DSpace at VNU: Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method

Several mesh-based numerical methods have been introduced to alleviate or overcome such difficulties in modeling crack growth problems, among which the extendedfinite element method XFEM[1

Crack growth modeling in elastic solids by the extended meshfree

Galerkin radial point interpolation method

Nha Thanh Nguyena, Tinh Quoc Buib,n,1, Chuanzeng Zhangb, Thien Tich Truonga

a

Department of Engineering Mechanics, Ho Chi Minh City University of Technology, Viet Nam

b Department of Civil Engineering, University of Siegen, Paul-Bonatz-Str 9-11, 57076 Siegen, Germany

a r t i c l e i n f o

Article history:

Received 18 November 2013

Received in revised form

1 March 2014

Accepted 25 April 2014

Keywords:

Extended meshfree method

Radial point interpolation method

Enrichment techniques

Crack propagation

Stress intensity factors

Fracture mechanics

a b s t r a c t

We present a new approach based on local partition of unity extended meshfree Galerkin method for modeling quasi-static crack growth in two-dimensional (2D) elastic solids The approach utilizing the local partition of unity as a priori knowledge on the solutions of the boundary value problems that can

be added into the approximation spaces of the numerical solutions It thus allows for extending the standard basis functions by enriching the asymptotic near crack-tipfields to accurately capture the singularities at crack-tips, and using a jump step function for the displacement discontinuity along the crack-faces The radial point interpolation method is used here for generating the shape functions The representation of the crack topology is treated by the aid of the vector level set technique, which handles only the nodal data to describe the crack We employ the domain-form of the interaction integral in conjunction with the asymptotic near crack-tipfield to extract the fracture parameters, while crack growth is controlled by utilizing the maximum circumferential stress criterion for the determina-tion of its propagating direcdetermina-tion The proposed method is accurate and efficient in modeling crack growths, which is demonstrated by several numerical examples with mixed-mode crack propagation and complex configurations

& 2014 Elsevier Ltd All rights reserved

1 Introduction

Advanced numerical methods have proved to be a useful tool in

modeling and simulating a wide range of engineering problems

Thefinite element method (FEM) has shown among others to be

very well suited for the modeling of fracture mechanics problems

However, it turns out that the FEM is difficult and cumbersome in

modeling the evolution of the discontinuous entities (e.g., crack

growth) The modification of the mesh topology during the

propagation of the crack is one of its big disadvantages Several

mesh-based numerical methods have been introduced to alleviate

or overcome such difficulties in modeling crack growth problems,

among which the extendedfinite element method (XFEM)[1]and

the boundary element methods (BEM) [2] are popular In the

contrary to the mesh-based approaches, meshfree methods have

been alternatively introduced and developed during the last two

decades, and successfully applied to a variety of engineering

problems including large deformation, crack propagation, high

gradient, damage, and so on, e.g., see[3–7]

An evident distinction between the based and the mesh-free methods is their discretization and approximation approaches Instead of working with elements or meshes in the mesh-based methods, a set of scattered nodes is used in the meshfree methods

to approximate thefield variables In recent years, different versions

of meshless methods have been developed including the smoothed particle hydrodynamics method (SPH)[8], the element free Galerkin method (EFG)[9,5–7], the meshless local Petrov–Galerkin method (MLPG)[10], the radial point interpolation method (RPIM)[11,4], the moving Kriging interpolation method (MK) [3], the reproducing kernel particle method (RKPM)[12], and many others

The enrichment techniques are integrated into the approxima-tion spaces in the meshfree methods, e.g., see[5,6], to accurately describe the discontinuities and the singularfield at the crack-tips

On the other hand, the vector level set method is also used as a useful tool in representing the crack geometry [6] Most of the previous works were based on the moving least square appro-ximation (MLS) shape functions, which do not satisfy the Kronecker-delta property, and thus, additional special techniques for treating the essential boundary conditions are required The RPIM shape functions as presented in[11], however, possess the Kronecner-delta function property automatically and hence can eliminate the need of additional techniques Apart from other applications of the RPIM, a recent development of the RPIM for extracting the crack-tip parameters of stationary cracks has been

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journal homepage:www.elsevier.com/locate/enganabound

Engineering Analysis with Boundary Elements

http://dx.doi.org/10.1016/j.enganabound.2014.04.021

0955-7997/& 2014 Elsevier Ltd All rights reserved.

n Corresponding author Tel.: þ 49 2717402836; fax: þ 49 2717404074.

E-mail address: tinh.buiquoc@gmail.com (T.Q Bui).

