A direct analytical method to extract mixed‐mode components of strain energy release rates from...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2013)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4542

A direct analytical method to extract mixed-mode components ofstrain energy release rates from Irwin’s integral using extended

finite element method

M. Lan1, H. Waisman1,*,† and I. Harari2

1Department of Civil Engineering & Engineering Mechanics, Columbia University, New York, NY 10027, USA2Department of Mechanical Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel

SUMMARY

A new analytical approach, within the extended finite element framework, is proposed to compute mixed-mode components of strain energy release rates directly from Irwin’s integral. Crack tip enrichment functionsin extended FEM allow for evaluation of integral quantities in closed form (for some crack configurationsstudied) and therefore resulting in a simple and accurate method.

Several benchmark examples on pure and mixed-mode problems are studied. In particular, we analyzethe effects of high-order enrichments, mesh refinement, and the integration limits of Irwin’s integral. Theresults indicate that high-order enrichment functions have significant effect on the convergence, in particularwhen the integral limits are finite. When the integral limits tend to zero, simpler strain energy release rateexpressions are obtained, and high-order terms vanish. Nonetheless, these terms contribute indirectly viacoefficients of first-order terms.

The numerical results show that high accuracy can be achieved with high-order enrichment terms andmesh refinement. However, the effect of the integral limits remains an open question, with finite integrationintervals chosen as h=2 tending to give more accurate results. Copyright © 2013 John Wiley & Sons, Ltd.

Received 8 December 2012; Revised 10 May 2013; Accepted 5 June 2013

KEY WORDS: extended finite element method; Irwin’s integral; mixed-mode fracture; stress intensityfactors; energy release rate; high-order asymptotic functions

1. INTRODUCTION

Stress intensity factors (SIFs) or equivalently strain energy release rates (SERRs) are importantquantities in linear elastic fracture mechanics (LFEM), used to predict the ‘stress intensity’ near thetip of a crack due to a remote load or residual stresses. In numerical simulations, these quantitiesare of significant importance as they are used to determine the stability and direction of crack prop-agation [1–3]. Hence, it is important to obtain accurate estimation of SIFs to predict accurate crackpaths and overall response of the structure.

Several methods have been proposed in the literature to extract SIFs. These methods can begrouped as direct and indirect methods. Direct methods are the simplest methods to compute SIFsand are mainly based on correlation of crack opening displacements, obtained directly from thefinite element solution with analytical solutions [4,5]. Indirect methods on the other hand are basedon integral or energy quantities to calculate SERRs and often require additional steps during thefinite element analysis process or a special post-processing technique, yet they yield more accurateresults [6, 7].

*Correspondence to: H. Waisman, Department of Civil Engineering & Engineering Mechanics, Columbia University,New York, NY 10027, USA.

†E-mail: waisman@civil.columbia.edu

M. LAN, H. WAISMAN AND I. HARARI

Some of the most well-known and used indirect methods are the J-integral method [8], itsdomain (area) variant [7, 9, 10] and the related M-integral (or interaction integral) method [11–13],the stiffness derivative method [14, 15], and the virtual crack closure technique (VCCT) [16]which is inspired by Irwin’s integral [17]. These methods are considered to be the cornerstone ofcomputational fracture mechanics, receiving significant attention in the past few decades.

Although Irwin’s contention is extremely important from a theoretical point of view, direct imple-mentation of the integrals in commercial software has been avoided, mainly because of numericalissues. For example, the stress fields obtained from the FEM, whether standard or quarter point ele-ments are employed [18], may not be sufficiently accurate due to numerical differentiation requiredto obtain the stress fields. In particular, these errors increase close to the singularity, where Irwin’sintegral is valid. To this end, Rybicki and Kanninen [16] proposed the VCCT method, where theyinterpret Irwin’s integral as the amount of work required to close the crack an amount�c and extractthe energy release rates through a penalty approach. The VCCT method has been implemented inmany commercial softwares because of its simplicity and effectiveness in computing the mixed-mode components of the energy release rates [19,20]. Extensive review of the method can be foundin [21].

Recently, an extended FEM (XFEM) has been proposed by Belytschko and co-workers [22, 23]for fracture problems without remeshing. In other words, one can model cracks without the needfor special conforming meshes, which imply that for crack propagation problems, remeshing maycompletely be avoided. The key idea of XFEM is to locally enrich the standard finite elementshape functions with Heaviside functions behind the crack tip to enable opening displacementsand four asymptotic functions (called branch functions) at the tip element. These branch functionsare obtained from Williams analytical solution [24] and incorporate the

pr terms in the displace-

ment field, which provides the stress singularity at the crack tip. We will review the XFEM in alater section.

Although the cracks in XFEM can be modeled without remeshing, it is still necessary to deter-mine the stability and direction of crack propagation. Most of the XFEM literature related to LEFMhas focused on the computation of SIFs by the J-integral method developed by Rice [8] and its vari-ants. A different approach that extends Parks classical stiffness derivative method [14] to XFEMhas been proposed by Waisman [25]. In that work, it was shown that the stiffness derivative can becomputed in a closed form during the analysis and thus the virtual crack extension, and the errorinherent in the finite difference scheme of the classical method can completely be avoided.

In the current paper, we propose a new analytical approach based on XFEM to compute SERRs,reverting back to the definition of Irwin’s integral. It is important to note that although the VCCTmethod works well for FEM, its extension to XFEM is not trivial as the extraction of forces at thecrack tip becomes more complicated. On the other hand, we prove that direct evaluation of Irwin’sintegral is straightforward with XFEM because more accurate asymptotic fields are employed toenrich the tip element, and hence closed form expressions are obtained and so there is no need forspecial post-processing.

For convenience, the derivation is carried out in polar coordinates and the integrals, obtained inclosed form, are verified via numerical integration. Several benchmark examples on pure and mixed-mode problems are studied. In particular, we analyze the effects of high-order enrichment functions,the mesh refinement, and the integration interval limits of Irwin’s integral. High-order terms havealso been studied in [26–31]. Liszka et al. [26] and Duarte et al. [27] have studied the enrichmentof the approximation space by the higher order terms of asymptotic field to extract SIFs directlyusing h-p clouds. In the interesting work of Liu et al. [28], the branch functions have been replacedwith high-order asymptotic terms, retaining not only the leading terms but also the associated coef-ficients. Employing this strategy, the authors show that SIFs can directly be extracted without anypost-processing. This approach has inspired Zamani et al. [29] and Réthoré et al. [31] to achievehigh accuracy by having appropriate modifications of the enrichment scheme and employing anoverlapping domain decomposition scheme.

