Accurate and efficient analysis of stationary and propagating crack problems by meshless methods

Accurate and efficient analysis of stationary and propagating crack problems by meshless methods

A. Khosravifard

^a,⇑

, M.R. Hematiyan

^a

, T.Q. Bui

^b,c,⇑

, T.V. Do

^d

aDepartment of Mechanical Engineering, Shiraz University, Shiraz 71936, Iran

bInstitute for Research and Development, Duy Tan University, Da Nang City, Viet Nam

cDepartment of Civil and Environmental Engineering, Tokyo Institute of Technology, 2-12-1-W8-22, O-okayama, Meguro-ku, Tokyo 152-8552, Japan

dDepartment of Mechanics, Le Quy Don Technical University, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:

Received 18 June 2016 Revised 11 October 2016 Accepted 11 October 2016 Available online 12 October 2016

Keywords:

Crack propagation Meshless methods Stress intensity factor Element free Galerkin (EFG) RPIM

a b s t r a c t

New numerical strategies based on meshless methods for the analysis of linear fracture mechanics prob- lems with minimum computational labor are presented. Stationary as well as propagating cracks can be accurately modeled and analyzed by these proposed meshless techniques. For numerical analysis of the problem, meshless methods based on global weak-form are used. In order to capture the singular stress field near the crack tip, two different approaches are adopted. In the first approach, the asymptotic dis- placement fields are added to the basis functions of the meshless method. In the second one, a few nodes are added in the vicinity of the crack tip, while regular basis functions are used. The accuracy and stability of the two methods for determination of the stress intensity factors are then compared. In this work, an accurate integration technique, i.e., the background decomposition method (BDM), is utilized for efficient evaluation of the domain integrals of the weak-form with minimum computational cost. The superior accuracy of the proposed techniques is assessed by virtue of several benchmark problems. Through the presented numerical results it is concluded that the proposed methods are promising for the analysis of linear fracture mechanics problems.

Ó2016 Elsevier Ltd. All rights reserved.

1. Introduction

Cracks can be found in many structural components and machine parts and greatly affect the performance and life-time of the parts. Analysis of the stress field developed in a cracked body and investigation of the circumferences under which cracks prop- agate are one of the main subjects of fracture mechanics. Fracture mechanics has been long used for derivation of analytical formulas predicting the stress intensity factors (SIFs) in simple geometries and loading conditions. Early studies on cracked bodies are cred- ited to Griffith[1]who developed a theory for the analysis of stress and displacement fields in a flat homogenous isotropic plate of uni- form thickness, containing a straight crack. Since then, many other closed-form solutions are presented for determination of the SIFs of cracks, mostly in simple geometries and under simple loading conditions[2]. Nevertheless, obtaining analytical solutions of frac-

ture mechanics problems in practical situations with complicated geometries and loading conditions is formidable. Consequently, accurate analysis of cracked bodies in practical circumstances lends itself to numerical methods[3].

Extensive research has been conducted on the area of numerical analysis of fracture mechanics and different techniques have been used successfully in this context. Finite element method (FEM) [4,5], boundary element method (BEM) [6,7], meshless methods [8–11], extended finite element method (XFEM)[12–14], numerical manifold method (NMM)[3], and the edge-based smoothed finite element method (ES-FEM)[15–24]are among the many numerical methods that have been used for the analysis of fracture mechanics problems. Since the present work focuses on the meshless tech- niques for analysis of stationary and propagating cracks, only a brief review of the literature on this subject is presented herein.

Belytschko et al. [25] implemented the element free Galerkin (EFG) method for the analysis of problems of fracture and static crack growth. They later extended the application of the EFG method for the analysis of dynamic fracture problems[8]. Organ et al. [26] developed continuous meshless approximations for domains with non-convex boundaries, with emphasis on cracks.

They developed two methods for obtaining smooth approximations http://dx.doi.org/10.1016/j.tafmec.2016.10.004

0167-8442/Ó2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding authors at: Department of Mechanical Engineering, Shiraz University, Shiraz 71936, Iran (A. Khosravifard), Institute for Research and Development, Duy Tan University, Da Nang City, Viet Nam (T.Q. Bui).

E-mail addresses:khosravifard@shirazu.ac.ir (A. Khosravifard), buiquoctinh@

duytan.edu.vn,tinh.buiquoc@gmail.com(T.Q. Bui).