1

Current address: Department of Mechanical and Environmental Informatics,

Graduate School of Information Science and Engineering, Tokyo Institute of

Technology, 2-12-1-W8-22, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan.

reported in [13,14] The main idea of [13,14] is to construct

enriched shape functions that can capture the singularity at the

crack-tip The method described in[13,14], however, is completely

different from that presented in this manuscript, and it is generally

limited, especially in modeling the crack propagation problems

The differences between the proposed method and the one

presented in[13,14]will be discussed in the following sections

In the present work, we present a partition of unity extended

meshfree Galerkin approach based on the radial point interpolation

method in conjunction with the vector level set method for modeling

the crack growth problems For the abbreviation purpose, the

method is named as X-RPIM Other than the enriched shape

functions in[13,14], the crack approximation is efficiently enriched

by using the Heaviside step function for the displacement

disconti-nuities along the crack-faces and the Westergard's solution near the

crack-tips This approach is known as an extrinsic enrichment

meshfree method and only the nodes surrounding the crack are

taken into account As a consequence this versatile X-RPIM is thus

suitable for crack growth simulation and has not been reported in the

literature yet The new approach utilizes not only the advantages of

the RPIM shape functions[4,11], but also the versatility of the vector

level set method[6,15] It should be noted here that the RPIM shape

functions possess the Kronecker-delta property, and thus completely

overcome the difficulty in imposing the essential boundary

condi-tions in most other existing meshfree methods, e.g., the MLS based

methods Additionally, as compared with lower-orderfinite elements

that are commonly applied and can capture only the linear

crack-openings, the meshfree methods however have the great advantage

to capture more realistic crack-openings due to the higher-order

continuity and non-local interpolation character[16]

In this contribution, the extrinsic enrichment approximation

with the fourfold enrichment functions is used The crack is

modeled by the crack-tip positions and a vector level set function

Only a narrow band surrounding the crack is to be considered

instead of the whole domain of the problem Since the vector

level set is based on the geometrical operations and the nodal

values at scattered nodes, it is completely independent of the

discretization of the problem domain and no visible cracks needs

to be defined For computing and extracting the fracture

para-meters, we adopt the domain-form of the interaction integral in

conjunction with the asymptotic crack-tip field For the

quasi-static crack propagation modeling, the crack growing direction is

determined from the maximum hoop-stress criterion The

accu-racy of the X-RPIM is demonstrated by a number of numerical

examples with single and mixed-mode cracks Quite complicated

configurations of the structures are considered The obtained

numerical results of the stress intensity factors and the crack

paths are compared with reference solutions available in the

literature Nevertheless, it should be stressed here that since our

main attention is to focus on modeling the propagation of cracks

and its accuracy, hence other topics such as the convergence and

error estimation will not be covered in this work They are

remaining our future research studies desirably with a

compre-hensive study on those mentioned issues

The outline of the manuscript is structured as follows The next

section presents the X-RPIM formulation for quasi-static crack

growth problems in elastic solids in which the shape functions and

their properties, the extended meshfree approximation, the vector

level set method with updating, the weak-form and discrete

equations, and the numerical implementation procedure are

described The crack growth criterion and the computation of

the stress intensity factors integrated into the method are briefly

introduced in Section 3 In Section 4, two numerical examples

involving single and mixed-mode cracks are investigated to

illus-trate the accuracy of the proposed method for evaluating the

fracture parameters Two numerical examples with complex

geometrical configurations for crack growth modeling are pre-sented and discussed in details in Section 5 Some conclusions drawn from the proposed method are reported in the last section

2 X-RPIM formulation for crack growth problems 2.1 Construction of the RPIM shape functions

Different from the FEM, meshfree shape functions rely only on the scattered nodes without the need for afinite element mesh The approximation of the distribution functions uðxiÞ within a sub-domainΩxDΩ can be performed based on all nodal values at xi, where i ¼ 1; …; n and n is the total number of nodes in the sub-domain The well-known RPIM interpolation uhðxiÞ; 8x AΩx is

defined as[4,11]

uhðxÞ ¼ ∑n

i ¼ 1

RiðrÞaiþ ∑m

j ¼ 1

pjðxÞbj¼ RT

a þ PTb ¼ ½ RT

PT a b



ð1Þ

where u ¼ uðx 1Þ uðx2Þ … uðxnÞT

is the vector of the nodal displacements; RiðrÞ is the radial basis functions (RBFs); pjðxÞ is the monomial in the 2D space coordinates xT¼ x; y½ , j ¼ 1; …; m where m is the number of polynomial basis functions The constants ai and bj are determined to construct the shape func-tions In this study, the thin plate spline function RiðrÞ ¼ rη

i, with

ri¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxxiÞ2þðyyiÞ2

q

and the shape parameterη ¼ 4:01, is used for constructing the RPIM shape functions unless otherwise stated