Our results indicate that high-order terms have significant effect when the integral limits are finite.However, when the integral limits tend to zero, the expressions are simplified as these terms vanishfrom the expression of SERRs and thus have only indirect contribution via coefficients of first-order

STRAIN ENERGY RELEASE RATES FROM IRWIN’S INTEGRAL USING XFEM

terms. The numerical results show that high accuracy can be achieved with high-order enrichmentterms and mesh refinement. However, the effect of the integral limits remains an open question, withfinite integration intervals chosen as h=2 tend to give more accurate results.

The reminder of the paper is organized as follows. In Section 2, we present a brief introductionto the XFEM. In Section 3, we discuss high-order enrichment XFEM and our proposed approach.Section 4 derives Irwin’s integral analytic formulae in terms of XFEM-based results followed byseveral benchmark examples on 2D pure and mixed-mode problems in Section 5 and conclusions.

2. MODELING CRACKS BY THE EXTENDED FINITE ELEMENT METHOD

2.1. Problem Statement

Consider a two-dimensional solid with an internal crack in the domain �, as illustrated in Figure 1.The solid is subjected to body forces b in �, traction loading Nt applied on �t , and displacementboundary conditions u D Nu on �u. In Cartesian coordinates, the displacement field is decom-posed into its components uD ¹ux ,uyº as illustrated. Additionally, the crack is defined by internalboundaries �c , which are assumed to be traction free.

The Galerkin approximation of the proposed problem is to seek a kinematically admissibledisplacement field uh 2 Uh, which is a finite dimensional subspace of the solution space U ,such that Z

�.uh/ W C W �.wh/d�D

b �wh d�C

Nt �wh d� 8wh 2 Uh0 (1)

where � and C are the standard strain and elasticity tensors. The weighting functions wh, whosevalues vanish on the Dirichlet boundary �u, belong to the finite dimensional subspace Uh0 .

The aim of this work is to compute mixed-mode components of SERRs directly from Irwin’sintegral, by employing the XFEM with high-order crack tip asymptotic functions. These functionsallow for the evaluation of the integral quantities in closed form and would therefore result in asimple and accurate method.

Figure 1. A solid with a plane crack.

2.2. Extended FEM overview

The key idea of XFEM is to locally enrich the standard finite element approximation with localpartitions of unity enrichment functions, which are chosen according to the physics of the prob-lem at hand. It follows that for crack problems, the mesh is independent of the crack orientation[32–34]. An excellent review on the XFEM can be found in [35, 36]. Similar enrichment methodsfor modeling cracks are based on the generalized FEM [37, 38].

Let uh 2 Uh be an extended finite element approximation to the discretized weak form of elas-ticity, where Uh is the appropriate Sobolev space [39]. The XFEM enriches the conventional shapefunction space with a set of functions H.x/ and Fj .x.r , �//, such that

uh.x/D

NI .x/uI C

nJXID1

NI .x/H.x/aI C

nTXID1

24NI .x/ nFXjD1

Fj .x.r , �//bjI

35 (2)

where x D ¹x,yºT are the spatial coordinates. n,nJ ,nT , and nF are the number of total nodalpoints, jump-enriched nodes, tip-enriched nodes, and enrichment functions, respectively. NI .x/ arethe standard finite element shape functions associated with standard DOF uI , whereas aI and bjIare the DOF associated with the enriched nodes. Typically, for linear elastic fracture problems, thecrack tip zone is enriched with the classical analytical solution for the near tip field [24], and onlythepr terms, which are given in Equation (10), are employed. Note that these functions are given

in polar coordinates .r , �/. Element nodes (behind the crack tip) that are fully cut by the crack areenriched with the Heaviside function

²C1 above �Cc�1 below ��c

where �Cc and ��c defines the edges of the discontinuity line that splits the element into two parts.Note that the

pr term in the displacement field is directly built into the equations, and hence the

stress singularity of 1pr

appears in the solution. The enriched nodes of inclined 2D cracks areillustrated in Figure 2.

Typically, these enriched nodes (and the corresponding elements) are easily obtained with levelset functions. These functions represent discontinuities and boundaries implicitly; however, they

Figure 2. Enrichment visualization of cracks in extended FEM. Crack lines are illustrated in red, blue circlesare jump-enriched nodes, and green squares are tip-enriched nodes.

have been found to be very useful for problems involving moving boundaries and are easily handledby the level set method [40,41]. In the context of finite elements, this approach has been found veryuseful as information of the crack geometry is only stored as additional nodal variables and theninterpolated to Gauss (integration) points with standard shape functions. It is also straightforwardto update these values during crack propagation. Moreover, the level set approach allows automaticdetection of enriched nodes and the representation of the enrichment functions in Equation (2) bymeans of level set values. For crack problems, one defines two level set functions ‰ and ˆ, whichare simply the distance in local crack coordinate system from any point to the crack front and cracksurface, respectively. Thus, the angle � and the distance r given in Equation (14), are obtained by

r Dp‰2Cˆ2 and � D tan�1

�(4)

For more details, the reader is referred to [34, 42].

