Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 978026 12 pageshttpdxdoiorg1011552013978026

Research ArticleA Phantom-Node Method with Edge-Based Strain Smoothing forLinear Elastic Fracture Mechanics

N Vu-Bac12 H Nguyen-Xuan3 L Chen4 C K Lee5 G Zi6 X Zhuang1

G R Liu7 and T Rabczuk28

1 Department of Geotechnical Engineering Tongji University Shanghai 200092 China2 Institute of Structural Mechanics Bauhaus-University Weimar Marienstraszlige 15 D-99423 Weimar Germany3Division of Computational Mechanics Ton Duc Thang University Ho Chi Minh City 70000 Vietnam4 School of Chemistry Physics and Mechanical Engineering Queensland University of Technology Brisbane QLD 4001 Australia5 School of Engineering Institute of Mechanics and Advanced Materials Theoretical and Computational MechanicsCardiff University Wales CF24 3AA UK

6Department of Civil Environmental amp Architectural Engineering Korea University 5 Ga 1 An-Am Dong Sung-Buk GuSeoul 136-701 Republic of Korea

7 Aerospace Systems Ohio Eminent Scholar University of Cincinnati Cincinnati OH 45221-0070 USA8 School of Civil Environmental and Architectural Engineering Korea University 5 Ga 1 Anam-dong Seongbuk-guSeoul 136-701 Republic of Korea

Correspondence should be addressed to X Zhuang xiaoyingzhuangtongjieducnand T Rabczuk timonrabczukuni-weimarde

Received 15 January 2013 Accepted 1 May 2013

Academic Editor Song Cen

Copyright copy 2013 N Vu-Bac et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a novel numerical procedure based on the combination of an edge-based smoothed finite element (ES-FEM)with a phantom-node method for 2D linear elastic fracture mechanics In the standard phantom-node method the cracks areformulated by adding phantom nodes and the cracked element is replaced by two new superimposed elements This approachis quite simple to implement into existing explicit finite element programs The shape functions associated with discontinuouselements are similar to those of the standard finite elements which leads to certain simplification with implementing in the existingcodesThe phantom-nodemethod allowsmodeling discontinuities at an arbitrary location in the meshThe ES-FEMmodel owns aclose-to-exact stiffness that is much softer than lower-order finite element methods (FEM) Taking advantage of both the ES-FEMand the phantom-nodemethod we introduce an edge-based strain smoothing technique for the phantom-nodemethodNumericalresults show that the proposed method achieves high accuracy compared with the extended finite element method (XFEM) andother reference solutions

1 Introduction

The extended finite element method (XFEM) [1] has becomea standard tool to model arbitrary crack growth Howeverthe implementation of XFEM in an existing finite elementcode requires severe modifications An alternative methodto model arbitrary crack growth was proposed by [2] sub-sequently implemented by [3] in a static setting and by [4]in a dynamic setting It was shown by [4 5] that the methodproposed by [2] is identical to a step-enriched XFEM [4]refer to this method as phantom node method The maindifference to the original XFEM is that the discontinuity

jump is not obtained by introducing additional unknownsbut by the so-called overlapping paired elements In otherwords a new ldquooverlappedrdquo element is introduced to handlethe crack kinematics when an underlying element is crackedIt is accomplished by integrating these overlapped elementsup to the crack Though it was shown by [4] that thecrack kinematics obtained with the phantom node methodare identical to the step-enriched XFEM and it has someadvantages over step-enriched XFEM

(1) As no additional degrees of freedom are introducedthe implementation of the phantom node method

2 Journal of Applied Mathematics

in an existing finite element code is simpler Forexample arbitrary crack growths for nonlinear mate-rials and cohesive zone models even for multiplecracks in two and three dimensions have already beenimplemented in ABAQUS [6] while an additionalplug-in [7] is required to model crack growth usingXFEM

(2) No mixed terms (K119906119886 and K119886119906) occur improvingconditioning

(3) Standard mass lumping schemes can be used dueto the absence of an enrichment There are severalcontributions to develop diagonalized mass matricesin standardXFEM [8 9] but they are based on certainassumptions

(4) The development of complex FE-formulations ismuch easier due to the lack of an enrichment Forexample when techniques such as EAS (enhancedassumed strain) or ANS (assumed natural strain)are used special attention is required in a standardXFEM-formulation particularly for problems withconstraints Those difficulties do not occur in thephantom node method [10]

The key drawback of the phantom node method com-pared to standard XFEM is its lower flexibility It was devel-oped for problems involving crack growth ldquoonlyrdquo Howeveravoiding a crack tip enrichment significantly facilitates theenrichment strategy and the crack tracking algorithm

(i) A crack tip enrichment introduces more additionalunknowns It is well known that a topological enrich-ment is needed for accuracy reasons [11] leadingto a substantial increase of additional unknowns(compared to ldquopurerdquo step-enriched formulations) andincreasing difficulties due to increasing the conditionnumber

(ii) The nonpolynomial (and singular) crack-tip enrich-ment complicates integration [12ndash15] and requiresspecial attention (blending)

(iii) The enrichment strategy and the crack growth algo-rithms are complicated in particular in 3D

Modeling crack growth with the phantom node methodon the other hand is quite simple Commonly plane cracksegments are introduced through the entire element thoughcrack tip elements were developed [16] that allow cracks toclose inside an element

Recently Liu et al constructed a new class of finiteelement methods based on strain smoothing Among thosemethods the so-called ES-FEM edge-based smoothed finiteelement method (ES-FEM) has been proven to be the mostefficient and accurate one In numerous application [17ndash19]it was shown that particularly low-order SFEM-formulationsare superior in terms of efficiency and accuracy over ldquostan-dardrdquo low-order finite element formulations In particular itwas shown formany applications [15 20] that results obtainedby triangular ES-FEM elements are of the same accuracy asstandard Q4-elements

Γt

Γu

Γc

Ω

Figure 1 A two-dimensional body containing a crack and boundaryconditions

Therefore we propose to couple the ES-FEM with thephantom-node method We name the new element edge-based phantom node method (ES-Phantom node) In thispaper we focus on two-dimensional problems in linear elasticfracture mechanics (LEFM) However our long term goal isto model fracture in nonlinear materials in 3D Numericalresults showhigh reliability of the presentmethod for analysisof fracture problems

This paper is organized as follows In Section 2 we brieflysummarize the basic theory of phantom-node method Abrief description of ES-FEM is called back in Section 3The combination between the phantom-node method andthe ES-FEM is elaborated in Section 4 Section 5 presentsthe integration technique Benchmark numerical problemstaken from linear elastic fracture mechanics are studied inSection 6 Finally we give some concluding remarks

2 A Brief Description ofPhantom-Node Method

Consider a deformable body occupying domainΩ inmotionsubjected to body forces b external applied traction t onboundary Γ

119905 and displacement boundary conditions u =

u on Γ119906containing a crack as shown in Figure 1 with the

corresponding finite element discretization In the phantom-node method a completely cracked element is replaced bytwo partially active superimposed elements 1 and 2 whosenodes consist of real nodes and phantom nodes markedby solid and empty circles respectively The active part ofelement 1 (Ω

1) u1(x) which holds for 119891(x) lt 0 and

the other active part (Ω2) u2(x) which holds for 119891(x) gt

0 The two parts of the model do not share nodes andtherefore they displace (deform) independently Areias andBelytschko [5] demonstrated that theHansbo andHansbo [2]formulation is equivalent to the XFEM formulation relyingon discontinuous enrichment with the Heaviside function

The displacement field within an element Ω119890in Figure 2

is rewritten as [4]

forallx isin Ω119890 u (x) = sum

119868isin1198781

u1119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u1(x)

119867(minus119891 (x))

+ sum

119868isin1198782

u2119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u2(x)

119867(119891 (x)) (1)

Journal of Applied Mathematics 3

f(x) = 0

3

f(x) lt 0

21

=

1 2Ω1

Element 1 Element 2

+

Real nodePhantom node

f(x) gt 0

f(x) lt 0

Ω2

3

f(x) gt 0

3

21

Figure 2 The decomposition of a cracked element into two super-imposed elements

A

B

C

D

EF

G

H

O

I

Boundary edge m Inner edge k

Field nodeCentroid of triangle

Ω(k)

Figure 3 Construction of edge-based strain smoothing domains

where 1198781and 119878

2are the nodes of superimposed elements 1

and 2 respectively As illustrated in Figure 2 each elementcontains real nodes and phantom nodes marked by solid andempty circles respectively 119873

119868is the finite element shape

function associated with node 119868 while u1119868and u2

119868are nodal

displacements of original nodes in superimposed element 1and 2 respectively

119867 is theHeaviside function given in [1 21ndash24] anddefinedby

119867(119909) = 1 119909 gt 0

0 119909 ⩽ 0(2)

Here we choose the physical domain up to the crackline Note that the crack line is a boundary in phantom nodemethod It is like the elements near the external boundarySo we avoid singularity in phantom node method Thecorresponding strain terms are written the same

The strain field is obtained as follows

forallx isin Ω119890 120598 (x) = sum

119868isin1198781

B119868(x) u1119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205981(x)

119867(minus119891 (x))

+ sum

119868isin1198782

B119868(x) u2119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205982(x)

119867(119891 (x)) (3)

where B119868is the standard strain-displacement matrix

Smoothing domain Smoothing domain 1 Smoothing domain 2

Ω(k)2

Ω(k)1

3

1 2

f(x) gt 0

f(x) lt 0

4

1 2

34

= +

Real nodePhantom node

3

21

4

Figure 4 The decomposition of a completely cracked smoothingdomain into two superimposed smoothing domains

Smoothing domain Smoothing domain 1 Smoothing domain 2

Real nodePhantom node

Ω(k)2

Ω(k)1

3

1 2

f(x) gt 0

f(x) lt 0

4

= +

3

1 2

4 3

2

4

1

Figure 5 The decomposition of a cracked smoothing domaincontaining crack tip into two superimposed smoothing domains

The jump in the displacement field across the crack iscalculated by

⟦u (x) ⟧ = u1 (x) minus u2 (x) on Γ119888

119868 is a phantom node in element 1 if 119891 (x

119868) gt 0

element 2 if 119891 (x119868) lt 0

(4)

In this paper the crack tip is forced to be located on theelementrsquos boundary

3 Brief on Edge-Based Strain SmoothingMethod in Finite Elements

31 Displacement and Strain Field In the ES-FEM [25] thedomain Ω is partitioned into a set of nonoverlapping no-gap smoothing domains constructed using element edgesof the triangular elements Ω(119896) satisfies the conditionsΩ = ⋃

119873119890

119896=1Ω(119896) and Ω(119894) cap Ω(119895) = 0 for all 119894 = 119895 in which

119873119890is the total number of edges of elements in the problem

domain In Figure 3 the smoothing domain the smoothingdomain corresponding to an inner edge 119896 and the smoothingdomain for a boundary edge119898 are illustrated

Numerical integration is implemented on chosen Gausspoints as illustrated in Figures 6 and 7 corresponding withsplit smoothing domain in Figure 4 and tip smoothingdomain in Figure 5 respectively

4 Journal of Applied Mathematics

1 2

34

5

6

78

Crack surface

Sub-sd1Sub-sd2

Gauss point on the boundary

6

4

Sub-sd1

2

4

5

65

5

4

2

Figure 6 The decomposition of a completely cracked smoothingdomain into two superimposed smoothing domains

1 2

34

5

6

78

Crack surface

Sub-sd1

Sub-sd2

6

4

Sub-sd1

24

5

6

5

Gauss point on the boundary

22

4

6Sub-sd2

Figure 7 The decomposition of a cracked smoothing domaincontaining crack tip into two superimposed smoothing domains

Distribution of the stress components 120590119909119909

and 120590119910119910

forand unstructured mesh are shown in Figures 27 and 28respectively

Introducing the edge-based smoothing operation thecompatible strain 120598 = nabla

119904uℎ119896is smoothed over cell Ω(119896)

associated with edge 119896 as follows

120598119896= intΩ(119896)

120598 (x) Φ119896(x) dΩ = int

Ω(119896)

nabla119904uℎ (x) Φ

119896(x) dΩ (5)

where Φ119896is a given normalized smoothing function that

satisfies

intΩ(119896)

Φ119896(x) dΩ = 1 (6)

Using the following constant smoothing function

Φ =

1

119860(119896) x isin Ω(119896)

0 x notin Ω(119896)

120598119896=

1

119860(119896)intΩ(119896)

nabla119904uℎ (x) dΩ =

1

119860(119896)intΓ(119896)

L119899uℎ (x) dΓ

(7)