Contents lists available atScienceDirect

Theoretical and Applied Fracture Mechanics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / t a f m e c

of the field variable in such domains; a transparency method and a diffraction technique. They concluded that such remedies are only required when enriched basis functions are used in the problem formulation. Krysl and Belytschko[27]presented the EFG method for modeling of dynamically propagating cracks in three-dimensional (3D) bodies. In that work, the crack surface was defined by a set of triangular elements. Rao and Rahman[28]

proposed a new weight function and a method for exact implemen- tation of essential boundary conditions in the EFG method for ana- lyzing linear-elastic cracked structures. Later, they extended their previous work for calculation of the stress intensity factors of sta- tionary cracks in two-dimensional (2D) functionally graded materi- als (FGMs) of arbitrary geometry [29]. They also presented two interaction integrals for mixed-mode fracture analysis of FGMs.

Duflot and Nguyen-Dang[30]made use of the EFG method for anal- ysis of fatigue growth of cracks in 2D bodies using Paris’ equation.

Enriched weight functions were integrated into the formulation of their method to capture the singularity at the crack tip. The propa- gation of cracks in 3D bodies was investigated by Duflot with the use of meshless methods[31]. The stress singularity along the crack front was taken into account by an enrichment of the shape func- tions of the meshless method by means of appropriate weight func- tions. The proposed algorithms were used for prediction of non-planar propagation of multiple cracks. Wen et al.[32]investi- gated FGMs containing cracks by the EFG method in conjunction with enriched radial basis functions. Sladek et al. [33]proposed the use of meshless local Petrov–Galerkin (MLPG) method for sta- tionary and transient crack analysis in 2D and 3D axisymmetric magneto-electric-elastic solids with continuously varying material properties. The complex variable meshless manifold method for fracture problems was presented by Gao and Cheng[34], based on the complex variable moving least-squares approximation and the finite cover theory. Zhuang et al.[35]mixed the ideas of visibil- ity criterion and diffraction method to associate the displacement jump with the crack surface. Their formulation was based on the use of level set coordinates and the EFG method. They later extended their work to account for propagation of cracks in three dimensions[36]. Using the strain smoothing technique, Liu et al.

[37] developed a cell-based radial point interpolation method (CS-RPIM) for analysis of fracture mechanics problems. Ghorashia et al. [38] investigated fracture analysis of orthotropic cracked media by applying the extended isogeometric analysis (XIGA) using the T-spline basis functions. Nguyen et al. [10]presented a new approach based on local partition of unity extended meshfree Galerkin method for modeling quasi-static crack growth in 2D elas- tic solids. They used the radial point interpolation method (RPIM) for generating the shape functions. In their work, representation of the crack topology was treated by the aid of the vector level set technique. Belytschko and Black[12]analyzed 2D dynamic stress concentration problems using the wavelet Galerkin method (WGM). They used a path independent J-integral to evaluate the dynamic stress intensity factor. A new meshless method based on Shepard function and Partition of Unity (MSPU) was proposed by Cai et al.[39]for calculation of crack SIFs and simulation of crack propagation. They made use of the virtual crack closure technique (VCCT) to capture the crack tip SIFs, and the crack propagation was determined based on the maximum circumferential stress criterion.

In all of the numerical methods for analysis of fracture mechan- ics problems the main focus is on accurate representation of the singular stress field near the crack tip. Classically, two different approaches have been adopted to model the singular behavior of the stress field at the crack tip. In the first approach, the basis func- tions of the meshless method are properly enriched to include a set of appropriate singular terms of the displacement and stress fields

[9]. In the second one, a nodal refinement is carried out in a local sub-domain in the vicinity of the crack tip[40]. Fleming et al.[9]

were among the first researchers who presented an enriched EFG formulation for the analysis of fracture problems. To obtain the enriched formulation, they proposed two different methods. In the first method, the asymptotic fields are added to the trial func- tion, while the basis functions are augmented by the asymptotic fields in the second method. Irrespective of the approach used for capturing the singular stress field at the crack tip, numerical evaluation of the domain integrals in fracture mechanics problems requires a special treatment[9]. The common approach is to use a background mesh for evaluation of the domain integrals. Finer integration cells are usually used in the vicinity of the crack tip, and a high order Gaussian quadrature method is utilized in that cells. As reported by Fleming et al.[9], for a propagating crack, the discontinuities which arise in the shape functions and their derivatives when the crack passes through a quadrature cell are neglected. As the result, in the conventional integration methods, the discontinuity of the displacement filed along the crack line leads to numerical errors and instabilities, unless a large number of quadrature points are used in the problem domain.