By enforcing uhðxÞ into Eq (1) to pass through all the nodal values at n nodes surrounding the point of interest x, a system of n linear algebraic equations is then obtained, one for each node, which can be written in the matrix form as

Usd¼ u 1 u2 … unT

where the moment matrix of the RBFs R0 and the polynomial moment matrix Pmare given by

R0¼

R1ðr1Þ R2ðr1Þ … Rnðr1Þ

R1ðr2Þ R2ðr2Þ … Rnðr2Þ

R1ðrnÞ R2ðrnÞ … RnðrnÞ

2 6 6 4

3 7 7 5

ðnnÞ

ð3Þ

PTm¼

pmðx1Þ … pmðxnÞ

2 6 6 6 4

3 7 7 7 5

ðmnÞ

ð4Þ

The vector of the coefficients for the RBFs and the vector of the coefficients for the polynomial are defined by

aT¼ a 1 a2 … anT

ð5Þ

bT¼ b 1 b2 … bmT

ð6Þ There are n þ m variables in Eq (2), so the following m constraint conditions are used as additional equations

∑n

i ¼ 1

pjðxiÞai¼ PT

Combining Eqs.(2)and(7)yields the following matrix form:

Usd¼ Usd

0

PTm 0

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

G0

a b



N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

The vector of the coefficients a0 can be obtained by the

following relation:

a0¼ a

b



¼ G 1

and substituting Eq.(9)into Eq.(1), we obtain

uhðxÞ ¼ Rh T PTi

G0 1Usd¼ ΦT

in which, the RPIM shape functions can be expressed as

ΦTðxÞ ¼ RT

PT

G0 1

¼ ϕh 1ðxÞ ϕ2ðxÞ … ϕnðxÞ ϕn þ 1ðxÞ … ϕn þ mðxÞi

ð11Þ The RPIM shape functions corresponding to the nodal

displace-ments can be written as

ΦTðxÞ ¼ ϕh 1ðxÞ ϕ2ðxÞ … ϕnðxÞi

ð12Þ Finally, Eq.(10)can be rewritten for the nodal displacements as

uhðxÞ ¼ ΦTðxÞUsd¼ ∑n

It must be noted that the RPIM shape functions possess the

Kronecker-delta function property regardless of the particular

form of the RBFs used As a result, no special techniques for

imposing the essential boundary conditions are required Another

key factor in the meshfree methods is the influence domain, which

is used to determine the number of field nodes within the

interpolation domain of interest Often, the following relation is

taken to determine the size of the support domain

with dc being the mean distance of the scattered node and αc

representing the scaling factor

2.2 Properties of the RPIM shape functions

The meshfree RPIM shape functions in general depend

uniquely on the distribution of the discretized nodes regardless

of the determined particular forms of the RBFs For the sake of

completeness, some specific properties of the RPIM shape

func-tions are briefly summarized as follows More details can be found

in[11,4]for instance

– The RPIM shape functions satisfy the Kronecker-delta function

properties

– The RPIM shape functions are of unity partition

∑n

– Reproducing properties

∑n

– Similar derivative of the RPIM shape functions

– Local compact support because the point interpolation is

carried out in an influence domain and each influence domain

is localized The system matrix obtained is thus sparse and

banded

2.3 Extended meshfree approximation with vector level set According to[6,15], the crack to be represented by the vector level set approach is modeled by the crack-tip position and a vector level set function The vector level set function is formed by the signed distance function, given by the closest point projection

to the crack-face and its gradient This function is then evaluated at points between the nodes by the vector extrapolation In practice, however, a narrow band surrounding the crack is considered for those functions to efficiently enhance the performance of the method instead of the whole problem domain An important advantage as compared with the classical level set approach is that only the nodal values for the level set needs to be updated during the crack growth process by geometrical operations on the data, and no evolution equation is introduced explicitly

The fundamental idea in capturing the crack is to enrich the approximation function Eq.(13)in terms of the signed distance function f and the distance from the crack-tip The approximation

is thus continuous in the whole problem domain but discontin-uous along the crack Finally, the displacement approximation is expressed as

uhðxÞ ¼ ∑

iA WðxÞϕiðxÞuiþ ∑

iA W b ðxÞϕiðxÞHðf ðxÞÞαiþ ∑

i A WSðxÞϕiðxÞ ∑4

j ¼ 1

ψjðxÞβij

! ð18Þ whereϕiis the RPIM shape functions while f ðxÞ denotes the signed distance from the crack line Again, here it is evident to see that our meshfree approximations as given by Eq (18) is completely different from that in[13,14]