120590 = 10

a = 05

b = 10

H=20

Figure 8 Sheet with edge crack under tension

where 119860(119896) = intΩ(119896)dΩ is the area of the smoothing domain

Ω(119896) Γ(119896) is the boundary of the smoothing domainΩ(119896) and

L119899is the outward unit normal matrix which can be expressed

as

L119899= [

[

119899119909

0

0 119899119910

119899119910119899119909

]

]

(8)

4 Edge-Based Strain Smoothing PhantomNode Method

41 Displacement and Strain Field The approximation of thedisplacement field is written similarly to (1)

uℎ (x) = sum

119868isin119878es-pht1

u1119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u1(x)

119867(119891 (x))

+ sum

119868isin119878es-pht2

u2119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u2(x)

119867(minus119891 (x)) (9)

where 119878es-pht1

and 119878es-pht2

are nodes associated with smoothingdomains 1 and 2 respectively consisting of real nodes andphantom nodes illustrated in Figures 4 and 5 The associatednodes of the inner smoothing domain Ω

(119896) (DEFG) andboundary smoothing domain Ω

(119898) (ABC) are shown inFigure 3

The connectivities of these superimposed smoothingdomains which are cracked completely and the correspond-ing active parts are shown in Figure 4

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 3 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 3 4]

The connectivity of a superimposed smoothing domaincontaining the crack tip and the corresponding active parts is

Journal of Applied Mathematics 5

0 500 1000 1500 2000 2500 3000 3500 4000 450019

2

21

22

23times10minus3

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 9 Strain energy for the sheet with edge crack under tension

shown in Figure 5 so that crack tip is guaranteed to locate onthe elementrsquos edge

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 4]

Using the strain smoothing operation the smoothedstrain associated with edge 119896 created from the displacementapproximation in (9) can be rewritten as

120598119896= sum

119868isin119878es-pht1

B119868(x119896) u1119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205981(x)

119867(minus119891 (x))

+ sum

119868isin119878es-pht2

B119868(x119896) u2119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205982(x)

119867(119891 (x)) (10)

where B119868(x119896) is the smoothed strain gradient matrix for the

standard ES-FEM part Those matrices write as follows

B119868(x119896) =

[[[

[

119887119868119909(x119896) 0

0 119887119868119910(x119896)

119887119868119910(x119896) 119887119868119909(x119896)

]]]

]

(11)

In (11) 119887119868ℎ(x119896) ℎ isin 119909 119910 is computed by

119887119868ℎ(x119896) =

1

119860119904

119896

intΓ119904

119896

119899ℎ(x)119873119894(x)119867 ((minus1)

119890

119891 (x)) dΓ (12)

Using Gauss-Legendre integration along the segments ofboundary Γ119904

119896 we have

119887119868ℎ=

1

119860119904

119896

119873seg

sum

119898=1

[

[

119873gauss

sum

119899=1

119908119898119899119873119894(x119898119899)119867

times ((minus1)119890

119891 (x119898119899)) 119899ℎ(x119898119899) ]

]

(13)

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(h)

log 1

0(e

e)

ES-Phantom R = 055Phantom-T3R = 050XFEM-T3(0t)R = 044

Figure 10The convergence in the energy norm versus ℎ (mesh size)for the sheet with an edge crack under tension

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus22

minus215

minus21

minus205

minus2

minus195

minus19

minus185

minus18

log10(h)

log 1

0(e

e)

XFEM R = 055

Figure 11 The convergence in the energy norm of XFEM versus ℎ(mesh size) for the sheet with an edge crack under tension

where 119873seg is the number of segments of the boundary Γ119904119896

119873gauss is the number ofGauss points used along each segment119908119898119899

are the corresponding Gauss weights x119898119899

is the 119899thGaussian point on the 119898th segment of the boundary Γ119904

119896

whose outward unit normal is denoted by 119899ℎ the subscript

ldquo119890rdquo is either 1 or 2 as shown in Figure 4 and the superscriptldquo119890rdquo indicates a domain restriction to element 119890

The stiffness matrix K associated with a smoothingdomain is assembled by a similar process as in the FEM

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

intΩ119904

119896

(B119906119868)119879

DB119906119869dΩ (14)

6 Journal of Applied Mathematics

ES-Phantom R = 049

Phantom-T3R = 047

XFEM-T3(0t)R = 041

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(h)

log 1

0(e

k)

Figure 12The convergence in the stress intensity factor119870119868versus ℎ

(mesh size) for the sheet with edge crack under tension

XFEM R = 050

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 13The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for the sheet with edge crack under tension

All entries in matrix B119868in (11) with triangular meshes are

constants over each smoothing domain the stiffness matrixin (14) is therefore calculated by

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

(B119906119868)119879

DB119906119869119860119904

119896 (15)

42 Weak Formulation and Discretized Equation We returnto the two-dimensional body in Figure 1 Since the smoothedstrain over smoothing domains is variationally consistent as

ES-Phantom R = 055

Phantom-T3R = 050

XFEM-T3(0t)R = 044

minus2 minus15 minus1 minus05 0 05

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(t)

log 1

0(e

e)

minus188

minus193

minus201

Figure 14 Computational efficiency of energy norm for the problemof a sheet with an edge crack under remote tension

ES-Phantom R = 050

Phantom-T3R = 047

XFEM-T3(0t)R = 042

minus048

minus058

minus069

minus04 minus02 0 02 04 06 08 1minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(t)

log 1

0(e

k)

Figure 15 Computational efficiency of mode 119868 SIF 119870119868for the

problem of a sheet with an edge crack under remote tension

proven in [26] and used by [27 28] the assumed displace-ment uℎ and the smoothed strains 120598 satisfy the ldquosmoothedrdquoGalerkin weak form find uℎ isin 119881 for all 120575uℎ isin 119881

0such that

intΩ

120575(120598 (uℎ))119879

D (120598 (uℎ)) dΩ minus intΩ

(120575uℎ)119879

bdΩ

minus intΓ

(120575uℎ)119879

tΓdΓ = 0

(16)

Journal of Applied Mathematics 7

120591 = 10

a = 35

b = 70H

=160

Figure 16 Sheet with edge crack under shear

0 500 1000 1500 2000 2500 3000 3500 4000 450074

76

78

8

82

84

86

88

9times10minus5

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 17 Strain energy for a sheet with an edge crack under shear

with 119881 = u | u isin 1198671(Ω Γ119888) u = u on Γ

119906 u discontinuous

on Γ119888 and 119881

0= 120575u | 120575u isin 119867

1

(Ω Γ119888) 120575u = 0 on Γ

119906 120575u

discontinuous on Γ119888

Substituting the trial and test functions into (16) wefinally obtain the familiar equation

Kd = f (17)

where f is the nodal force vector that is identical to that in thestandard phantomnode The edge-based smoothed stiffnessmatrix K for all subcells follows (15)

The smoothed stress 120590ℎ is obtained in the same way fromthe as 120598ℎ in FEM which is constant over a smoothing cell

ES-Phantom R = 061

Phantom-T3R = 055

XFEM-T3(0t)R = 049

minus08 minus07 minus06 minus05 minus04 minus03minus295

minus29

minus285

minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 18The convergence in the energy norm versus ℎ (mesh size)for a sheet with an edge crack under shear

XFEM R = 050

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 19 The convergence in the energy norm of XFEM versus ℎ(mesh size) for a sheet with an edge crack under shear

In particular for linear elastic problems 120590ℎ = D120598ℎ iscalculated on the level of the smoothing cell

43 Crack Growth and Stress Intensity Factor Fractureparameters such as mode 119868 and mode 119868119868 stress intensityfactors (SIFs) are determined using the domain form [29 30]of the interaction integral [31] All the finite elements withina radius of 119903

119889= 119903119896ℎ119890from the crack-tip are used Herein ℎ

119890

is the crack tip element size and 119903119896is a scalar

In this paper crack growth is governed by the maxi-mum hoop stress criterion [32 33] which assumes that thecrack will propagate from its tip in the direction 120579

119888 where

8 Journal of Applied Mathematics

ES-Phantom R = 056

XFEM-T3(0t)R = 043

minus08 minus07 minus06 minus05 minus04 minus03minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Phantom-T3R = 050

Figure 20 The convergence in the stress intensity factor 119870119868versus

ℎ (mesh size) for a sheet with an edge crack under shear

XFEM R = 058

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 21The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for a sheet with an edge crack under shear

the circumferential (hoop) stress 120590120579120579is maximum The angle

of crack propagation satisfies the following equation

119870119868sin (120579119888) + 119870119868119868(3 cos (120579

119888) minus 1) = 0 (18)

Solving this equation we have

120579119888= 2 arctan[[

[

minus2 (119870119868119868119870119868)

1 + radic1 + 8(119870119868119868119870119868)2

]]

]

(19)

Once119870119868and119870

119868119868are known (19) may be used to compute the

direction of propagation

ES-Phantom R = 043

Phantom-T3R = 024

XFEM-T3(0t)R = 046

minus08 minus07 minus06 minus05 minus04 minus03

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Figure 22 The convergence in the stress intensity factor 119870119868119868versus

ℎ (mesh size) for sheet with edge crack under shear

XFEM R = 021

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus078

minus076

minus074

minus072

minus07

minus068

log10(h)

log 1

0(e

k)

Figure 23 The convergence in the stress intensity factor 119870119868119868

ofXFEM versus ℎ (mesh size) for sheet with edge crack under shear

5 Numerical Examples

In all numerical examples we are not using near-tip enrich-ment that is only discontinuous enrichment is used Thismeans that the best convergence rate attainable is 12 in the1198671normand 1 in the119871

2norm (O(ℎ12) andO(ℎ) respectively

where ℎ is the mesh size)

51 Sheet with an Edge Crack under Uniaxial Tension Con-sider a sheet under uniaxial tension as shown in Figure 8Thedimensions of the sheet are in unit of [mm] The materialparameters are Youngrsquos modulus 119864 = 3 times 107 Pa Poissonrsquos

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

2 Journal of Applied Mathematics

in an existing finite element code is simpler Forexample arbitrary crack growths for nonlinear mate-rials and cohesive zone models even for multiplecracks in two and three dimensions have already beenimplemented in ABAQUS [6] while an additionalplug-in [7] is required to model crack growth usingXFEM

(2) No mixed terms (K119906119886 and K119886119906) occur improvingconditioning

(3) Standard mass lumping schemes can be used dueto the absence of an enrichment There are severalcontributions to develop diagonalized mass matricesin standardXFEM [8 9] but they are based on certainassumptions

(4) The development of complex FE-formulations ismuch easier due to the lack of an enrichment Forexample when techniques such as EAS (enhancedassumed strain) or ANS (assumed natural strain)are used special attention is required in a standardXFEM-formulation particularly for problems withconstraints Those difficulties do not occur in thephantom node method [10]

The key drawback of the phantom node method com-pared to standard XFEM is its lower flexibility It was devel-oped for problems involving crack growth ldquoonlyrdquo Howeveravoiding a crack tip enrichment significantly facilitates theenrichment strategy and the crack tracking algorithm

(i) A crack tip enrichment introduces more additionalunknowns It is well known that a topological enrich-ment is needed for accuracy reasons [11] leadingto a substantial increase of additional unknowns(compared to ldquopurerdquo step-enriched formulations) andincreasing difficulties due to increasing the conditionnumber

(ii) The nonpolynomial (and singular) crack-tip enrich-ment complicates integration [12ndash15] and requiresspecial attention (blending)

(iii) The enrichment strategy and the crack growth algo-rithms are complicated in particular in 3D

Modeling crack growth with the phantom node methodon the other hand is quite simple Commonly plane cracksegments are introduced through the entire element thoughcrack tip elements were developed [16] that allow cracks toclose inside an element

Recently Liu et al constructed a new class of finiteelement methods based on strain smoothing Among thosemethods the so-called ES-FEM edge-based smoothed finiteelement method (ES-FEM) has been proven to be the mostefficient and accurate one In numerous application [17ndash19]it was shown that particularly low-order SFEM-formulationsare superior in terms of efficiency and accuracy over ldquostan-dardrdquo low-order finite element formulations In particular itwas shown formany applications [15 20] that results obtainedby triangular ES-FEM elements are of the same accuracy asstandard Q4-elements

Γt

Γu

Γc

Ω

Figure 1 A two-dimensional body containing a crack and boundaryconditions

Therefore we propose to couple the ES-FEM with thephantom-node method We name the new element edge-based phantom node method (ES-Phantom node) In thispaper we focus on two-dimensional problems in linear elasticfracture mechanics (LEFM) However our long term goal isto model fracture in nonlinear materials in 3D Numericalresults showhigh reliability of the presentmethod for analysisof fracture problems

This paper is organized as follows In Section 2 we brieflysummarize the basic theory of phantom-node method Abrief description of ES-FEM is called back in Section 3The combination between the phantom-node method andthe ES-FEM is elaborated in Section 4 Section 5 presentsthe integration technique Benchmark numerical problemstaken from linear elastic fracture mechanics are studied inSection 6 Finally we give some concluding remarks