2.3. High-order enrichment functions

For traction-free cracks in local crack coordinates (Figure 1), the asymptotic fields for thedisplacement components u and v near the crack tip are given by Williams [24], as

u.r , �/D1XiD1

2�np2�

²KI i

�� C

2C .�1/i

�cos

2� �

2� 2

��

�(5)

CKII i

�� C

2� .�1/i

�sin

2� �

2� 2

��

�³v.r , �/D

2�ip2�

²KI i

��

2� .�1/i

�sin

2� C

2� 2

��

�(6)

CKII i

�� C

2� .�1/i

�cos

2� �

2� 2

��

where KI i and KII i are coefficients and � and � are the shear modulus and Kolosov constant,respectively. The Kolosov constant is defined as

²3� 4, plane strain3��1C�

, plane stress(7)

where is Poisson’s ratio.The expressions in Equations (5) and (6) may be truncated such that

u.r , �/D4XiD1

ui .ri=2, �/CO.r5=2/ (8)

v.r , �/D4XiD1

vi .ri=2, �/CO.r5=2/ (9)

where the coefficients u1, ...,u4 and v1, ..., v4 in Equations (8) and (9) are given in Appendix A.1.As opposed to [43], here we incorporate the high-order terms in a traditional fashion, considering

only the space spanned by high-order functions. See Appendix A.2 for a complete derivation. Thus,

the enrichment functions for different orders of r functions are

pr W F1 D

²pr sin

2,pr cos

2,pr sin � sin

2,pr sin � cos

³(10)

r W F2 D ¹r cos � , r sin �º (11)

r3=2 W F3 D²r3=2 cos

2, r3=2 sin

2, r3=2 sin � sin

2, r3=2 sin � cos

³(12)

r2 W F4 D ¹r2, r2 sin 2� , r2 cos 2�º (13)

and the full set of enrichments, with 13 terms, used in our analysis is therefore

F .r , �/D ¹F1, F2, F3, F4º (14)

3. EXTRACTION OF MIXED-MODE COMPONENTS OF STRAIN ENERGY RELEASERATES USING IRWIN’S INTEGRAL AND EXTENDED FEM

Irwin’s integral [17] is an important theoretical concept in LEFM. However, previous attempts toevaluate this integral directly using the FEM required a two-stage analysis [44], and sometimes theerrors reported were as high as 20% even with high-order elements [45]. Alternatively, the Irwinintegral inspired Rybicki and Kanninen to develop the VCCT method [16], avoiding the numericaldifficulties associated with direct implementation of Irwin’s integral. Nevertheless, in this paper, weshow that the use of XFEM provides a new way for evaluating Irwin’s integral directly in closedform, leading to accurate results.

3.1. Analytical expansion of Irwin’s integral

According to Irwin [17], the work required to extend a crack by an infinitesimal distance�c is equalto the work required to close the crack to its original length. Thus, the SERRs for a mixed-modestate, expressed in a polar coordinate system .r , �/ with the origin at the crack tip, is defined by

G D GI C GII (15)

where G is the total energy release rate, additively decomposed into individual components GI andGII due to Mode I and Mode II deformations, which are given by

GI D lim�c!C0

Z �c

�� .�c � r , 0/ Nu� .r/dr (16)

GII D lim�c!C0

Z �c

r� .�c � r , 0/ Nur.r/dr (17)

�� and r� are the normal and shear stresses in polar coordinates, Nur and Nu� are relative slidingand opening displacements between corresponding points on crack surfaces, and �c is the crackextension at the crack tip. The sliding and opening displacement jumps are defined by

ur.r/D ur.r ,��/� ur.r ,�/ (18)

u� .r/D u� .r ,��/� u� .r ,�/ (19)

The transformation of displacement fields from polar coordinates to Cartesian coordinates isgiven as

ur D u cos � C v sin � (20)

u� D�u sin � C v cos � (21)

Thus, under the assumption that the local crack coordinate system is aligned with the crackaxis (Figure 1), the relations are simplified and the sliding and opening displacement jumps,Equations (18)–(19), become

ur.r/D u .r ,�/� u .r ,��/ (22)

u� .r/D v .r ,�/� v .r ,��/ (23)

Substituting Williams expansion, Equations (5)–(6), into Equations (18)–(19), and retaining onlythe first three leading terms, yields

ur.r/D

ui .ri=2,�/�

ui .ri=2,��/D

mIIi r2i�12 CO

�r72

�(24)

u� .r/D

vi .ri=2,�/�

vi .ri=2,��/D

mIi r2i�12 CO

�r72

�(25)

The kinematic strain-displacement relations in polar coordinates are defined as

�rr D@ur

@r(26)

�� Dur

@�(27)

�r� D1

@�C@u�

@r�u�

�(28)

where �rr , �� , and �r� are the radial, tangential, and shear strain, respectively. Considering a planestrain state, the stresses �� and r� are given as

�� DE

.1C /.1� 2/.�rr C .1� /�� / (29)

r� DE

.1C /�r� (30)

Combining the stresses with strains, Equations (26)–(29) and plugging in Williams solutions,Equations (5)–(6), we arrive at the definition of normal and shear stresses ahead of the crack tip

�� .r , 0/DE

.1C /.1� 2/

11XiD4

mIi ri�52 CO

�r72

�(31)

r� .r , 0/DE

2.1C /

11XiD4

mIIi ri�52 CO

�r72

�(32)

Finally, substituting the opening displacements in Equations (24)–(25) and stresses inEquations (31)–(32) into the definition of SERRs, GI in Equation (35) and GII in Equation (36)and integrating, we have

eGI.�c/DE

.1C /.1� 2/

11XiD0

˛Ii �ci2 CO

��c6

eGII.�c/DE

2.1C /

11XiD0

˛IIi �ci2 CO

��c6

Here,eGI andeGII have been defined as

eGI D1

Z �c

�� .�c � r , 0/ Nu� .r/dr (35)

fGII D1

Z �c

r� .�c � r , 0/ Nur.r/dr (36)

so that

GI D lim�c!C0

eGI.�c/ (37)

GII D lim�c!C0

eGII.�c/ (38)

The coefficients ˛Ii and ˛IIi are listed in Appendix B.Note that a simple expression is obtained in the limit when �c! 0 because all high-order terms

vanish, that is,

eGI.�c D 0/DE

.1C /.1� 2/˛I0 D

.1C /.1� 2/

I4 (39)

eGII.�c D 0/DE

2.1C /˛II0 D

2.1C /

4mII1 m

II4 (40)

Our goal is to find the coefficients ˛Ii and ˛IIi analytically, using the enrichment functions inXFEM.

3.2. Extended FEM realization of Irwin’s integral

To obtain expressions for the coefficients ˛Ii and ˛IIi , we consider a generic rectangular ele-ment with a horizontal crack and tip at the center of the element, as shown in Figure 3. In polarcoordinates, any point .xp ,yp/ in the element domain is defined by

xp D r cos � yp D r sin � (41)

The nodal coordinates of the element are therefore .xI ,yI /, xI D ˙hx=2,yI D ˙hy=2, I D1, ..., 4, where hx and hy are the length of the edges in the x-direction and the y-direction,respectively.

