188] The meshfree approximation is similar to the FEMformulation


119873119869(X)u

119869 (26)

where 119873119869(X) denotes the meshfree shape function and

119906119869are the nodal parameters Most meshfree methods are

not interpolatory complicating the imposition of essentialboundary conditions Popular meshfree methods include thesmoothed particle hydrodynamics (SPH) method [189] theelement-free galerkin (EFG) method [190] the reproducingkernel particle method (RKPM) [191] the point interpolationmethod (PIM) [192] the meshless local-petrov galerkinmethod (MLPG) [193] or the Finite Point Method [194]among others


Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp

Figure 18 The principle of the phantom node method in which the hatched area is integrated to build the discrete momentum equation thesolid circles represent real nodes and the empty ones phantom nodes

Most meshfree shape functions depend on a weighting orkernel function which is denoted by 119882

119869(X) Usually radial

kernel functions are employed where 119882119869(X) = 119882(119903

0) is

expressed in terms of 1199030= X minusX

119869 The kernel functions are

chosen to have compact support that is 119882119869(X) = 0 if X minus

X119869 gt ℎ

119869 where ℎ

119869is the ldquointerpolationrdquo radius often called

dilation parameter The kernel function can be expressed interms of material coordinates X (Lagrangian kernel) or spa-tial coordinates x (Eulerian kernel) Formulations based onLagrangian kernels are more efficient as the shape functionscan be constructed at the beginning of the simulation whileEulerian kernels require updating during the course of thesimulation More details on meshfree methods can be foundin the literature for example in several review articles andbooks [195ndash202] Before discussing different possibilities toincorporate fracture into MMs the shape functions of theEFG-method will be given as it is helpful for the explanationsin the subsequent sections They are constructed as follows

119873119869(X) = p119879 (X) sdot A(X)minus1 sdotD (X

119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)

where p is the polynomial basis Usually a linear polynomialbasis is choosen given in 2D by

p = [1 119883 119884] (28)

371 ldquoNaturalrdquo Fracture Due to the absence of a meshfracture in meshfree methods can occur naturally [38 203204] In contrast to element erosion there is no need toerode particles Fracture occurs when particles move outsidethe domain of influence of neighboring particles Liberskyet al [203] presented impressive results of dynamic fractureand fragmentation predicting fragment distributions quiteaccurately Dilts [205] presented an algorithm to detectfracture surfaces within the SPHmethod in 2D He extendedthe approach later to three dimensions [206] and solvedimpressive impact problems However care has to be takenas fracture can occur artificially as shown for example byRabczuk et al [207] This is mostly the case when Euleriankernels are employed Formulations based on Lagrangiankernels do not suffer from numerical fracture Howeveras the neighbor list does not change during the course ofthe simulation fracture cannot be modeled ldquonaturallyrdquo anymore Moreover the Lagrangian kernel has to be consistentlyupdated for problems involving large deformations A veryefficient algorithm to treat dynamic fracture and fragmenta-tion in meshfree methods based on Lagrangian kernels wasproposed byMaurel and Combescure [208] and Combescureet al [209] In their work connectivities between neighboringparticles were broken when their distance exceeds a giventhreshold They solved complex problems including fluid-structure interaction and fragmentation [210 211] Anothereffective and simple method to treat fracture in meshfreemethods is the material point method (MPM) [212] Theadvantage of theMPMovermost SPHmethods for fracture istheir computational efficiency [213]While thosemethods arequite attractive tomodel dynamic fracture and fragmentationinvolving an enormous amount of cracks they are not well


suited for static fracture with few cracks due to their lessaccurate representation of the crack kinematics

372 Visibility Method The visibility method is the firstapproach that introduces a discrete crack into the meshfreediscretization In the visibility method the crack boundary isconsidered to be opaque Thus the displacement discontinu-ity is modeled by excluding the particles on the opposite sideof the crack in the approximation of the displacement fieldsee Figure 19

uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)

This is identical to setting the shape function across the crackto zero as shown in Figure 19 The jump in the displacementis then computed by

[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)

Difficulties arise for particles close to the crack tip sinceundesired interior discontinuities occur (Figure 20) due tothe abrupt cut of the shape function see Figure 19 Neverthe-less Krysl and Belytschko [214] showed convergence for thevisibility method For linear complete EFG shape functionsthey even showed that the convergence rate is not affectedby the discontinuity The length and size of the (undesired)discontinuities depend on the nodal spacing near the crackboundary If the nodal spacing approaches zero the length ofthe discontinuities tends to zero With this argument and thetheory of nonconforming finite elements convergence of thediscontinuous displacement field can be shown

It should also be noted that the visibility criterion leadsto discontinuities in shape functions near nonconvex bound-aries such as kinks crack edges and holes as shown inFigure 20 in two dimensions

An efficient implementation of the visibility method in2D is given for example in [215]

38 The Diffraction Method The diffraction method is animprovement of the visibility method It eliminates theundesired interior discontinuities see Figure 21 (see alsoFigure 19) The diffraction method is also suitable for non-convex crack boundaries It is motivated by the way lightdiffracts around a sharp corner but the equations used inconstructing the domain of influence and theweight functionbear almost no relationship to the equation of diffractionThemethod is only applicable to radial basis kernel functionswitha single parameter ℎ

1198680

The idea of the diffraction method is not only to treatthe crack as opaque but also to evaluate the length ofthe ray ℎ

0by a path which passes around the corner of

the discontinuity A typical weight function is shown inFigure 19 It should be noted that the shape function of thediffraction method is quite complex with several areas ofrapidly varying derivatives that complicates quadrature of thediscrete Galerkin form [200] Moreover the extension of the

diffraction method into three dimensions is complex Nev-ertheless several authors have presented implementations ofthe diffraction method in 3D [216 217]

39TheTransparencyMethod Thetransparencymethodwasdeveloped as an alternative to the diffraction method byOrgan et al [178] The transparency method is easier extend-able into three dimensions than the diffractionmethod In thetransparency method the crack is made transparent near thecrack tipThe degree of transparency is related to the distancefrom the crack tip to the point of intersection see Figure 19

An additional requirement is usually imposed for parti-cles close to the crack Since the angle between the crack andthe ray from the node to the crack tip is small a sharp gradientin the weight function across the line ahead of the crack isintroduced In order to reduce this effect Organ et al [178]imposed that all nodes have a minimum distance from thecrack surface that is the normal distance to the crack surfacemust be larger than 120574ℎwith 0 lt 120574 lt 1 they suggested 120574 = 14

310 The ldquoSee Throughrdquo and ldquoContinuous Linerdquo MethodThe ldquosee-throughrdquo method was proposed by Terry [218]for constructing continuous approximations near nonconvexboundariesTherefore the boundary was considered as com-pletely transparent such that the discontinuity is removedThough the ldquosee-throughrdquo method works well for capturingfeatures such as interior holes it is not well suited to modelcracks

In the continuous linemethod fromKrysl and Belytschko[214] and Duarte and Oden [219] the crack is completelytransparent at the crack tip In other words particles with apartially cut domain of influence can see through the crackThis drastically shortens the crack If no special techniquesare introduced the crack also does not close at the crack tipthat leads to inaccurate solutions Crack closure at the cracktip can be enforced with Lagrangemultiplier or by decreasingthe domain of influence of nodes close to the crack tip

Belytschko and Fleming [220] suggest to combine differ-ent methods depending on the convexity of the crack bound-ary for example to use the visibility for convex boundariesand other methods for non-convex crack boundaries Theysuggest a criterion based on the angle of the wedge that can bewritten in terms of the surface normal see Figure 22 Whenn119860 sdot n119861 le 120573 with 120573 = 0∘ as a cutoff value the boundary canbe considered as convex otherwise non-convex The cutoffvalue of 120573 = 0

∘ corresponds to the wedge angle of 120596 = 90∘ inFigure 22

311 Extended Meshfree Methods Enrichment in meshfreemethods was used before the development of XFEM A goodoverview paper on enrichedmeshfreemethods in the contextof LEFM is given by Fleming et al [221] see also the reviewarticles by Belytschko et al [200] Nguyen et al [201] andHuerta et al [202]The enrichment schemes are classified intointrinsic enrichment and extrinsic enrichment An intrinsicenrichment introduces enrichment functions as additional


Crack

Visibility criterion

Crack

Crack

Transparency criterion

05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053

Diffraction criterion

119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868

Figure 19 Principle of the visibility diffraction and transparencymethods with corresponding shape functions from Belytschko et al [200]

functions to the polynomial basis compare to (28) as thefollowing

p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)

This concept was later also used in the intrinsic XFEM[222] The major drawback of this approach is its highercomputational cost as the enrichment needs to be employed

everywhere to avoid artificial discontinuities Techniques andblending enriched approximations to ldquostandardrdquo approxima-tions have been used but they introduce additional complex-ity Duflot and Nguyen-Dang [223] proposed an alternativeintrinsic enrichment by enriching the kernel functionsTheirapproach does not require blending techniques and is com-putationally cheaper However it is also less accurate Theextrinsic enrichment can be categorized into extrinsic MLSenrichment and extrinsic PU enrichment The extrinsic MLSenrichment introduces enrichment parameters that can beinterpreted as stress-intensity factors (SIFs) at least whenthe enrichment is introduced globally Hence the SIFs can


Crack

Interdiscontinuities

(a)

Crack

Domain of influence

119868

(b)

Figure 20 (a) Undesired introduced discontinuities by the visibilitymethod (b) Difficulties with the visibility method for concaveboundaries and kinks

be obtained directly from the numerical analysis withoutcomputing the J-integral or interaction integral However itwas shown by Zamani et al [224 225] that such approachesare less accurate The extrinsic PU enrichment is comparableto an XFEMenrichment It is believed that it is the best choicein terms of accuracy efficiency and flexibility

An extended element-free Galerkin (XEFG) methodbased on vector level sets was first proposed by Ventura etal [226] in the context of LEFM It was later extended tononlinear materials and cohesive cracks by Rabczuk and Zi[227] Rabczuk et al [228] In contrast to XFEM due to theheavily overlapping shape functions a crack tip enrichmenthas to be employed to ensure crack closure at the cracktip though alternative methods have been developed thatavoid the use of tip enrichments [229 230] However thoseapproaches are cumbersome in three dimensions

312 Cracking Particles Method While all previous meshfreeapproaches for fracture require crack path continuity thecracking particles method [231ndash235] does not represent

Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)

Figure 21 The diffraction method

the crack as continuous surface This makes the methodparticularly useful for 3D applications including complexcrack patterns such as crack branching and crack coalescenceIn contrast to methods that enforce crack path continuitycrack branching and crack coalescence are a (more) naturaloutcome of the analysis in the cracking particles method

Tomodel cracks in the cracking particlesmethod the dis-placement is decomposed into continuous and discontinuousparts

u (X) = ucont (X) + uenr (X) (32)

where X are the material coordinates ucont denotes thecontinuous displacement anduenr denotes the discontinuouspart

The crack is modelled by a set of discrete cracks as shownin Figure 23 These discrete cracks are restricted to lie on theparticles that is each crack plane (or line in 2D) always passes


119860

119861120596

n

n

Figure 22 Domain of influence near a wedge-shaped non-convexboundary The boundary is enforced if n119860 sdot n119861 le 120573

(a)

(b)

Figure 23 Schematic on the right shows a crackmodel for the crackon the left

through a particle Since the crack geometry is described bythe set of cracked particles a representation for the geometryof the crack is not needed

The approximation of the displacement field in thecracking particles method is given by

uℎ (X) = sum119868isinN

119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)

Figure 24 (a) Spurious cracking and (b) improved crack pattern

whereN is the total set of nodes in the model andN119888is the

set of cracked nodes 119904(119891119868(X)) is the step function In general

different shape functions can be used for the continuous anddiscontinuous part of the displacement field

The jump in the displacement across the crack dependsonly on the discontinuous part of the displacement fielduenr(X) and hence the enrichment parameters q

119868 It can be

shown from (33) that

u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)

The factor 2 comes from the fact that the step function is usedthat is minus1 on one side of the crack and +1 on the other side ofthe crack

The cracking particlesmethodmight suffer from spuriouscrack patterns [236 237] However they can be eliminated bya set of simple crack injection rules [184] Firstly one has todistinguish between crack propagation and crack initiationSince the crack is not considered as continuous surface crackpropagation is assumed when no cracked node is detected inthe vicinity (in a circle [sphere in 3D] with radius 119903 = 120572ℎ120572 ge 15) of a sampling point where cracking is detectedFor propagating cracks spurious cracks as shown in Figure 24are avoided by introducing a prevention zone Figure 25 Forexample for particle 119871 the crack prevention zone is giventhrough the angle 120572

100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)

For branching cracks it is often observed that the orientationof neighboring cracks severely differ see node119870 in Figure 25


120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n

Figure 25 Crack prevention nodes 119869 and119870 are located in the crackprevention zone node 119868 is located in the crack propagation zone

Therefore cracks are also allowed in the prevention zonewhen the following criterion is fulfilled

n119871sdot n119870lt cos120573 (36)

with 120572 = 2Π9 and 120573 = Π3 Similar criteria are usedwithin the cracking particles method in H X Wang and SX Wang [238] Zhang [237] reported that cracking rules canbe avoided by the use of rotating crack segments

Recently the cracking particles method has been pre-sented without enrichment to model fracture in continua[235] and structures [239] This results in less degrees offreedom and facilitates diagonalizing the mass matrix whenthe method is used in explicit dynamics

313 Boundary Element Method In the Boundary ElementMethod (BEM) the weak form is formulated in boundaryintegral form It reduces therefore the dimension of theproblem for example a three-dimensional problem reducesto two dimensions However the BEM is only applicableto problems where Greenrsquos functions can be computedrestricting the application range of the method Due tothe dimension-reduction the remeshing procedure is fairlysimple as the crack is part of the boundary Interestingapplications of the BEM to fracture can be found for examplein [240ndash257]

314 Isogeometric Analysis for Fracture Isogeometric analysis(IGA) [258] was developed to unify CAD and CAE Ver-hoosel et al [130] were the first who incorporated fracturein isogeometric analysis based on T-splines Therefore theyused the concept of knot insertion to introduce discontinu-ities in the displacement approximation This method suffersfrom similar problems as interelement-separation methodsMoreover for complex crack patterns with slightly deviatingangles between the crack branches it is expected that themesh will be extremely distorted De Luycker et al [259]

Ghorashi et al [260] and recently Tambat and Subbarayan[261] combined XFEM with isogeometric analysis (XIGA)allowing crack growth without remeshing also in the contextof IGA

315The Variational Approach to Fracture Variational meth-ods in fracturemechanics are a relatively new development inthe field The underlying theory was laid down by Francfortand Marigo [262] who proposed the idea that cracks shouldpropagate along a path of least energyThe goal of variationalmethods is to circumvent several importantweaknesses of theclassical theory emanating from the work of Griffith in 1920

In classical fracture mechanics crack propagation isassumed to occur under thermodynamic equilibrium Inparticular for crack formation occurring under constant loadthe externalwork done by the applied loading is equal to twicethe energy in the bulk so that the total energy of the systemmay be expressed as 119880 = minus119882

119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where

119880119878is a postulated quantity known as the surface (dissipation)

energy of the crack system Thermodynamic equilibriumwith respect to crack growth then requires that 119889119880119889119886 =

0 where 119886 is the crack length The previous expressionprovides the critical condition for fracture The simplestcase to consider is an infinite solid with a straight crack oflength 2119886 and subjected to far-end tractions of magnitude120590 directed perpendicular to the crack For this problem theexact solutions for the stresses and strains are available dueto Inglis so that an analytic form of 119880

119864may be obtained

equal to 12058711988621205902119871119864 for the case of plane stress We now assume

that the surface energy to be 119880119878

= 4120574119886 where 120574 is thesurface energy per unit area which is considered a materialproperty Enforcement of thermodynamic equilibrium yieldsthe expression for the critical applied stress required for crackpropagation 120590

119891radic119886 = radic2119864120574120587 According to the classical

theory the existing crack will not propagate for 120590 lt 120590119891and

will do so for 120590 = 120590119891

Despite improvements made by subsequent authors toaccount for nonlinearity and inelasticity the original form ofthe fracture criterion remained that is 120590

119891radic119886 = constant

As pointed out in [262] this inverse proportionality of thecritical stress to the square root of the initial crack lengthmeans that classical fracture mechanics is unable to predictcrack initiation since for a body without an initial crack(119886 = 0) 120590

119891becomes infinite Other important weaknesses

of the classical theory listed in [262] are its inadequacy forpredicting the direction of the crack path and its inabilityto handle crack jumps (brutal cracking) The former arisesfrom the fact that Griffithrsquos criterion gives only one constraintwhereas the description of the crack tip propagation requiresa number of functions equal to the dimensionality of theproblem The latter weakness stems from the classical theorybeing silent on what happens when 120590 gt 120590

119891(unstable

cracking) Griffithrsquos criterion is also incapable of predictingwhen a crack will branch so that additional criteria arerequired in order to evaluate the possibility of this kind ofresponse

In order to address the above mentioned weaknessesof classical fracture mechanics Francfort and Marigo [262]


proposed an alternative formulation of brittle fracture basedon energy minimization concepts For a body denoted by Ω

inD-dimensional space energy (similar to the surface energyconcept of Griffith) is assigned to the crack system Γ isin Ω andexpressed as