2 A Brief Description ofPhantom-Node Method

Consider a deformable body occupying domainΩ inmotionsubjected to body forces b external applied traction t onboundary Γ

119905 and displacement boundary conditions u =

u on Γ119906containing a crack as shown in Figure 1 with the

corresponding finite element discretization In the phantom-node method a completely cracked element is replaced bytwo partially active superimposed elements 1 and 2 whosenodes consist of real nodes and phantom nodes markedby solid and empty circles respectively The active part ofelement 1 (Ω

1) u1(x) which holds for 119891(x) lt 0 and

the other active part (Ω2) u2(x) which holds for 119891(x) gt

0 The two parts of the model do not share nodes andtherefore they displace (deform) independently Areias andBelytschko [5] demonstrated that theHansbo andHansbo [2]formulation is equivalent to the XFEM formulation relyingon discontinuous enrichment with the Heaviside function

The displacement field within an element Ω119890in Figure 2

is rewritten as [4]

forallx isin Ω119890 u (x) = sum

119868isin1198781

u1119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u1(x)

119867(minus119891 (x))

+ sum

119868isin1198782

u2119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u2(x)

119867(119891 (x)) (1)

Journal of Applied Mathematics 3

f(x) = 0

3

f(x) lt 0

21

=

1 2Ω1

Element 1 Element 2

+

Real nodePhantom node

f(x) gt 0

f(x) lt 0

Ω2

3

f(x) gt 0

3

21

Figure 2 The decomposition of a cracked element into two super-imposed elements

A

B

C

D

EF

G

H

O

I

Boundary edge m Inner edge k

Field nodeCentroid of triangle

Ω(k)

Figure 3 Construction of edge-based strain smoothing domains

where 1198781and 119878

2are the nodes of superimposed elements 1

and 2 respectively As illustrated in Figure 2 each elementcontains real nodes and phantom nodes marked by solid andempty circles respectively 119873

119868is the finite element shape

function associated with node 119868 while u1119868and u2

119868are nodal

displacements of original nodes in superimposed element 1and 2 respectively

119867 is theHeaviside function given in [1 21ndash24] anddefinedby

119867(119909) = 1 119909 gt 0

0 119909 ⩽ 0(2)

Here we choose the physical domain up to the crackline Note that the crack line is a boundary in phantom nodemethod It is like the elements near the external boundarySo we avoid singularity in phantom node method Thecorresponding strain terms are written the same

The strain field is obtained as follows

forallx isin Ω119890 120598 (x) = sum

119868isin1198781

B119868(x) u1119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205981(x)

119867(minus119891 (x))

+ sum

119868isin1198782

B119868(x) u2119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205982(x)

119867(119891 (x)) (3)

where B119868is the standard strain-displacement matrix

Smoothing domain Smoothing domain 1 Smoothing domain 2

Ω(k)2

Ω(k)1

3

1 2

f(x) gt 0

f(x) lt 0

4

1 2

34

= +

Real nodePhantom node

3

21

4

Figure 4 The decomposition of a completely cracked smoothingdomain into two superimposed smoothing domains

Smoothing domain Smoothing domain 1 Smoothing domain 2

Real nodePhantom node

Ω(k)2

Ω(k)1

3

1 2

f(x) gt 0

f(x) lt 0

4

= +

3

1 2

4 3

2

4

1

Figure 5 The decomposition of a cracked smoothing domaincontaining crack tip into two superimposed smoothing domains

The jump in the displacement field across the crack iscalculated by

⟦u (x) ⟧ = u1 (x) minus u2 (x) on Γ119888

119868 is a phantom node in element 1 if 119891 (x

119868) gt 0

element 2 if 119891 (x119868) lt 0

(4)

In this paper the crack tip is forced to be located on theelementrsquos boundary

3 Brief on Edge-Based Strain SmoothingMethod in Finite Elements

31 Displacement and Strain Field In the ES-FEM [25] thedomain Ω is partitioned into a set of nonoverlapping no-gap smoothing domains constructed using element edgesof the triangular elements Ω(119896) satisfies the conditionsΩ = ⋃

119873119890

119896=1Ω(119896) and Ω(119894) cap Ω(119895) = 0 for all 119894 = 119895 in which

119873119890is the total number of edges of elements in the problem

domain In Figure 3 the smoothing domain the smoothingdomain corresponding to an inner edge 119896 and the smoothingdomain for a boundary edge119898 are illustrated

Numerical integration is implemented on chosen Gausspoints as illustrated in Figures 6 and 7 corresponding withsplit smoothing domain in Figure 4 and tip smoothingdomain in Figure 5 respectively

4 Journal of Applied Mathematics

1 2

34

5

6

78

Crack surface

Sub-sd1Sub-sd2

Gauss point on the boundary

6

4

Sub-sd1

2

4

5

65

5

4

2

Figure 6 The decomposition of a completely cracked smoothingdomain into two superimposed smoothing domains

1 2

34

5

6

78

Crack surface

Sub-sd1

Sub-sd2

6

4

Sub-sd1

24

5

6

5

Gauss point on the boundary

22

4

6Sub-sd2

Figure 7 The decomposition of a cracked smoothing domaincontaining crack tip into two superimposed smoothing domains

Distribution of the stress components 120590119909119909

and 120590119910119910

forand unstructured mesh are shown in Figures 27 and 28respectively

Introducing the edge-based smoothing operation thecompatible strain 120598 = nabla

119904uℎ119896is smoothed over cell Ω(119896)

associated with edge 119896 as follows

120598119896= intΩ(119896)

120598 (x) Φ119896(x) dΩ = int

Ω(119896)

nabla119904uℎ (x) Φ

119896(x) dΩ (5)

where Φ119896is a given normalized smoothing function that

satisfies

intΩ(119896)

Φ119896(x) dΩ = 1 (6)

Using the following constant smoothing function

Φ =

1

119860(119896) x isin Ω(119896)

0 x notin Ω(119896)

120598119896=

1

119860(119896)intΩ(119896)

nabla119904uℎ (x) dΩ =

1

119860(119896)intΓ(119896)

L119899uℎ (x) dΓ

(7)

120590 = 10

a = 05

b = 10

H=20

Figure 8 Sheet with edge crack under tension

where 119860(119896) = intΩ(119896)dΩ is the area of the smoothing domain

Ω(119896) Γ(119896) is the boundary of the smoothing domainΩ(119896) and

L119899is the outward unit normal matrix which can be expressed

as

L119899= [

[

119899119909

0

0 119899119910

119899119910119899119909

]

]

(8)

4 Edge-Based Strain Smoothing PhantomNode Method

41 Displacement and Strain Field The approximation of thedisplacement field is written similarly to (1)

uℎ (x) = sum

119868isin119878es-pht1

u1119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u1(x)

119867(119891 (x))

+ sum

119868isin119878es-pht2

u2119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u2(x)

119867(minus119891 (x)) (9)

where 119878es-pht1

and 119878es-pht2

are nodes associated with smoothingdomains 1 and 2 respectively consisting of real nodes andphantom nodes illustrated in Figures 4 and 5 The associatednodes of the inner smoothing domain Ω

(119896) (DEFG) andboundary smoothing domain Ω

(119898) (ABC) are shown inFigure 3

The connectivities of these superimposed smoothingdomains which are cracked completely and the correspond-ing active parts are shown in Figure 4

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 3 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 3 4]

The connectivity of a superimposed smoothing domaincontaining the crack tip and the corresponding active parts is

Journal of Applied Mathematics 5

0 500 1000 1500 2000 2500 3000 3500 4000 450019

2

21

22

23times10minus3

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 9 Strain energy for the sheet with edge crack under tension

shown in Figure 5 so that crack tip is guaranteed to locate onthe elementrsquos edge

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 4]

Using the strain smoothing operation the smoothedstrain associated with edge 119896 created from the displacementapproximation in (9) can be rewritten as

120598119896= sum

119868isin119878es-pht1

B119868(x119896) u1119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205981(x)

119867(minus119891 (x))

+ sum

119868isin119878es-pht2

B119868(x119896) u2119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205982(x)

119867(119891 (x)) (10)

where B119868(x119896) is the smoothed strain gradient matrix for the

standard ES-FEM part Those matrices write as follows

B119868(x119896) =

[[[

[

119887119868119909(x119896) 0

0 119887119868119910(x119896)

119887119868119910(x119896) 119887119868119909(x119896)

]]]

]

(11)

In (11) 119887119868ℎ(x119896) ℎ isin 119909 119910 is computed by

119887119868ℎ(x119896) =

1

119860119904

119896

intΓ119904

119896

119899ℎ(x)119873119894(x)119867 ((minus1)

119890

119891 (x)) dΓ (12)

Using Gauss-Legendre integration along the segments ofboundary Γ119904

119896 we have

119887119868ℎ=

1

119860119904

119896

119873seg

sum

119898=1

[

[

119873gauss

sum

119899=1

119908119898119899119873119894(x119898119899)119867

times ((minus1)119890

119891 (x119898119899)) 119899ℎ(x119898119899) ]

]

(13)

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(h)

log 1

0(e

e)

ES-Phantom R = 055Phantom-T3R = 050XFEM-T3(0t)R = 044

Figure 10The convergence in the energy norm versus ℎ (mesh size)for the sheet with an edge crack under tension

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus22

minus215

minus21

minus205

minus2

minus195

minus19

minus185

minus18

log10(h)

log 1

0(e

e)

XFEM R = 055

Figure 11 The convergence in the energy norm of XFEM versus ℎ(mesh size) for the sheet with an edge crack under tension

where 119873seg is the number of segments of the boundary Γ119904119896

119873gauss is the number ofGauss points used along each segment119908119898119899

are the corresponding Gauss weights x119898119899

is the 119899thGaussian point on the 119898th segment of the boundary Γ119904

119896

whose outward unit normal is denoted by 119899ℎ the subscript

ldquo119890rdquo is either 1 or 2 as shown in Figure 4 and the superscriptldquo119890rdquo indicates a domain restriction to element 119890

The stiffness matrix K associated with a smoothingdomain is assembled by a similar process as in the FEM

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

intΩ119904

119896

(B119906119868)119879

DB119906119869dΩ (14)

6 Journal of Applied Mathematics

ES-Phantom R = 049

Phantom-T3R = 047

XFEM-T3(0t)R = 041

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(h)

log 1

0(e

k)

Figure 12The convergence in the stress intensity factor119870119868versus ℎ

(mesh size) for the sheet with edge crack under tension

XFEM R = 050

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 13The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for the sheet with edge crack under tension

All entries in matrix B119868in (11) with triangular meshes are

constants over each smoothing domain the stiffness matrixin (14) is therefore calculated by

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

(B119906119868)119879

DB119906119869119860119904

119896 (15)

42 Weak Formulation and Discretized Equation We returnto the two-dimensional body in Figure 1 Since the smoothedstrain over smoothing domains is variationally consistent as

ES-Phantom R = 055

Phantom-T3R = 050

XFEM-T3(0t)R = 044

minus2 minus15 minus1 minus05 0 05

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(t)

log 1

0(e

e)

minus188

minus193

minus201

Figure 14 Computational efficiency of energy norm for the problemof a sheet with an edge crack under remote tension

ES-Phantom R = 050

Phantom-T3R = 047

XFEM-T3(0t)R = 042

minus048

minus058

minus069

minus04 minus02 0 02 04 06 08 1minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(t)

log 1

0(e

k)

Figure 15 Computational efficiency of mode 119868 SIF 119870119868for the

problem of a sheet with an edge crack under remote tension

proven in [26] and used by [27 28] the assumed displace-ment uℎ and the smoothed strains 120598 satisfy the ldquosmoothedrdquoGalerkin weak form find uℎ isin 119881 for all 120575uℎ isin 119881

0such that

intΩ

120575(120598 (uℎ))119879

D (120598 (uℎ)) dΩ minus intΩ

(120575uℎ)119879

bdΩ

minus intΓ

(120575uℎ)119879

tΓdΓ = 0

(16)

Journal of Applied Mathematics 7

120591 = 10

a = 35

b = 70H

=160

Figure 16 Sheet with edge crack under shear

0 500 1000 1500 2000 2500 3000 3500 4000 450074

76

78

8

82

84

86

88

9times10minus5

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 17 Strain energy for a sheet with an edge crack under shear

with 119881 = u | u isin 1198671(Ω Γ119888) u = u on Γ

119906 u discontinuous

on Γ119888 and 119881

0= 120575u | 120575u isin 119867

1

(Ω Γ119888) 120575u = 0 on Γ

119906 120575u

discontinuous on Γ119888

Substituting the trial and test functions into (16) wefinally obtain the familiar equation