In Eq.(18), the jump enrichment functions Hðf ðxÞÞ is defined as

[6,15]

Hðf ðxÞÞ ¼ þ1 if f ðxÞ40

1 if f ðxÞo0

(

ð19Þ

while the vector of the fourfold enrichment functionsψjðxÞ (j¼1, 2,

3, 4) is given by ψðxÞ ¼ pffiffiffir

sinφ

2;pffiffiffir cosφ

2;pffiffiffir sinφ

2 sinφ;pffiffiffir

cos φ

2sinφ

ð20Þ with r being the distance from the crack-tip xTIPto x, andφ being the angle between the tangent to the crack-line and the segment

x  xTIP as depicted in Fig 1 In Eq (18), Wb denotes the set of nodes whose support contains the point x and is bisected by the crack-line (seeFig 2, the circle on the left-hand side) and WS is the set of nodes whose support contains the point x and is split by

Fig 1 Distance r and angle φ.

Fig 2 Definition of the sets of the nodes W b and W S , respectively.

the crack-line and contains the crack-tip (seeFig 2, the circle on

the right-hand side) Also, αi and βij in Eq (18) are additional

unknowns in the variational formulation

It is interesting to note that the enrichment approach is

generally suitable for crack propagation simulation because the

vector level set is based on the geometrical operations and

the nodal values at scattered nodes, completely independent of

the discretization of the problem domain and no visible crack

needs to be defined

2.4 Updating of the signed distance functions

In contrast to stationary crack problems, the domain geometry

is changing during the evolution of the crack because of the

discontinuity of crack-line Thus, updating the nodal values of the

signed distance functions to perform the enrichment in each

calculation step is required A detailed description of the vector

level set method for crack growth modeling can be found in[6,15]

Basically, the two nodal sets of Wband WS in Eq (18) must be

refined and their data must be updated appropriately during the

propagation of the crack

Let Sn be the set of nodes for which the signed distance

function f is defined at the current step n while Sn  1

be the set

at the previous step n 1 Let Sabe the set of nodes for which the

signed distance is updated at the current step The two vectors tn

and tn  1respectively are the crack-tip advance vectors at the step

n and step n  1

The definition of the nodal sets is depicted inFig 3 The set

Sn  1is denoted by the area behind the segment 1 and the signed

distance values from these nodes to the crack-line are determined

in the previous step and keep unchanged during the whole

process if the previous crack-face does not alter The shaded area

contains nodes that belong to the set Sa, which lie behind the

segment 2 and do not belong to the set Sn  1to ensure that each

point always has a unique projection onto the crack advance

vector Let Stip be the set of nodes whose distance from the

crack-tip is less than or equal to rf, Stip¼ x : ‖xxTIP‖rrf

, then the set Sais determined by the following three conditions[6]:

Sa¼ x AStip

: x =2Sn  1

; ðx xn TIPÞUtnr0; ðxxn  1

TIP ÞUtn  140

ð21Þ Thefirst condition means that the nodes belong to Stip

but do not belong to Sn  1, the second condition is used to pick nodes lie

behind the segment 2 and the third one ensures that these nodes

lie in front of the segment 1 Once the set Sahas been determined,

the following procedure is applied to compute the new values of

the level set function for these nodes

Let nn denotes the counterclockwise normal vector to tn and

^nn

¼ nn=‖nn‖ is its unit vector (Fig 4) The closest projection vectors of the nodes fðxIÞ belong to Sa

on to the crack-tip advance vector is computed as follows[6]:

fðxIÞ ¼ ^nn

½ðxIxn  1 TIP ÞU ^nn

 if ðxIxn  1

TIP ÞUtnZ0

ðxIxn  1 TIP Þ if ðxIxn  1

TIP ÞUtno0

(

ð22Þ and the Heaviside function value of a node xIASa

is determined by Hðf ðxIÞÞ ¼ sign½ðxIxn  1

2.5 Weak-form and discrete equations

We consider a 2D, small strain, and linear elastic problem in the domainΩ bounded by Γ, and subjected to the body force vector b

in the domain and traction t on Γt The weak-form of the equilibrium equations can be expressed as[22]

Z

Ω∇sδu :rdΩZ

ΩδuTb dΩZ

where∇s is the symmetric gradient operator andris the stress tensor for a displacement field u The discrete form can be obtained by using Eq (18) as an approximation for u and δu Finally, this leads to the linear system of algebraic equations

where d ¼ fu; α; βgT is the vector of unknown In Eq (25), the stiffness matrix K is dependent on either enriched or non-enriched nodes in the domain For non-enriched nodes, the stiffness matrix is determined as