119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)

where H119873minus1 denotes the (119873 minus 1)-D Hausdorff measureThe function 119896(119909) represents the energy required to createan infinitesimal crack at the point 119909 and is strictly positiveand bounded away from zero The above definition is ratheraccommodating in terms of crack geometries and in additionto the usual ldquosharprdquo crack surfaces of classical fracturemechanics it also admits entities like point clouds and edgecracks However cracks having 119873-D Hausdorff measure(termed as ldquofatrdquo cracks in [262]) are not allowed for examplea crack with computable area in a 2D or volume in 3D sincethese will require an infinite amount of energy to propagateThe bulk energy is defined as

119864119889(Γ 119880) = inf

VisinC(Γ119880)intΩΓ

119882(119909 120598 (V (119909))) 119889119909 (38)

in which C(Γ 119880) is the set of kinematically admissibledisplacement fields The total energy of the body is the sumof these two terms

119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)

The problem then is to find the mapping Γ(119905) such that atsome given time 119905 gt 0 the crack geometry Γ(119905) is the onethat minimizes 119864(119880 Γ(119905)) and furthermore contains all Γ(119904)for 119904 lt 119905 (condition of irreversibility) The authors warnthat the use of global energy minimization to drive the crackevolution is not based on thermodynamic principles but isadapted as ldquo a rather convenient postulate which providesuseful insight into a variety of behaviorsrdquo [262] They showalso that this new formulation is free of the earlier identifiedweaknesses of the classical theory

Numerical experiments on the proposed formulationwere carried out by Bourdin et al [263] who pointed to thesimilarity of the new formulation to models of image seg-mentation obtained by minimization of the Mumford-Shahfunctional This meant that the inapplicability of standardnumerical methods owing to the fact that the formulationallowed for jump sets of the displacement field (representingthe cracks) whose locus was unknown a priori Twomethodsof solution were demonstrated based on the mathematicalconcept of Γ-convergence In one of these methods a secondfield was included in the energy functional in addition tothe primary displacement field This auxiliary scalar fieldacted as a regularizer for the jump sets of the displacementfield and took on values of 0 on the crack and 1 awayfrom it The regularizing variable was referred to by laterresearchers as the phase field and it allowed for the treatmentof the global energy minimization as a standard variationalproblem for which classical FEM is up to the task albeitwith the restriction that the characteristic length of the mesh

should tend to zero faster than the characteristic length of theregularization so as not to overestimate the surface energy ofthe crack Due to the inclusion of a second field a coupledsystem of equations must now be solved consisting of theoriginal equilibriumlinear momentum equations and theevolution PDE of the phase field

A phase field model for mode III dynamic fracture wasdevised by Karma et al [264] which made use of a phasefield evolution equation based on the standard two-minimumGinzburg-Landau form that is

120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)

where 119892 is a function obeying the constraints 119892(0) = 0119892(1) = 1 and 1198921015840(0) = 1198921015840(1) = 0 The KKL phasefield model was subsequently utilized by Hakim and Karma[265] to analyze the laws of quasistatic crack tip motionAmong other things they showed that for kinked cracks inanisotropic media force-balance gives predictions that aresignificantly different from those obtained using the principleof maximum energy release rate and that the predictionsobtained from force-balance hold even when fracture ismodeled as irreversible It should bementioned that up to thispoint phase field implementations featured isotropic damagewherein fracture occurred both in tension and compressionThis sometimes resulted in unphysical response such assample interpenetration in the crack branching simulation inBourdin et al [263]

Thermodynamically consistent phase field models offracture were developed byMiehe et al [266 267] along withcorresponding incremental variational principles andnumer-ical implementation within a finite element frameworkTheirimplementation of the phase field was also slightly differentfrom that of previous authors whereas Bourdin et al [263]and subsequent papers utilized the convention that the phasefield took values of 0 at cracks and 1 at unbroken states (hencea pseudomaterial density function) andMiehe et al reversedthe convention by assigning to the phase field values of 1 and0 for fully cracked and fully unbroken states respectively Asa consequence the functional associated with the phase fieldmay now be seen to represent the crack surface itself Theythen derived the evolution equation for the phase field basedon the assumption that the solution is negative exponential innature Furthermore they extended the application of phasefields to fracture simulations involving viscous overforceresponse via a time-regularized three-field formulation andshowed that the coupled system (linear momentum + phasefield evolution) could be solved either monolithically or via arobust staggered-update scheme [266 267]

An important addition of Miehe et al [266 267] to theexisting theory was the modelling of anisotropic degradationby using an additive decomposition of the stored energy ofthe undamaged solid to come up with an anisotropic energyfunction of the form

Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)


where the positive and negative parts of the energies aredefined by

Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)

and ⟨119909⟩plusmnare ramp functions defined as ⟨119909⟩

plusmn= (12)[|119909| plusmn

119909] The energy decomposition allows for the case wherefracture occurs in tension only as opposed to both tensionand compression and effectively fixes the sample inter-penetration issue encountered by Bourdin in crack branchingsimulations Following the reversed convention on the phasefield the constraints on 119892(120601) are 119892(0) = 1 119892(1) = 0 and1198921015840(1) = 0 To maintain numerical stability a small positive

constant 119896 is included in the formulation so that the materialretains some residual stiffness even when it attains a fullydamaged state

One challenge with the use of phase field models forfracture is the computational expense associated with meshsize requirements since the use of a mesh having a charac-teristic length that is not small enough compared to the crackregularization parameter yields erroneous results with regardto the energy Based on numerical experiments utilizing thestationary phase field equation Miehe et al [266 267] foundthat the characteristic length of the mesh should be no morethan half the value of the regularization parameter at leastpresumably in the vicinity of the regularized crack In aneffort to reduce computational expense Kuhn and Muller[268 269] (see also their work in [270]) introduced speciallyengineered FEM shape functions of an exponential nature todiscretize the phase fieldThese shape functions have the form

119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)

in 1D for the 2-node line element 2D shape functions forthe 4-node quadrilateral are obtained as a tensor productsof the above with certain tweaks The shape functions areparametrized by 120575 which is the ratio between the elementsize and phase field regularization parameter Their resultsshowed that with the special element shape functions accu-rate prediction of surface energy associated with the phasefield is possible with a much lower level of refinementcompared to standard FEM shape functions The drawbackto their approach is that some information regarding crackorientation is necessary for the proper construction of theexponential shape functions so that numerical exampleswere confined to cases where the general direction of thecrack path is known a priori

4 Tracking the Crack Path

Methods that ensure crack path continuity such as XFEMrequire the representation of the crack surface and algorithmsto track the crack path While those tasks are relatively easy

to implement in two dimensions their implementation in 3Dis challenging

The topology of the crack surface is commonly repre-sented either explicitly by piece-wise planar crack segmentsor implicitly by level set functions Meshfree methods andthe phantom node method usually use the former methodwhile many XFEM-implementations are based on an implicitrepresentation of the crack surface using level sets

There are three major approaches to track the evolvingcrack surface

(i) local methods(ii) global methods(iii) level set method

A review article of those methods including implementationdetails in the context of partition of unity methods is given byRabczuk et al [271]

41 Local Methods With local crack tracking algorithmsthe alignment of the crack surfaces is enforced with respectto its neighborhood Local crack tracking algorithms areusually characterized by recursively ldquocuttingrdquo elements (orbackground cells in meshfree methods) and are especiallyeffective in three dimensions Local methods have difficultiesto ensure a continuous crack surface in 3D Jager et al [272]report for this reason they are not well suited to modelcurved cracks However adaptive refinement and smoothingtechniques allow also the representation of curved surfaces(modelled by piece-wise planar crack segments) [186 232]Local methods were pursued in XFEM for example byGasser and Holzapfel [273 274] and Areias and Belytschko[147] or in the context of meshfree methods by Rabczuk andBelytschko [186 232] Krysl and Belytschko [275] amongothers

42 Global Methods In global methods a linear heat con-duction problem is solved each load step for the mechanicalproblem that needs to fulfill the condition

a sdot n0= b sdot n

0= 0 (44)

where a and b are two vector fields and n0denotes the

vector of the normal to the crack surface The family ofsurfaces enveloping both vector fields a and b are describedby a temperature-like function 119879(X) when the followingconditions hold

a sdot nabla0119879 = nabla

0119879 sdot a = 0 in Ω

0

b sdot nabla0119879 = nabla

0119879 sdot b = 0 in Ω

0

(45)

Equation (45) can be rephrased as an anisotropic heatconduction problem [101 276 277] The drawback of thismethod is that the heat conduction problem has to be solvedat every time step making global methods computationallyexpensive

Feist and Hofstetter [114] proposed the domain cracktracking algorithm a slight variant of the global method Intheir algorithm the scalar field 119879 is not constructed globally


but only for a subset of the domain that is affected by thecrack Jager et al [272] report that global algorithm are wellsuited to track curved cracks but less suitable for straightcracks

43 Level Set Method The level set method was originallydeveloped to track interfaces that propagate orthogonal totheir surface To update the interface the Hamilton-Jacobiequation was solved with respect to the level set It makesthe method particularly attractive for problems in fluidmechanics However the original level set method is not wellsuited to track crack surfaces for three reasons as noted forexample by Ventura et al [278] and Duflot [279]

(i) The zero isobar of the level set must be updatedbehind the crack front to account for the fact once amaterial point is cracked it remains cracked

(ii) The crack surface is an open surface that extends dur-ing the crack propagation It requires the introductionof another level set function (orthogonal to the levelset function describing the crack surface) in order touniquely determine the position of a material pointwith respect to the crack surface When the crackpropagates this level set function needs to be updatedas well Moreover both level set functions mustbe periodically reinitialized to the signed-distancefunctions to preserve stability [280 281]

(iii) The level set functions are not updated with the speedof the interface in the normal direction and hence theHamilton-Jacobi equation cannot be used Insteadthe level set function propagates with the speed of thecrack front

Crack propagation with level sets can be modelled by differ-ent techniques that can be classified into four groups [279]In the first group the level set is updated by the solutionof differential equations The second group is defined onalgebraic relations between the coordinates of a given pointthe coordinates of the crack front and the crack advancevector see for example Stolarska et al [282] This methodis not easily extendable into three dimensions The Vectorlevel set method [226 278] is defined in terms of geometrictransformations In the vector level set method the distanceto the crack surface is stored in addition to the signed distancefunction This facilitates the implementation since there isno need to solve a PDE to update the level set The lastclass of methods are based on algebraic and trigonometricequations involving the initial value of the level set functionsand the crack advance vector Some of them also requirethe description of the crack front They are well suitedfor three-dimensional applications Duflot [279] suggested amixed method defined with differential and non-differentialequationsHe also gives an excellent state-of-the-art overviewon level set techniques for cracks

A very efficient and elegant crack propagation and trackcracking algorithm in the context of XFEM was recentlypresented by Fries and Baydoun [283] and in the context ofmeshfree methods by Zhuang et al [284 285]

5 Fracture Criteria

The fracture criterion is needed to determine whether or nota crack will propagate or nucleate Moreover the fracturecriterion should provide the orientation and the ldquolengthrdquoof the crack as well as whether or not cracks branch orjoin Methods that ensure crack path continuity need todistinguish between crack nucleation and crack propagationDetecting branching cracks in dynamic problems is particu-larly difficult in those methodsThe fracture criterion is oftenmet at several quadrature points in front of the crack tip andreliable criteria to branch cracks in practice are still missingThe best results are obtained when the crack branches areknown in advance

In most applications the ldquolengthrdquo of the crack is con-trolled Usually it is correlated to the underlying discretiza-tion To the best knowledge of the author all of the methodsassume a straightplanar extension of an existing cracksurface

In the following section different cracking criteria toobtain the orientation of a newly (nucleated or propagated)crack segment are reviewed

51 Fracture Mechanics-Based Criteria Besides approachesbased on configurational forces [286ndash289] there are fourmajor cracking criteria in LEFM

(i) Maximum hoop stress criterion or maximum princi-pal stress criterion

(ii) Minimum strain energy density criterion Sih [290](iii) Maximum energy release rate criterion Wu [291](iv) The zero 119870

119868119868criterion (Vanishing in-plane SIF (119870

119868119868)

in shear mode for infinitesimally small crack exten-sion) Goldstein and Salganik [292]

The first two criteria predict the direction of the crack tra-jectory from the stress state prior to the crack extension Thelast two criteria require stress analysis for virtually extendedcracks in various directions to find the appropriate crack-growth direction Duflot and Hung [223] report for mediummixed-mode problems all criteria yield almost identicalresults However according to Shen and Stephansson [293]only the maximum energy release rate criterion allows topredict secondary cracks in compressed specimensThe crackis propagated in an angle of 120579

119888from the crack tip Note that

LEFM can only deal with crack propagation but not withcrack nucleation It also can not handle crack branching

In themaximumhoop stress ormaximumprincipal stresscriterion the maximum circumferential stress 120590

120579120579 often

called hoop stress in the polar coordinate system around thecrack tip corresponds to the maximum principal stress and isgiven for a crack propagating with constant velocity V

119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)

where the functions 119891119868ℎand 119891119868119868

ℎrepresent the angular varia-

tion of the stress for different values of the crack-tip speedV119888 When the maximum hoop stress is larger equal a critical


hoop stress 120590119888120579120579

then the crack is propagated in the directionperpendicular to the maximum hoop stress For pure mode 119868fracture 120590119888

120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)

with the fracture toughness 119870119888119868

In LEFM the local direction of the crack growth isdetermined by the condition that the local shear stress is zerothat leads to the condition

119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)

that results in the crack propagation angle

120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)

The minimum strain energy density criterion is based ona critical strain-energy-density factor 119878119888

119903 The basic assump-

tion is that fracture occurs when the interior minimum of thestrain-energy-density factor 119878 reached 119878

119888

119903 The strain energy

density factor 119878 represents the strength of the elastic energyfield in the vicinity of the crack tip which is singular of theorder 119903minus1The factor 119878119888

119903is assumed to be a material parameter

and can be used as a measure of the fracture toughness undermixed mode conditions For pure mode I fracture 119878119888

119903can be

directly expressed in terms of119870119888119868

119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)

where 119866 denotes the shear modulus and 120581 is the kosolovconstant For general mixed mode fracture 119878119888

119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)

with minus120587 lt 1205790lt 120587 and

11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622

=(120581 + 1) (1 minus cos 120579) + (1 + cos 120579) (3 cos 120579 minus 1)

16119866

(52)

The parameters 119886119894119895

are obtained from the strain-energydensity function For further details see Sih [290] Notethat the minimum strain energy-density criterion is a localcriterion since fracture occurs when the energy density ina volume element near the crack tip reaches a critical valuewhile the classical fracture mechanics theory is based onglobal energy balance

Crack

Crack extension

Figure 26 Crack propagation with the energy release rate criterion

In the maximum energy release rate criterion the crackpropagates in the direction defined by the angle 120572 = 120572

119888(120590)

where 120572119888satisfies the condition

120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)

where 120572 is defined in Figure 26 and G denotes the energyrelease rate Crack propagation occurs at a stress state 120590 =

120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material

parameter According toWu [291] the energy release rate canbe decomposed into an antiplane (antiplane shear load) partG119860and a plane (plane load) partG

119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)

For the antiplane shear loadG119860is given by

G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)

For plane strain conditions

G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)

52 Rankine Criterion For a Rankine material a crack isintroduced when the principal tensile stress reaches theuniaxial tensile strength The crack is initiated perpendicularto the direction of the principal tensile stress Usually somekind of smoothing technique is applied that either averagesthe crack normal or the stress tensor [176 228 273 294295] This is done in order to improve the reliability of thecomputed stresses in front of the crack tip For exampleWellsand Sluys [294] and Mergheim et al [176] use an averagedstress tensor of the form

120590119898= int

Ω119888

119908120590 119889Ω (57)

where Ω119888is a certain domain around the crack tip and 119908 is a

weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)


where 119897 determines how fast the weight function decays fromthe crack tip Gasser and Holzapfel [273] and Rabczuk et al[228] smooth the crack normal instead of the stress tensorThe Rankine criterion is applicable to brittle materials andworks well for mode I-fracture

53 Loss of Material Stability Fracture is caused by a mate-rial instability A classical definition of material stability isbased on the so-called Legendre-Hadamard condition whichestablishes that for any non zero vectors 119899 and ℎ the followingpointwise inequality must hold

119890 = minnh (n otimes h) A (n otimes h) gt 0 (59)

where A = C119905 + 120590 otimes 120575C119905 is the constitutive tangentoperator and 119890 is the so-called ellipticity indicator For a linearcomparison solid if non-propagating singular surfaces donot occur the Legendre-Hadamard condition is identical tothe strong-ellipticity condition used for example in Simo etal [58] In case (59) is no longer fulfilled the material losesstability and n defines the direction of propagation and h isthe polarization of the wave This condition ensures that thespeed of propagating waves in a solid remains real Equalityin expression (59) is the necessary condition for stationarywaves Belytschko et al [31] give a textbook description forobtaining condition (59) by means of a stability analysisof the momentum equation when a perturbation of theform u = h exp(119894120596119905 + 119896n sdot x) is applied The Legendre-Hadamard condition is occasionally called strong ellipticityof the constitutive relation (see Marsden and Hughes [296])In the dynamical case the Legendre-Hadamard conditionimplies the hyperbolicity of the IBVP For a rate-dependentmaterial we refer to it as loss of material stability The readeris referred to Silhavy [25] or Ogden [297] for details about theconcepts mentioned above