Kd = f (17)

where f is the nodal force vector that is identical to that in thestandard phantomnode The edge-based smoothed stiffnessmatrix K for all subcells follows (15)

The smoothed stress 120590ℎ is obtained in the same way fromthe as 120598ℎ in FEM which is constant over a smoothing cell

ES-Phantom R = 061

Phantom-T3R = 055

XFEM-T3(0t)R = 049

minus08 minus07 minus06 minus05 minus04 minus03minus295

minus29

minus285

minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 18The convergence in the energy norm versus ℎ (mesh size)for a sheet with an edge crack under shear

XFEM R = 050

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 19 The convergence in the energy norm of XFEM versus ℎ(mesh size) for a sheet with an edge crack under shear

In particular for linear elastic problems 120590ℎ = D120598ℎ iscalculated on the level of the smoothing cell

43 Crack Growth and Stress Intensity Factor Fractureparameters such as mode 119868 and mode 119868119868 stress intensityfactors (SIFs) are determined using the domain form [29 30]of the interaction integral [31] All the finite elements withina radius of 119903

119889= 119903119896ℎ119890from the crack-tip are used Herein ℎ

119890

is the crack tip element size and 119903119896is a scalar

In this paper crack growth is governed by the maxi-mum hoop stress criterion [32 33] which assumes that thecrack will propagate from its tip in the direction 120579

119888 where

8 Journal of Applied Mathematics

ES-Phantom R = 056

XFEM-T3(0t)R = 043

minus08 minus07 minus06 minus05 minus04 minus03minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Phantom-T3R = 050

Figure 20 The convergence in the stress intensity factor 119870119868versus

ℎ (mesh size) for a sheet with an edge crack under shear

XFEM R = 058

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 21The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for a sheet with an edge crack under shear

the circumferential (hoop) stress 120590120579120579is maximum The angle

of crack propagation satisfies the following equation

119870119868sin (120579119888) + 119870119868119868(3 cos (120579

119888) minus 1) = 0 (18)

Solving this equation we have

120579119888= 2 arctan[[

[

minus2 (119870119868119868119870119868)

1 + radic1 + 8(119870119868119868119870119868)2

]]

]

(19)

Once119870119868and119870

119868119868are known (19) may be used to compute the

direction of propagation

ES-Phantom R = 043

Phantom-T3R = 024

XFEM-T3(0t)R = 046

minus08 minus07 minus06 minus05 minus04 minus03

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Figure 22 The convergence in the stress intensity factor 119870119868119868versus

ℎ (mesh size) for sheet with edge crack under shear

XFEM R = 021

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus078

minus076

minus074

minus072

minus07

minus068

log10(h)

log 1

0(e

k)

Figure 23 The convergence in the stress intensity factor 119870119868119868

ofXFEM versus ℎ (mesh size) for sheet with edge crack under shear

5 Numerical Examples

In all numerical examples we are not using near-tip enrich-ment that is only discontinuous enrichment is used Thismeans that the best convergence rate attainable is 12 in the1198671normand 1 in the119871

2norm (O(ℎ12) andO(ℎ) respectively

where ℎ is the mesh size)

51 Sheet with an Edge Crack under Uniaxial Tension Con-sider a sheet under uniaxial tension as shown in Figure 8Thedimensions of the sheet are in unit of [mm] The materialparameters are Youngrsquos modulus 119864 = 3 times 107 Pa Poissonrsquos

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Journal of Applied Mathematics 3

f(x) = 0

3

f(x) lt 0

21

=

1 2Ω1

Element 1 Element 2

+

Real nodePhantom node

f(x) gt 0

f(x) lt 0

Ω2

3

f(x) gt 0

3

21

Figure 2 The decomposition of a cracked element into two super-imposed elements

A

B

C

D

EF

G

H

O

I

Boundary edge m Inner edge k

Field nodeCentroid of triangle

Ω(k)

Figure 3 Construction of edge-based strain smoothing domains

where 1198781and 119878

2are the nodes of superimposed elements 1

and 2 respectively As illustrated in Figure 2 each elementcontains real nodes and phantom nodes marked by solid andempty circles respectively 119873

119868is the finite element shape

function associated with node 119868 while u1119868and u2

119868are nodal

displacements of original nodes in superimposed element 1and 2 respectively

119867 is theHeaviside function given in [1 21ndash24] anddefinedby

119867(119909) = 1 119909 gt 0

0 119909 ⩽ 0(2)

Here we choose the physical domain up to the crackline Note that the crack line is a boundary in phantom nodemethod It is like the elements near the external boundarySo we avoid singularity in phantom node method Thecorresponding strain terms are written the same

The strain field is obtained as follows

forallx isin Ω119890 120598 (x) = sum

119868isin1198781

B119868(x) u1119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205981(x)

119867(minus119891 (x))

+ sum

119868isin1198782

B119868(x) u2119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205982(x)

119867(119891 (x)) (3)

where B119868is the standard strain-displacement matrix

Smoothing domain Smoothing domain 1 Smoothing domain 2

Ω(k)2

Ω(k)1

3

1 2

f(x) gt 0

f(x) lt 0

4

1 2

34

= +

Real nodePhantom node

3

21

4

Figure 4 The decomposition of a completely cracked smoothingdomain into two superimposed smoothing domains

Smoothing domain Smoothing domain 1 Smoothing domain 2

Real nodePhantom node

Ω(k)2

Ω(k)1

3

1 2

f(x) gt 0

f(x) lt 0

4

= +

3

1 2

4 3

2

4

1

Figure 5 The decomposition of a cracked smoothing domaincontaining crack tip into two superimposed smoothing domains

The jump in the displacement field across the crack iscalculated by

⟦u (x) ⟧ = u1 (x) minus u2 (x) on Γ119888

119868 is a phantom node in element 1 if 119891 (x

119868) gt 0

element 2 if 119891 (x119868) lt 0

(4)

In this paper the crack tip is forced to be located on theelementrsquos boundary

3 Brief on Edge-Based Strain SmoothingMethod in Finite Elements

31 Displacement and Strain Field In the ES-FEM [25] thedomain Ω is partitioned into a set of nonoverlapping no-gap smoothing domains constructed using element edgesof the triangular elements Ω(119896) satisfies the conditionsΩ = ⋃

119873119890

119896=1Ω(119896) and Ω(119894) cap Ω(119895) = 0 for all 119894 = 119895 in which

119873119890is the total number of edges of elements in the problem

domain In Figure 3 the smoothing domain the smoothingdomain corresponding to an inner edge 119896 and the smoothingdomain for a boundary edge119898 are illustrated

Numerical integration is implemented on chosen Gausspoints as illustrated in Figures 6 and 7 corresponding withsplit smoothing domain in Figure 4 and tip smoothingdomain in Figure 5 respectively

4 Journal of Applied Mathematics

1 2

34

5

6

78

Crack surface

Sub-sd1Sub-sd2

Gauss point on the boundary

6

4

Sub-sd1

2

4

5

65

5

4

2

Figure 6 The decomposition of a completely cracked smoothingdomain into two superimposed smoothing domains

1 2

34

5

6

78

Crack surface

Sub-sd1

Sub-sd2

6

4

Sub-sd1

24

5

6

5

Gauss point on the boundary

22

4

6Sub-sd2

Figure 7 The decomposition of a cracked smoothing domaincontaining crack tip into two superimposed smoothing domains

Distribution of the stress components 120590119909119909

and 120590119910119910

forand unstructured mesh are shown in Figures 27 and 28respectively

Introducing the edge-based smoothing operation thecompatible strain 120598 = nabla

119904uℎ119896is smoothed over cell Ω(119896)

associated with edge 119896 as follows

120598119896= intΩ(119896)

120598 (x) Φ119896(x) dΩ = int

Ω(119896)

nabla119904uℎ (x) Φ

119896(x) dΩ (5)

where Φ119896is a given normalized smoothing function that

satisfies

intΩ(119896)

Φ119896(x) dΩ = 1 (6)

Using the following constant smoothing function

Φ =

1

119860(119896) x isin Ω(119896)

0 x notin Ω(119896)

120598119896=

1

119860(119896)intΩ(119896)

nabla119904uℎ (x) dΩ =

1

119860(119896)intΓ(119896)

L119899uℎ (x) dΓ

(7)

120590 = 10

a = 05

b = 10

H=20

Figure 8 Sheet with edge crack under tension

where 119860(119896) = intΩ(119896)dΩ is the area of the smoothing domain

Ω(119896) Γ(119896) is the boundary of the smoothing domainΩ(119896) and

L119899is the outward unit normal matrix which can be expressed

as

L119899= [

[

119899119909

0

0 119899119910

119899119910119899119909

]

]

(8)

4 Edge-Based Strain Smoothing PhantomNode Method

41 Displacement and Strain Field The approximation of thedisplacement field is written similarly to (1)

uℎ (x) = sum

119868isin119878es-pht1

u1119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u1(x)

119867(119891 (x))

+ sum

119868isin119878es-pht2

u2119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u2(x)

119867(minus119891 (x)) (9)

where 119878es-pht1

and 119878es-pht2

are nodes associated with smoothingdomains 1 and 2 respectively consisting of real nodes andphantom nodes illustrated in Figures 4 and 5 The associatednodes of the inner smoothing domain Ω

(119896) (DEFG) andboundary smoothing domain Ω

(119898) (ABC) are shown inFigure 3

The connectivities of these superimposed smoothingdomains which are cracked completely and the correspond-ing active parts are shown in Figure 4

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 3 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 3 4]

The connectivity of a superimposed smoothing domaincontaining the crack tip and the corresponding active parts is

Journal of Applied Mathematics 5

0 500 1000 1500 2000 2500 3000 3500 4000 450019

2

21

22

23times10minus3

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 9 Strain energy for the sheet with edge crack under tension

shown in Figure 5 so that crack tip is guaranteed to locate onthe elementrsquos edge

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 4]

Using the strain smoothing operation the smoothedstrain associated with edge 119896 created from the displacementapproximation in (9) can be rewritten as

120598119896= sum

119868isin119878es-pht1

B119868(x119896) u1119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205981(x)

119867(minus119891 (x))

+ sum

119868isin119878es-pht2

B119868(x119896) u2119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205982(x)

119867(119891 (x)) (10)

where B119868(x119896) is the smoothed strain gradient matrix for the

standard ES-FEM part Those matrices write as follows

B119868(x119896) =

[[[

[

119887119868119909(x119896) 0

0 119887119868119910(x119896)

119887119868119910(x119896) 119887119868119909(x119896)

]]]

]

(11)

In (11) 119887119868ℎ(x119896) ℎ isin 119909 119910 is computed by

119887119868ℎ(x119896) =

1

119860119904

119896

intΓ119904

119896

119899ℎ(x)119873119894(x)119867 ((minus1)

119890

119891 (x)) dΓ (12)

Using Gauss-Legendre integration along the segments ofboundary Γ119904

119896 we have

119887119868ℎ=

1

119860119904

119896

119873seg

sum

119898=1

[

[

119873gauss

sum

119899=1

119908119898119899119873119894(x119898119899)119867

times ((minus1)119890

119891 (x119898119899)) 119899ℎ(x119898119899) ]

]

(13)

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(h)

log 1

0(e

e)

ES-Phantom R = 055Phantom-T3R = 050XFEM-T3(0t)R = 044

Figure 10The convergence in the energy norm versus ℎ (mesh size)for the sheet with an edge crack under tension

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus22

minus215

minus21

minus205

minus2

minus195

minus19

minus185

minus18

log10(h)

log 1

0(e

e)

XFEM R = 055

Figure 11 The convergence in the energy norm of XFEM versus ℎ(mesh size) for the sheet with an edge crack under tension

where 119873seg is the number of segments of the boundary Γ119904119896

119873gauss is the number ofGauss points used along each segment119908119898119899

are the corresponding Gauss weights x119898119899

is the 119899thGaussian point on the 119898th segment of the boundary Γ119904

119896

whose outward unit normal is denoted by 119899ℎ the subscript

ldquo119890rdquo is either 1 or 2 as shown in Figure 4 and the superscriptldquo119890rdquo indicates a domain restriction to element 119890

The stiffness matrix K associated with a smoothingdomain is assembled by a similar process as in the FEM

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

intΩ119904

119896

(B119906119868)119879

DB119906119869dΩ (14)

6 Journal of Applied Mathematics

ES-Phantom R = 049

Phantom-T3R = 047

XFEM-T3(0t)R = 041

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(h)

log 1

0(e

k)

Figure 12The convergence in the stress intensity factor119870119868versus ℎ

(mesh size) for the sheet with edge crack under tension

XFEM R = 050

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 13The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for the sheet with edge crack under tension

All entries in matrix B119868in (11) with triangular meshes are

constants over each smoothing domain the stiffness matrixin (14) is therefore calculated by

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

(B119906119868)119879

DB119906119869119860119904

119896 (15)