Figure 3. Tip element illustration with crack tip at center: crack line is illustrated in red.

Tip elements in XFEM consider near tip asymptotic fields, and hence the displacement field inEquation (2) may be simplified and written in polar coordinates as

u.r , �/D4XID1

0@uI C nFXjD1

1A (42)

In polar coordinates, the standard linear shape functions are

NI .r , �/D1

�1C 4

xI r cos �

� 1C 4

yI r sin �

!I D 1, ..., 4 (43)

and the derivatives of the displacement field @u@r

, @u@�

and the shape functions @NI@r

, @NI@�

are given inAppendix A.3.

Assuming a general case where all enrichment functions, including high-order functions inEquation (14), are used to enrich the solution space (i.e., nF D 13), we substituteEquations (42)–(43) into Equations (18)–(19) and arrive at

ur.r/D

NI .r ,�/

0@uxI C 13XjD1

Fj .r ,�/bxjI

1A� 4XID1

NI .r ,��/

0@uxI C 13XjD1

Fj .r ,��/bxjI

�1� 4

��2prbx1I C 2r

32 bx8I

�(44)

u� .r/D

NI .r ,�/

0@uyI C 13XjD1

Fj .r ,�/byjI

1A� 4XID1

NI .r ,��/

0@uyI C 13XjD1

Fj .r ,��/byjI

�1� 4

��2prby1I C 2r

32 by8I

�(45)

Similarly, the stress �� .r , 0/ is obtained by plugging Equations (A.12)–(A.15) intoEquations (31)–(32) and using the derivatives of the enrichment function in Equations(A.10)–(A.11), giving the expressions

�� .r , 0/DE

.1C /.1� 2/

�uxI C

prbx2I C rbx5I C r

32 bx7I C r

2.bx11I C bx13I /�

�1C 4

��1

2prbx2I C bx5I C

prbx7I C 2r.bx11I C bx13I /

CE.1� /

.1C /.1� 2/

�h2x C 4 xI r

h2xh2y

!��uyI C

prby2I C rby5I C r

32 by7I C r

2.by11I C by13I /�

�1C 4

��pr

2by1I C by4I

Crby6I C r32

2by8I C by10I

�C 2r2by12I

�!(46)

and similarly the stress r� .r , 0/

r� .r , 0/DE

2.1C /

�h2x C 4 xI r

h2xh2y

��uxI C

prbx2I C rbx5I C r

32 bx7I C r

2.bx11I C bx13I /�C1

�1C 4

��

2bx1I C bx4I

�C rbx6I C r

2bx8I C bx10I

�C 2r2bx12I

h2x.uyI C

prby2I C rby5I C r

32 by7I C r

2.by11I C by13I //

�1C 4

��1

2prby2I C by5I C

prby7I C 2r.by11I C by13I /

�!(47)

The final step is to match all coefficients of leading order terms for opening displacement jumpsin Equation (44) with (24) and Equation (45) with (25). Similarly, the coefficients for stresses inEquations (46)–(47) are matched with those in (31)–(32). These constants are listed in Appendix Band the general solution for mixed-mode components of SERRs with finite integration limits isobtained in closed form.

In the special case of �c ! 0, high-order terms vanish, and only constants mI1 , mI4 and mII1 ,mII4 remain. These are given as

2by1I mII1 D

2bx1I (48)

8C .1� /

�by1I

8Cby4I

�(49)

mII4 D

�bx1I

8Cbx4I

4Cby2I

�(50)

where the coefficients bxj and byj are obtained from the solution of the algebraic system ofequations. Thus, SERRs can be computed in closed form and obtained directly without the needfor special post-processing techniques, and are given by

eGI.�c D 0/D�

.1C /.1� 2/

2by1I C .1� /

�by1I

8Cby4I

� 4XID1

eGII.�c D 0/D�

2.1C /

�bx1I

8Cbx4I

4Cby2I

�!(52)

Although the SERRs computation with �c equal to 0, as in Equations (51) and (52), does notdepend directly on the DOF of the higher order asymptotic functions, using these functions in theenrichment also affects the DOF of the branch functions and hence the computation of the SERRs.

In Section 4, we present results for both �c D 0 and �c D hx2

. SIFs are related to SERRs andcan directly be obtained by [16]

sG1E�

sG2E�

²1� 2, plane strain1, plane stress

and E is the Young modulus.

4. NUMERICAL EXAMPLES

The proposed approach is studied on a Mode I: Single Edge Notch Tension (SENT) panel prob-lem, Mode II problem, and mixed-mode benchmark problem. All numerical examples shown in thissection are plane strain problems. To alleviate computational issues related to tip element integra-tion, we used the trapezoidal integration rule with 200 � 200 equally spaced quadrature points forcomputation of the element stiffness matrix.

The SERRs are computed using Irwin’s integral. In all examples, we consider the effect of high-order terms, mesh refinement, and effect of integration interval �c. In particular, we considerthe following: (i) integration step of �c D 0, computed by Equations (51) and (52) and (ii) anintegration step of �c D hx

2, computed by Equations via (33) and (34).

All results related to the derivation in Section 3 have been verified by numerical integration (1DGauss Quadrature rule) of Irwin’s integral, as shown in Figure 4. Stress points in front of the cracktip and the corresponding displacement opening behind the tip have been computed numerically bythe following approximation

eGI.�c/D1

Z �c

�� .�c � r , 0/ Nu� .r/dr �1

ngpXiD1

wi��c � r�i , 0

Nu��r�i

eGII.�c/D1

Z �c

r� .�c � r , 0/ Nur.r/dr �1

ngpXiD1

wir��c � r�i , 0

Nur�r�i

Figure 4. Numerical integration of Irwin’s integral used to verify the analytical results in Section 3. Samecolor ‘x’ symbols indicate stress and displacement opening couples used in the integration.

where ngp is the number of Gauss quadrature points, r�i are the Gauss point coordinates, and wiare the weights associated with the integration rule. The pairs of stress and displacement openingused in Equations (55) and (56) to compute SERRs are illustrated by the same color ‘x’-symbol inFigure 4. The analytical derivation in Section 3 is verified and found to be in excellent agreementwith the numerical integration.