Loss of hyperbolicity (or loss of ellipticity or loss ofmaterial stability resp) is determined by minimizing 119890 withrespect to n and h if 119890 is negative for any combination of nand h the material has lost stability at that material pointIn 2D 119890 can be expressed as a function of the two angles 120579and 120601 where n = [cos(120579) sin(120579)] and h = [cos(120601) sin(120601)]Therefore based on expression in (59) let us define for a givenmaterial point of a solid at a given time

Q=n sdot A sdot n (60)

where Q is the acoustic tensor with components119876119894119896=119899

119895119899119897119860119894119895119896119897

We say that a material point is stablewhenever the minimum eigenvalue of Q is strictly positiveand unstable otherwise It can be shown that this conditionis equivalent to condition (59)

One difficulty is that the analysis of the acoustic tensor forisotropic materials generally will yield two directions n fromwhich one direction has to be chosen At times the loss ofmaterial stability criterion becomes ambiguous an exampleof the minimum eigenvector of Q as a function of the angle120579 is shown in Figure 28 simplified for a two-dimensionalexample For uniaxial tension two angles are obtained butthey correspond to the same plane In other stress states four

0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928

Figure 27Minimum eigenvalue of the acoustic tensor as a functionof the two angles for one material point

angles corresponding to two planes are sometimes obtainedA typical eigenvalue landscape as function of the crack anglesin three dimension is shown in Figure 27 for one materialpoint and isotropic 119869

2-plasticity with strain softening

In [232] we chose the direction of themaximumdisplace-ment gradient by maximizing

119892 = max⏟⏟⏟⏟⏟⏟⏟119897

(n119879119897sdot (nablau sdot h

119897)) 119897 = 1 2 (61)

where the normals n119897correspond to minima of Q (60) h is

the corresponding eigenvector of Q Oliver et al [298] haveshown that an anisotropic continuum model will lead to aunique angle 120579 in the material stability analysis

Since the normal n = [cos120572 cos120593 cos120572 sin120593 sin120572]

depends on two angles in three dimensions the proce-dure of finding the minimum eigenvalue of Q can becomecomputationally expensive A bisection method can reducecomputational cost Another efficient way to compute theeigenvalues of the acoustic tensor is given by Ortiz et al [77]

54 Rank-One Stability Criterion Belytschko et al [299]suggest to check the rank-one stability criterion

F A

F le 0 forall

F (62)

before carrying out a material stability analysis at everyquadrature point They point out that it is a stricter butcomputational cheaper criterion than the loss-of materialstability criterion However the rank-one stability criteriondoes not provide the orientation of the crack


5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)

Figure 28 Minimum eigenvalue of the acoustic tensor Q versusdifferent angles of the orientation of n for (a) uniaxial tension (b)an arbitrarily directed load

55 Energy Criteria The orientation of the crack can beobtained by global energy minimization For different ori-entations of the crack the global energy is computed andthe crack is propagated in the direction where the globalenergy has its minimum This global fracture criterion canbe expressed as

I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)

where G119905is the surface toughness that characterizes the

surface energy necessary to create the new crack surfaceSuch criteria were used in the context of XFEM

by Meschke and Dummerstorf [300] Dummerstorf andMeschke [301] It was presented in a two-dimensional settingfor cohesive and cohesionless cracks As pointed out bythe authors an extension to 3D is cumbersome Miehe andGurses [289] presented a similar method in the contextof an r-adaptive interelement-separation method in twodimension The crack orientation as well as the crack lengthis introduced as additional unknowns in the variationalformulation

56 Computation of the Crack ldquoLengthrdquo Asmentioned abovethe easiest way is to control the ldquolengthrdquo of the crack In the

context of XFEM Belytschko et al [302] suggested to takeadvantage of the level set method to obtain the new positionof the crack tip The key assumption is that the hyperbolicityindicator 119890 = h sdotQ sdot hmust vanish at the crack tip

120597119890

120597119905+ V

119888

sdot nabla119890 = 0 with V119888

= V119888s (64)

where V119888 is the crack speed s gives its direction whichmust fulfill the condition n sdot s = 0 To obtain the cracklength (64) has to be solved for V119888 Due to the hyperboliccharacter of the differential equation the standard BubnovGalerkin formulation leads to instabilities and stabilizationtechniques such as upwinding Petrov-Galerkin Galerkinleast squares streamline upwind Petrov Galerkin or finiteincrement calculus Stabilization is needed as pointed out forexample in [285]

6 Future Perspectives and Conclusion

Numerous computational methods for fracture have beendeveloped in the past two decades Advances in partition ofunity methods have provided effective and reliable tools toanalyze fracture for numerous applications Improvementswere made particularly concerning the accuracy (throughenrichment) and the efficiency (no remeshing) Someof thosemethods are already available in commercial CAE softwarepackages and can be used for commercial applications aspointed out previouslyMost of the partition of unitymethodsfor fracture are well suited for static fracture when amoderatenumber of cracks occur Many applications (of partition ofunity methods) focus on crack propagation mainly for prop-agation of single crack surfaces without branching cracks andcrack coalescence While the majority of the computationalmethods discussed above are capable of handling complexphenomena such as branching cracks reliable criteria tobranch a crack in practical FE-simulations are still missingThose fracture criteria can be categorized in local criteria andglobal criteria Local criteria are based on the stress state inthe vicinity of the crack tip while global criteria are basedon energy minimization Both criteria should in principlepredict complex fracture pattern naturally For example it isknown that the ellipticity indicator 119890 should be zero at thecrack tip and if one could find the locations where 119890 = 0 thepositions of the new crack tip are obtained naturally Besidessome efforts to seaming-less model the transition fromcontinuum to discontinuum the combination of the fracturecriterion and the computational method remains a challengeA very interesting approach is the variational approach tofracture [262] as the crack path is the natural outcomeof the analysis It also circumvents the implementation ofcomplex crack tracking algorithms and the need to describethe topology of the crack surface Moreover the analysis ofcoupled problems seems to be simpler as the enrichmentin XFEM can become quite cumbersome However mostimplementations of the (already computational expensive)variational approach to fracture require a high resolution ofthe crack (with several elements) increasing computationalcost further Combining the advantages of partition of unity


methods for cracks (such as XFEM) and the variationalapproach to fracture does not seem to be easy but it couldlead to a new efficient and reliable pathway tomodel complexfracture patterns

Whilemdashas stated abovemdashthe majority of the applica-tions of computational methods for fracture focus on crackpropagation problems far less studies are concerned withcrack nucleation Methods as XFEM seem not well suitedfor dynamic fracture and fragmentation involving the nucle-ation growth and coalescence of an enormous number ofcracks An alternative pathway to dynamic fracture offermethods such as the distinct element method (DEM) [303304] discontinuous deformation analysis (DDA) [305 306]the particle flow code (PFC) [307] or peridynamics [308] inbrief discrete methods (not discussed in this paper) that arenot based on continuum approaches

The exact prediction of complex fracture patterns isnearly impossible for many applications due to its stochasticnature The onset of crack nucleation and in fact also thedirection in crack propagation depends on numerous factorssuch as the micro-structure of the material (imperfectionsvoids microcracks etc) or loading conditions that are notknown exactly The author has made the experience thatresults are particularly sensitive with respect to boundaryconditions for example Most of the novel computationalmethods for fracture are based on deterministic approachesThere are far less contributions on statistical computationalmethods for fracture [309ndash314] The development of efficientand reliable stochastic computational methods is one of thekey challenges in the future Two pathways can be followed

(i) deterministic methods in a stochastic setting and(ii) full stochastic methodsThe first approach seems simpler as no modifications of

the existing computational methods are needed Samplingmethods belong to the first category while variance reductionmethods for example belong to the second category Onemajor challenge will be the design of computational efficientmethods

The key objective of computational methods is theirapplication to ldquoreal-worldrdquo problems for example in orderto support the design of new products The choice of themethod depends on the application Most effort in the pastwas devoted to develop new methods However there hasbeen minimal effort to assess those methods In view ofthe growing number of computational methods for frac-ture it would be of high practical relevance to provide aframework to quantitatively assess the quality of existingcomputational methods with respect to their accuracy reli-ability and robustness for a specific application For examplecomputational methods (and constitutive models) have anumber of uncertain input parameters such as the dilationparameter inmeshfreemethods It is of utmost importance toquantitatively determine the sensitivity of the uncertain inputparameters with respect to a quantity of interest (the output)It will also help to quantify the predictive capabilities of thecomputational methods

One major application of computational methods forfracture is their use in the design of newmaterials Obtaining

Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)

Figure 29 Schematic of a (a) hierarchical (b) semiconcurrent and(c) concurrent multiscale methods [315]

a fundamental understanding of how materials fail wasis amain research direction in materials science In failure theresponse of a structure is driven by fine scale features (nano-or micro-structure of the material) For an accurate predic-tion of material failure it is important to account for finescale features in the process zone Therefore an importantfuture research direction is the development of multiscalemethods for fracture While numerous multiscale methods(see eg [316 317]) were developed for intact materials farfewer methods are applicable for fracture simulations thoughthe effort is rapidly increasing


Multiscale methods can be categorized into hierarchicalsemiconcurrent and concurrent methods [318] In hierarchi-cal multiscale methods information is passed from the fine-scale to the coarse scale but not vice versa Computationalhomogenization [319] is a classical hierarchical upscalingtechniqueHowever hierarchicalmultiscale approaches seemless suitable to model fracture One basic assumption for theapplication of homogenization theories is the existence ofdisparate length scales [320]LCr ≪ LRVE ≪ LSpec whereLCrLRVE andLSpec are the crack length the representativevolume element (RVE) and specimen-size respectively Forproblems involving fracture the first condition is violated asLCr is of the order of LRVE Moreover periodic boundaryconditions (PBCs) often used at the fine-scale cannot be usedwhen a crack touches a boundary as the displacement jumpin that boundary violates the PBC Nevertheless fine-scalefeatures can also be incorporated ldquohierarchicallyrdquo throughenrichment For example Gracie et al [321 322] Belytschkoand Gracie [323] developed special enrichment functions fordislocations

The basic idea of semiconcurrent multiscale methodsis illustrated in Figure 29(b) In semi-concurrent multiscalemethods information is passed from the fine-scale to thecoarse-scale and vice versa Semi-concurrent multiscalemethods are of the same computational efficiency as concur-rent multiscale methods [324] The key advantage of semi-concurrent multiscale methods over concurrent multiscalemethods is their flexibility that is their ability to coupletwo different software packages for example MD softwareto FE software Parallelization is generally simple as well Apopular semi-concurrent multiscale method is the FE2 [325]originally developed for intact materials Kouznetsova [319]was the first who extended thismethod to problems involvingmaterial failure see also Kouznetsova et al [326] or recentcontribution byNguyen et al [327] Verhoosel et al [328] andBelytschko et al [299]

Numerous concurrent multiscale methods [329ndash332]have been developed that can be classified into ldquointerfacerdquocouplingmethods and ldquoHandshakerdquo couplingmethods Inter-face coupling methods seem to be less effective for dynamicapplications as avoiding spurious wave reflections at theldquoartificialrdquo interface seem to be more problematic Some ofthe concurrent multiscale methods have been extended tomodeling fracture [318 333ndash335]

Future challenges will lie in the development of efficientmethods to transfer length scales when fracture occurs toadaptively choose the discretization and the model basedon error estimation and to bridge disparate time scaleTo overcome the high computational cost will be anotherchallenge in particular when these methods will be appliedto ldquoreal-worldrdquo applications for example in computationalmaterials design

References

[1] S LiW K Liu A J Rosakis T Belytschko andWHao ldquoMesh-free Galerkin simulations of dynamic shear band propagationand failure mode transitionrdquo International Journal of Solids andStructures vol 39 no 5 pp 1213ndash1240 2002

[2] S Li and B C Simonson ldquoMeshfree simulation of ductilecrack propagationrdquo International Journal of ComputationalEngineering Science vol 6 pp 1ndash25 2005

[3] B C Simonsen and S Li ldquoMesh-free simulation of ductilefracturerdquo International Journal for Numerical Methods in Engi-neering vol 60 no 8 pp 1425ndash1450 2004

[4] D C Simkins Jr and S Li ldquoMeshfree simulations of thermo-mechanical ductile fracturerdquo Computational Mechanics vol 38no 3 pp 235ndash249 2006

[5] B Ren and S Li ldquoMeshfree simulations of plugging failures inhigh-speed impactsrdquo Computers and Structures vol 88 no 15-16 pp 909ndash923 2010

[6] B Ren J Qian X Zeng A K Jha S Xiao and S Li ldquoRecentdevelopments on thermo-mechanical simulations of ductilefailure bymeshfreemethodrdquoComputerModeling in Engineeringand Sciences vol 71 no 3 pp 253ndash277 2011

[7] B Ren and S Li ldquoModeling and simulation of large-scale ductilefracture in plates and shellsrdquo International Journal of Solids andStructures vol 49 no 18 pp 2373ndash2393 2012

[8] E E Gdoutos Fracture Mechanics An Introduction vol 123Kluwer Academic 2005

[9] J Planas and M Elices ldquoNonlinear fracture of cohesive materi-alsrdquo International Journal of Fracture vol 51 no 2 pp 139ndash1571991

[10] A Carpinteri ldquoPost-peak and post-bifurcation analysis ofcohesive crack propagationrdquo Engineering Fracture Mechanicsvol 32 no 2 pp 265ndash278 1989

[11] NMoes andT Belytschko ldquoExtendedfinite elementmethod forcohesive crack growthrdquo Engineering FractureMechanics vol 69no 7 pp 813ndash833 2002

[12] A Carpinteri ldquoNotch sensitivity in fracture testing of aggrega-tivematerialsrdquoEngineering FractureMechanics vol 16 no 4 pp467ndash481 1982

[13] Z P Bazant and G Pijaudier-Cabot ldquoNonlocal continuumdamage localization instabilities and convergencerdquo Journal ofApplied Mechanics Transactions ASME vol 55 no 2 pp 287ndash293 1988

[14] Z P Bazant and M Jirasek ldquoNonlocal integral formulations ofplasticity and damage survey of progressrdquo Journal of Engineer-ing Mechanics vol 128 no 11 pp 1119ndash1149 2002

[15] R De Borst J Pamin and M G D Geers ldquoOn coupledgradient-dependent plasticity and damage theories with a viewto localization analysisrdquo European Journal of Mechanics A vol18 no 6 pp 939ndash962 1999

[16] R de Borst M A Gutierrez G N Wells J J C Remmers andHAskes ldquoCohesive-zonemodels higher-order continuum the-ories and reliabilitymethods for computational failure analysisrdquoInternational Journal for Numerical Methods in Engineering vol60 no 1 pp 289ndash315 2004

[17] R De Borst ldquoFracture in quasi-brittle materials a review ofcontinuum damage-based approachesrdquo Engineering FractureMechanics vol 69 no 2 pp 95ndash112 2001

[18] J Fish Q Yu and K Shek ldquoComputational damage mechanicsfor composite materials based on mathematical homogeniza-tionrdquo International Journal for Numerical Methods in Engineer-ing vol 45 no 11 pp 1657ndash1679 1999

[19] D Krajcinovic and S Mastilovic ldquoSome fundamental issues ofdamage mechanicsrdquo Mechanics of Materials vol 21 no 3 pp217ndash230 1995

[20] D Krajcinovic Damage Mechanics North Holland Series inApplied Mathematics and Mechanics Elsevier 1996


[21] J Lemaitre and H Lippmann A Course on Damage Mechanicsvol 2 Springer Berlin Germany 1996

[22] R H J Peerlings R De Borst W A M Brekelmans and JH P De Vree ldquoGradient enhanced damage for quasi-brittlematerialsrdquo International Journal for Numerical Methods inEngineering vol 39 no 19 pp 3391ndash3403 1996

[23] J F Shao and J W Rudnicki ldquoMicrocrack-based continuousdamage model for brittle geomaterialsrdquoMechanics of Materialsvol 32 no 10 pp 607ndash619 2000

[24] J C Simo and J W Ju ldquoStrain- and stress-based continuumdamage models-I Formulationrdquo International Journal of Solidsand Structures vol 23 no 7 pp 821ndash840 1987

[25] M SilhavyThe Mechanics and Thermodynamics of ContinuousMedia Springer Berlin Germany 1997

[26] L M Kachanov ldquoTime of the rupture process under creepconditionsrdquo Izvestiya Akademii Nauk SSSR Otdelenie Teknichesvol 8 pp 26ndash31 1958

[27] R Hill ldquoAcceleration waves in solidsrdquo Journal of the Mechanicsand Physics of Solids vol 10 pp 1ndash16 1962

[28] B Loret and J H Prevost ldquoDynamic strain localization inelasto-(visco-)plastic solids Part 1 General formulation andone-dimensional examplesrdquo Computer Methods in AppliedMechanics and Engineering vol 83 no 3 pp 247ndash273 1990

[29] J H Prevost and B Loret ldquoDynamic strain localization inelasto-(visco-)plastic solids part 2 plane strain examplesrdquoComputer Methods in Applied Mechanics and Engineering vol83 no 3 pp 275ndash294 1990