42 Weak Formulation and Discretized Equation We returnto the two-dimensional body in Figure 1 Since the smoothedstrain over smoothing domains is variationally consistent as

ES-Phantom R = 055

Phantom-T3R = 050

XFEM-T3(0t)R = 044

minus2 minus15 minus1 minus05 0 05

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(t)

log 1

0(e

e)

minus188

minus193

minus201

Figure 14 Computational efficiency of energy norm for the problemof a sheet with an edge crack under remote tension

ES-Phantom R = 050

Phantom-T3R = 047

XFEM-T3(0t)R = 042

minus048

minus058

minus069

minus04 minus02 0 02 04 06 08 1minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(t)

log 1

0(e

k)

Figure 15 Computational efficiency of mode 119868 SIF 119870119868for the

problem of a sheet with an edge crack under remote tension

proven in [26] and used by [27 28] the assumed displace-ment uℎ and the smoothed strains 120598 satisfy the ldquosmoothedrdquoGalerkin weak form find uℎ isin 119881 for all 120575uℎ isin 119881

0such that

intΩ

120575(120598 (uℎ))119879

D (120598 (uℎ)) dΩ minus intΩ

(120575uℎ)119879

bdΩ

minus intΓ

(120575uℎ)119879

tΓdΓ = 0

(16)

Journal of Applied Mathematics 7

120591 = 10

a = 35

b = 70H

=160

Figure 16 Sheet with edge crack under shear

0 500 1000 1500 2000 2500 3000 3500 4000 450074

76

78

8

82

84

86

88

9times10minus5

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 17 Strain energy for a sheet with an edge crack under shear

with 119881 = u | u isin 1198671(Ω Γ119888) u = u on Γ

119906 u discontinuous

on Γ119888 and 119881

0= 120575u | 120575u isin 119867

1

(Ω Γ119888) 120575u = 0 on Γ

119906 120575u

discontinuous on Γ119888

Substituting the trial and test functions into (16) wefinally obtain the familiar equation

Kd = f (17)

where f is the nodal force vector that is identical to that in thestandard phantomnode The edge-based smoothed stiffnessmatrix K for all subcells follows (15)

The smoothed stress 120590ℎ is obtained in the same way fromthe as 120598ℎ in FEM which is constant over a smoothing cell

ES-Phantom R = 061

Phantom-T3R = 055

XFEM-T3(0t)R = 049

minus08 minus07 minus06 minus05 minus04 minus03minus295

minus29

minus285

minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 18The convergence in the energy norm versus ℎ (mesh size)for a sheet with an edge crack under shear

XFEM R = 050

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 19 The convergence in the energy norm of XFEM versus ℎ(mesh size) for a sheet with an edge crack under shear

In particular for linear elastic problems 120590ℎ = D120598ℎ iscalculated on the level of the smoothing cell

43 Crack Growth and Stress Intensity Factor Fractureparameters such as mode 119868 and mode 119868119868 stress intensityfactors (SIFs) are determined using the domain form [29 30]of the interaction integral [31] All the finite elements withina radius of 119903

119889= 119903119896ℎ119890from the crack-tip are used Herein ℎ

119890

is the crack tip element size and 119903119896is a scalar

In this paper crack growth is governed by the maxi-mum hoop stress criterion [32 33] which assumes that thecrack will propagate from its tip in the direction 120579

119888 where

8 Journal of Applied Mathematics

ES-Phantom R = 056

XFEM-T3(0t)R = 043

minus08 minus07 minus06 minus05 minus04 minus03minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Phantom-T3R = 050

Figure 20 The convergence in the stress intensity factor 119870119868versus

ℎ (mesh size) for a sheet with an edge crack under shear

XFEM R = 058

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 21The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for a sheet with an edge crack under shear

the circumferential (hoop) stress 120590120579120579is maximum The angle

of crack propagation satisfies the following equation

119870119868sin (120579119888) + 119870119868119868(3 cos (120579

119888) minus 1) = 0 (18)

Solving this equation we have

120579119888= 2 arctan[[

[

minus2 (119870119868119868119870119868)

1 + radic1 + 8(119870119868119868119870119868)2

]]

]

(19)

Once119870119868and119870

119868119868are known (19) may be used to compute the

direction of propagation

ES-Phantom R = 043

Phantom-T3R = 024

XFEM-T3(0t)R = 046

minus08 minus07 minus06 minus05 minus04 minus03

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Figure 22 The convergence in the stress intensity factor 119870119868119868versus

ℎ (mesh size) for sheet with edge crack under shear

XFEM R = 021

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus078

minus076

minus074

minus072

minus07

minus068

log10(h)

log 1

0(e

k)

Figure 23 The convergence in the stress intensity factor 119870119868119868

ofXFEM versus ℎ (mesh size) for sheet with edge crack under shear

5 Numerical Examples

In all numerical examples we are not using near-tip enrich-ment that is only discontinuous enrichment is used Thismeans that the best convergence rate attainable is 12 in the1198671normand 1 in the119871

2norm (O(ℎ12) andO(ℎ) respectively

where ℎ is the mesh size)

51 Sheet with an Edge Crack under Uniaxial Tension Con-sider a sheet under uniaxial tension as shown in Figure 8Thedimensions of the sheet are in unit of [mm] The materialparameters are Youngrsquos modulus 119864 = 3 times 107 Pa Poissonrsquos

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

4 Journal of Applied Mathematics

1 2

34

5

6

78

Crack surface

Sub-sd1Sub-sd2

Gauss point on the boundary

6

4

Sub-sd1

2

4

5

65

5

4

2

Figure 6 The decomposition of a completely cracked smoothingdomain into two superimposed smoothing domains

1 2

34

5

6

78

Crack surface

Sub-sd1

Sub-sd2

6

4

Sub-sd1

24

5

6

5

Gauss point on the boundary

22

4

6Sub-sd2

Figure 7 The decomposition of a cracked smoothing domaincontaining crack tip into two superimposed smoothing domains

Distribution of the stress components 120590119909119909

and 120590119910119910

forand unstructured mesh are shown in Figures 27 and 28respectively

Introducing the edge-based smoothing operation thecompatible strain 120598 = nabla

119904uℎ119896is smoothed over cell Ω(119896)

associated with edge 119896 as follows

120598119896= intΩ(119896)

120598 (x) Φ119896(x) dΩ = int

Ω(119896)

nabla119904uℎ (x) Φ

119896(x) dΩ (5)

where Φ119896is a given normalized smoothing function that

satisfies

intΩ(119896)

Φ119896(x) dΩ = 1 (6)

Using the following constant smoothing function

Φ =

1

119860(119896) x isin Ω(119896)

0 x notin Ω(119896)

120598119896=

1

119860(119896)intΩ(119896)

nabla119904uℎ (x) dΩ =

1

119860(119896)intΓ(119896)

L119899uℎ (x) dΓ

(7)

120590 = 10

a = 05

b = 10

H=20

Figure 8 Sheet with edge crack under tension

where 119860(119896) = intΩ(119896)dΩ is the area of the smoothing domain

Ω(119896) Γ(119896) is the boundary of the smoothing domainΩ(119896) and

L119899is the outward unit normal matrix which can be expressed

as

L119899= [

[

119899119909

0

0 119899119910

119899119910119899119909

]

]

(8)

4 Edge-Based Strain Smoothing PhantomNode Method

41 Displacement and Strain Field The approximation of thedisplacement field is written similarly to (1)

uℎ (x) = sum

119868isin119878es-pht1

u1119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u1(x)

119867(119891 (x))

+ sum

119868isin119878es-pht2

u2119868119873119868(x)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

u2(x)

119867(minus119891 (x)) (9)

where 119878es-pht1

and 119878es-pht2

are nodes associated with smoothingdomains 1 and 2 respectively consisting of real nodes andphantom nodes illustrated in Figures 4 and 5 The associatednodes of the inner smoothing domain Ω

(119896) (DEFG) andboundary smoothing domain Ω

(119898) (ABC) are shown inFigure 3

The connectivities of these superimposed smoothingdomains which are cracked completely and the correspond-ing active parts are shown in Figure 4

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 3 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 3 4]

The connectivity of a superimposed smoothing domaincontaining the crack tip and the corresponding active parts is

Journal of Applied Mathematics 5

0 500 1000 1500 2000 2500 3000 3500 4000 450019

2

21

22

23times10minus3

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 9 Strain energy for the sheet with edge crack under tension

shown in Figure 5 so that crack tip is guaranteed to locate onthe elementrsquos edge

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 4]

Using the strain smoothing operation the smoothedstrain associated with edge 119896 created from the displacementapproximation in (9) can be rewritten as

120598119896= sum

119868isin119878es-pht1

B119868(x119896) u1119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205981(x)

119867(minus119891 (x))

+ sum

119868isin119878es-pht2

B119868(x119896) u2119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205982(x)

119867(119891 (x)) (10)

where B119868(x119896) is the smoothed strain gradient matrix for the

standard ES-FEM part Those matrices write as follows

B119868(x119896) =

[[[

[

119887119868119909(x119896) 0

0 119887119868119910(x119896)

119887119868119910(x119896) 119887119868119909(x119896)

]]]

]

(11)

In (11) 119887119868ℎ(x119896) ℎ isin 119909 119910 is computed by

119887119868ℎ(x119896) =

1

119860119904

119896

intΓ119904

119896

119899ℎ(x)119873119894(x)119867 ((minus1)

119890

119891 (x)) dΓ (12)

Using Gauss-Legendre integration along the segments ofboundary Γ119904

119896 we have

119887119868ℎ=

1

119860119904

119896

119873seg

sum

119898=1

[

[

119873gauss

sum

119899=1

119908119898119899119873119894(x119898119899)119867

times ((minus1)119890

119891 (x119898119899)) 119899ℎ(x119898119899) ]

]

(13)

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(h)

log 1

0(e

e)

ES-Phantom R = 055Phantom-T3R = 050XFEM-T3(0t)R = 044

Figure 10The convergence in the energy norm versus ℎ (mesh size)for the sheet with an edge crack under tension

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus22

minus215

minus21

minus205

minus2

minus195

minus19

minus185

minus18

log10(h)

log 1

0(e

e)

XFEM R = 055

Figure 11 The convergence in the energy norm of XFEM versus ℎ(mesh size) for the sheet with an edge crack under tension

where 119873seg is the number of segments of the boundary Γ119904119896

119873gauss is the number ofGauss points used along each segment119908119898119899

are the corresponding Gauss weights x119898119899

is the 119899thGaussian point on the 119898th segment of the boundary Γ119904

119896

whose outward unit normal is denoted by 119899ℎ the subscript

ldquo119890rdquo is either 1 or 2 as shown in Figure 4 and the superscriptldquo119890rdquo indicates a domain restriction to element 119890

The stiffness matrix K associated with a smoothingdomain is assembled by a similar process as in the FEM

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

intΩ119904

119896

(B119906119868)119879

DB119906119869dΩ (14)

6 Journal of Applied Mathematics

ES-Phantom R = 049

Phantom-T3R = 047

XFEM-T3(0t)R = 041

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(h)

log 1

0(e

k)

Figure 12The convergence in the stress intensity factor119870119868versus ℎ

(mesh size) for the sheet with edge crack under tension

XFEM R = 050

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 13The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for the sheet with edge crack under tension

All entries in matrix B119868in (11) with triangular meshes are

constants over each smoothing domain the stiffness matrixin (14) is therefore calculated by

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

(B119906119868)119879

DB119906119869119860119904

119896 (15)

42 Weak Formulation and Discretized Equation We returnto the two-dimensional body in Figure 1 Since the smoothedstrain over smoothing domains is variationally consistent as

ES-Phantom R = 055

Phantom-T3R = 050

XFEM-T3(0t)R = 044

minus2 minus15 minus1 minus05 0 05

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(t)

log 1

0(e

e)

minus188

minus193

minus201

Figure 14 Computational efficiency of energy norm for the problemof a sheet with an edge crack under remote tension

ES-Phantom R = 050

Phantom-T3R = 047

XFEM-T3(0t)R = 042

minus048

minus058

minus069

minus04 minus02 0 02 04 06 08 1minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(t)

log 1

0(e

k)

Figure 15 Computational efficiency of mode 119868 SIF 119870119868for the

problem of a sheet with an edge crack under remote tension

proven in [26] and used by [27 28] the assumed displace-ment uℎ and the smoothed strains 120598 satisfy the ldquosmoothedrdquoGalerkin weak form find uℎ isin 119881 for all 120575uℎ isin 119881

0such that

intΩ

120575(120598 (uℎ))119879

D (120598 (uℎ)) dΩ minus intΩ

(120575uℎ)119879

bdΩ

minus intΓ

(120575uℎ)119879

tΓdΓ = 0

(16)