4.1. Mode I: Single Edge Notch Tension problem

We study first a Mode I problem: a SENT panel [46]. The problem studied consists of an edge crackof length a in a rectangular domain with dimensions h �W under unit tractions on its upper andbottom edges in opposite direction, as shown in Figure 5. Young’s modulus and Poisson’s ratio aretaken as E D 107 and D 0.3, respectively.

Figure 5. Mode I: Single Edge Notch Tension (SENT) benchmark problem [46].

Table I. Results for KI of Single Edge Notch Tension problem.

Order Mesh KI.�c D 0/ Relative error KI.�c D hx=2/ Relative error

1=2 9� 19 5.891 47.45% 8.128 27.50%19� 39 6.127 45.35% 8.474 24.42%29� 59 6.201 44.69% 8.577 23.49%39� 79 6.237 44.37% 8.627 23.05%

1 9� 19 10.406 7.18% 10.216 8.88%19� 39 10.659 4.93% 10.478 6.54%29� 59 10.729 4.30% 10.552 5.88%39� 79 10.762 4.01% 10.587 5.57%

3=2 9� 19 10.459 6.71% 10.834 3.37%19� 39 10.639 5.10% 11.009 1.81%29� 59 10.699 4.57% 11.060 1.35%39� 79 10.728 4.31% 11.084 1.13%

2 9� 19 10.490 6.43% 10.871 3.04%19� 39 10.684 4.71% 11.029 1.63%29� 59 10.740 4.20% 11.075 1.22%39� 79 10.767 3.97% 11.096 1.03%

The dimension h=W is set equal to 2, and a=W is set equal to 1=2. For this benchmark problem,KI can be expressed as [46]

�1.12� 0.23

� aW

�C 10.56

� aW

�2� 21.74

� aW

�3C 30.42

� aW

�4�p�a (57)

The computed results are summarized in Table I. It can be seen from the table that the pro-posed approach yields convergent results with higher order enrichment at the tip element. Setting�c D hx=2 generally produces more accurate results, whereas setting �c D 0 results in com-putational simplicity. Nonetheless, both methods compute the SIFs in closed form and no specialpost-processing is required. The proposed approach provides sufficient accuracy with order of 3=2,whereas higher order r2 performs slightly better.

4.2. Mode II example

A Mode II example shown in Figure 6 is studied. The problem consists of a square domain withh D W D 10, with a=W D 1=2 under p D 1 unit traction load on its left edge. Young’s modulus

Figure 6. Mode II example problem.

Table II. Results for KII of pure mode II problem.

Order Mesh KII.�c D 0/ Relative error KII.�c D hx=2/ Relative error

1=2 9� 9 0.593 52.3% 0.662 46.8%19� 19 0.768 38.3% 0.854 31.4%29� 29 0.826 33.6% 0.919 26.1%39� 39 0.855 31.3% 0.952 23.5%

1 9� 9 0.758 39.1% 0.768 38.3%19� 19 0.974 21.7% 0.981 21.1%29� 29 1.047 15.9% 1.054 15.3%39� 39 1.083 12.9% 1.090 12.4%

3=2 9� 9 0.820 34.1% 0.820 34.1%19� 19 1.061 14.8% 1.056 15.1%29� 29 1.140 8.4% 1.134 8.8%39� 39 1.179 5.3% 1.173 5.7%

2 9� 9 0.846 32.0% 0.825 33.7%19� 19 1.080 13.2% 1.059 14.9%29� 29 1.160 6.8% 1.137 8.6%39� 39 1.200 3.6% 1.176 5.5%

and Poisson’s ratios are taken as E D 107 and D 0.3, respectively. The traction acts in oppositedirections, as indicated in the figure to allow for Mode II sliding deformation. No analytical resultsare available for this problem; hence, we use the J-integral method with a very fine mesh (99� 99)as the reference solution, which gives

KII D 1.244 (58)

The results are summarized in Table II. It is interesting to note that for this example, the choice of�c D 0 provides the most accurate results. Similar to Example 4.1, the proposed approach withorder of 3=2 provides sufficient accuracy, whereas higher order r2 performs slightly better.

4.3. Mixed-mode benchmark problem

We study the performance of the proposed approach on a benchmark mixed-mode problem [43,47].The problem considered consists of a rectangular domain with dimensions 7� 16 units subjected to

(a) (b)

Figure 7. Mixed-mode edge crack example problem: (a) geometric definition of the problem with boundaryconditions and (b) enriched jump and tip nodes.

Table III. Results for KI of mixed-mode case.

Order Mesh KI.�c D 0/ Relative error KI.�c D hx=2/ Relative error

1=2 13� 23 19.863 41.6% 24.824 27.0%25� 45 20.318 40.2% 25.815 24.1%37� 67 20.462 39.8% 26.132 23.1%49� 89 20.532 39.6% 26.287 22.7%

1 13� 23 31.575 7.1% 31.125 8.5%25� 45 32.229 5.2% 31.835 6.4%37� 67 32.422 4.6% 32.048 5.7%49� 89 32.512 4.4% 32.147 5.4%

3=2 13� 23 32.529 4.3% 33.126 2.6%25� 45 33.064 2.8% 33.645 1.0%37� 67 33.238 2.2% 33.796 0.6%49� 89 33.322 2.0% 33.865 0.4%

2 13� 23 32.326 5.0% 33.083 2.7%25� 45 32.928 3.2% 33.591 1.2%37� 67 33.100 2.6% 33.732 0.8%49� 89 33.179 2.4% 33.797 0.6%

Table IV. Results for KII of mixed-mode case.