[30] J W Rudnicki and J R Rice ldquoConditions for the localization ofdeformation in pressure-sensitive dilatant materialsrdquo Journal ofthe Mechanics and Physics of Solids vol 23 no 6 pp 371ndash3941975

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

[32] Z P Bazant and T B Belytschko ldquoWave propagation in a strain-softening bar exact solutionrdquo Journal of EngineeringMechanicsvol 111 no 3 pp 381ndash389 1985

[33] R De Borst andH BMuehlhaus ldquoGradient-dependent plastic-ity formulation and algorithmic aspectsrdquo International Journalfor NumericalMethods in Engineering vol 35 no 3 pp 521ndash5391992

[34] N A Fleck and J W Hutchinson ldquoA phenomenological theoryfor strain gradient effects in plasticityrdquo Journal of the Mechanicsand Physics of Solids vol 41 no 12 pp 1825ndash1857 1993

[35] R H J Peerlings R de Borst W A M Brekelmans and M GD Geers ldquoLocalisation issues in local and nonlocal continuumapproaches to fracturerdquo European Journal of Mechanics A vol21 no 2 pp 175ndash189 2002

[36] Z P Bazant ldquoWhy continuum damage is nonlocal microme-chanics argumentsrdquo Journal of Engineering Mechanics vol 117no 5 pp 1070ndash1087 1991

[37] G Etse and K Willam ldquoFailure analysis of elastoviscoplasticmaterial modelsrdquo Journal of EngineeringMechanics vol 125 no1 pp 60ndash68 1999

[38] T Rabczuk and J Eibl ldquoSimulation of high velocity concretefragmentation using SPHMLSPHrdquo International Journal forNumericalMethods in Engineering vol 56 no 10 pp 1421ndash14442003

[39] D S Dugdale ldquoYielding of steel sheets containing slitsrdquo Journalof the Mechanics and Physics of Solids vol 8 no 2 pp 100ndash1041960

[40] G I Barenblatt ldquoThe mathematical theory of equilibriumcracks in brittle fracturerdquo Advances in Applied Mechanics vol7 pp 55ndash129 1962

[41] AHillerborgMModeer and P E Petersson ldquoAnalysis of crackformation and crack growth in concrete by means of fracturemechanics and finite elementsrdquo Cement and Concrete Researchvol 6 no 6 pp 773ndash781 1976

[42] K Keller S Weihe T Siegmund and B Kroplin ldquoGeneralizedCohesive Zone Model incorporating triaxiality dependentfailure mechanismsrdquo Computational Materials Science vol 16no 1ndash4 pp 267ndash274 1999

[43] F Zhou J F Molinari and T Shioya ldquoA rate-dependentcohesive model for simulating dynamic crack propagation inbrittle materialsrdquo Engineering Fracture Mechanics vol 72 no9 pp 1383ndash1410 2005

[44] I Carol and P C Prat ldquoA statically constrained microplanemodel for the smeared analysis of concrete crackingrdquo inComputer Aided Analysis and Design of Concrete StructuresN Bicanic and H Mang Eds pp 919ndash930 Pinedidge PressSwansea UK 1990

[45] J CervenkaDiscrete crackmodeling in concrete structures [PhDthesis] University of Colorado 1994

[46] G T Camacho and M Ortiz ldquoComputational modelling ofimpact damage in brittle materialsrdquo International Journal ofSolids and Structures vol 33 no 20ndash22 pp 2899ndash2938 1996

[47] A Pandolfi P Krysl andMOrtiz ldquoFinite element simulation ofring expansion and fragmentation the capturing of length andtime scales through cohesive models of fracturerdquo InternationalJournal of Fracture vol 95 no 1ndash4 pp 279ndash297 1999

[48] X Liu S Li and N Sheng ldquoA cohesive finite element for quasi-continuardquoComputationalMechanics vol 42 no 4 pp 543ndash5532008

[49] S Li X Zeng B Ren J Qian J Zhang and A K Jha ldquoAnatomistic-based interphase zone model for crystalline solidsrdquoComputer Methods in Applied Mechanics and Engineering vol229ndash232 pp 87ndash109 2012

[50] J Qian and S Li ldquoApplication of multiscale cohesive zonemodel to simulate fracture in polycrystalline solidsrdquo Journalof Engineering Materials and Technology Transactions of theASME vol 133 no 1 Article ID 011010 2011

[51] X Zeng and S Li ldquoA multiscale cohesive zone model and sim-ulations of fracturesrdquo Computer Methods in Applied Mechanicsand Engineering vol 199 no 9ndash12 pp 547ndash556 2010

[52] X Zeng and S Li ldquoApplication of a multiscale cohesive zonemethod to model composite materialsrdquo International Journalof Multiscale Computational Engineering vol 10 pp 391ndash4052012

[53] L Liu and S Li ldquoA finite temperature multiscale interphasefinite element method and simulations of fracturerdquo ASMEJournal of EngineeringMaterials andTechnology vol 134ArticleID 03014 pp 1ndash12 2012

[54] M He and S Li ldquoAn embedded atom hyperelastic constitutivemodel and multiscale cohesive finite element methodrdquo Compu-tational Mechanics vol 49 no 3 pp 337ndash355 2012

[55] M Elices G V Guinea J Gomez and J Planas ldquoThe cohesivezone model advantages limitations and challengesrdquo Engineer-ing Fracture Mechanics vol 69 no 2 pp 137ndash163 2001

[56] K D Papoulia C H Sam and S A Vavasis ldquoTime continuityin cohesive finite element modelingrdquo International Journal forNumerical Methods in Engineering vol 58 no 5 pp 679ndash7012003


[57] Z P Bazant and B H Oh ldquoCrack band theory for fracture ofconcreterdquoMateriaux et Constructions vol 16 no 3 pp 155ndash1771983

[58] J C Simo J Oliver and F Armero ldquoAn analysis of strongdiscontinuities induced by strain-softening in rate-independentinelastic solidsrdquo Computational Mechanics vol 12 no 5 pp277ndash296 1993

[59] M Jirasek and T Zimmermann ldquoAnalysis of rotating crackmodelrdquo Journal of Engineering Mechanics vol 124 no 8 pp842ndash851 1998

[60] M asek and T Zimmermann ldquoRotating crack model withtransition to scalar damagerdquo Journal of Engineering Mechanicsvol 124 no 3 pp 277ndash284 1998

[61] T Rabczuk J Akkermann and J Eibl ldquoA numerical model forreinforced concrete structuresrdquo International Journal of Solidsand Structures vol 42 no 5-6 pp 1327ndash1354 2005

[62] F Ohmenhauser S Weihe and B Kroplin ldquoAlgorithmic imple-mentation of a generalized cohesive crack modelrdquo Computa-tional Materials Science vol 16 no 1ndash4 pp 294ndash306 1999

[63] A Carpinteri B Chiaia and P Cornetti ldquoA scale-invariantcohesive crack model for quasi-brittle materialsrdquo EngineeringFracture Mechanics vol 69 no 2 pp 207ndash217 2001

[64] M Francois and G Royer-Carfagni ldquoStructured deformationof damaged continua with cohesive-frictional sliding roughfracturesrdquo European Journal of Mechanics A vol 24 no 4 pp644ndash660 2005

[65] R de Borst J J C Remmers and A Needleman ldquoMesh-independent discrete numerical representations of cohesive-zone modelsrdquo Engineering Fracture Mechanics vol 73 no 2 pp160ndash177 2006

[66] G R Johnson and R A Stryk ldquoEroding interface and improvedtetrahedral element algorithms for high-velocity impact com-putations in three dimensionsrdquo International Journal of ImpactEngineering vol 5 no 1ndash4 pp 411ndash421 1987

[67] T Belytschko and J I Lin ldquoA three-dimensional impact-penetration algorithm with erosionrdquo International Journal ofImpact Engineering vol 5 no 1ndash4 pp 111ndash127 1987

[68] S R Beissel G R Johnson and C H Popelar ldquoAn element-failure algorithm for dynamic crack propagation in generaldirectionsrdquo Engineering Fracture Mechanics vol 61 no 3-4 pp407ndash425 1998

[69] R Fan and J Fish ldquoThe 119903119904-method for material failuresimulationsrdquo International Journal for Numerical Methods inEngineering vol 73 no 11 pp 1607ndash1623 2008

[70] J H SongHWang andT Belytschko ldquoA comparative study onfinite element methods for dynamic fracturerdquo ComputationalMechanics vol 42 no 2 pp 239ndash250 2008

[71] A Pandolfi and M Ortiz ldquoAn eigenerosion approach to brittlefracturerdquo International Journal for Numerical Methods in Engi-neering vol 92 no 8 pp 694ndash714 2012

[72] B Schmidt F Fraternali andMOrtiz ldquoEigenfracture an eigen-deformation approach to variational fracturerdquo SIAMMultiscaleModeling and Simulation vol 7 no 3 pp 1237ndash1266 2008

[73] T Boslashrvik O S Hopperstad and K O Pedersen ldquoQuasi-brittlefracture during structural impact of AA7075-T651 aluminiumplatesrdquo International Journal of Impact Engineering vol 37 no5 pp 537ndash551 2010

[74] M Negri ldquoA finite element approximation of the Griffithrsquosmodel in fracture mechanicsrdquo Numerische Mathematik vol 95no 4 pp 653ndash687 2003

[75] X P Xu and A Needleman ldquoNumerical simulations of fastcrack growth in brittle solidsrdquo Journal of the Mechanics andPhysics of Solids vol 42 no 9 pp 1397ndash1434 1994

[76] X P Xu and A Needleman ldquoVoid nucleation by inclusiondebonding in a crystal matrixrdquo Modelling and Simulation inMaterials Science and Engineering vol 1 no 2 pp 111ndash132 1993

[77] M Ortiz Y Leroy and A Needleman ldquoA finite elementmethodfor localized failure analysisrdquo Computer Methods in AppliedMechanics and Engineering vol 61 no 2 pp 189ndash214 1987

[78] F Cirak M Ortiz and A Pandolfi ldquoA cohesive approach tothin-shell fracture and fragmentationrdquo Computer Methods inApplied Mechanics and Engineering vol 194 no 21ndash24 pp2604ndash2618 2005

[79] A Pandolfi P R Guduru M Ortiz and A J Rosakis ldquoThreedimensional cohesive-element analysis and experiments ofdynamic fracture in C300 steelrdquo International Journal of Solidsand Structures vol 37 no 27 pp 3733ndash3760 2000

[80] F Zhou and J F Molinari ldquoDynamic crack propagation withcohesive elements a methodology to address mesh depen-dencyrdquo International Journal for Numerical Methods in Engi-neering vol 59 no 1 pp 1ndash24 2004

[81] M L Falk A Needleman and J R Rice ldquoA critical evaluationof cohesive zonemodels of dynamic fracturerdquo Journal of Physicsvol 11 pp Pr5-43ndashPr5-50 2001

[82] H D Espinosa P D Zavattieri and G L Emore ldquoAdaptiveFEM computation of geometric and material nonlinearitieswith application to brittle failurerdquo Mechanics of Materials vol29 no 3-4 pp 275ndash305 1998

[83] S Knell A numerical modeling approach for the transientresponse of solids at the mesoscale [PhD thesis] Univeritat derBundeswehr Munchen 2011

[84] H D Espinosa P D Zavattieri and S K Dwivedi ldquoA finitedeformation continuumdiscrete model for the description offragmentation and damage in brittle materialsrdquo Journal of theMechanics and Physics of Solids vol 46 no 10 pp 1909ndash19421998

[85] S Roshdy and R S Barsoum ldquoTriangular quarter-pointelements as elastic and perfectly-plastic crack tip elementsrdquoInternational Journal For Numerical Methods in Engineeringvol 11 no 1 pp 85ndash98 1977

[86] R S Barsoum ldquoApplication of quadratic isoparametric finiteelements in linear fracture mechanicsrdquo International Journal ofFracture vol 10 no 4 pp 603ndash605 1974

[87] R S Barsoum ldquoFurther application of quadratic isoparametricfinite elements to linear fracturemechanics of plate bending andgeneral shellsrdquo International Journal of Fracture vol 11 no 1 pp167ndash169 1975

[88] R S Barsoum ldquoOn the use of isoparametric finite elements inlinear fracture mechanicsrdquo International Journal for NumericalMethods in Engineering vol 10 no 1 pp 25ndash37 1976

[89] G R LiuNNourbakhshnia andYWZhang ldquoAnovel singularES-FEM method for simulating singular stress fields near thecrack tips for linear fracture problemsrdquo Engineering FractureMechanics vol 78 no 6 pp 863ndash876 2011

[90] G R Liu N Nourbakhshnia L Chen and Y W ZhangldquoA novel general formulation for singular stress field usingthe ES-FEM method for the analysis of mixed-mode cracksrdquoInternational Journal of Computational Methods vol 7 no 1 pp191ndash214 2010

[91] L Chen G R Liu Y Jiang K Zeng and J Zhang ldquoA singularedge-based smoothed finite element method (ES-FEM) for


crack analyses in anisotropic mediardquo Engineering FractureMechanics vol 78 no 1 pp 85ndash109 2011

[92] L Chen G R Liu N Nourbakhsh-Nia and K Zeng ldquoA singu-lar edge-based smoothed finite element method (ES-FEM) forbimaterial interface cracksrdquo Computational Mechanics vol 45no 2-3 pp 109ndash125 2010

[93] Y Jiang G R Liu Y W Zhang L Chen and T E Tay ldquoAsingular ES-FEM for plastic fracture mechanicsrdquo ComputerMethods in AppliedMechanics and Engineering vol 200 no 45-46 pp 2943ndash2955 2011

[94] H Nguyen-Xuan G R Liu N Nourbakhshnia and L Chen ldquoAnovel singular es-fem for crack growth simulationrdquo EngineeringFracture Mechanics vol 84 pp 41ndash66 2012

[95] N Nourbakhshnia and G R Liu ldquoA quasi-static crack growthsimulation based on the singular ES-FEMrdquo International Jour-nal for NumericalMethods in Engineering vol 88 no 5 pp 473ndash492 2011

[96] T Belytschko J Fish and B E Engelmann ldquoA finite ele-ment with embedded localization zonesrdquo Computer Methods inApplied Mechanics and Engineering vol 70 no 1 pp 59ndash891988

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

[98] M Jirasek ldquoComparative study on finite elements with embed-ded discontinuitiesrdquo Computer Methods in Applied Mechanicsand Engineering vol 188 no 1 pp 307ndash330 2000

[99] H R Lotfi and P B Shing ldquoEmbedded representation offracture in concrete with mixed finite elementsrdquo InternationalJournal for Numerical Methods in Engineering vol 38 no 8 pp1307ndash1325 1995

[100] M Klisinski K Runesson and S Sture ldquoFinite element withinner softening bandrdquo Journal of Engineering Mechanics ASCEvol 117 pp 575ndash587 1991

[101] E Samaniego X Oliver and A Huespe Contributions tothe continuum modelling of strong discontinuities in two-dimensional solids [PhD thesis] International Center forNumerical Methods in Engineering Barcelona Spain 2003Monograph CIMNE No 72

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

[103] J Oliver A E Huespe M D G Pulido and E SamaniegoldquoOn the strong discontinuity approach in finite deformationsettingsrdquo International Journal for Numerical Methods in Engi-neering vol 56 no 7 pp 1051ndash1082 2003

[104] J Oliver ldquoOn the discrete constitutive models induced bystrong discontinuity kinematics and continuum constitutiveequationsrdquo International Journal of Solids and Structures vol 37no 48ndash50 pp 7207ndash7229 2000

[105] J Oliver M Cervera and O Manzoli ldquoStrong discontinuitiesand continuum plasticity models the strong discontinuityapproachrdquo International Journal of Plasticity vol 15 no 3 pp319ndash351 1999

[106] J Oliver ldquoModelling strong discontinuities in solid mechanicsvia strain softening constituitive equations part 1 fundamen-tals part 2 numerical simulationrdquo International Journal ForNumerical Methods in Engineering vol 39 pp 3575ndash3624 1996

[107] C Linder and C Miehe ldquoEffect of electric displacementsaturation on the hysteretic behavior of ferroelectric ceramicsand the initiation and propagation of cracks in piezoelectricceramicsrdquo Journal of theMechanics and Physics of Solids vol 60no 5 pp 882ndash903 2012

[108] C D Foster R I Borja and R A Regueiro ldquoEmbedded strongdiscontinuity finite elements for fractured geomaterials withvariable frictionrdquo International Journal for Numerical Methodsin Engineering vol 72 no 5 pp 549ndash581 2007

[109] C Linder D Rosato and C Miehe ldquoNew finite elements withembedded strong discontinuities for the modeling of failurein electromechanical coupled solidsrdquo Computer Methods inApplied Mechanics and Engineering vol 200 no 1ndash4 pp 141ndash161 2011

[110] F Armero and K Garikipati ldquoAn analysis of strong discontinu-ities in multiplicative finite strain plasticity and their relationwith the numerical simulation of strain localization in solidsrdquoInternational Journal of Solids and Structures vol 33 no 20ndash22pp 2863ndash2885 1996

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

[112] F Armero and C Linder ldquoNumerical simulation of dynamicfracture using finite elements with embedded discontinuitiesrdquoInternational Journal of Fracture vol 160 no 2 pp 119ndash1412009