In all of the aforementioned works, a somewhat complex proce- dure is utilized in order to model the singular behavior of the stress field near the crack tip. In this paper, a simple yet robust technique is proposed and it is shown that the crack tip SIFs and the crack propagation path can be handled accurately and with minimum burden. To this end, the background decomposition method (BDM)[41]is used for evaluation of the domain integrals of the weak-form. The BDM is a numerical domain integration technique which is specially designed for evaluation of integrals of the func- tions with severe variations in the problem domain. The BDM has been previously used for evaluating the domain integrals in mesh- less methods for the analysis of 2D and 3D elasto-statics problems with no cracks. Using the BDM, not only the severe variations of the integrand at the crack tip are grasped, but also the discontinu- ity of the shape functions and their derivatives are taken into account accurately. Consequently, this approach leads to very stable and accurate solutions for the stationary and propagating crack problems. The basic ideas of the present work were previ- ously presented in a paper for modeling of stationary cracks[42].

Herein, these ideas are extended to the analysis of propagating cracks.

The proposed methodology of the present work can be imple- mented with any meshless method based on the weak formulation.

Herein, the EFG method with enriched basis functions and also the RPIM with nodal refinement are adopted. The EFG was first pro- posed by Belytschko et al. for the analysis of heat conduction and thermo-elasticity problems[40]. Since then, the method has found many other uses and proved to be a robust technique for the anal- ysis of many engineering problems. The RPIM is another meshless technique that has found widespread use in many branches of science and engineering. The meshless RPIM was originally intro- duced by Liu and Gu[43,44]for the analysis of solid mechanics problems and was later developed for the analysis of many other engineering problems, including fracture mechanics[37].

The rest of this paper is structured as follows: In Section2, the background decomposition method for evaluation of the integrals associated with the weak-form of the fracture mechanics prob- lems is presented in detail. In Section3, some basic ideas for deter- mination of the SIFs with theJ-integral method and determination of the crack trajectory are reviewed. This section is followed by a concise review of the meshless formulation of the fracture mechanics, in Section4. Several numerical examples are presented to assess the accuracy and efficiency of the proposed techniques in

Section5. Some conclusions drawn from the study are presented in Section6.

2. The background decomposition method for evaluation of domain integrals in crack problems

The issue of evaluation of domain integrals in the meshless methods is still an active research area[41,45–48]. This issue is even more crucial for the meshless analysis of fracture mechanics problems. The reason for the importance of domain integrals in fracture problems is twofold: firstly, the stress field near the crack tip has a singular behavior. And the second reason is that the dis- placement field is discontinuous across the crack faces. The con- ventional approach for evaluation of domain integrals of fracture problems is to use a large number of quadrature points near the crack tip. Nevertheless, this approach does not consider the discon- tinuity of the shape functions and their derivatives[9]. Cai et al.

[39] proposed the use of the Cartesian transformation method (CTM) for evaluation of integrals of cracked domains. The CTM is a domain integration scheme which is capable of evaluation of integrals without any quadrature cells[45]. The CTM can take into account the discontinuity of the shape functions across the crack face. However it does not consider the density of the nodal points or the singular behavior of the integrand. The BDM favors the potentials of the CTM, while at the same time, the behavior of the integrand and the density of the nodal points are also taken into account[41].

Hematiyan et al.[41]proposed the BDM for efficient evaluation of domain integrals in meshless methods, which are based on weak formulation. The BDM accounts for the local density of the nodes in the problem domain when determining the position of integration points. The BDM has been found to be mostly suitable for problems in which the density of the nodal points varies in different parts of the domain or in cases where the integrand has a singular behavior at some parts of the domain. Consequently, analysis of cracked domains would be an interesting application of the BDM.

Evaluation of an integral by the BDM in a domain where the nodal density varies drastically follows a four-step procedure. In the first step, the scattered nodes are divided into some groups, based on their local density. The domain is then decomposed into some portions in the second step. In the third step, integration points and weights of each partition are determined. Finally, global vectors for evaluation of the domain integral are calculated in the fourth step. A brief discussion of these four steps for fracture prob- lems is given in the following paragraphs.

2.1. Grading the nodes based on their local density

For determining the local density of the nodal points in a domain, the average spacing of each node to its neighboring nodes is calculated first, and a spacing value (s) is assigned to each node in this step. The maximum and minimum values ofsin the domain

are denoted bysmaxandsmin, respectively. If these values are rela- tively close to each other, the nodes need not be classified into dif- ferent groups. Otherwise, the interval [smin smax] is divided into some sub-intervals. A grade of nodal spacing is associated with each sub-interval, i.e.,

grade 1:s2 ½s_min s₂,s₁¼ ðs_minþs₂Þ=2 grade 2:s2 ½s2 s3,s2¼ ðs2þs3Þ=2 ...

gradeR:s2 ½s_R s_max,s_R¼ ðs_Rþs_maxÞ=2

The values ofsiare selected such that 26siþ1=si63.