Kij¼

Kuu

ij Kuαij Kuijβ

Kαuij Kααij Kαβij

Kβuij Kβαij Kββij

2 6 6

3 7

where

Kuu

ij ¼R

ΩðBu

iÞTDBu

j dΩ; Kuα

ij ¼R

ΩðBu

iÞTDBαj dΩ ¼ ðKαu

ji ÞT;

Kuijβ¼Z

ΩðBu

iÞT

DBβj dΩ ¼ ðKβu

jiÞT

; Kαα

ij ¼Z

ΩðBα

iÞT

DBαj dΩ;

Kαβij ¼Z

ΩðBα

iÞTDBβj dΩ ¼ ðKβα

jiÞT; Kββ

ij ¼Z

ΩðBβ

iÞTDBβj dΩ ð27Þ while for non-enriched ones

Kij¼ Kuu

The external force vector f in Eq (25) for enriched nodes is explicitly given by

fi¼ ffu

i; fα

i; fβ i1; fβ i2; fβ i3; fβ

Fig 3 Geometrical description for the set S a

Fig 4 The projection of a point x I belonging to S a onto the advance vector t n N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

whereas for non-enriched nodes

fi¼ fu

with

fui ¼

Z

Ωϕib dΩþ

Z

fαi ¼

Z

ΩϕiHb dΩþ

Z

fβij¼

Z

Ωϕiψjb dΩþ

Z

Γtϕiψjt dΓ; with ðj ¼ 14Þ ð33Þ Additionally, the displacement gradient matrices Bui; Bα

i; Bβ

i and the matrix of the elastic constants D dependent on Young's modulus E

and Poisson's ratioν are given respectively by

Bui ¼

ϕi;x 0

0 ϕi;y

ϕi;y ϕi;x

2

6

3

7

5; Bα

i ¼

ðϕiHÞ;x 0

0 ðϕiHÞ;y

ðϕiHÞ;y ðϕiHÞ;x

2 6

3 7 5;

Bβi¼

ðϕiψjÞ;x 0

0 ðϕiψjÞ;y

ðϕiψjÞ;y ðϕiψjÞ;x

2

6

3 7

ð1þνÞð12νÞ

2

6

3 7

5 ðplane strainÞ ð35Þ

To integrate the stiffness matrix and the force vector arising in

the discrete equations and enrichments, Gaussian quadrature is

used over the background elements Note also that the background

elements or cells are independent of the nodal arrangement, for

more information, e.g., see[9,16]

2.6 Numerical implementation procedure

The key steps of the numerical implementation procedure for

the X-RPIM are outlined as follows:

(1) Divide the problem domain into a set of scattered nodes and

obtain the information on node coordinates Define material

properties and loading

(2) Detect nodes and store them into different sets corresponding

to nodes used for boundary conditions, loading conditions,

and so on

(3) Set up integration cells with a set of quadrature points covering

the domain

(4) Define parameters used for the meshless shape functions such

as the coefficients of the weight function, the size and the

shape of the support domain

(5) Define the initial crack information as a line by specifying the

starting and ending points as their crack-tip nodes

(6) Loop over the number of incremental steps

a Vector level set initialization and selecting/updating

enriched nodes appropriately and store them into different

sets including Wband WS

b Loop over the quadrature points

i At each quadrature point, a support domain is defined

to collect a set of scattered nodes surrounding this point

of interest

ii Compute shape functions in Eq (18) for every node

located inside the support domain

iii Compute the stiffness matrix as defined in Eqs.(27)and

(28)

iv Compute the force vector as defined in Eqs.(31)–(33)

v Assemble the stiffness matrix and load vector into the global stiffness matrix and force vector

c End the loop over the quadrature points

d Imposing the essential boundary conditions as in the FEM

e Solve the linear system of algebraic equations to obtain the nodal displacements

f Recovery of the stress and strainfields

g Calculate the SIFs using the interaction integral method

h Compute the crack propagation angle based on the infor-mation of the computed SIFs to determine the crack growth direction

i Specify a given size of crack growth, update the new crack-line including the crack path and tips using the vector level set method as described inSection 2.4

(7) End the loop over the propagation steps

(8) Visualization of the results

3 Crack growth simulation and the SIFs implementation

In the simulation of the crack propagation problems an appro-priate criterion to detect the direction of the growing crack must

be used The crack growth direction is commonly determined based on several criteria including the maximum circumferential stress, the maximum energy release rate and the minimum strain energy density In this study we adopt the maximum hoop-stress criterion[17]to evaluate the crack growth direction Basically, the criterion states that the crack will grow from its tip in a radial direction at a critical angle, so that the maximum circumferential stress reaches a critical material strength The critical angleθc is calculated based on the mixed-mode stress intensity factor as follows[17,1]:

θc¼ 2 tan 1 1 

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 8ðKII=KIÞ2

q 4ðKII=KIÞ

0

@

1

The SIFs are computed using the domain-form of the interac-tion integral[18,19] The coordinates are assumed here to be the local tip coordinates with the x-axis parallel to the crack-faces The path-independent J-integral is expressed as

J ¼ Z

Γ

wδ1jsij∂ui

∂x1

where w ¼ 1=2ðsijεijÞ is the strain energy density and njis the jth component of the outward unit vector normal to an arbitrary contourΓ enclosing the crack-tip For general mixed-mode crack problems, the relationship between the value of the J-integral and the SIFs is

J ¼1

where ~E ¼ E is for plane-stress and ~E ¼ E=ð1ν2Þ for plane-strain problems, and KIand KIIare the two SIFs corresponding to mode-I and mode-II crack-openings, respectively

To evaluate the J-integral, the contour integral in Eq (37) is transformed into an equivalent domain-form by applying the Green's theorem associated with an arbitrary smoothing weight function q Two states of the cracked body are considered State 1 (superscript 1) is the one corresponding to the actual state and state 2 (superscript 2) is the auxiliary state which is often chosen

as the known asymptotic crack-tipfield for mode-I or mode-II The interaction integral can be then expressed as

Ið1;2Þ¼Z

A sð1Þ ij

∂uð2Þi

∂x1

þsð2Þ ij

∂uð1Þi

∂x1

Wð1;2Þδ1j

∂q

∂xj

where Wð1;2Þis the interaction strain energy determined by

Wð1;2Þ¼1ðsð1Þ

ij εð2Þ

ij þsð2Þ

ij εð1Þ

and the smoothing weight function

q ¼ 1 2 xj j

c

1 2 y 

c

ð41Þ

is chosen in this work with c being the length of the square area

for computing the interaction integral

The stress intensity factor for a mode-I crack can then be

evaluated by the interaction integral from Eq (39) with the

auxiliary mode-I crack-tipfield as follows:

KI¼~E

2I

The stress intensity factor for a mode-II crack can also be

obtained in the same way

4 Accuracy study

It is necessary to investigate the precision of the X-RPIM along

the crack-line discontinuity and in the vicinity of the crack-tip We

begin by considering two numerical examples with single and

mixed-mode cracks, in order to show the accuracy of the present

X-RPIM approach The accuracy of the method is estimated by

comparing the SIFs results calculated by the X-RPIM with respect

to the analytical solutions The effect of the scaling factor on the

SIFs is also investigated Background cells with Gaussian

quad-rature are used for evaluating the stiffness matrix and the force

vector Generally, a 4  4 quadrature is adequate but except in the

elements/cells surrounding the crack-tip where a 8  8 Gaussian

quadrature is used instead

4.1 Mode-I: an edge-cracked plate under a tensile loading Let us consider a rectangular plate with an edge-crack sub-jected to a uniformed tensile loadings¼ 1:0 as depicted inFig 5a The plane strain condition is assumed and the plate is determined

by the following configuration parameters L ¼ 2W ¼ 16 and a crack-length a The Young's modulus E ¼ 1000 and the Poisson's ratio ν ¼ 0:3 are chosen The accuracy of the computed SIFs obtained by the proposed X-RPIM is compared with the analytical solutions of this particular example, which is given by[20]

Kexact

I ¼spffiffiffiffiffiffi 1:120:23Waþ10:55 Wa 221:72 Wa 3þ30:39 Wa 4

ð34Þ First, the finite size effect on the SIFs is investigated numeri-cally using the present X-RPIM by altering the crack-length/weight ratio a=W from 0.2 to 0.6, respectively The scaling factor α ¼ 1:8 in

Eq (14) is taken Two different sets of 15  30 and 20  40 regularly scattered nodes are used, and the obtained results of the KI factor for various ratios a=W are presented inTable 1 in

0 2 4 6 8 10 12 14 16

Fig 5 (a) Schematic configuration of an edge-cracked plate subjected to a uniform tensile loading (b) Discretization of the problem and definition of enriched nodes near the crack-faces (star) and at the crack-tip (empty circle).

Table 1 Comparison of the K I factor between the analytical and the proposed X-RPIM solutions.