Order Mesh KII.�c D 0/ Relative error KII.�c D hx=2/ Relative error

1=2 13� 23 3.170 30.3% 3.550 22.0%25� 45 3.260 28.4% 3.610 20.7%37� 67 3.293 27.6% 3.631 20.2%49� 89 3.310 27.2% 3.642 20.0%

1 13� 23 4.128 9.3% 4.128 9.3%25� 45 4.150 8.8% 4.155 8.7%37� 67 4.156 8.7% 4.179 8.1%49� 89 4.159 8.6% 4.182 8.1%

3=2 13� 23 4.527 0.5% 4.503 1.0%25� 45 4.537 0.3% 4.519 0.7%37� 67 4.540 0.2% 4.524 0.6%49� 89 4.542 0.2% 4.526 0.5%

2 13� 23 4.648 2.2% 4.509 0.9%25� 45 4.659 2.4% 4.527 0.5%37� 67 4.660 2.4% 4.531 0.4%49� 89 4.660 2.4% 4.533 0.4%

a unit shear traction on its upper edge and is fixed on the bottom, as shown in Figure 7(a). The cracklength is 3.5 units, and the enriched jump and tip nodes are illustrated in Figure 7(b). We employ theproposed method to compute the SIFs KI and KII. Young’s modulus and Poisson’s ratio are takenas E D 3� 107 and D 0.25, respectively.

The analytical solution for this problem, given in [43, 47], is

KI D 34.00I KII D 4.55 (59)

Tables III and IV present numerical results for Mode I and Mode II, respectively. The most sig-nificant improvement in accuracy is observed when adding order r terms to the enrichment space.Nonetheless, r3=2 terms also improves the accuracy of the solution. However, when enriching withr2 terms, the results actually deteriorate. A similar trend has also been reported in [28, 29] whenhigh-order functions were employed.

5. CONCLUSIONS

We propose a new analytical approach to compute mixed-mode components of SERRs by directevaluation of Irwin’s integral and in the framework of the XFEM. High-order enrichment functionsin XFEM are employed and closed form expressions for SERRs are obtained. Hence, special post-processing procedures are avoided, and the SERRs are obtained directly from coefficients of thealgebraic system of equations.

Several benchmark examples on pure and mixed-mode problems are studied where the effectsof high-order enrichments, mesh refinement and integration limits of Irwin’s integral are inves-tigated. The results indicate that high-order enrichment functions have significant effect on theconvergence, in particular when the integral limits are finite. When the integral limits tend to zero,simpler SERR expressions are obtained and high-order terms vanish. Nonetheless, these terms stillcontribute indirectly via coefficients of first-order terms.

The approach is found to be simple and accurate. Nonetheless, the optimal choice of integrationlimits remains an open question and will be studied in future work.

APPENDIX A

A.1. Expansion of leading high-order functions

Terms of orderpr

u1 DKI1r

2�p2�

2�.� � 1/C sin

2sin �

³CKII1r

2�p2�

2�.� C 1/C cos

2sin �

³(A.1)

v1 DKI1r

2�p2�

2�.� C 1/� cos

2sin �

³CKII1r

2�p2�

2�.1� �/C sin

2sin �

³(A.2)

Terms of order r

u2 DKI2r

4�p2�

cos �¹� C 1º CKII2r

4�p2�

sin �¹� C 1º (A.3)

v2 DKI2r

4�p2�

sin �¹� � 3º CKII2r

4�p2�

cos �¹�� 1º (A.4)

Terms of order r3=2

u3 DKI3r

6�p2�

2�.� � 1/� 2 sin � sin

�� C

�³(A.5)

CKII3r

6�p2�

2�.�� 1/C 2 sin � cos

�� C

v3 DKI3r

6�p2�

²2 sin � cos

��

�� sin

2�.� C 1/

³(A.6)

CKII3r

6�p2�

²�2 sin � sin

�� C

�C cos

2�.1� �/

³Terms of order r2

u4 DKI4r

8�p2�¹.� C 3/ cos 2� � 2º C

KII4r2

8�p2�¹.� C 1/ sin 2�º (A.7)

v4 DKI4r

8�p2�¹.� � 3/ sin 2�º C

KII4r2

8�p2�¹.�� C 1/ cos 2� � 2º (A.8)

A.2. Derivatives of high-order functions used for extended FEM implementation

The crack tip functions in Equation (14) are given by8̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂<̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂:

F1 Dpr sin �

F2 Dpr cos �

F3 Dpr sin � sin �

F4 Dpr sin � cos �

2F5 D r cos �F6 D r sin �F7 D r3=2 cos �

F8 D r3=2 sin �2

F9 D r3=2 sin � sin �2

F10 D r3=2 sin � cos �

2F11 D r

F12 D r2 sin 2�

F13 D r2 cos 2�

Their first derivatives with respect to r and � are given as8̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂<̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂:

f r1 D @F1@rD 1

sin �2

f r2 D @F2@rD 1

cos �2

f r3 D @F3@rD 1

sin �2

sin �

f r4 D @F4@rD 1

cos �2

sin �

f r5 D @F5@rD cos �

f r6 D @F6@rD sin �

f r7 D @F7@rD 3

pr cos �

f r8 D @F8@rD 3

pr sin �

f r9 D @F9@rD 3

pr sin �

2sin �

f r10 D@F10@rD 3

pr cos �

2sin �

f r11 D@F11@rD 2r

f r12 D@F12@rD 2r sin 2�

f r13 D@F13@rD 2r cos 2�

(A.10)

8̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂<̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂:̂

f �1 D @F1@�Dpr

2cos �

f �2 D @F2@�D�

2sin �

f �3 D @F3@�Dpr

2cos �

2sin � C

pr sin �

2cos �

f �4 D @F4@�Dpr

2cos �

2cos � �

pr sin �

2sin �

f �5 D @F5@�D�r sin �

f �6 D @F6@�D r cos �

f �7 D @F7@�D� r

2sin �

f �8 D @F8@�D r3=2

2cos �

f �9 D @F9@�D r3=2

2cos �

2sin � C r3=2 sin �

2cos �

f �10 D@F10@�D� r

2sin �

2sin � C r3=2 cos �

2cos �

f �11 D@F11@�D 0

f �12 D@F12@�D 2r2 cos 2�

f �13 D@F13@�D�2r2 sin 2�

(A.11)

A.3. Derivatives of displacement field and shape function in polar coordinates

The derivatives of the displacement field in Equation (42) with respect to r and � are

@ruI C

nFXjD1

Fj bjI C

nFXjD1

@rbjI (A.12)