2.2. Partitioning of the integration domain

Based on the grading of the nodal points, the integration domain is divided into some partitions such that either all of the nodes in a partition belong to a single grade, or there are at most four nodes in a partition. For partitioning of the domain the well-known quadtree partitioning technique is used. In this tech- nique, a square that covers the original domain is considered.

The square is recursively subdivided into four child quadrants.

Each child square is subdivided into four new squares unless all the nodes in each square belong to the same grade or the number of nodes in the square is equal or less than four. InFig. 1(a), a typ- ical cracked domain is schematically depicted. In this domain, the nodal points are concentrated at the crack tip.Fig. 1(b) shows the partitions of the integration domain.

2.3. Determination of the integration points and weights of the BDM In this step, using a ray sweep method, the positions of integra- tion points of each partition are determined. An integration weight is associated to each integration point, and the value of the integral is evaluated by the weighted summation of the value of the inte- grand at the integration points, i.e.,

Ik¼X^Nk

j¼1

W^K_ifðxi;yiÞ ð1Þ

whereIkrefers to the value of the integralR

Xkfðx;yÞdXon thek-th partition.W^K_i are the integration weights, andðxi;y_iÞare the coordi- nates of the integration points. Nkis the number of integration points in the partition and is selected according to the grade of the nodal points in the partition. This means that, more integration points are located in partitions with a fine nodal arrangement. In this way, each part of the domain receives as much integration point as is required for accurate evaluation of the domain integral.

If an integration partition is fully inside the original integration domain, the standard Gaussian quadrature method is used for determination of the integration weights and points of that parti-

Fig. 1.A typical cracked domain along with (a) the meshless nodal points, (b) the integration partitions.

tion. Otherwise, the ray sweep method is used.Fig. 2 shows an integration partition which is partially inside the original domain.

The integral of a functionfðx;yÞover this partition can be written as:

Ik¼ Z

Xk

fðx;yÞdX¼ Z d

c

Z b a

gðx;yÞdx

! dy¼

Z d c

lðyÞdy ð2Þ

where lðyÞ ¼

Z b a

gðx;yÞdx ð3Þ

and

gðx;yÞ ¼ fðx;yÞ ðx;yÞ 2Xk

0 ðx;yÞ R Xk

ð4Þ The line integral on the right hand side of Eq.(2), as well as the inte- gral in Eq.(3)are evaluated by the composite Gaussian quadrature method, i.e.,

Ik¼Xⁿ

i¼1

Z y_iþ1 y_i

lðyÞdy

!

¼Xⁿ

i¼1

Z1 1

lðyð

g

ÞÞJid

g

ð5Þ whereJ_i¼dy=d

g

¼ ðy_iþ1y_iÞ=2 is the Jacobian of thei-th integra- tion interval. Each of the integrals on the right hand side of Eq.

(5)is evaluated by them-point Gaussian method to give Ik¼Xⁿ

i¼1

X^m

j¼1

J_iwjlðy_jÞ ð6Þ

wherey_j¼yð

g

_jÞ, andwjand

g

jare the integration weight and inte- gration point of the Gaussian quadrature method. In order to eval- uateIkby Eq.(6), the value oflðy_jÞshould be computed by Eq.(3).

The integral in Eq.(3)is evaluated along the liney¼y_j. This line is referred to as an integration ray and is depicted inFig. 3. From this figure it can be inferred that the number of intersection points of each ray with the boundary is always even. Therefore,lðy_jÞcan be rewritten as follows:

lðy_jÞ ¼ Z b

a

gðx;y_jÞdx

¼ Z x2

x1

fðx;y_jÞdxþ Z x4

x3

fðx;y_jÞdxþ þ Z x2q

x2q1

fðx;y_jÞdx ð7Þ

If the composite Gaussian quadrature method is used for evalu- ation of each integral on the right hand side of Eq.(7), the following formula is obtained:

Zx2i x_2i1

fðx;y_jÞdx¼Xⁿ⁰

r¼1

Z 1 1

FðnÞJ⁰_rdn¼Xⁿ⁰

r¼1

X^m0

s¼1

J⁰_rwsFðnsÞ ð8Þ

whereFðnÞ ¼fðxðnÞ;y_jÞandJ⁰_r¼^x2ix_2n²ⁱ¹0 .