[113] J Oliver A E Huespe and P J Sanchez ldquoA comparativestudy on finite elements for capturing strong discontinuities E-FEM vs X-FEMrdquo Computer Methods in Applied Mechanics andEngineering vol 195 no 37ndash40 pp 4732ndash4752 2006

[114] C Feist and G Hofstetter ldquoThree-dimensional fracture simu-lations based on the SDArdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 31 no 2 pp 189ndash212 2007

[115] J M Sancho J Planas A M Fathy J C Galvez and D ACendon ldquoThree-dimensional simulation of concrete fractureusing embedded crack elements without enforcing crack pathcontinuityrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 31 no 2 pp 173ndash187 2007

[116] J Mosler and G Meschke ldquoEmbedded crack vs smeared crackmodels a comparison of elementwise discontinuous crack pathapproaches with emphasis on mesh biasrdquo Computer Methods inAppliedMechanics andEngineering vol 193 no 30ndash32 pp 3351ndash3375 2004

[117] 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

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

[119] J M Melenk and I Babuska ldquoThe partition of unity finiteelement method basic theory and applicationsrdquo ComputerMethods in AppliedMechanics and Engineering vol 139 no 1ndash4pp 289ndash314 1996

[120] T Strouboulis K Copps and I Babuska ldquoThe generalizedfinite element method an example of its implementationand illustration of its performancerdquo International Journal forNumerical Methods in Engineering vol 47 no 8 pp 1401ndash14172000


[121] T Strouboulis I Babuska and K Copps ldquoThe design andanalysis of the generalized finite element methodrdquo ComputerMethods in Applied Mechanics and Engineering vol 181 no 1ndash3pp 43ndash69 2000

[122] J Chessa and T Belytschko ldquoAn enriched finite elementmethod and level sets for axisymmetric two-phase flow withsurface tensionrdquo International Journal for Numerical Methods inEngineering vol 58 no 13 pp 2041ndash2064 2003

[123] J Chessa andT Belytschko ldquoAn extendedfinite elementmethodfor two-phase fluidsrdquo Journal of AppliedMechanics TransactionsASME vol 70 no 1 pp 10ndash17 2003

[124] A Zilian and A Legay ldquoThe enriched space-time finite elementmethod (EST) for simultaneous solution of fluid-structureinteractionrdquo International Journal for Numerical Methods inEngineering vol 75 no 3 pp 305ndash334 2008

[125] U M Mayer A Gerstenberger andW A Wall ldquoInterface han-dling for three-dimensional higher-order XFEM-computationsin fluid-structure interactionrdquo International Journal for Numer-ical Methods in Engineering vol 79 no 7 pp 846ndash869 2009

[126] R Duddu S Bordas D Chopp and B Moran ldquoA combinedextended finite element and level set method for biofilmgrowthrdquo International Journal for Numerical Methods in Engi-neering vol 74 no 5 pp 848ndash870 2008

[127] D Rabinovich D Givoli and S Vigdergauz ldquoXFEM-basedcrack detection schemeusing a genetic algorithmrdquo InternationalJournal for Numerical Methods in Engineering vol 71 no 9 pp1051ndash1080 2007

[128] D Rabinovich D Givoli and S Vigdergauz ldquoCrack identifi-cation by lsquoarrival timersquo using XFEM and a geneticalgorithmrdquo International Journal for Numerical Methods inEngineering vol 77 no 3 pp 337ndash359 2009

[129] E BechetM Scherzer andMKuna ldquoApplication of theX-FEMto the fracture of piezoelectric materialsrdquo International Journalfor Numerical Methods in Engineering vol 77 no 11 pp 1535ndash1565 2009

[130] C V Verhoosel J J C Remmers and M A Gutierrez ldquoApartition of unity-based multiscale approach for modellingfracture in piezoelectric ceramicsrdquo International Journal forNumerical Methods in Engineering vol 82 no 8 pp 966ndash9942010

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

[132] P M A Areias and T Belytschko ldquoTwo-scale shear bandevolution by local partition of unityrdquo International Journal forNumerical Methods in Engineering vol 66 no 5 pp 878ndash9102006

[133] C A Duarte L G Reno and A Simone ldquoA high-ordergeneralized FEM for through-the-thickness branched cracksrdquoInternational Journal for Numerical Methods in Engineering vol72 no 3 pp 325ndash351 2007

[134] C A Duarte O N Hamzeh T J Liszka andWW TworzydloldquoA generalized finite element method for the simulationof three-dimensional dynamic crack propagationrdquo ComputerMethods in Applied Mechanics and Engineering vol 190 no 15ndash17 pp 2227ndash2262 2001

[135] C A Duarte and D-J Kim ldquoAnalysis and applications of ageneralized finite elementmethodwith global-local enrichmentfunctionsrdquo Computer Methods in Applied Mechanics and Engi-neering vol 197 no 6ndash8 pp 487ndash504 2008

[136] J P Pereira C A Duarte X Jiao and D Guoy ldquoGeneralizedfinite element method enrichment functions for curved sin-gularities in 3D fracture mechanics problemsrdquo ComputationalMechanics vol 44 no 1 pp 73ndash92 2009

[137] C A Duarte D-J Kim and I Babuska ldquoA global-localapproach for the construction of enrichment functions forthe generalized FEM and its application to three-dimensionalcracksrdquo in Advances in Meshfree Techniques vol 5 pp 1ndash26Springer Dordrecht The Netherlands 2007

[138] D J Kim J P Pereira and C A Duarte ldquoAnalysis ofthree-dimensional fracture mechanics problems a two-scaleapproach using coarse-generalized FEMmeshesrdquo InternationalJournal for Numerical Methods in Engineering vol 81 no 3 pp335ndash365 2010

[139] D-J Kim C A Duarte and N A Sobh ldquoParallel simulationsof three-dimensional cracks using the generalized finite elementmethodrdquo Computational Mechanics vol 47 no 3 pp 265ndash2822011

[140] A Menk and S P A Bordas ldquoA robust preconditioning tech-nique for the extended finite element methodrdquo InternationalJournal for Numerical Methods in Engineering vol 85 no 13pp 1609ndash1632 2011

[141] I Babuska and U Banerjee ldquoStable generalized finite elementmethod (SGFEM)rdquo Computer Methods in Applied Mechanicsand Engineering vol 201ndash204 pp 91ndash111 2012

[142] G Zi and T Belytschko ldquoNew crack-tip elements for XFEMand applications to cohesive cracksrdquo International Journal forNumericalMethods in Engineering vol 57 no 15 pp 2221ndash22402003

[143] G Zi J H Song E Budyn S H Lee and T Belytschko ldquoAmethod for growing multiple cracks without remeshing and itsapplication to fatigue crack growthrdquo Modelling and Simulationin Materials Science and Engineering vol 12 no 5 pp 901ndash9152004

[144] E Budyn G Zi N Moes and T Belytschko ldquoA method formultiple crack growth in brittle materials without remeshingrdquoInternational Journal for Numerical Methods in Engineering vol61 no 10 pp 1741ndash1770 2004

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

[146] C Daux N Moes J Dolbow N Sukumar and T BelytschkoldquoArbitrary branched and intersecting cracks with the extendedfinite element methodrdquo International Journal for NumericalMethods in Engineering vol 48 no 12 pp 1741ndash1760 2000

[147] P M A Areias and T Belytschko ldquoAnalysis of three-dimensional crack initiation and propagation using theextended finite element methodrdquo International Journal forNumerical Methods in Engineering vol 63 no 5 pp 760ndash7882005

[148] P M A Areias J H Song and T Belytschko ldquoAnalysisof fracture in thin shells by overlapping paired elementsrdquoComputer Methods in Applied Mechanics and Engineering vol195 no 41ndash43 pp 5343ndash5360 2006

[149] J A Sethian Level Set Methods and Fast Marching MethodsEvolving Interfaces in Computational Geometry Fluid Mechan-ics Computer Vision andMaterials Science Cambridge Univer-sity Press Cambridge UK 1999


[150] 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

[151] B Prabel A Combescure A Gravouil and S Marie ldquoLevelset X-FEMnon-matchingmeshes application to dynamic crackpropagation in elastic-plastic mediardquo International Journal forNumerical Methods in Engineering vol 69 no 8 pp 1553ndash15692007

[152] P Laborde J Pommier Y Renard and M Salaun ldquoHigh-order extended finite element method for cracked domainsrdquoInternational Journal for Numerical Methods in Engineering vol64 no 3 pp 354ndash381 2005

[153] G Ventura ldquoOn the elimination of quadrature subcells for dis-continuous functions in the extended finite-element methodrdquoInternational Journal for Numerical Methods in Engineering vol66 no 5 pp 761ndash795 2006

[154] G Ventura R Gracie and T Belytschko ldquoFast integrationand weight function blending in the extended finite elementmethodrdquo International Journal for Numerical Methods in Engi-neering vol 77 no 1 pp 1ndash29 2009

[155] R Gracie H Wang and T Belytschko ldquoBlending in theextended finite element method by discontinuous Galerkin andassumed strain methodsrdquo International Journal for NumericalMethods in Engineering vol 74 no 11 pp 1645ndash1669 2008

[156] S P A Bordas T Rabczuk N X Hung et al ldquoStrain smoothingin FEM and XFEMrdquo Computers and Structures vol 88 no 23-24 pp 1419ndash1443 2010

[157] K W Cheng and T P Fries ldquoHigher-order XFEM for curvedstrong and weak discontinuitiesrdquo International Journal forNumerical Methods in Engineering vol 82 no 5 pp 564ndash5902010

[158] A Nagarajan and S Mukherjee ldquoA mapping method fornumerical evaluation of two-dimensional integrals with 1119903

singularityrdquo Computational Mechanics vol 12 no 1-2 pp 19ndash26 1993

[159] E Bechet H Minnebo N Moes and B Burgardt ldquoImprovedimplementation and robustness study of the X-FEM for stressanalysis around cracksrdquo International Journal for NumericalMethods in Engineering vol 64 no 8 pp 1033ndash1056 2005

[160] J Chessa H Wang and T Belytschko ldquoOn the constructionof blending elements for local partition of unity enrichedfinite elementsrdquo International Journal for Numerical Methods inEngineering vol 57 no 7 pp 1015ndash1038 2003

[161] F Stazi E Budyn J Chessa and T Belytschko ldquoXFEM forfracture mechanics with quadratic elementsrdquo ComputationalMechanics vol 31 pp 38ndash48 2003

[162] 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

[163] J E Tarancon A Vercher E Giner and F J FuenmayorldquoEnhanced blending elements for XFEM applied to linearelastic fracture mechanicsrdquo International Journal for NumericalMethods in Engineering vol 77 no 1 pp 126ndash148 2009

[164] J Bellec and J E Dolbow ldquoA note on enrichment functionsfor modelling crack nucleationrdquo Communications in NumericalMethods in Engineering vol 19 no 12 pp 921ndash932 2003

[165] B L Karihaloo and Q Z Xiao ldquoModelling of stationary andgrowing cracks in FE framework without remeshing a state-of-the-art reviewrdquo Computers and Structures vol 81 no 3 pp 119ndash129 2003

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

[167] T Belytschko R Gracie and G Ventura ldquoA review ofextendedgeneralized finite elementmethods formaterial mod-elingrdquo Modelling and Simulation in Materials Science andEngineering vol 17 no 4 Article ID 043001 2009

[168] S Mohammadi Extended Finite Element Method for FractureAnalysis of Structures Blackwell Publishing Oxford UK 2008

[169] A Hansbo and P Hansbo ldquoA finite elementmethod for the sim-ulation of strong and weak discontinuities in solid mechanicsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 33ndash35 pp 3523ndash3540 2004

[170] J H Song P M A Areias and T Belytschko ldquoA methodfor dynamic crack and shear band propagation with phantomnodesrdquo International Journal for Numerical Methods in Engi-neering vol 67 no 6 pp 868ndash893 2006

[171] T Menouillard J Rethore A Combescure and H BungldquoEfficient explicit time stepping for the extended finite elementmethod (X-FEM)rdquo International Journal for NumericalMethodsin Engineering vol 68 no 9 pp 911ndash939 2006

[172] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamicsapplication to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[173] H Talebi C Samaniego E Samaniego and T Rabczuk ldquoOnthe numerical stability and masslumping schemes for explicitenriched meshfree methodsrdquo International Journal for Numeri-cal Methods in Engineering vol 89 pp 1009ndash1027 2012

[174] T Chau-Dinh G Zi P S Lee T Rabczuk and J H SongldquoPhantom-nodemethod for shell models with arbitrary cracksrdquoComputers and Structures vol 92-93 pp 242ndash256 2012

[175] T Rabczuk G Zi A Gerstenberger and W A Wall ldquoA newcrack tip element for the phantom-node method with arbitrarycohesive cracksrdquo International Journal for NumericalMethods inEngineering vol 75 no 5 pp 577ndash599 2008

[176] J Mergheim E Kuhl and P Steinmann ldquoA finite elementmethod for the computational modelling of cohesive cracksrdquoInternational Journal for Numerical Methods in Engineering vol63 no 2 pp 276ndash289 2005

[177] J Mergheim and P Steinmann ldquoA geometrically nonlinear FEapproach for the simulation of strong andweak discontinuitiesrdquoComputer Methods in Applied Mechanics and Engineering vol195 no 37ndash40 pp 5037ndash5052 2006

[178] D OrganM Fleming T Terry and T Belytschko ldquoContinuousmeshless approximations for nonconvex bodies by diffractionand transparencyrdquo Computational Mechanics vol 18 no 3 pp225ndash235 1996

[179] T Belytschko D Organ and M Tabbara ldquoNumerical sim-ulations of mixed mode dynamic fracture in concrete usingelement-free Galerkin methodsrdquo in Proceedings of the Interna-tional Conference on Environmental Systems (ICES rsquo95) 1995

[180] T Belytschko Y Y Lu L Gu and M Tabbara ldquoElement-freegalerkinmethods for static and dynamic fracturerdquo InternationalJournal of Solids and Structures vol 32 no 17-18 pp 2547ndash25701995

[181] T Belytschko Y Y Lu and L Gu ldquoCrack propagation byelement-free Galerkin methodsrdquo Engineering Fracture Mechan-ics vol 51 no 2 pp 295ndash315 1995


[182] J J C Remmers R De Borst and A Needleman ldquoA cohesivesegments method for the simulation of crack growthrdquo Compu-tational Mechanics vol 31 no 1-2 pp 69ndash77 2003

[183] J J C Remmers R de Borst and A Needleman ldquoThesimulation of dynamic crack propagation using the cohesivesegments methodrdquo Journal of the Mechanics and Physics ofSolids vol 56 no 1 pp 70ndash92 2008

[184] J-H Song and T Belytschko ldquoCracking node method fordynamic fracture with finite elementsrdquo International Journal forNumerical Methods in Engineering vol 77 no 3 pp 360ndash3852009

[185] Y You J S Chen and H Lu ldquoFilters reproducing kernel andadaptive meshfree methodrdquo Computational Mechanics vol 31no 3-4 pp 316ndash326 2003

[186] T Rabczuk and T Belytschko ldquoAdaptivity for structured mesh-free particle methods in 2D and 3Drdquo International Journal forNumerical Methods in Engineering vol 63 no 11 pp 1559ndash15822005

[187] T Rabczuk and E Samaniego ldquoDiscontinuous modelling ofshear bands using adaptive meshfree methodsrdquo ComputerMethods in AppliedMechanics and Engineering vol 197 no 6ndash8pp 641ndash658 2008

[188] T-P Fries A Byfut A Alizada K W Cheng and A SchroderldquoHanging nodes and XFEMrdquo International Journal for Numeri-cal Methods in Engineering vol 86 no 4-5 pp 404ndash430 2011

[189] R A Gingold and J J Monaghan ldquoSmoothed particle hydrody-namicstheory and applications to non-spherical starsrdquoMonthlyNotices of the Royal Astronomical Society vol 181 pp 375ndash3891977

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

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

[192] G R Liu and Y T Gu ldquoA point interpolation method fortwo-dimensional solidsrdquo International Journal For NumericalMethods in Engineering vol 50 pp 937ndash951 2001

[193] S N Atluri The Meshless Local Petrov-Galerkin (MLPG)Method Tech Science Press 2002

[194] E Onate S Idelsohn O C Zienkiewicz and R L Taylor ldquoAfinite point method in computational mechanics Applicationsto convective transport and fluid flowrdquo International Journal forNumerical Methods in Engineering vol 39 no 22 pp 3839ndash3866 1996

[195] G R Liu and M B Liu Smoothed Particle Hydrodynamics AMeshfree Particle Method 2003

[196] G R LiuMesh FreeMethods Moving Beyond the Finite ElementMethod CRC Press Boca Raton Fla USA 2002

[197] G R Liu An Introduction to Meshfree Methods and TheirProgramming Springer 2006

[198] S Li and W K Liu ldquoMeshfree and particle methods and theirapplicationsrdquoAppliedMechanics Reviews vol 55 no 1 pp 1ndash342002

[199] S Li andW K LiuMeshfree Particle Methods Springer BerlinGermany 2004

[200] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[201] V P Nguyen T Rabczuk S Bordas and M Duflot ldquoMeshlessmethods a review and computer implementation aspectsrdquoMathematics and Computers in Simulation vol 79 no 3 pp763ndash813 2008

[202] A Huerta T Belytschko S Fernandez-Mendez and TRabczuk Encyclopedia of Computational Mechanics chapterMeshfree Methods John Wiley and Sons 2004