@�uI C

nFXjD1

Fj bjI C

nFXjD1

@�bjI (A.13)

where the derivatives of the shape functions are

�1C 4

xI cos �

� 1C 4

yI r sin �

�1C 4

xI r cos �

� 1C 4

yI sin �

!(A.14)

@�D1

�1� 4

xI r sin �

� 1C 4

yI r sin �

�1C 4

xI r cos �

� 1C 4

yI r cos �

!(A.15)

APPENDIX B: COEFFICIENTS OF IRWIN’S INTEGRAL EXPANSION

The coefficients of Irwin’s integral expansion in Equations (33) and (34) are

8̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂<̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂:

˛r0 D14mr1m

˛r1 D13mr1m

˛r2 D316mr2m

r4� C

116mr1m

˛r3 D215mr1m

15mr2m

˛r4 D132mr1m

r8� C

132mr2m

r6� C

532mr3m

˛r5 D8105mr1m

235mr2m

17mr3m

˛r6 D3256mr2m

r8� C

5256mr3m

r6� C

5256mr1m

r10� r D I , II

˛r7 D8315mr2m

263mr3m

16315mr1m

˛r8 D3512mr2m

r10� C

3512mr3m

˛r9 D8693mr3m

161155

mr2mr11

˛r10 D5

2048mr3m

˛r11 D163003

mr3mr11�

The coefficients of m are

8̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂<̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂:

mI1 DP4ID1

12by1I

mI2 DP4ID1

�by8I2�2by1IxI

�mI3 D

��2xI by8I

�mI4 D

bx2I8C .1� /

�by1I8C

�mI5 D

�xIuxIh2xC bx5I

�C .1� /

�yIuyI

�mI6 D

�32xI bx2Ih2xC 3

�C.1� /

�yI by2I

xI by1I

xI by4I

by8I8C

by10I4

�mI7 D

�2xI bx5Ih2x

C bx11I2C bx13I

�C.1� /

�4xIyIuyI

h2xh2y

CyI by5I

xI by6I

by12I2

�mI8 D

52xI bx7Ih2x

C.1� /P4ID1

�4xIyI by2I

h2xh2y

CyI by7I

h2yC 1

xI by8I

xI by10I

�mI9 D

�3xI bx11I

h2xC 3xI bx13I

�C.1� /

�4xIyI by5I

h2xh2y

CyI by11I

yI by13I

2xI by12I

�mI10 D .1� /

4xIyI by7I

h2xh2y

mI11 D .1� /P4ID1

�4xIyI by11I

h2xh2y

C4xIyI by13I

h2xh2y

8̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂<̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂̂̂ˆ̂̂̂̂:

mII1 DP4ID1

12bx1I

mII2 DP4ID1

�bx8I2� 2bx1IxI

�mII3 D

��2xI bx8I

�mII4 D

�bx1I8C bx4I

�mII5 D

�yIuxIh2yC

h2xC bx6I

�mII6 D

�yI bx2Ih2yC xI bx1I

2h2xC xI bx4I

3xI by2I

2h2xC bx8I

8C bx10I

3by7I8

�mII7 D

�4xIyIuxIh2xh

C yI bx5Ih2yC xI bx6I

2xI by5I

by11I2C

by13I2C bx12I

�mII8 D

�4xIyI bx2Ih2xh

C yI bx7Ih2yC xI bx8I

2h2xC xI bx10I

5xI by7I

�mII9 D

�4xIyI bx5Ih2xh

C yI bx11Ih2y

C yI bx13Ih2y

C 2xI bx12Ih2x

C3xI by11I

3xI by13I

�mII10 D

4xIyI bx7Ih2xh

mII11 DP4ID1

�4xIyI bx11I

h2xh2y

C 4xIyI bx13Ih2xh

�(B.3)

REFERENCES

1. Erdogan F, Sih G. On the crack extension in plane loading and transverse shear. Journal of Basic Mechanics 1963;85:519–527.

2. Hussain M, Pu S, Underwood J. Strain energy release rate for a crack under combined mode I and mode II. FractureAnalysis ASTM STP 1974; 560:2–28.

3. Maiti S, Smith R. Comparison of the criteria for mixed mode brittle fracture based on the pre-instability stress-strainfield. Part II: pure shear and uniaxial loading. Journal of Fracture 1984; 24:5–22.

4. Chan SK, Tuba IS, Wilson WK. On the finite element method in linear fracture mechanics. Engineering FractureMechanics 1970; 2:1–17.

5. Shih CF, Delorenzi HG, German MD. Crack extension modeling with singular quadratic isoparametric elements.International Journal of Fracture 1976; 1:647–651.

6. Banks-Sills L, Sherman D. Comparison of methods for calculating stress intensity factors with quarter-pointelements. International Journal of Fracture 1986; 32:127–140.

7. Li FZ, Shih CF, Needleman A. A comparison of methods for calculating energy release rates. Engineering FractureMechanics 1985; 21(2):405–421.

8. Rice JR. A path independent integral and the approximate analysis of strain concentration by notches and cracks.Journal of Applied Mechanics 1968; 35:379–386.

9. Destuynder P, Lescure S, Djaoua M. Some remarks on elastic fracture mechanics. Journal de mécanique théoriqueet appliquée 1983; 2(1):113–135.

10. Moran B, Shih CF. A general treatment of crack tip contour integrals. International Journal of Fracture 1987;35:295–310.

11. Yau JF, Wang SS, Corten HT. A mixed-mode crack analysis of isotropic solids using conservation-laws of elasticity.Journal of Applied Mechanics 1980; 47(2):335–341.

12. Wang SS, Yau JF, Corten HT. A mixed-mode crack analysis of rectilinear anisotropic solids using conservation lawsof elasticity. International Journal of Fracture 1980; 16:247–259.

13. Banks-Sills L, Dolev O. The conservative M-integral for thermal-elastic problems. International Journal of Fracture2004; 125(1):149–170.

14. Parks DM. A stiffness derivative finite element technique for determination of crack tip stress intensity factors.International Journal of Fracture 1974; 10(4):487–502.

15. Hellen TK. On the method of virtual crack extensions. International Journal for Numerical Methods in Engineering1975; 9(1):187–207.