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comparison with the exact solution[20] The distribution of the

20  40 scattered nodes of the cracked plate is depicted inFig 5b

The percentage errors of the SIFs compared to the analytical

solution are also estimated As seen in Table 1 the KI results

derived from the X-RPIM are in good agreement with the

analy-tical solution, especially for the fine nodal set, i.e., the 20  40

nodes It is also interesting to see that the amplitude of the SIFs

increases with increasing the crack-length As a consequence the

influence of the finite size of the plate on the solutions of the

considered crack problem is evident and significant Furthermore,

the deformed shape of the cracked plate subjected to a tensile

loading obtained using a regular set of 20  40 scattered nodes is

visualized inFig 6 This deformation is reasonable as compared

with that in[13]

Next, we numerically investigate the influence of the support

domain size on the SIFs The support domain size is determined

via the scaling factor As well-known that there are no exact rules

for determining the domain size in the meshfree methods, but

most of the previous studies have found numerically that a scaling

factor in around of 1.8 is appropriate As a result we explore the

effect of the support domain size on the SIFs by considering

several specified values of the scaling factor, e.g., α ¼ 1:6; 1:7;

1:8; 1:9 and 2.0 The crack-length is chosen as a¼3.5 and a set of

scattered nodes of 20  40 is taken Two radial basis functions are

used in the X-RPIM including the thin plate splines (TPS) and the

multi-quadrics (MQ) [4,11] Additional results derived from the Element Free Galerkin with moving least square shape function (EFG-MLS) are also given for the comparison purpose All the results for the SIFs are then presented in Table 2, and a good agreement among each other can be found Our particular numer-ical experiments have found that the scaling factor should be selected in a small range as stated above, i.e., around 1.8, which could in general result in an acceptable solution Moreover, as compared with the exact solutions, the errors obtained by the MLS-EFG are slightly larger than that delivered by the proposed X-RPIM with both the TPS and the MQ functions Therefore, we simply decide to take a scaling factor of 1.8 for the rest of the numerical investigations unless stated otherwise

4.2 Mixed-mode: an edge-cracked plate under a uniform shear loading

A mixed-mode crack problem is examined in this subsection

A rectangular plate with an edge-crack subjected to a uniform shear loading as shown inFig 7is considered The geometrical and material parameters used for this example are L ¼ 16; W ¼ 7;

−2

0

2

4

6

8

10

12

14

16

18

Fig 6 The deformed shape of the cracked plate with 20  40 nodes, a ¼3.5,

enlarged by a factor of 50.

Table 2

Influence of the support domain size on the SIFs.

Fig 7 Schematic configuration of a mixed-mode edge-crack in a rectangular plate under a uniform shear loading.

Table 3 Comparison of the SIFs for an edge-cracked plate under a shear loading obtained by the FEM and the X-RPIM.

a ¼ 3:5; E ¼ 1000; ν ¼ 0:25; and τ ¼ 1 The analytical SIFs results

KI¼ 34 and KII¼ 4:55 are available in[5]for the case a/W¼ 0.5

Other reference solutions utilizing the FEM are given in [13]

Similar to the previous example, thefinite size effect on the SIFs

through a variation of the crack-length/weight ratio a=W from

0.3 to 0.5 is investigated Different sets of the scattered nodes are

considered The computed results for the SIFs for both mode-I and

mode-II SIFs for different a/W ratios are tabulated in Table 3in

comparison with the FEM The X-RPIM method has been shown to

work well for this mixed-mode crack problem As expected, the

SIFs for both crack modes calculated by the proposed X-RPIM

match well with those obtained by the FEM The exact solutions

are particularly considered for the ratio a/W¼0.5 and a good

agreement among different methods can be found as well Again,

as in the previous example the SIFs increase with increasing the

crack-length Consequently the effect of the finite size of the

cracked plate on the SIFs is significant Similarly, the deformed

shapes of the cracked plate subjected to the uniform shear loading

are shown inFig 8for two different sets of the scattered nodes,

respectively The deformations look reasonable as well

5 Numerical examples for crack growth problems

In this section, numerical examples are presented to show the

applicability and the accuracy of the developed X-RPIM method in

modeling crack growth problems with complex geometries For

this purpose, two rather complex numerical examples in 2D elastic

solids are considered The crack path is simulated numerically In

thefirst example, the crack growth path from a fillet in a structural

member considering two different types of boundary conditions is

simulated, and the second example shows the crack growth

modeling in a perforated panel with a circular hole

5.1 Crack growth from afillet

This crack growth problem from a fillet was designed and

studied experimentally in[21]to investigate the influence of the

thickness of the lower I-beam on the crack growth This example

has been analyzed previously by several authors using the MLS

meshfree methods [5,6] Detailed information on the original

model can be found in[21]but here we only simulate a simplified model as stated in[5,6] The specimen to be modeled is depicted

inFig 9representing two types of the boundary conditions In the first type (type-1) boundary conditions, as shown inFig 9a, the displacements along the entire bottom edge are fully constrained, whereas only both ends of the bottom one arefixed for the second type (type-2) boundary conditions as depicted in Fig 9b To prevent rigid body translation in the horizontal direction, an additional degree of freedom isfixed at the right end corner for both types of the boundary conditions The problems are con-sidered in the plane strain condition and linear elasticity with Young's modulus E ¼ 2  1011Pa and Poisson's ratioν ¼ 0:3 The initial crack-length is set to be a ¼ 5 mm and the applied load is