[203] L D Libersky P W Randles T C Carney and D L Dickin-son ldquoRecent improvements in SPH modeling of hypervelocityimpactrdquo International Journal of Impact Engineering vol 20 no6ndash10 pp 525ndash532 1997

[204] T Rabczuk and J Eibl ldquoModelling dynamic failure of concretewith meshfree methodsrdquo International Journal of Impact Engi-neering vol 32 no 11 pp 1878ndash1897 2006

[205] G A Dilts ldquoMoving least-squares particle hydrodynamicsII Conservation and boundariesrdquo International Journal forNumericalMethods in Engineering vol 48 no 10 pp 1503ndash15242000

[206] A Haque and G A Dilts ldquoThree-dimensional boundary detec-tion for particlemethodsrdquo Journal of Computational Physics vol226 no 2 pp 1710ndash1730 2007

[207] T Rabczuk T Belytschko and S P Xiao ldquoStable particlemethods based on Lagrangian kernelsrdquo Computer Methods inAppliedMechanics and Engineering vol 193 no 12ndash14 pp 1035ndash1063 2004

[208] B Maurel and A Combescure ldquoAn SPH shell formulation forplasticity and fracture analysis in explicit dynamicsrdquo Interna-tional Journal for Numerical Methods in Engineering vol 76 no7 pp 949ndash971 2008

[209] A Combescure BMaurel and S Potapov ldquoModelling dynamicfracture of thin shells filled with fluid a fully SPH modelrdquoMecanique et Industries vol 9 no 2 pp 167ndash174 2008

[210] B Maurel S Potapov J Fabis and A Combescure ldquoFullSPH fluid-shell interaction for leakage simulation in explicitdynamicsrdquo International Journal for Numerical Methods inEngineering vol 80 no 2 pp 210ndash234 2009

[211] S Potapov B Maurel A Combescure and J Fabis ldquoModelingaccidental-type fluid-structure interaction problems with theSPH methodrdquo Computers and Structures vol 87 no 11-12 pp721ndash734 2009

[212] D Sulsky Z Chen and H L Schreyer ldquoA particle methodfor history-dependent materialsrdquo Computer Methods in AppliedMechanics and Engineering vol 118 no 1-2 pp 179ndash196 1994

[213] S Ma X Zhang and X M Qiu ldquoComparison study ofMPM and SPH in modeling hypervelocity impact problemsrdquoInternational Journal of Impact Engineering vol 36 no 2 pp272ndash282 2009

[214] P Krysl and T Belytschko ldquoElement-free Galerkin methodconvergence of the continuous and discontinuous shape func-tionsrdquo Computer Methods in Applied Mechanics and Engineer-ing vol 148 no 3-4 pp 257ndash277 1997

[215] T Rabczuk and T Belytschko ldquoAn adaptive continuumdiscretecrack approach for meshfree particle methodsrdquo Latin AmericanJournal of Solids and Structures vol 1 pp 141ndash166 2003

[216] N Sukumar B Moran T Black and T Belytschko ldquoAnelement-free Galerkin method for three-dimensional fracturemechanicsrdquo Computational Mechanics vol 20 no 1-2 pp 170ndash175 1997

[217] M Duflot ldquoA meshless method with enriched weight functionsfor three-dimensional crack propagationrdquo International Journalfor Numerical Methods in Engineering vol 65 no 12 pp 1970ndash2006 2006


[218] T G Terry Fatigue crack propagationmodeling using the elementfree galerkin method [MS thesis] Northwestern University1994

[219] C A Duarte and J T Oden ldquoAn h-p adaptive methodusing cloudsrdquo Computer Methods in Applied Mechanics andEngineering vol 139 no 1ndash4 pp 237ndash262 1996

[220] T Belytschko and M Fleming ldquoSmoothing enrichment andcontact in the element-free Galerkin methodrdquo Computers ampStructures vol 71 no 2 pp 173ndash195 1999

[221] M Fleming Y A Chu B Moran and T Belytschko ldquoEnrichedelement-free Galerkin methods for crack tip fieldsrdquo Interna-tional Journal for Numerical Methods in Engineering vol 40 no8 pp 1483ndash1504 1997

[222] T P Fries and T Belytschko ldquoThe intrinsic XFEM a methodfor arbitrary discontinuities without additional unknownsrdquoInternational Journal for Numerical Methods in Engineering vol68 no 13 pp 1358ndash1385 2006

[223] M Duflot and H Nguyen-Dang ldquoA meshless method withenriched weight functions for fatigue crack growthrdquo Interna-tional Journal for Numerical Methods in Engineering vol 59 no14 pp 1945ndash1961 2004

[224] A Zamani R Gracie and M R Eslami ldquoHigher order tipenrichment of extended finite element method in thermoelas-ticityrdquo Computational Mechanics vol 46 no 6 pp 851ndash8662010

[225] A Zamani R Gracie and M R Eslami ldquoCohesive andnon-cohesive fracture by higher-order enrichment of xfemrdquoInternational Journal For Numerical Methods in Engineeringvol 90 pp 452ndash483 2012

[226] G Ventura J X Xu and T Belytschko ldquoA vector levelset method and new discontinuity approximations for crackgrowth by EFGrdquo International Journal for Numerical Methodsin Engineering vol 54 no 6 pp 923ndash944 2002

[227] T Rabczuk and G Zi ldquoA meshfree method based on thelocal partition of unity for cohesive cracksrdquo ComputationalMechanics vol 39 no 6 pp 743ndash760 2007

[228] T Rabczuk S Bordas and G Zi ldquoA three-dimensional mesh-free method for continuous multiple-crack initiation propa-gation and junction in statics and dynamicsrdquo ComputationalMechanics vol 40 no 3 pp 473ndash495 2007

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

[230] G Zi T Rabczuk and W Wall ldquoExtended meshfree methodswithout branch enrichment for cohesive cracksrdquoComputationalMechanics vol 40 no 2 pp 367ndash382 2007

[231] T Rabczuk and T Belytschko ldquoCracking particles a simplifiedmeshfree method for arbitrary evolving cracksrdquo InternationalJournal for Numerical Methods in Engineering vol 61 no 13 pp2316ndash2343 2004

[232] T Rabczuk and T Belytschko ldquoA three-dimensional largedeformation meshfree method for arbitrary evolving cracksrdquoComputer Methods in Applied Mechanics and Engineering vol196 no 29-30 pp 2777ndash2799 2007

[233] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1ndash4 pp 19ndash49 2006

[234] T Rabczuk P M A Areias and T Belytschko ldquoA simplifiedmesh-free method for shear bands with cohesive surfacesrdquo

International Journal for Numerical Methods in Engineering vol69 no 5 pp 993ndash1021 2007

[235] T Rabczuk G Zi S Bordas and H Nguyen-Xuan ldquoA simpleand robust three-dimensional cracking-particle method with-out enrichmentrdquo Computer Methods in Applied Mechanics andEngineering vol 199 no 37ndash40 pp 2437ndash2455 2010

[236] L Chen and Y Zhang ldquoDynamic fracture analysis using dis-crete cohesive crack methodrdquo International Journal for Numeri-cal Methods in Biomedical Engineering vol 26 no 11 pp 1493ndash1502 2010

[237] Y Y Zhang ldquoMeshless modelling of crack growth with discreterotating crack segmentsrdquo International Journal ofMechanics andMaterials in Design vol 4 no 1 pp 71ndash77 2008

[238] H X Wang and S X Wang ldquoAnalysis of dynamic fracture withcohesive crack segment methodrdquo CMES Computer Modeling inEngineering amp Sciences vol 35 no 3 pp 253ndash274 2008

[239] F Caleyron A Combescure V Faucher and S PotapovldquoDynamic simulation of damagefracture transition insmoothed particles hydrodynamics shellsrdquo InternationalJournal For Numerical Methods in Engineering vol 90 pp707ndash738 2012

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

[241] Y Mi and M H Aliabadi ldquoThree-dimensional crack growthsimulation using BEMrdquo Computers and Structures vol 52 no5 pp 871ndash878 1994

[242] E Pan ldquoA general boundary element analysis of 2-D linearelastic fracture mechanicsrdquo International Journal of Fracturevol 88 no 1 pp 41ndash59 1997

[243] E Pan ldquoABEManalysis of fracturemechanics in 2D anisotropicpiezoelectric solidsrdquo Engineering Analysis with Boundary Ele-ments vol 23 no 1 pp 67ndash76 1999

[244] Y Ryoji andC Sang-Bong ldquoEfficient boundary element analysisof stress intensity factors for interface cracks in dissimilarmaterialsrdquo Engineering Fracture Mechanics vol 34 no 1 pp179ndash188 1989

[245] E Pan and F G Yuan ldquoBoundary element analysis of three-dimensional cracks in anisotropic solidsrdquo International JournalFor Numerical Methods in Engineering vol 48 pp 211ndash2372000

[246] M Doblare F Espiga L Gracia and M Alcantud ldquoStudyof crack propagation in orthotropic materials by using theboundary element methodrdquo Engineering Fracture Mechanicsvol 37 no 5 pp 953ndash967 1990

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

[248] E Schnack ldquoHybrid bem modelrdquo International Journal forNumerical Methods in Engineering vol 24 no 5 pp 1015ndash10251987

[249] J Sladek V Sladek and Z P Bazant ldquoNon-local boundary inte-gral formulation for softening damagerdquo International Journal forNumerical Methods in Engineering vol 57 no 1 pp 103ndash1162003

[250] XW Gao C Zhang J Sladek and V Sladek ldquoFracture analysisof functionally graded materials by a BEMrdquo Composites Scienceand Technology vol 68 no 5 pp 1209ndash1215 2008

[251] F Garcıa-Sanchez R Rojas-Dıaz A Saez and C ZhangldquoFracture of magnetoelectroelastic composite materials using


boundary element method (BEM)rdquo Theoretical and AppliedFracture Mechanics vol 47 no 3 pp 192ndash204 2007

[252] T A Cruse ldquoBIE fracture mechanics analysis 25 years ofdevelopmentsrdquo Computational Mechanics vol 18 no 1 pp 1ndash11 1996

[253] R N Simpson S P A Bordas J Trevelyan and T Rabczuk ldquoAtwo-dimensional isogeometric boundary element method forelastostatic analysisrdquo Computer Methods in Applied Mechanicsand Engineering vol 209212 pp 87ndash100 2012

[254] A P Cisilino and M H Aliabadi ldquoThree-dimensional bound-ary element analysis of fatigue crack growth in linear and non-linear fracture problemsrdquo Engineering Fracture Mechanics vol63 no 6 pp 713ndash733 1999

[255] R Simpson and J Trevelyan ldquoEvaluation of j 1 and j 2integrals for curved cracks using an enriched boundary elementmethodrdquo Engineering Fracture Mechanics vol 78 pp 623ndash6372011

[256] R Simpson and J Trevelyan ldquoA partition of unity enricheddual boundary element method for accurate computations infracture mechanicsrdquo Computer Methods in Applied Mechanicsand Engineering vol 200 no 1ndash4 pp 1ndash10 2011

[257] G E Bird J Trevelyan and C E Augarde ldquoA coupledBEMscaled boundary FEM formulation for accurate computa-tions in linear elastic fracture mechanicsrdquo Engineering Analysiswith Boundary Elements vol 34 no 6 pp 599ndash610 2010

[258] T J R Hughes J A Cottrell and Y Bazilevs ldquoIsogeometricanalysis CAD finite elements NURBS exact geometry andmesh refinementrdquoComputer Methods in AppliedMechanics andEngineering vol 194 no 39ndash41 pp 4135ndash4195 2005

[259] E De Luycker D J Benson T Belytschko Y Bazilevs and MC Hsu ldquoX-FEM in isogeometric analysis for linear fracturemechanicsrdquo International Journal for Numerical Methods inEngineering vol 87 no 6 pp 541ndash565 2011

[260] S S Ghorashi N Valizadeh and S Mohammadi ldquoExtendedisogeometric analysis for simulation of stationary and propa-gating cracksrdquo International Journal For Numerical Methods inEngineering vol 89 pp 1069ndash1101 2012

[261] A Tambat and G Subbarayan ldquoIsogeometric enriched fieldapproximationsrdquo Computer Methods in Applied Mechanics andEngineering vol 245ndash246 pp 1ndash21 2012

[262] G A Francfort and J-J Marigo ldquoRevisiting brittle fracture asan energy minimization problemrdquo Journal of the Mechanics andPhysics of Solids vol 46 no 8 pp 1319ndash1342 1998

[263] B Bourdin G A Francfort and J-J Marigo ldquoNumerical exper-iments in revisited brittle fracturerdquo Journal of theMechanics andPhysics of Solids vol 48 no 4 pp 797ndash826 2000

[264] A Karma D A Kessler and H Levine ldquoPhase-field model ofmode III dynamic fracturerdquo Physical Review Letters vol 87 no4 Article ID 045501 4 pages 2001

[265] V Hakim and A Karma ldquoLaws of crack motion and phase-field models of fracturerdquo Journal of the Mechanics and Physicsof Solids vol 57 no 2 pp 342ndash368 2009

[266] CMieheMHofacker and FWelschinger ldquoA phase fieldmodelfor rate-independent crack propagation robust algorithmicimplementation based on operator splitsrdquo Computer Methodsin Applied Mechanics and Engineering vol 199 no 45ndash48 pp2765ndash2778 2010

[267] C Miehe F Welschinger and M Hofacker ldquoThermodynam-ically consistent phase-field models of fracture variationalprinciples and multi-field FE implementationsrdquo InternationalJournal for Numerical Methods in Engineering vol 83 no 10pp 1273ndash1311 2010

[268] C Kuhn and R Muller ldquoA new finite element technique for aphase field model of brittle fracturerdquo Journal of Theoretical andApplied Mechanics vol 49 pp 1115ndash1133 2011

[269] C Kuhn and RMuller ldquoExponential finite element shape func-tions for a phase field model of brittle fracturerdquo in Proceedingsof the 11th International Conference on Computational Plasticity(COMPLAS rsquo11) pp 478ndash489 2011

[270] C Kuhn and R Muller ldquoA continuum phase field model forfracturerdquo Engineering Fracture Mechanics vol 77 no 18 pp3625ndash3634 2010

[271] T Rabczuk S Bordas and G Zi ldquoOn three-dimensionalmodelling of crack growth using partition of unity methodsrdquoComputers and Structures vol 88 no 23-24 pp 1391ndash1411 2010

[272] P Jager P Steinmann and E Kuhl ldquoOn local tracking algo-rithms for the simulation of three-dimensional discontinuitiesrdquoComputational Mechanics vol 42 no 3 pp 395ndash406 2008

[273] T C Gasser and G A Holzapfel ldquoModeling 3D crack prop-agation in unreinforced concrete using PUFEMrdquo ComputerMethods in Applied Mechanics and Engineering vol 194 no 25-26 pp 2859ndash2896 2005

[274] T C Gasser and G A Holzapfel ldquo3D Crack propagationin unreinforced concrete A two-step algorithm for tracking3D crack pathsrdquo Computer Methods in Applied Mechanics andEngineering vol 195 no 37ndash40 pp 5198ndash5219 2006

[275] P Krysl and T Belytschko ldquoThe element free Galerkin methodfor dynamic propagation of arbitrary 3-D cracksrdquo InternationalJournal for Numerical Methods in Engineering vol 44 no 6 pp767ndash800 1999

[276] J Oliver A E Huespe E Samaniego and E W V ChavesldquoContinuum approach to the numerical simulation of materialfailure in concreterdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 28 no 7-8 pp 609ndash632 2004

[277] P Jager P Steinmann and E Kuhl ldquoTowards the treatment ofboundary conditions for global crack path tracking in three-dimensional brittle fracturerdquoComputational Mechanics vol 45no 1 pp 91ndash107 2009

[278] G Ventura E Budyn and T Belytschko ldquoVector level sets fordescription of propagating cracks in finite elementsrdquo Interna-tional Journal for Numerical Methods in Engineering vol 58 no10 pp 1571ndash1592 2003

[279] M Duflot ldquoA study of the representation of cracks with levelsetsrdquo International Journal for Numerical Methods in Engineer-ing vol 70 no 11 pp 1261ndash1302 2007

[280] D L Chopp ldquoComputing minimal surfaces via level set curva-ture flowrdquo Journal of Computational Physics vol 106 no 1 pp77ndash91 1993

[281] D L Chopp and J A Sethian ldquoFlow under curvature singular-ity formation minimal surfaces and geodesicsrdquo ExperimentalMathematics vol 2 no 4 pp 235ndash255 1993

[282] 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

[283] T-P Fries and M Baydoun ldquoCrack propagation with theextended finite element method and a hybrid explicit-implicitcrack descriptionrdquo International Journal for Numerical Methodsin Engineering vol 89 no 12 pp 1527ndash1558 2012

[284] X Zhuang C Augarde and S Bordas ldquoAccurate fracturemodelling using meshless methods the visibility criterion andlevel sets formulation and 2Dmodellingrdquo International Journal


for Numerical Methods in Engineering vol 86 no 2 pp 249ndash268 2011

[285] X Zhuang C Augarde and K M Mathisen ldquoFracture mod-eling using meshless methods and level sets in 3d frameworkand modelingrdquo International Journal for Numerical Methods inEngineering vol 92 no 11 pp 969ndash998 2012