16. Rybicki EF, Kanninen MF. A finite element calculation of stress intensity factors by a modified crack closure integral.Engineering Fracture Mechanics 1977; 9:931–938.

17. Irwin G. Fracture. In Handbuck der Physic. Bd. 6, Fluegge S (ed.), Elastizitaet und Plastizitaet. Springer: Berlin,1958; 551–590.

18. Barsoum RS. One the use of isoparametric finite elements in linear fracture mechanics. International Journal forNumerical Methods in Engineering 1976; 10:25–37.

19. Shivakumar KN, Tan PW, Newman JC. A virtual crack-closure technique for calculating stress intensity factors forcracked three dimensional bodies. International Journal of Fracture 1988; 36:R43–R50.

20. Fawaz S. Application of the virtual crack closure technique to calculate stress intensity factors for through crackswith an elliptical crack front. Engineering Fracture Mechanics 1998; 59(3):327 –342.

21. Krueger R. The virtual crack closure technique: history, approach and applications. Technical Report, NASA/CR-2002-211628, ICASE Report No. 2002-10, NASA, 2002.

22. Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journalfor Numerical Methods in Engineering 1999; 46:131–150.

23. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal forNumerical Methods in Engineering 1999; 45:601–620.

24. Williams ML. On the stress distribution at the base of a stationary crack. Journal of Applied Mechanics 1957;24:1957.

25. Waisman H. An analytical stiffness derivative extended finite element technique for extraction of crack tip strainenergy release rates. Engineering Fracture Mechanics 2010; 77(16):3204–3215.

26. Liszka T, Duarte C, Tworzydlo W. hp-Meshless Cloud Method. Computer Methods in Applied Mechanics andEngineering 1996; 139(14):263–288.

27. Duarte C, Oden J. An h-p adaptive method using clouds. Computer Methods in Applied Mechanics and Engineering1996; 139(14):237–262.

28. Liu X, Xiao Q, Karihaloo B. XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials.International Journal for Numerical Methods in Engineering 2004; 59(8):1103–1118.

29. Zamani A, Gracie R, Reza Eslami M. Cohesive and non-cohesive fracture by higher-order enrichment of XFEM.International Journal for Numerical Methods in Engineering 2012; 90(4):452–483.

30. Zamani A, Gracie R, Eslami M. Higher order tip enrichment of eXtended Finite Element Method in thermoelasticity.Computational Mechanics 2010; 46(6):851–866.

31. Réthoré J, Roux S, Hild F. Hybrid analytical and extended finite element method (HAX-FEM): a new enrichmentprocedure for cracked solids. International Journal for Numerical Methods in Engineering 2010; 81(3):269–285.

32. Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journalfor Numerical Methods in Engineering 1999; 46:131–150.

33. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal forNumerical Methods in Engineering 1999; 45:601–620.

34. Sukumar N, Chopp D. L, Moës N, Belytschko T. Modeling holes and inclusions by level sets in the extendedfinite-element method. Computer Methods in Applied Mechanics and Engineering 2000; 190(46–47):6183–6200.

35. Belytschko T, Gracie R, Ventura G. A review of extended/generalized finite element methods for material modeling.Modelling and Simulation in Materials Science and Engineering 2009; 17(4):043001.

36. Fries TP, Belytschko T. The extended/generalized finite element method: an overview of the method and itsapplications. International Journal for Numerical Methods in Engineering 2010; 84(3):253–304.

37. Duarte CA, Babuska I, Oden JT. Generalized finite element methods for three-dimensional structural mechanicsproblems. Computers & Structures 2000; 77(2):215–232.

38. Duarte CA, Hamzeh ON, Liszka TJ, Tworzydlo WW. A generalized finite element method for the simulation ofthree-dimensional dynamic crack propagation. Computer Methods in Applied Mechanics and Engineering 2001;190(15–17):2227–2262.

39. Hughes TJR. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Inc.:Englewood Cliffs, New Jersey, 1987.

40. Osher S, Sethian JA. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobiformulations. Journal of Computation Physics 1988; 79:12–49.

41. Sethian JA. Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry, FluidMechanics, Computer Vision, and Materials Science. Cambridge University Press: Cambridge, UK, 1999.

42. Moës N, Gravouil A, Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets—partI: mechanical model. International Journal for Numerical Methods in Engineering 2002; 53(11):2549–2568.

43. Liu XY, Xiao QZ, Karihaloo BL. XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials.International Journal for Numerical Methods in Engineering 2004; 59:1103–1118.

44. Smith F, Alavi M. Stress intensity factors for a part-circular surface flow. 1st International Conference on PressureVessel Technology, Delft, 1969, ASME/KIvI, paper II-62.

45. Hellen TK. The finite-element calculation of stress-intensity factors using energy techniques. 2nd InternationalConference on Structural Mechanics in Reactor Technology, Berlin, Germany, 10-14 Sept. 1973, paper G5/3.

46. Ewalds HL, Wanhill RJH. Fracture Mechanics. Edward Arnold: London, UK, 1984.47. Wilson WK. Combined mode fracture mechanics. Ph.D. Thesis, University of Pittsburgh, 1969.

A direct analytical method to extract mixed‐mode components of strain energy release rates from...

Documents

Transcript of A direct analytical method to extract mixed‐mode components of strain energy release rates from...

Electrochemistry extract

ESTUDIO INTEGRAL TESA:

Calculo Integral Granville Solucionario

Problemario de calculo integral

La Gerencia Integral - elmayorportaldegerencia.com

Computation of a general integral of Fermi-Dirac distribution by McDougall-Stoner method

A Boundary Integral—Normal Mode Method

5 THE INTEGRAL - OnCourse

Barrons Integral AK.pdf

Hepatomegalia - Pediatría integral

INTEGRAL UNIVERSITY, LUCKNOW

Integral definida1 2015-02 UDP integral definida - area

Fungsi Khusus Bentuk Integral

What Is Integral Spirituality?

FISICA integral

A fast, efficient and automated method to extract vessels from fundus images

Integral Accumulator гидроаккумуляторы

Download Extract - ReadingZone

integral definida

Identifying sources of chlorinated aliphatic hydrocarbons in a residential area in Italy using the integral pumping test method