0 2 4 6 8 10 12 14 16 18 20

0 2 4 6 8 10 12 14 16 18 20

Fig 8 Distributed nodes and deformed shape of the cracked plate subjected to a uniform shear loading (a) 10  20 and (b) 20  40 nodes enlarged by 10 times.

Fig 9 Schematic configuration of a fillet with a crack Type-1 (a) and type-2 (b) boundary conditions.

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taken as P ¼ 1 N The specimen for both types of loading is

discretized using the same set of 1115 irregularly scattered nodes

as depicted inFig 10orFig 11 The problem is solved

incremen-tally by increasing the crack sizeΔa ¼ 5 mm in each step In all

numerical simulations, a total number of 14 steps are performed

Figs 10 and 11visualize the crack growth paths from thefillet

for the type-1 and type-2 boundary conditions, respectively It is

emphasized again that the same set of scattered nodes is used for

both types of the boundary conditions A close-up view to the

crack paths for both types of the boundary conditions at the

vicinity of thefillet is additionally depicted inFig 12 The results

shown are consistent with both the experimental [21] and the

previous numerical predictions using the meshfree methods, see

Figs 20 and 21 in[5], and Figs 19 and 20 in[6] As observed in[5]

and nonetheless again found here in this study that the crack

growth curves for the type-2 boundary condition sharply

down-ward and propagates todown-ward the bottom of the structure (see

Figs 11 and 12) In contrast, the crack grows almost directly

toward the oppositefillet for the type-1 boundary condition (see

Figs 10–12)

5.2 Crack growth in a perforated panel with a circular hole

Finally, we consider a perforated plate with afixed bottom edge

and subjected to a uniform tensile load P at the top edge as

sketched in Fig 13 We take this example because of the avail-ability of the reference solutions, which have been obtained by the BEM and the FEM[23] The plate is assumed to be linear elastic with Young's modulus E ¼ 3  107kN=m2 and Poisson's ratio

ν ¼ 0:2 The geometrical parameters are the same as used in[23]

with H ¼ 3 m; H1¼ 1:2 m; H2¼ 1:5 m; D ¼ 0:4 m; W1¼ 0:7 m;

0 50 100 150 200 250 300 350 400 0

50 100 150

Fig 11 Distributed nodes and evolution of the crack path from a fillet: type-2 boundary condition.

type−2 boundary condition

type−1 boundary condition

Fig 12 Close-up of the crack paths for both types of the boundary conditions at the vicinity of the fillet.

0 50 100 150 200 250 300 350 400

0

50

100

150

Fig 10 Distributed nodes and evolution of the crack path from a fillet: type-1

boundary condition.

W2¼ 0:3 m and the initial crack length is a ¼ 0:1 m The applied tensile load is to P ¼ 5 kN=m2in this model The problem domain

is discretized with a set of 1986 irregularly scattered nodes as shown inFig 14 The step-size for the crack growth is chosen as

Δa ¼ 0:03 m and a total number of 24 steps have been used The crack growth paths obtained by the proposed X-RPIM are visualized inFig 14 As expected, we found that the propagation of the crack path is well consistent with that predicted by both the BEM and the FEM, e.g., see Fig 5 in[23] As already stated in[23]

and again found in this study that the crack tip approaches the hole, it turns towards the hole andfinally collapses with the hole

Fig 15shows a close-up view at the crack path in the perforated panel with a circular hole subjected to a uniform tensile loading

0 0.5 1 1.5 2 2.5 3

Fig 14 Distributed nodes and the propagation of the crack path in a perforated panel with a circular hole subjected to a uniform tensile loading.

Fig 15 Close-up of the crack paths of a perforated panel in a circular hole subjected to a uniform tensile loading.

Fig 13 Schematic configuration of a perforated plate with a circle hole subjected

to a uniform tensile loading.

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the crack- line discontinuity and in the vicinity of the crack- tip We

begin by considering two numerical examples with single... examples in 2D elastic

solids are considered The crack path is simulated numerically In

the? ??rst example, the crack growth path from a fillet in a structural

member considering... again found in this study that the crack tip approaches the hole, it turns towards the hole andfinally collapses with the hole

Fig 15shows a close-up view at the crack path in the perforated