[286] J Mosler ldquoA variationally consistent approach for crack propa-gation based on configurational forcesrdquo in IUTAM Symposiumon Progress in the Theory and Numerics of ConfigurationalMechanics vol 17 of IUTAM Bookseries pp 239ndash247 2009

[287] M E Gurtin and P Podio-Guidugli ldquoConfigurational forcesand the basic laws for crack propagationrdquo Journal of theMechanics and Physics of Solids vol 44 no 6 pp 905ndash927 1996

[288] M E Gurtin and P Podio-Guidugli ldquoConfigurational forcesand a constitutive theory for crack propagation that allows forkinking and curvingrdquo Journal of the Mechanics and Physics ofSolids vol 46 no 8 pp 1343ndash1378 1998

[289] C Miehe and E Gurses ldquoA robust algorithm forconfigurational-force-driven brittle crack propagation with r-adaptive mesh alignmentrdquo International Journal for NumericalMethods in Engineering vol 72 no 2 pp 127ndash155 2007

[290] G C Sih ldquoStrain-energy-density factor applied to mixed modecrack problemsrdquo International Journal of Fracture vol 10 no 3pp 305ndash321 1974

[291] C H Wu ldquoFracture under combined loads by maximumenergy release rate criterionrdquo Journal of Applied MechanicsTransactions ASME vol 45 no 3 pp 553ndash558 1978

[292] R VGoldstein andR L Salganik ldquoBrittle fracture of solids witharbitrary cracksrdquo International Journal of Fracture vol 10 no 4pp 507ndash523 1974

[293] B Shen and O Stephansson ldquoModification of the G-criterionfor crack propagation subjected to compressionrdquo EngineeringFracture Mechanics vol 47 no 2 pp 177ndash189 1994

[294] G N Wells and L J Sluys ldquoA new method for modellingcohesive cracks using finite elementsrdquo International Journal forNumerical Methods in Engineering vol 50 no 12 pp 2667ndash2682 2001

[295] S Mariani and U Perego ldquoExtended finite element methodfor quasi-brittle fracturerdquo International Journal for NumericalMethods in Engineering vol 58 no 1 pp 103ndash126 2003

[296] J E Marsden and T J R HughesMathematical Foundations ofElasticity Dover Publications New York NY USA 1983

[297] R W Ogden Non-Linear Elastic Deformations Halsted PressNew York NY USA 1984

[298] J Oliver D L Linero A E Huespe and O L Manzoli ldquoTwo-dimensional modeling ofmaterial failure in reinforced concreteby means of a continuum strong discontinuity approachrdquoComputer Methods in Applied Mechanics and Engineering vol197 no 5 pp 332ndash348 2008

[299] T Belytschko S Loehnert and J-H Song ldquoMultiscale aggre-gating discontinuities a method for circumventing loss ofmaterial stabilityrdquo International Journal for Numerical Methodsin Engineering vol 73 no 6 pp 869ndash894 2008

[300] GMeschke andPDumstorff ldquoEnergy-basedmodeling of cohe-sive and cohesionless cracks via X-FEMrdquo Computer Methodsin Applied Mechanics and Engineering vol 196 no 21ndash24 pp2338ndash2357 2007

[301] P Dummerstorf and G Meschke ldquoCrack propagation crite-ria in the framework of X-FEM-based structural analysesrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 31 no 2 pp 239ndash259 2007

[302] T Belytschko H Chen J Xu and G Zi ldquoDynamic crack prop-agation based on loss of hyperbolicity and a new discontinuousenrichmentrdquo International Journal for Numerical Methods inEngineering vol 58 no 12 pp 1873ndash1905 2003

[303] P A Cundall and R D Hart ldquoDevelopment of generalized 2-dand 3-d distinct element programs for modeling jointed rockrdquoMisc Paper SL-85-1 US Army Corps of Engineers 1985

[304] P A Cundall and O D L Strack ldquoA discrete numerical modelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[305] G H Shi and R E Goodman ldquoTwo dimensional discontinuousdeformation analysisrdquo International Journal for Numerical ampAnalytical Methods in Geomechanics vol 9 no 6 pp 541ndash5561985

[306] G-H Shi and R E Goodman ldquoGeneralization of two-dimensional discontinuous deformation analysis for forwardmodellingrdquo International Journal for Numerical amp AnalyticalMethods in Geomechanics vol 13 no 4 pp 359ndash380 1989

[307] P A Cundall and H Konietzky ldquoPfc-ein neues werkzeug furnumerische modellierungenrdquo Bautechnik vol 73 no 8 1996

[308] R W Macek and S A Silling ldquoPeridynamics via finite elementanalysisrdquo Finite Elements in Analysis and Design vol 43 no 15pp 1169ndash1178 2007

[309] W K Liu D Qian S Gonella S LiW Chen and S ChirputkarldquoMultiscale methods for mechanical science of complex mate-rials bridging from quantum to stochastic multiresolutioncontinuumrdquo International Journal for Numerical Methods inEngineering vol 83 no 8-9 pp 1039ndash1080 2010

[310] A Nouy A Clement F Schoefs and N Moes ldquoAn extendedstochastic finite element method for solving stochastic partialdifferential equations on random domainsrdquo Computer Methodsin Applied Mechanics and Engineering vol 197 no 51-52 pp4663ndash4682 2008

[311] J Grasa J A Bea and M Doblare ldquoA probabilistic extendedfinite element approach application to the prediction of bonecrack propagationrdquoKey EngineeringMaterials vol 348-349 pp77ndash80 2007

[312] J Grasa J A Bea J F Rodrıguez and M Doblare ldquoTheperturbation method and the extended finite element methodAn application to fracture mechanics problemsrdquo Fatigue andFracture of Engineering Materials and Structures vol 29 no 8pp 581ndash587 2006

[313] I Arias S Serebrinsky and M Ortiz ldquoA cohesive modelof fatigue of ferroelectric materials under electro-mechanicalcyclic loadingrdquo in Smart Structures and Materials 2004 ActiveMaterials Behaviour and Mechanics Proceedings of SPIE pp371ndash378 San Diego Calif USA March 2004

[314] L J Lucas H Owhadi and M Ortiz ldquoRigorous verificationvalidation uncertainty quantification and certification throughconcentration-of-measure inequalitiesrdquo Computer Methods inApplied Mechanics and Engineering vol 197 no 51-52 pp 4591ndash4609 2008

[315] T Belytschko and J H Song ldquoCoarse-graining of multiscalecrack propagationrdquo International Journal for Numerical Meth-ods in Engineering vol 81 no 5 pp 537ndash563 2010

[316] M F Horstemeyer ldquoMultiscale modeling a reviewrdquo PracticalAspects of Computational Chemistry pp 87ndash135 2010

[317] J Fish and Z Yuan ldquoMultiscale enrichment based on partitionof unityrdquo International Journal for Numerical Methods in Engi-neering vol 62 no 10 pp 1341ndash1359 2005


[318] R Gracie and T Belytschko ldquoConcurrently coupled atomisticand XFEM models for dislocations and cracksrdquo InternationalJournal for Numerical Methods in Engineering vol 78 no 3 pp354ndash378 2009

[319] V Kouznetsova Computational homogenization for the multi-scale analysis of multi-phase materials [PhD thesis] Nether-lands Institute for Metals Research Amsterdam The Nether-lands 2002

[320] S Nemat-Nasser and M Hori Micromechanics Overall Prop-erties of Heterogeneous Materials Elsevier Amsterdam TheNetherlands 1993

[321] R Gracie J Oswald and T Belytschko ldquoOn a new extendedfinite element method for dislocations core enrichment andnonlinear formulationrdquo Journal of the Mechanics and Physics ofSolids vol 56 no 1 pp 200ndash214 2008

[322] R Gracie G Ventura and T Belytschko ldquoA new fast finite ele-mentmethod for dislocations based on interior discontinuitiesrdquoInternational Journal for Numerical Methods in Engineering vol69 no 2 pp 423ndash441 2007

[323] T Belytschko and R Gracie ldquoOnXFEM applications to disloca-tions and interfacesrdquo International Journal of Plasticity vol 23no 10-11 pp 1721ndash1738 2007

[324] M Xu R Gracie and T Belytschko ldquoMultiscale modeling withextended bridging domain methodrdquo in Bridging the Scales inScience and Engineering J Fish Ed Oxford University Press2002

[325] F Feyel and J L Chaboche ldquoFE 2 multiscale approach formodelling the elastoviscoplastic behaviour of long fibre SiCTicomposite materialsrdquo Computer Methods in Applied Mechanicsand Engineering vol 183 no 3-4 pp 309ndash330 2000

[326] V Kouznetsova M G D Geers and W A M BrekelmansldquoMulti-scale constitutive modelling of heterogeneous materi-als with a gradient-enhanced computational homogenizationschemerdquo International Journal for Numerical Methods in Engi-neering vol 54 no 8 pp 1235ndash1260 2002

[327] V P Nguyen O Lloberas-Valls M Stroeven and L J SluysldquoHomogenization-based multiscale crack modelling frommicro-diffusive damage tomacro-cracksrdquoComputerMethods inAppliedMechanics and Engineering vol 200 no 9ndash12 pp 1220ndash1236 2011

[328] C V Verhoosel J J C Remmers M A Gutierrez and Rde Borst ldquoComputational homogenization for adhesive andcohesive failure in quasi-brittle solidsrdquo International Journal forNumericalMethods in Engineering vol 83 no 8-9 pp 1155ndash11792010

[329] E B Tadmor M Ortiz and R Phillips ldquoQuasicontinuumanalysis of defects in solidsrdquo Philosophical Magazine A vol 73no 6 pp 1529ndash1563 1996

[330] R E Miller and E B Tadmor ldquoThe Quasicontinuum methodoverview applications and current directionsrdquo Journal ofComputer-Aided Materials Design vol 9 no 3 pp 203ndash2392002

[331] H B Dhia ldquoThe arlequin method a partition of models forconcurrentmultiscale analysesrdquo in Proceedings of the Challengesin Computational Mechanics Workshop 2006

[332] F F Abraham J Q Broughton N Bernstein and E KaxirasldquoSpanning the length scales in dynamic simulationrdquo Computa-tional Physics vol 12 pp 538ndash546 1998

[333] S Loehnert and T Belytschko ldquoAmultiscale projection methodfor macromicrocrack simulationsrdquo International Journal forNumericalMethods in Engineering vol 71 no 12 pp 1466ndash14822007

[334] P A Guidault O Allix L Champaney and C Cornuault ldquoAmultiscale extended finite element method for crack propaga-tionrdquo Computer Methods in AppliedMechanics and Engineeringvol 197 no 5 pp 381ndash399 2008

[335] P Aubertin J Rethore and R de Borst ldquoEnergy conservationof atomisticcontinuum couplingrdquo International Journal forNumericalMethods in Engineering vol 78 no 11 pp 1365ndash13862009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Mode I Mode II Mode III

Figure 1 Different failure modes

in the absence of plasticity we may call the failure quasi-brittle The stress-strain curve for quasi-brittle materials suchas concrete ceramics and rock is characterized by a dropof the stress after the peak stress is reached This softeningcan be explained by damage mechanics due to the initiationgrowth and coalescences of microcracks that reduce theeffective cross-sectional area Figure 2 For more details thereader is referred to numerous articles and books on damagemechanics [13ndash26]

The use of strain-softening material models leads toan ill-posed boundary value problem (BVP) [27ndash31] Thesolution of ill-posed BVPs with computationalmethods leadsto certain difficulties Bazant and Belytschko [32] showedthat the deformations localize in a set of measure zero in1D the localization will occur in a point in 2D in a lineand in 3D in a surface They further showed that the energydissipation approaches zero with increasing refinement of thediscretization yielding physically meaningless results

The difficulties that emerge with the material instabilitiesdue to the ill-posedness of the BVP can be avoided by regular-ization techniques such as higher order continuum modelsgradient based models and polar theories (the most well-known theory is probably the Cosserat continuum) nonlocalmodels viscous models or cohesive zone models Theseregularization techniques introduce a characteristic lengthinto the discretization In other words the local character ofthe deformation is lost and the fracture is ldquosmearedrdquo over acertain domain involving several elements Therefore a veryfine mesh has to be used in order to resolve the crack As thecrack introduces a jump in the displacement field methodsthat smear the crack over a certain region are not capableof representing the correct crack kinematics Moreover thedifferent length scales (of the structure and the characteristiclength) may significantly increase the computational costA lot of effort has been devoted to develop methods thatare capable of capturing the crack and take advantage ofregularization techniques that can be coupled to these strongdiscontinuity approaches Modeling fracture with strong-discontinuity approaches require two key ingredients

(1) a method that is capable of capturing the crackkinematics and

(2) a cracking criterion that determines the orientationand the length of the crack

Computational methods that describe the topology of thecrack as continuous surface require also methods to describethe crack surface and track the crack path The paper isstructured as follows In the next section popular regulariza-tion techniques are summarized including gradient enhancedmodels non-local models viscous models cohesive zonemodels and smeared crack models Section 3 gives anoverview of the state-of-the-art computational methods for(discrete) fracture of brittle and quasi-brittle solids Section 4is devoted to crack tracking procedures as it is a majorchallenge for computational methods preserving crack pathcontinuity Section 5 discusses cracking criterion that deter-mine the transition from the continuum to the discontinuumThe article finishes with future research directions and someconcluding remarks

2 Regularization

21 Gradient-Enhanced Models Gradient-enhanced modelsor briefly called gradient models are typically describedby differential equations that contain higher order spatialderivativesThe coefficientsmultiplying the terms of differentorder have different physical dimensions It is possible todeduce the characteristic length from their ratio Gradientmodels can be incorporated into plasticity format [33 34]damage format [22 35] or in a combination of both [15] Sincethey require higher order spatial derivatives higher orderelements need to be used Gradient models are sometimesreferred to asweakly non-localmodels see Bazant and Jirasek[14] though some recent models [22] are also classified asstrongly non-local

22 Nonlocal Models Strongly non-local models are modelsof the integral type see Figure 3 In non-local formulationsthe stress at a given material point does not only depend onthat local strain but also on the strain of the neighboringpoints This effect is obtained by averaging certain internalvariables in a given neighborhood Figure 3 The size ofthis neighborhood is a material parameter that dependsalso on the geometry of the structure of interest In otherwords changing the dimensions of the problem requiresrecalibration of the parameters of the non-local model Theycan be obtained by an inverse analysis Bazant and Jirasek [14]



120601(119863)

120590 120576120590eff 120576eff


1

Volume equivalence

119877 119877119903

119871119889120572119899119897

119883119877

119903

119878




119888ℎassociated


119897119888ℎ

=120578

radic119864984858 (2)


0=








Cohesive zone

Macrocrack


120596119898119886119909








119908max

0


Cohesive traction

Crack opening

(a)

Cohesive traction

Crack opening

(b)

Cohesive traction

Crack opening

(c)

Cohesive traction

Crack opening

(d)

Cohesive traction

Crack opening

(e)

Cohesive traction

Crack opening

(f)






120590

120576pre 120576post

120576

(a)

120590

120576pre 120576post

120576

(b)

120590

120576

(c)











119910

2

119909

Φ119900

(1)Φ119903

1




(a)


(b)




(a)

(b)









(a)

(b)

119878

Ω+0Ωminus0

(c)





(a)

(b)





119873119868(X) u

119869+M

(119890)

119904(X)

u(119890)119868

(119883)

(3)




119904

is given by

M(119890)

119904(X) =

0 forall (119890) notin 119878

119867(119890)


120588(119890) =

119873+

119890

sum119868=1

119873+

119868(X)

(4)


119873+




120598ℎ



119868)119878

minus (nabla120588(119890)

otimes

u(119890)119868

(119883)

)119878

+120578(119890)119904

119896(

u(119890)119868

(119883)

otimes n)

119878

(5)



119904is


120578(119890)

119904=


119890

0 forallX notin 119878119896119890

(6)


[K(119890) K

(119890)

K(119890) K

(119890)

]u(119890)u(119890)119868

=

Fext119868

0 (7)

with

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0


0

K(119890)

= intΩ0


0

(8)






levelu(119890)119868

= minus[K(119890)

]minus1

K(119890)u(119890) (9)




Ku = f (10)

with

K = K










119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1


119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)










120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)


X119873isinΓ(119873)119888

(15)




119868



120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120601(119863)

120590 120576120590eff 120576eff


1

Volume equivalence

119877 119877119903

119871119889120572119899119897

119883119877

119903

119878




119888ℎassociated


119897119888ℎ

=120578

radic119864984858 (2)


0=








Cohesive zone

Macrocrack


120596119898119886119909








119908max

0


Cohesive traction

Crack opening

(a)

Cohesive traction

Crack opening

(b)

Cohesive traction

Crack opening

(c)

Cohesive traction

Crack opening

(d)

Cohesive traction

Crack opening

(e)

Cohesive traction

Crack opening

(f)






120590

120576pre 120576post

120576

(a)

120590

120576pre 120576post

120576

(b)

120590

120576

(c)











119910

2

119909

Φ119900

(1)Φ119903

1




(a)


(b)




(a)

(b)









(a)

(b)

119878

Ω+0Ωminus0

(c)





(a)

(b)





119873119868(X) u

119869+M

(119890)

119904(X)

u(119890)119868

(119883)