Journal of Applied Mathematics 7

120591 = 10

a = 35

b = 70H

=160

Figure 16 Sheet with edge crack under shear

0 500 1000 1500 2000 2500 3000 3500 4000 450074

76

78

8

82

84

86

88

9times10minus5

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 17 Strain energy for a sheet with an edge crack under shear

with 119881 = u | u isin 1198671(Ω Γ119888) u = u on Γ

119906 u discontinuous

on Γ119888 and 119881

0= 120575u | 120575u isin 119867

1

(Ω Γ119888) 120575u = 0 on Γ

119906 120575u

discontinuous on Γ119888

Substituting the trial and test functions into (16) wefinally obtain the familiar equation

Kd = f (17)

where f is the nodal force vector that is identical to that in thestandard phantomnode The edge-based smoothed stiffnessmatrix K for all subcells follows (15)

The smoothed stress 120590ℎ is obtained in the same way fromthe as 120598ℎ in FEM which is constant over a smoothing cell

ES-Phantom R = 061

Phantom-T3R = 055

XFEM-T3(0t)R = 049

minus08 minus07 minus06 minus05 minus04 minus03minus295

minus29

minus285

minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 18The convergence in the energy norm versus ℎ (mesh size)for a sheet with an edge crack under shear

XFEM R = 050

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 19 The convergence in the energy norm of XFEM versus ℎ(mesh size) for a sheet with an edge crack under shear

In particular for linear elastic problems 120590ℎ = D120598ℎ iscalculated on the level of the smoothing cell

43 Crack Growth and Stress Intensity Factor Fractureparameters such as mode 119868 and mode 119868119868 stress intensityfactors (SIFs) are determined using the domain form [29 30]of the interaction integral [31] All the finite elements withina radius of 119903

119889= 119903119896ℎ119890from the crack-tip are used Herein ℎ

119890

is the crack tip element size and 119903119896is a scalar

In this paper crack growth is governed by the maxi-mum hoop stress criterion [32 33] which assumes that thecrack will propagate from its tip in the direction 120579

119888 where

8 Journal of Applied Mathematics

ES-Phantom R = 056

XFEM-T3(0t)R = 043

minus08 minus07 minus06 minus05 minus04 minus03minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Phantom-T3R = 050

Figure 20 The convergence in the stress intensity factor 119870119868versus

ℎ (mesh size) for a sheet with an edge crack under shear

XFEM R = 058

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 21The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for a sheet with an edge crack under shear

the circumferential (hoop) stress 120590120579120579is maximum The angle

of crack propagation satisfies the following equation

119870119868sin (120579119888) + 119870119868119868(3 cos (120579

119888) minus 1) = 0 (18)

Solving this equation we have

120579119888= 2 arctan[[

[

minus2 (119870119868119868119870119868)

1 + radic1 + 8(119870119868119868119870119868)2

]]

]

(19)

Once119870119868and119870

119868119868are known (19) may be used to compute the

direction of propagation

ES-Phantom R = 043

Phantom-T3R = 024

XFEM-T3(0t)R = 046

minus08 minus07 minus06 minus05 minus04 minus03

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Figure 22 The convergence in the stress intensity factor 119870119868119868versus

ℎ (mesh size) for sheet with edge crack under shear

XFEM R = 021

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus078

minus076

minus074

minus072

minus07

minus068

log10(h)

log 1

0(e

k)

Figure 23 The convergence in the stress intensity factor 119870119868119868

ofXFEM versus ℎ (mesh size) for sheet with edge crack under shear

5 Numerical Examples

In all numerical examples we are not using near-tip enrich-ment that is only discontinuous enrichment is used Thismeans that the best convergence rate attainable is 12 in the1198671normand 1 in the119871

2norm (O(ℎ12) andO(ℎ) respectively

where ℎ is the mesh size)

51 Sheet with an Edge Crack under Uniaxial Tension Con-sider a sheet under uniaxial tension as shown in Figure 8Thedimensions of the sheet are in unit of [mm] The materialparameters are Youngrsquos modulus 119864 = 3 times 107 Pa Poissonrsquos

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Journal of Applied Mathematics 5

0 500 1000 1500 2000 2500 3000 3500 4000 450019

2

21

22

23times10minus3

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 9 Strain energy for the sheet with edge crack under tension

shown in Figure 5 so that crack tip is guaranteed to locate onthe elementrsquos edge

nodes of smoothing domain 1 (Ω(119896)1) = [1 2 4]

nodes of smoothing domain 2 (Ω(119896)2) = [1 2 4]

Using the strain smoothing operation the smoothedstrain associated with edge 119896 created from the displacementapproximation in (9) can be rewritten as

120598119896= sum

119868isin119878es-pht1

B119868(x119896) u1119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205981(x)

119867(minus119891 (x))

+ sum

119868isin119878es-pht2

B119868(x119896) u2119868⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1205982(x)

119867(119891 (x)) (10)

where B119868(x119896) is the smoothed strain gradient matrix for the

standard ES-FEM part Those matrices write as follows

B119868(x119896) =

[[[

[

119887119868119909(x119896) 0

0 119887119868119910(x119896)

119887119868119910(x119896) 119887119868119909(x119896)

]]]

]

(11)

In (11) 119887119868ℎ(x119896) ℎ isin 119909 119910 is computed by

119887119868ℎ(x119896) =

1

119860119904

119896

intΓ119904

119896

119899ℎ(x)119873119894(x)119867 ((minus1)

119890

119891 (x)) dΓ (12)

Using Gauss-Legendre integration along the segments ofboundary Γ119904

119896 we have

119887119868ℎ=

1

119860119904

119896

119873seg

sum

119898=1

[

[

119873gauss

sum

119899=1

119908119898119899119873119894(x119898119899)119867

times ((minus1)119890

119891 (x119898119899)) 119899ℎ(x119898119899) ]

]

(13)

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(h)

log 1

0(e

e)

ES-Phantom R = 055Phantom-T3R = 050XFEM-T3(0t)R = 044

Figure 10The convergence in the energy norm versus ℎ (mesh size)for the sheet with an edge crack under tension

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus22

minus215

minus21

minus205

minus2

minus195

minus19

minus185

minus18

log10(h)

log 1

0(e

e)

XFEM R = 055

Figure 11 The convergence in the energy norm of XFEM versus ℎ(mesh size) for the sheet with an edge crack under tension

where 119873seg is the number of segments of the boundary Γ119904119896

119873gauss is the number ofGauss points used along each segment119908119898119899

are the corresponding Gauss weights x119898119899

is the 119899thGaussian point on the 119898th segment of the boundary Γ119904

119896

whose outward unit normal is denoted by 119899ℎ the subscript

ldquo119890rdquo is either 1 or 2 as shown in Figure 4 and the superscriptldquo119890rdquo indicates a domain restriction to element 119890

The stiffness matrix K associated with a smoothingdomain is assembled by a similar process as in the FEM

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

intΩ119904

119896

(B119906119868)119879

DB119906119869dΩ (14)

6 Journal of Applied Mathematics

ES-Phantom R = 049

Phantom-T3R = 047

XFEM-T3(0t)R = 041

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(h)

log 1

0(e

k)

Figure 12The convergence in the stress intensity factor119870119868versus ℎ

(mesh size) for the sheet with edge crack under tension

XFEM R = 050

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 13The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for the sheet with edge crack under tension

All entries in matrix B119868in (11) with triangular meshes are

constants over each smoothing domain the stiffness matrixin (14) is therefore calculated by

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

(B119906119868)119879

DB119906119869119860119904

119896 (15)

42 Weak Formulation and Discretized Equation We returnto the two-dimensional body in Figure 1 Since the smoothedstrain over smoothing domains is variationally consistent as

ES-Phantom R = 055

Phantom-T3R = 050

XFEM-T3(0t)R = 044

minus2 minus15 minus1 minus05 0 05

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(t)

log 1

0(e

e)

minus188

minus193

minus201

Figure 14 Computational efficiency of energy norm for the problemof a sheet with an edge crack under remote tension

ES-Phantom R = 050

Phantom-T3R = 047

XFEM-T3(0t)R = 042

minus048

minus058

minus069

minus04 minus02 0 02 04 06 08 1minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(t)

log 1

0(e

k)

Figure 15 Computational efficiency of mode 119868 SIF 119870119868for the

problem of a sheet with an edge crack under remote tension

proven in [26] and used by [27 28] the assumed displace-ment uℎ and the smoothed strains 120598 satisfy the ldquosmoothedrdquoGalerkin weak form find uℎ isin 119881 for all 120575uℎ isin 119881

0such that

intΩ

120575(120598 (uℎ))119879

D (120598 (uℎ)) dΩ minus intΩ

(120575uℎ)119879

bdΩ

minus intΓ

(120575uℎ)119879

tΓdΓ = 0

(16)

Journal of Applied Mathematics 7

120591 = 10

a = 35

b = 70H

=160

Figure 16 Sheet with edge crack under shear

0 500 1000 1500 2000 2500 3000 3500 4000 450074

76

78

8

82

84

86

88

9times10minus5

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 17 Strain energy for a sheet with an edge crack under shear

with 119881 = u | u isin 1198671(Ω Γ119888) u = u on Γ

119906 u discontinuous

on Γ119888 and 119881

0= 120575u | 120575u isin 119867

1

(Ω Γ119888) 120575u = 0 on Γ

119906 120575u

discontinuous on Γ119888

Substituting the trial and test functions into (16) wefinally obtain the familiar equation

Kd = f (17)

where f is the nodal force vector that is identical to that in thestandard phantomnode The edge-based smoothed stiffnessmatrix K for all subcells follows (15)

The smoothed stress 120590ℎ is obtained in the same way fromthe as 120598ℎ in FEM which is constant over a smoothing cell

ES-Phantom R = 061

Phantom-T3R = 055

XFEM-T3(0t)R = 049

minus08 minus07 minus06 minus05 minus04 minus03minus295

minus29

minus285

minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 18The convergence in the energy norm versus ℎ (mesh size)for a sheet with an edge crack under shear

XFEM R = 050

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 19 The convergence in the energy norm of XFEM versus ℎ(mesh size) for a sheet with an edge crack under shear

In particular for linear elastic problems 120590ℎ = D120598ℎ iscalculated on the level of the smoothing cell

43 Crack Growth and Stress Intensity Factor Fractureparameters such as mode 119868 and mode 119868119868 stress intensityfactors (SIFs) are determined using the domain form [29 30]of the interaction integral [31] All the finite elements withina radius of 119903

119889= 119903119896ℎ119890from the crack-tip are used Herein ℎ

119890

is the crack tip element size and 119903119896is a scalar

In this paper crack growth is governed by the maxi-mum hoop stress criterion [32 33] which assumes that thecrack will propagate from its tip in the direction 120579

119888 where

8 Journal of Applied Mathematics

ES-Phantom R = 056

XFEM-T3(0t)R = 043

minus08 minus07 minus06 minus05 minus04 minus03minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Phantom-T3R = 050

Figure 20 The convergence in the stress intensity factor 119870119868versus

ℎ (mesh size) for a sheet with an edge crack under shear

XFEM R = 058

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 21The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for a sheet with an edge crack under shear

the circumferential (hoop) stress 120590120579120579is maximum The angle

of crack propagation satisfies the following equation

119870119868sin (120579119888) + 119870119868119868(3 cos (120579

119888) minus 1) = 0 (18)

Solving this equation we have

120579119888= 2 arctan[[

[

minus2 (119870119868119868119870119868)

1 + radic1 + 8(119870119868119868119870119868)2

]]

]

(19)

Once119870119868and119870

119868119868are known (19) may be used to compute the

direction of propagation

ES-Phantom R = 043

Phantom-T3R = 024

XFEM-T3(0t)R = 046

minus08 minus07 minus06 minus05 minus04 minus03

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Figure 22 The convergence in the stress intensity factor 119870119868119868versus

ℎ (mesh size) for sheet with edge crack under shear

XFEM R = 021

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus078

minus076

minus074

minus072

minus07

minus068

log10(h)

log 1

0(e

k)

Figure 23 The convergence in the stress intensity factor 119870119868119868

ofXFEM versus ℎ (mesh size) for sheet with edge crack under shear

5 Numerical Examples

In all numerical examples we are not using near-tip enrich-ment that is only discontinuous enrichment is used Thismeans that the best convergence rate attainable is 12 in the1198671normand 1 in the119871

2norm (O(ℎ12) andO(ℎ) respectively

where ℎ is the mesh size)

51 Sheet with an Edge Crack under Uniaxial Tension Con-sider a sheet under uniaxial tension as shown in Figure 8Thedimensions of the sheet are in unit of [mm] The materialparameters are Youngrsquos modulus 119864 = 3 times 107 Pa Poissonrsquos