(3)




119904

is given by

M(119890)

119904(X) =

0 forall (119890) notin 119878

119867(119890)


120588(119890) =

119873+

119890

sum119868=1

119873+

119868(X)

(4)


119873+




120598ℎ



119868)119878

minus (nabla120588(119890)

otimes

u(119890)119868

(119883)

)119878

+120578(119890)119904

119896(

u(119890)119868

(119883)

otimes n)

119878

(5)



119904is


120578(119890)

119904=


119890

0 forallX notin 119878119896119890

(6)


[K(119890) K

(119890)

K(119890) K

(119890)

]u(119890)u(119890)119868

=

Fext119868

0 (7)

with

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0


0

K(119890)

= intΩ0


0

(8)






levelu(119890)119868

= minus[K(119890)

]minus1

K(119890)u(119890) (9)




Ku = f (10)

with

K = K










119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1


119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)










120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)


X119873isinΓ(119873)119888

(15)




119868



120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Cohesive zone

Macrocrack


120596119898119886119909








119908max

0


Cohesive traction

Crack opening

(a)

Cohesive traction

Crack opening

(b)

Cohesive traction

Crack opening

(c)

Cohesive traction

Crack opening

(d)

Cohesive traction

Crack opening

(e)

Cohesive traction

Crack opening

(f)






120590

120576pre 120576post

120576

(a)

120590

120576pre 120576post

120576

(b)

120590

120576

(c)











119910

2

119909

Φ119900

(1)Φ119903

1




(a)


(b)




(a)

(b)









(a)

(b)

119878

Ω+0Ωminus0

(c)





(a)

(b)





119873119868(X) u

119869+M

(119890)

119904(X)

u(119890)119868

(119883)

(3)




119904

is given by

M(119890)

119904(X) =

0 forall (119890) notin 119878

119867(119890)


120588(119890) =

119873+

119890

sum119868=1

119873+

119868(X)

(4)


119873+




120598ℎ



119868)119878

minus (nabla120588(119890)

otimes

u(119890)119868

(119883)

)119878

+120578(119890)119904

119896(

u(119890)119868

(119883)

otimes n)

119878

(5)



119904is


120578(119890)

119904=


119890

0 forallX notin 119878119896119890

(6)


[K(119890) K

(119890)

K(119890) K

(119890)

]u(119890)u(119890)119868

=

Fext119868

0 (7)

with

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0


0

K(119890)

= intΩ0


0

(8)






levelu(119890)119868

= minus[K(119890)

]minus1

K(119890)u(119890) (9)




Ku = f (10)

with

K = K










119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1


119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)










120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)


X119873isinΓ(119873)119888

(15)




119868



120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120590

120576pre 120576post

120576

(a)

120590

120576pre 120576post

120576

(b)

120590

120576

(c)











119910

2

119909

Φ119900

(1)Φ119903

1




(a)


(b)




(a)

(b)









(a)

(b)

119878

Ω+0Ωminus0

(c)





(a)

(b)





119873119868(X) u

119869+M

(119890)

119904(X)

u(119890)119868

(119883)

(3)




119904

is given by

M(119890)

119904(X) =

0 forall (119890) notin 119878

119867(119890)


120588(119890) =

119873+

119890

sum119868=1

119873+

119868(X)

(4)


119873+




120598ℎ



119868)119878

minus (nabla120588(119890)

otimes

u(119890)119868

(119883)

)119878

+120578(119890)119904

119896(

u(119890)119868

(119883)

otimes n)

119878

(5)



119904is


120578(119890)

119904=


119890

0 forallX notin 119878119896119890

(6)


[K(119890) K

(119890)

K(119890) K

(119890)

]u(119890)u(119890)119868

=

Fext119868

0 (7)

with

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0


0

K(119890)

= intΩ0


0

(8)






levelu(119890)119868

= minus[K(119890)

]minus1

K(119890)u(119890) (9)




Ku = f (10)

with

K = K










119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1


119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)










120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)


X119873isinΓ(119873)119888

(15)




119868



120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119910

2

119909

Φ119900

(1)Φ119903

1




(a)


(b)




(a)

(b)









(a)

(b)

119878

Ω+0Ωminus0

(c)





(a)

(b)





119873119868(X) u

119869+M

(119890)

119904(X)

u(119890)119868

(119883)

(3)




119904

is given by

M(119890)

119904(X) =

0 forall (119890) notin 119878

119867(119890)


120588(119890) =

119873+

119890

sum119868=1

119873+

119868(X)

(4)


119873+




120598ℎ



119868)119878

minus (nabla120588(119890)

otimes

u(119890)119868

(119883)

)119878

+120578(119890)119904

119896(

u(119890)119868

(119883)

otimes n)

119878

(5)



119904is


120578(119890)

119904=


119890

0 forallX notin 119878119896119890

(6)


[K(119890) K

(119890)

K(119890) K

(119890)

]u(119890)u(119890)119868

=

Fext119868

0 (7)

with

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0


0

K(119890)

= intΩ0


0

(8)






levelu(119890)119868

= minus[K(119890)

]minus1

K(119890)u(119890) (9)




Ku = f (10)

with

K = K










119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1


119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)










120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)


X119873isinΓ(119873)119888

(15)




119868



120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(a)

(b)

119878

Ω+0Ωminus0

(c)





(a)

(b)





119873119868(X) u

119869+M

(119890)

119904(X)

u(119890)119868

(119883)

(3)




119904

is given by

M(119890)

119904(X) =

0 forall (119890) notin 119878

119867(119890)


120588(119890) =

119873+

119890

sum119868=1

119873+

119868(X)

(4)


119873+




120598ℎ



119868)119878

minus (nabla120588(119890)

otimes

u(119890)119868

(119883)

)119878

+120578(119890)119904

119896(

u(119890)119868

(119883)

otimes n)

119878

(5)



119904is


120578(119890)

119904=


119890

0 forallX notin 119878119896119890

(6)


[K(119890) K

(119890)

K(119890) K

(119890)

]u(119890)u(119890)119868

=

Fext119868

0 (7)

with

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0


0

K(119890)

= intΩ0


0

(8)






levelu(119890)119868

= minus[K(119890)

]minus1

K(119890)u(119890) (9)




Ku = f (10)

with

K = K










119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1


119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)










120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)


X119873isinΓ(119873)119888

(15)




119868



120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

(b)





119873119868(X) u

119869+M

(119890)

119904(X)

u(119890)119868

(119883)

(3)




119904

is given by

M(119890)

119904(X) =

0 forall (119890) notin 119878

119867(119890)


120588(119890) =

119873+

119890

sum119868=1

119873+

119868(X)

(4)


119873+




120598ℎ



119868)119878

minus (nabla120588(119890)

otimes

u(119890)119868

(119883)

)119878

+120578(119890)119904

119896(

u(119890)119868

(119883)

otimes n)

119878

(5)



119904is


120578(119890)

119904=


119890

0 forallX notin 119878119896119890

(6)


[K(119890) K

(119890)

K(119890) K

(119890)

]u(119890)u(119890)119868

=

Fext119868

0 (7)

with

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0

B119879CB 119889Ω0

K(119890)

= intΩ0


0

K(119890)

= intΩ0


0

(8)






levelu(119890)119868

= minus[K(119890)

]minus1

K(119890)u(119890) (9)




Ku = f (10)

with

K = K










119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1


119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)










120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)


X119873isinΓ(119873)119888

(15)




119868



120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Ku = f (10)

with

K = K










119905

crack tips reads


119873119868(X)u

119868+

119899119888

sum119873=1


119873119868(X) 120595(119873)

119868a(119873)119868

+

119898119905

sum119872=1


119873119868(X)

119873119870

sum119870=1

120601(119872)

119870119868b(119872)119870119868

(13)










120595(119873)

119868= sign [119891

(119873)

(X)] minus sign [119891(119873)

(X119868)] (14)

with

119891(119873)


X119873isinΓ(119873)119888

(15)




119868



120601(119872)

119870119868= [radic119903 sin(

120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(16)


1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4Crack

Shiing crack

1198732(X) (X)1198733

1198732(X)120595(119891(X))

(X)120595(119891(X))1198733

1198733(X)(120595(119891(X)) minus 120595(119891(X3)))

1198732(X)(120595(119891(X)) minus 120595(119891(X2)))


Crack

119891 = 0

119892 = 0119891 gt 0

119891 lt 0

120579119903



120601(119872)

119868= 119903

119904 sin(120579

2) (17)







Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Crack

(a)

Crack

(b)

Crack

(c)



119891ℎ


119873119868(X) 119891

119868 119892

ℎ


119873119868(X) 119892

119868 (18)



Crack







Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Ωblnd

ΩenrΩstd





[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

u119869

a119869

b119869119870

=

fext119868

fext119868

fext119868119870

(19)

or

K119868119869d119869= fext

119868 (20)


K119868119869=[[

[

K119906119906119868119869

K119906119886119868119869

K119906119887119868119869119870

K119886119906119868119869

K119886119886119868119869

K119886119887119868119869119870

K119887119906119868119869119870

K119887119886119868119869119870

K119887119887119868119869119870

]]

]

(21)



fext119868

= f119906119868f119886119868f119887119870119868119879


=

f1198871119868

f1198872119868

f1198873119868

f1198874119868 and

f119906119868= int

Ω

119873119868(X) b 119889Ω + int

Γ119905

119873119868(X) t 119889Γ

f119886119868= int

Ω

119873119868(X) 120595

119868b 119889Ω + int

Γ119905

119873119868(X) 120595

119868t 119889Γ

f119887119870119868

= intΩ

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) b 119889Ω

+ intΓ119905

119873119868(X) (120601

119870119868(X) minus 120601

119870119868(X

119868)) t 119889Γ

(22)


K119894119895119868119869= int

Ω

(B119894119868)119879

CB119895119869119889Ω 119894 119895 = 119906 119886 119887 (23)


B119906119868

= nablaN119868(X)

B119886119868

= nablaN119868(X) 120595

119868

B119887119868

= nabla [N119868(X) (120601119897

119870119868(X) minus 120601

119897

119870119868(X

119868)])

(24)









Crack

1

(a)

Crack

1 2

34

(b)

Crack










Crack

Step-enriched nodes












uℎ (X 119905) = sum


u119868(119905)119873

119868(X)119867 (119891 (X))

+ sum


u119869(119905)119873

119869(X)119867 (minus119891 (X))

(25)


0Wminus

0W+

P andWminus


0Ωminus

0Ω+P and


0andΩminus

0 We note





119873119869(X)u

119869 (26)





Ω0

Ω0

Ω+0

minus

+

Ωminus0

119905

119910

Γ119906

Γ119905

119891(119883) gt 0119891(119883) lt 0

Ωp

Ωp





0) is




X119869 gt ℎ

119869 where ℎ




119869)

A (X) = sum119869

p (X119869) p119879 (X

119869)119882 (119903

119869 ℎ)

D (X119869) = sum

119869

p (X119869)119882 (119903

119869 ℎ119869)

(27)


p = [1 119883 119884] (28)





uℎ (X) = sum

119868isinS+

119873119868(X) u

119868 (29)


[[u (X)]] = ( sum

119868isinS+

119873119868(X) u

119868) minus ( sum

119868isinSminus

119873119868(X)u

119868) (30)





1198680













Crack


Crack

Crack


05040302010

1

08

06

04

02

0

055053

051049

047 045047

049051

053

Crack line

Crack line

Crack line

5305151

04444444449 049055111111111111

0

055053

051049

047

055053

051049

047

05040302010

045047

049051

053

045047

049051

053


119883119884

119885

119883119884

119885

119883119884119885

119868

119868

119868



p = [1119883 119884radic119903 sin(120579

2) radic119903 cos(120579

2) radic119903 sin(

120579

2) sin 120579

radic119903 cos(120579

2) sin 120579]

(31)




Crack


(a)

Crack

Domain of influence

119868

(b)





Crack

119868

(a)

Crack

119909119868

1199040(119909)

1199041

119909119888

1199042(119909)

119909

(b)




u (X) = ucont (X) + uenr (X) (32)




119860

119861120596

n

n


(a)

(b)





119873119868(X) u

119868+ sum119868isinN119888

119873119868(X) 119904 (119891

119868(X)) q

119868 (33)

(a)

(b)






119868 It can be


u (X) = 2 sum119868isinN119888

119873119868(X) q

119868 (34)



100381610038161003816100381610038161003816100381610038161003816n sdot

x minus x119871

1003817100381710038171003817x minus x119871

1003817100381710038171003817

100381610038161003816100381610038161003816100381610038161003816lt cos(Π

2minus 120572) (35)



120572

120572

119869

119871

119868

119870

119868

119870

L

119869

n

n

n

n



n119871sdot n119870lt cos120573 (36)








119871+ 119880

119864+ 119880

119878= minus119880

119864+ 119880

119878 where










will do so for 120590 = 120590119891






119891(unstable






119864119904(Γ) = int

Γ

119896 (119909) 119889H119873minus1

(119909) (37)


119864119889(Γ 119880) = inf


119882(119909 120598 (V (119909))) 119889119909 (38)


119864 (Γ 119880) = 119864119889(Γ 119880) + 119864

119904(Γ) (39)





120591120597119905120601 (x 119905) = 119863

120601nabla2

120601 minus119889

119889120601(1

41206012

(1 minus 1206012

))

minus120583

21198921015840

(120601) (120598 120598 minus 1205982

119888)

(40)




Ψ (120598 120601) = [119892 (120601) + 119896]Ψ+

0(120598) + Ψ

minus

0(120598) (41)



Ψplusmn

0(120598) =

120582

2⟨1205981+ 120598

2+ 120598

3⟩2

plusmn+ 120583 (⟨120598

1⟩2

plusmn+ ⟨120598

2⟩2

plusmn+ ⟨120598

3⟩2

plusmn)

(42)






119873119890

1(120585 120575) = 1 minus

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

119873119890

2(120585 120575) =

exp (minus120575 (1 + 120585) 4) minus 1

exp (1205752) minus 1

(43)











a sdot n0= b sdot n

0= 0 (44)




0119879 sdot a = 0 in Ω

0


0119879 sdot b = 0 in Ω

0

(45)











5 Fracture Criteria








119868119868)






120579120579 often


119888by

120590120579120579

=119870119868

radic2120587119903119891119868

ℎ(120579 V

119888) +

119870119868119868

radic2120587119903119891119868119868

ℎ(120579 V

119888) (46)





hoop stress 120590119888120579120579


120579120579is given by

120590119888

120579120579=

119870119888119868

radic2120587119903 (47)



119870119868sin 120579

119888+ 119870

119868119868(3 cos 120579

119888minus 1) = 0 (48)


120579119888= 2 arctan(

119870119868minus radic1198702

119868+ 81198702

119868119868

4119870119868119868

) (49)




119888





119903can be


119878119888

119903= (120581 minus 1)

1205872(119870119888119868)2

8119866 (50)


119903is

119878119888

119903= 119886

111198702

119868+ 2119886

12119870119868119870119868119868+ 119886

221198702

119868119868forall120579 = 120579

0 (51)


11988611

=(1 + cos 120579) (120581 minus cos 120579)

16119866

11988612

=(2 cos 120579 minus 120581 + 1) sin 120579

16119866

11988622


16119866

(52)

The parameters 119886119894119895


Crack

Crack extension



119888(120590)


120597G

120597120572

10038161003816100381610038161003816100381610038161003816120572=120572119888

= 0

1205972G

1205971205722

100381610038161003816100381610038161003816100381610038161003816120572=120572119888

le 0

(53)


120590119888when G(120590

119888 120572119888(120590119888)) = G

119892 where G

119892is a material


119875

G (120590 120572) = G119875(120590 120572) +G

119860(120590 120572) (54)


G119860(120590 120572) =

120587

21205902

23(1 minus 120572

1 + 120572

120572

) (55)


G119875(120590 120572) =

1 minus ]

21205902

23(119870

2

119868+ 119870

2

119868119868) (56)


120590119898= int

Ω119888

119908120590 119889Ω (57)


weighting function

119908 =1

(2120587)32

1198973exp(minus 1199032

21198972) (58)









119895119899119897119860119894119895119896119897



0

2120587

2120587

120573

120572M

inim

um ei

genv

alue

of119928





119892 = max⏟⏟⏟⏟⏟⏟⏟119897


119897)) 119897 = 1 2 (61)






F A

F le 0 forall

F (62)



5900

4900

2900

3900

1900

900

minus1000 60 120 180 240 300 360

Angle (deg)

Min

imum

eige

nval

ue o

f119928

(a)

18000

13000

8000

minus2000 0 60 120 180 240 300 360

3000

Angle (deg )

Min

imum

eige

nval

ue o

f119928

(b)



I = intΩ

119882(u) 119889Ω + intΓ

G119905119889Γ (63)






120597119890

120597119905+ V

119888


= V119888s (64)












Coarse scale

Hot spots

PM (FM)

PM (FM)

(a)

Coarse scale

Fine scale

Cell 1

Cell 2 Cell 3

Cell 4

Cell 5 Cell 6

Hotspot 1

Hotspot 2

Hotspot 3

Hotspot 4

Hotspot 5

Hotspot 6

PMPM

PM

PM

PM PM

FM

FM

FM

FM FM

FM

(b)

Coarse scale

Fine scale

(c)








References






































































































































































































































































































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Review Article Computational Methods for Fracture in