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

6 Journal of Applied Mathematics

ES-Phantom R = 049

Phantom-T3R = 047

XFEM-T3(0t)R = 041

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1 minus09minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(h)

log 1

0(e

k)

Figure 12The convergence in the stress intensity factor119870119868versus ℎ

(mesh size) for the sheet with edge crack under tension

XFEM R = 050

minus17 minus16 minus15 minus14 minus13 minus12 minus11 minus1minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 13The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for the sheet with edge crack under tension

All entries in matrix B119868in (11) with triangular meshes are

constants over each smoothing domain the stiffness matrixin (14) is therefore calculated by

K119868119869=

119873119904

sum

119896=1

K119904119868119869119896

=

119873119904

sum

119896=1

(B119906119868)119879

DB119906119869119860119904

119896 (15)

42 Weak Formulation and Discretized Equation We returnto the two-dimensional body in Figure 1 Since the smoothedstrain over smoothing domains is variationally consistent as

ES-Phantom R = 055

Phantom-T3R = 050

XFEM-T3(0t)R = 044

minus2 minus15 minus1 minus05 0 05

minus21

minus205

minus2

minus195

minus19

minus185

minus18

minus175

minus17

log10(t)

log 1

0(e

e)

minus188

minus193

minus201

Figure 14 Computational efficiency of energy norm for the problemof a sheet with an edge crack under remote tension

ES-Phantom R = 050

Phantom-T3R = 047

XFEM-T3(0t)R = 042

minus048

minus058

minus069

minus04 minus02 0 02 04 06 08 1minus085

minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

minus04

minus035

log10(t)

log 1

0(e

k)

Figure 15 Computational efficiency of mode 119868 SIF 119870119868for the

problem of a sheet with an edge crack under remote tension

proven in [26] and used by [27 28] the assumed displace-ment uℎ and the smoothed strains 120598 satisfy the ldquosmoothedrdquoGalerkin weak form find uℎ isin 119881 for all 120575uℎ isin 119881

0such that

intΩ

120575(120598 (uℎ))119879

D (120598 (uℎ)) dΩ minus intΩ

(120575uℎ)119879

bdΩ

minus intΓ

(120575uℎ)119879

tΓdΓ = 0

(16)

Journal of Applied Mathematics 7

120591 = 10

a = 35

b = 70H

=160

Figure 16 Sheet with edge crack under shear

0 500 1000 1500 2000 2500 3000 3500 4000 450074

76

78

8

82

84

86

88

9times10minus5

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 17 Strain energy for a sheet with an edge crack under shear

with 119881 = u | u isin 1198671(Ω Γ119888) u = u on Γ

119906 u discontinuous

on Γ119888 and 119881

0= 120575u | 120575u isin 119867

1

(Ω Γ119888) 120575u = 0 on Γ

119906 120575u

discontinuous on Γ119888

Substituting the trial and test functions into (16) wefinally obtain the familiar equation

Kd = f (17)

where f is the nodal force vector that is identical to that in thestandard phantomnode The edge-based smoothed stiffnessmatrix K for all subcells follows (15)

The smoothed stress 120590ℎ is obtained in the same way fromthe as 120598ℎ in FEM which is constant over a smoothing cell

ES-Phantom R = 061

Phantom-T3R = 055

XFEM-T3(0t)R = 049

minus08 minus07 minus06 minus05 minus04 minus03minus295

minus29

minus285

minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 18The convergence in the energy norm versus ℎ (mesh size)for a sheet with an edge crack under shear

XFEM R = 050

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 19 The convergence in the energy norm of XFEM versus ℎ(mesh size) for a sheet with an edge crack under shear

In particular for linear elastic problems 120590ℎ = D120598ℎ iscalculated on the level of the smoothing cell

43 Crack Growth and Stress Intensity Factor Fractureparameters such as mode 119868 and mode 119868119868 stress intensityfactors (SIFs) are determined using the domain form [29 30]of the interaction integral [31] All the finite elements withina radius of 119903

119889= 119903119896ℎ119890from the crack-tip are used Herein ℎ

119890

is the crack tip element size and 119903119896is a scalar

In this paper crack growth is governed by the maxi-mum hoop stress criterion [32 33] which assumes that thecrack will propagate from its tip in the direction 120579

119888 where

8 Journal of Applied Mathematics

ES-Phantom R = 056

XFEM-T3(0t)R = 043

minus08 minus07 minus06 minus05 minus04 minus03minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Phantom-T3R = 050

Figure 20 The convergence in the stress intensity factor 119870119868versus

ℎ (mesh size) for a sheet with an edge crack under shear

XFEM R = 058

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 21The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for a sheet with an edge crack under shear

the circumferential (hoop) stress 120590120579120579is maximum The angle

of crack propagation satisfies the following equation

119870119868sin (120579119888) + 119870119868119868(3 cos (120579

119888) minus 1) = 0 (18)

Solving this equation we have

120579119888= 2 arctan[[

[

minus2 (119870119868119868119870119868)

1 + radic1 + 8(119870119868119868119870119868)2

]]

]

(19)

Once119870119868and119870

119868119868are known (19) may be used to compute the

direction of propagation

ES-Phantom R = 043

Phantom-T3R = 024

XFEM-T3(0t)R = 046

minus08 minus07 minus06 minus05 minus04 minus03

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Figure 22 The convergence in the stress intensity factor 119870119868119868versus

ℎ (mesh size) for sheet with edge crack under shear

XFEM R = 021

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus078

minus076

minus074

minus072

minus07

minus068

log10(h)

log 1

0(e

k)

Figure 23 The convergence in the stress intensity factor 119870119868119868

ofXFEM versus ℎ (mesh size) for sheet with edge crack under shear

5 Numerical Examples

In all numerical examples we are not using near-tip enrich-ment that is only discontinuous enrichment is used Thismeans that the best convergence rate attainable is 12 in the1198671normand 1 in the119871

2norm (O(ℎ12) andO(ℎ) respectively

where ℎ is the mesh size)

51 Sheet with an Edge Crack under Uniaxial Tension Con-sider a sheet under uniaxial tension as shown in Figure 8Thedimensions of the sheet are in unit of [mm] The materialparameters are Youngrsquos modulus 119864 = 3 times 107 Pa Poissonrsquos

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Journal of Applied Mathematics 7

120591 = 10

a = 35

b = 70H

=160

Figure 16 Sheet with edge crack under shear

0 500 1000 1500 2000 2500 3000 3500 4000 450074

76

78

8

82

84

86

88

9times10minus5

Number of nodes

Stra

in en

ergy

ES-PhantomPhantom-T3

XFEM-T3(0t)Ref sol

Figure 17 Strain energy for a sheet with an edge crack under shear

with 119881 = u | u isin 1198671(Ω Γ119888) u = u on Γ

119906 u discontinuous

on Γ119888 and 119881

0= 120575u | 120575u isin 119867

1

(Ω Γ119888) 120575u = 0 on Γ

119906 120575u

discontinuous on Γ119888

Substituting the trial and test functions into (16) wefinally obtain the familiar equation

Kd = f (17)

where f is the nodal force vector that is identical to that in thestandard phantomnode The edge-based smoothed stiffnessmatrix K for all subcells follows (15)

The smoothed stress 120590ℎ is obtained in the same way fromthe as 120598ℎ in FEM which is constant over a smoothing cell

ES-Phantom R = 061

Phantom-T3R = 055

XFEM-T3(0t)R = 049

minus08 minus07 minus06 minus05 minus04 minus03minus295

minus29

minus285

minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 18The convergence in the energy norm versus ℎ (mesh size)for a sheet with an edge crack under shear

XFEM R = 050

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus28

minus275

minus27

minus265

minus26

minus255

minus25

minus245

log 1

0(e

e)

log10(h)

Figure 19 The convergence in the energy norm of XFEM versus ℎ(mesh size) for a sheet with an edge crack under shear

In particular for linear elastic problems 120590ℎ = D120598ℎ iscalculated on the level of the smoothing cell

43 Crack Growth and Stress Intensity Factor Fractureparameters such as mode 119868 and mode 119868119868 stress intensityfactors (SIFs) are determined using the domain form [29 30]of the interaction integral [31] All the finite elements withina radius of 119903

119889= 119903119896ℎ119890from the crack-tip are used Herein ℎ

119890

is the crack tip element size and 119903119896is a scalar

In this paper crack growth is governed by the maxi-mum hoop stress criterion [32 33] which assumes that thecrack will propagate from its tip in the direction 120579

119888 where

8 Journal of Applied Mathematics

ES-Phantom R = 056

XFEM-T3(0t)R = 043

minus08 minus07 minus06 minus05 minus04 minus03minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Phantom-T3R = 050

Figure 20 The convergence in the stress intensity factor 119870119868versus

ℎ (mesh size) for a sheet with an edge crack under shear

XFEM R = 058

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 21The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for a sheet with an edge crack under shear

the circumferential (hoop) stress 120590120579120579is maximum The angle

of crack propagation satisfies the following equation

119870119868sin (120579119888) + 119870119868119868(3 cos (120579

119888) minus 1) = 0 (18)

Solving this equation we have

120579119888= 2 arctan[[

[

minus2 (119870119868119868119870119868)

1 + radic1 + 8(119870119868119868119870119868)2

]]

]

(19)

Once119870119868and119870

119868119868are known (19) may be used to compute the

direction of propagation

ES-Phantom R = 043

Phantom-T3R = 024

XFEM-T3(0t)R = 046

minus08 minus07 minus06 minus05 minus04 minus03

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Figure 22 The convergence in the stress intensity factor 119870119868119868versus

ℎ (mesh size) for sheet with edge crack under shear

XFEM R = 021

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus078

minus076

minus074

minus072

minus07

minus068

log10(h)

log 1

0(e

k)

Figure 23 The convergence in the stress intensity factor 119870119868119868

ofXFEM versus ℎ (mesh size) for sheet with edge crack under shear

5 Numerical Examples

In all numerical examples we are not using near-tip enrich-ment that is only discontinuous enrichment is used Thismeans that the best convergence rate attainable is 12 in the1198671normand 1 in the119871

2norm (O(ℎ12) andO(ℎ) respectively

where ℎ is the mesh size)

51 Sheet with an Edge Crack under Uniaxial Tension Con-sider a sheet under uniaxial tension as shown in Figure 8Thedimensions of the sheet are in unit of [mm] The materialparameters are Youngrsquos modulus 119864 = 3 times 107 Pa Poissonrsquos

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

8 Journal of Applied Mathematics

ES-Phantom R = 056

XFEM-T3(0t)R = 043

minus08 minus07 minus06 minus05 minus04 minus03minus1

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Phantom-T3R = 050

Figure 20 The convergence in the stress intensity factor 119870119868versus

ℎ (mesh size) for a sheet with an edge crack under shear

XFEM R = 058

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus075

minus07

minus065

minus06

minus055

minus05

minus045

log10(h)

log 1

0(e

k)

Figure 21The convergence in the stress intensity factor119870119868of XFEM

versus ℎ (mesh size) for a sheet with an edge crack under shear

the circumferential (hoop) stress 120590120579120579is maximum The angle

of crack propagation satisfies the following equation

119870119868sin (120579119888) + 119870119868119868(3 cos (120579

119888) minus 1) = 0 (18)

Solving this equation we have

120579119888= 2 arctan[[

[

minus2 (119870119868119868119870119868)

1 + radic1 + 8(119870119868119868119870119868)2

]]

]

(19)

Once119870119868and119870

119868119868are known (19) may be used to compute the

direction of propagation

ES-Phantom R = 043

Phantom-T3R = 024

XFEM-T3(0t)R = 046

minus08 minus07 minus06 minus05 minus04 minus03

minus09

minus08

minus07

minus06

minus05

minus04

minus03

log10(h)

log 1

0(e

k)

Figure 22 The convergence in the stress intensity factor 119870119868119868versus

ℎ (mesh size) for sheet with edge crack under shear

XFEM R = 021

minus08 minus075 minus07 minus065 minus06 minus055 minus05 minus045 minus04 minus035 minus03minus08

minus078

minus076

minus074

minus072

minus07

minus068

log10(h)

log 1

0(e

k)

Figure 23 The convergence in the stress intensity factor 119870119868119868

ofXFEM versus ℎ (mesh size) for sheet with edge crack under shear

5 Numerical Examples

In all numerical examples we are not using near-tip enrich-ment that is only discontinuous enrichment is used Thismeans that the best convergence rate attainable is 12 in the1198671normand 1 in the119871

2norm (O(ℎ12) andO(ℎ) respectively

where ℎ is the mesh size)

51 Sheet with an Edge Crack under Uniaxial Tension Con-sider a sheet under uniaxial tension as shown in Figure 8Thedimensions of the sheet are in unit of [mm] The materialparameters are Youngrsquos modulus 119864 = 3 times 107 Pa Poissonrsquos

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Journal of Applied Mathematics 9

P = 1

a = 2 H=2

b = 6

P = 1

Figure 24 Double cantilever beam with an edge crack

ratio ] = 03 The plane strain condition is assumed Thereference mode 119868 SIF is given by

119870exact119868

= 119865(119886

119887) 120590radic119886120587 = 16118Paradicmm (20)

where 119886 = 05 is the crack length 119887 is the sheet width and119865(119886119887) is given by

119865(119886

119887) = 112 minus 0231 (

119886

119887) + 1055 (

119886

119887)

2

minus 2172 (119886

119887)

3

+ 3039 (119886

119887)

4

(21)

The strain energy and the error in the energy norm aredefined as

119864(Ω)

=1

2intΩ

120598119879D120598dΩ

119890119890=

1003816100381610038161003816100381610038161003816100381610038161003816

119864num(Ω)

minus 119864ref(Ω)

119864ref(Ω)

1003816100381610038161003816100381610038161003816100381610038161003816

12

119890119896=

1003816100381610038161003816100381610038161003816100381610038161003816

119870numsif minus 119870

refsif

119870refsif

1003816100381610038161003816100381610038161003816100381610038161003816

12

times 100 sif = 119868 119868119868

(22)

where the superscript ldquoref rdquo denotes the exact or referencesolution and ldquonumrdquo denotes the numerical solution

The results of the ES-Phantom node are compared withthose of the standard Phantom-node using triangularmeshesand the XFEM-T3(0t) (the ldquostandardrdquo XFEM formulationwithout tip enrichment that only employs the Heavisideenrichment of (2)) Figure 9 shows that the strain energyof the ES-Phantom node method is more accurate thanboth the original Phantom-node and the XFEM-T3(0t) Theconvergence rates in terms of the strain energy and thestress intensity factor 119870

119868are depicted in Figures 10 and 12

respectively Furthermore Phantom-node using triangularmeshes (Phantom-T3) is superior to XFEM-T3(0t) althoughthey are equivalent to each other We also have includedtwo more figures comparing the phantom-node method to

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23minus015

minus01

0

005

01

015Domain discretized with T3 elements

minus005

(b)

Figure 25 (a) Deformed shape of the double cantilever beam(structured mesh) and (b) crack path simulated by ES-Phantomnode method (structured mesh) after ten-step growing in which thefilled circles are the new crack tip after each step

the tip-enriched XFEM in Figures 11 and 13 as a referencealthough it would not be fair to compare a method thatincludes the asymptotic crack tip enrichment to a methodthat models the crack in a much simpler way

Note that the proposed method leads to a similar con-vergence rate to the standard XFEM and standard phantom-node which is close to optimal (12) given the lack of tipenrichment Also note that the error level of the proposedmethod is a fifth of an order of magnitude lower than themethod compared with

The computational efficiency in terms of the error in theenergy norm and the relative error of119870

119868versus computation

time (119904) is compared for the ES-Phantom the standardPhantom and the XFEM-T3(0t) The results are plotted inFigures 14 and 15 respectively It is clear that the presentmethod always produces higher computational efficiencythat is accuracy to computational time ratio compared tothe other methods The accuracy of the present method is

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

10 Journal of Applied Mathematics

0 1 2 3 4 5 6

minus2

minus15

minus1

minus05

0

05

1

15

2Numerical deformed configuration

(a)

195 2 205 21 215 22 225 23

minus015

minus01

minus005

0

005

01

015

(b)

Figure 26 (a) Deformed shape of the double cantilever beam(unstructured mesh) and (b) Crack path simulated by ES-Phantomnode method (unstructured mesh) after ten-step growing in whichthe filled circles are the new crack tip after each step

approximate (1) (ES-Phantom10minus188)(Phantom10minus193) =112 times as much as that of the standard Phantom(ES-Phantom10minus188)(XFEM-T3(0t)10minus201) = 134 timesof the XFEM-T3(0t) in term of error in energy norm(2) (ES-Phantom10minus048)(Phantom10minus058) = 126 timesas much as that of the standard Phantom and (ES-Phantom10minus048)(XFEM-T3(0t)10minus058) = 162 times ofthe XFEM-T3(0t) in term of relative error for119870

119868

52 Sheet with Edge-Crack under Shear In this example weconsider the edge crack geometry subjected to a shear loadas shown in Figure 16 The material parameters are Youngrsquosmodulus119864 = 3times107 Pa and Poissonrsquos ratio ] = 025The exactstress intensity factors for this load case are given by [31]

119870119868= 340Paradicmm 119870

119868119868= 455Paradicmm (23)

The results shown in Figures 16 17 18 19 20 21 and 22show that ES-Phantom node results are more accurate than

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

10

20

30

40

50

(b)

Figure 27 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(structured mesh) after the crack propagates

both those of the standard Phantom-node and the XFEM-T3(0t) ES-Phantom node maintains slight superconvergentsolutions in the strain energy Furthermore Phantom-nodeusing triangular meshes (Phantom-T3) is superior to XFEM-T3(0t) although they are equivalent to each other with respectto the convergence in energy norm and the stress intensityfactor119870

119868 Figures 19 21 and 23 again are shown as a reference

for readers

53 Crack Growth Simulation in a Double Cantilever BeamIn this section the ES-Phantom node with structured andunstructured meshes is used for crack grow simulation Thedimensions of the double cantilever beam Figure 24 are6mm times 2mm and an initial pre-crack with length 119886 =

2mm is considered Plane stress conditions are assumed withYoungrsquos modulus 119864 = 100MPa as well as the Poisson ratio] = 3 and the load119875 is taken to be unity By symmetry a crackon the mid-plane of the cantilever beam is dominated bypure mode 119868 and the crack would propagate in a straight line

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Journal of Applied Mathematics 11

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

minus10

0

10

20

30

40

50

(a)

1 2 3 4 5

minus25

minus2

minus15

minus1

minus05

0

05

1

15

2

25

0

5

10

15

20

25

30

35

40

45

(b)

Figure 28 Stress (a) 120590119909119909

and (b) 120590119910119910

contours in the sheet(unstructured mesh) after the crack propagates

We also have included simulations with a ldquodistortedrdquo meshand showed that the crack path does not change

The crack growth increment Δ119886 is taken so that thetip is always located at an elementrsquos edge and the crackgrowth is simulated for 10 steps The domain is discretizedwith a structured and unstructured mesh of 2730 nodesThe crack path is simulated using both the proposed ES-Phantom node method and XFEM-T3(0t) and Figures 25and 26 show the deformed shape of the double cantileverbeam with the magnification factor of 25 times 104 used to enablea clear description and the evolution of the crack path Theresult shows that the crack path for an initial angle 120579

119888= 0

agrees with the published results [1]

6 Conclusions

A numerical Phantom-node method for analysis of two-linear elastic fracture problems was developed in frameworkof the ES-FEM to create the novel ES-Phantom nodemethod

In this method a cracked element is replaced by two super-imposed elements and a set of additional phantomnodesThetwo first examples were performed to investigate convergencerate in terms of strain energy and stress intensity factorsThe results have shown that the ES-Phantom node is ableto produce superconvergent solutions Meanwhile the lastexample has demonstrated the capability of the method todeal with the growing crack

Future applications of this method may deal with theinteractions among a large number of cracks in linear elasticsolids with the purpose of obtaining the higher accuracy andefficiency in solving complicated crack interactions as shownin [34] This will be studied in the future

Acknowledgments

The authors gratefully acknowledge the support by theDeutscher Akademischer Austausch Dienst (DAAD)X Zhuang acknowledges the supports from the NSFC(51109162) the National Basic Research Program of China(973 Program 2011CB013800) and the Shanghai PujiangProgram (12PJ1409100)

References

[1] 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

[2] 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

[3] 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

[4] 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

[5] P M A Areias and T Belytschko ldquoA comment on the articleldquoA finite element method for simulation of strong and weakdiscontinuities in solid mechanicsrdquordquo Computer Methods inApplied Mechanics and Engineering vol 195 no 9ndash12 pp 1275ndash1276 2006

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

[7] M Duflot ldquoIndustrial applications of XFEM for 3D crackpropagation with MorfeoCrack and Abaqusrdquo in ECCOMASThematic Conference on XFEM Cardiff UK June 2011

[8] 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

[9] T Menouillard J Rethore N Moes A Combescure and HBung ldquoMass lumping strategies for X-FEM explicit dynamics

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

12 Journal of Applied Mathematics

application to crack propagationrdquo International Journal forNumerical Methods in Engineering vol 74 no 3 pp 447ndash4742008

[10] 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 242ndash246 2012

[11] 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

[12] 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

[13] 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

[14] 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

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

[16] 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

[17] G R Liu K Y Dai and T T Nguyen ldquoA smoothed finiteelement method for mechanics problemsrdquo ComputationalMechanics vol 39 no 6 pp 859ndash877 2007

[18] G R Liu T T Nguyen K Y Dai and K Y Lam ldquoTheoreticalaspects of the smoothed finite element method (SFEM)rdquo Inter-national Journal for Numerical Methods in Engineering vol 71no 8 pp 902ndash930 2007

[19] N Nguyen-Thanh T Rabczuk H Nguyen-Xuan and S P ABordas ldquoA smoothed finite element method for shell analysisrdquoComputer Methods in Applied Mechanics and Engineering vol198 no 2 pp 165ndash177 2008

[20] S P A Bordas and S Natarajan ldquoOn the approximation in thesmoothed finite elementmethod (SFEM)rdquo International Journalfor Numerical Methods in Engineering vol 81 no 5 pp 660ndash670 2010

[21] T Rabczuk PM A Areias and T Belytschko ldquoAmeshfree thinshell method for non-linear dynamic fracturerdquo InternationalJournal for Numerical Methods in Engineering vol 72 no 5 pp524ndash548 2007

[22] T Rabczuk and T Belytschko ldquoApplication of particle methodsto static fracture of reinforced concrete structuresrdquo Interna-tional Journal of Fracture vol 137 no 1-4 pp 19ndash49 2006

[23] 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

[24] S Bordas P V Nguyen C Dunant A Guidoum and HNguyen-Dang ldquoAn extended finite element libraryrdquo Interna-tional Journal for Numerical Methods in Engineering vol 71 no6 pp 703ndash732 2007

[25] G R Liu T Nguyen-Thoi and K Y Lam ldquoAn edge-basedsmoothed finite element method (ES-FEM) for static free

and forced vibration analyses of solidsrdquo Journal of Sound andVibration vol 320 no 4-5 pp 1100ndash1130 2009

[26] G R Liu ldquoAG space theory and aweakenedweak (W2) form fora unified formulation of compatible and incompatible methodspart I theoryrdquo International Journal for Numerical Methods inEngineering vol 81 no 9 pp 1093ndash1126 2010

[27] L Chen T Rabczuk S P A Bordas G R Liu K Y Zeng andP Kerfriden ldquoExtended finite element method with edge-basedstrain smoothing (ESm-XFEM) for linear elastic crack growthrdquoComputer Methods in Applied Mechanics and Engineering vol209ndash212 pp 250ndash265 2012

[28] N Vu-Bac H Nguyen-Xuan L Chen et al ldquoA node-basedsmoothed extended finite element method (NS-XFEM) forfracture analysisrdquo CMES Computer Modeling in Engineeringand Sciences vol 73 no 4 pp 331ndash355 2011

[29] F Z Li C F Shih and A Needleman ldquoA comparison of meth-ods for calculating energy release ratesrdquo Engineering FractureMechanics vol 21 no 2 pp 405ndash421 1985

[30] B Moran and C F Shih ldquoCrack tip and associated domainintegrals from momentum and energy balancerdquo EngineeringFracture Mechanics vol 27 no 6 pp 615ndash642 1987

[31] S S Wang J F Yau and H T Corten ldquoA mixed-mode crackanalysis of rectilinear anisotropic solids using conservation lawsof elasticityrdquo International Journal of Fracture vol 16 no 3 pp247ndash259 1980

[32] F Erdogan and G Sih ldquoOn the crack extension in sheets underplane loading and transverse shearrdquo Journal Basic Engineeringvol 85 no 6 pp 519ndash527 1963

[33] A Menk and S Bordas ldquoCrack growth calculations in solderjoints based on microstructural phenomena with X-FEMrdquoComputational Materials Science vol 50 no 3 pp 1145ndash11562011

[34] D F Li C F Li S Q Shu Z X Wang and J Lu ldquoA fastand accurate analysis of the interacting cracks in linear elasticsolidsrdquo International Journal of Fracture vol 151 pp 169ndash1852008

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal

Impact Factor 173028 Days Fast Track Peer ReviewAll Subject Areas of ScienceSubmit at httpwwwtswjcom

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawi Publishing Corporation httpwwwhindawicom Volume 2013

The Scientific World Journal