As mentioned previously, the discontinuity of the shape func- tions across a crack line is taken into account by the BDM proce- dure. This is due to the use of the ray sweep method. Fig. 4 depicts the integration partitions that are crossed by a typical crack. In this figure, the red¹ lines represent the crack face and the cross marks represent the integration points. It is seen that dif- ferent integration intervals are selected on the two sides of the crack line and the distribution of the integration points conform to the shape of the domain. Upon substitution of Eqs.(7) and (8)into Eq.

(6), the final form of the formula for evaluation of the integral in thek-th partition is obtained (see Eq.(1)).

2.4. Computation of the global vectors

After obtaining the value of the domain integral in each parti- tion, the total value of the domain integral can be obtained by add- ing the values obtained from each partition:

I¼X^P

k¼1

Ik¼X^P

k¼1

X^Nk

j¼1

W^k_jfðQ^k_jÞ ¼X^N

n¼1

WnfðQ_nÞ ¼W^TFint ð9Þ

wherepis the total number of partitions, andQ_iare the coordinates of the integration points. The vectorWcollects the values of the integration weights and the vectorF_int includes the values of the integrand at the integration points.Fig. 5depicts the integration points of the BDM for the domain shown inFig. 1. It is clear that the density of the integration points conform to that of the nodal points.

As mentioned earlier, the BDM is also a robust tool for evalua- tion of domain integrals with integrands that exhibit singular behavior. For instance, the BDM can be used for evaluation of domain integrals that appear in the formulation of the meshless methods that use enriched basis functions. In such cases, except for the first step, the general procedure of the BDM is similar to the above mentioned steps. Instead of categorizing the nodes, based on their local density, the distance of a node to the crack Fig. 2.Illustration of an integration partition which is partially inside the original

domain.

Fig. 3.Illustration of an integration ray and its intersection with the boundary of a domain.

1For interpretation of color in Fig. 4, the reader is referred to the web version of this article.

tip is considered. In this way, parts of the region that are close to the crack tip will receive a denser distribution of integration points.

The presented formulation of the BDM in this paper is for 2D problems. However, using the Octree partitioning technique it can be applied for the analysis of 3D problems, as well. More details for 3D implementation of this technique can be found in [41].

3. Evaluation of fracture parameters for stationary and propagating cracks

Apart from the distribution of the displacement and stress fields in the whole domain, the local intensity of the stress field at the crack tip is of great importance when studying a cracked body.

The local state of stress at the crack tip is expressed by means of the stress intensity factors (SIFs). The concept of SIF was first intro- duced by Irwin[49]. Since then, the SIF is the most important frac- ture parameter of a cracked medium. Not only the SIF is used to describe the stress state at the crack tip, but it is also used to estab- lish different failure criteria due to fracture. In this section, a brief review of the computational methods for evaluation of the SIFs and prediction of the crack propagation path is performed.

3.1. Numerical calculation of the stress intensity factors

Different techniques are used to numerically calculate the stress intensity factors of a crack. The path independentJ- and interaction integral and its domain counterpart[10], the virtual crack closure technique (VCCT)[39], and the displacement correlation method [50]are among these techniques. In the present work, the domain form of the interaction integrals are utilized for evaluation of the SIFs. To this end, the path independentJ-integral is defined as:

J¼ Z

C

wd1j

r

1j

@ui

@x1

njdC ð10Þ

wherew¼1=2

r

ij

e

ijis the strain energy density,uiare components of the displacement field,

r

ijare the stress components,

e

ijare the strain components,niare the components of the unit vector normal to the domain boundary, andCis an arbitrary closed contour encir- cling the crack tip. In Eq.(10), all of the quantities are defined in the local crack-tip coordinate system, with thex1axis parallel to the crack line. Value of the J-integral in mixed-mode conditions is related to the SIFs, according to the following relation:

J¼1

EðK²_I þK²_IIÞ ð11Þ

KIandKIIare the stress intensity factors corresponding to the open- ing and sliding modes, respectively, andEis defined as:

Fig. 4.Typical integration partitions which are crossed by a crack and the corresponding integration points.

Fig. 5.Distribution of the BDM integration points for the domain shown inFig. 1.

E¼ E plane stress

E

1t² plane strain (

ð12Þ In order to separate the contributions of the SIFs to the value of theJ-integral, the so called interaction integral is used:

M^ð1;2Þ¼ Z

C

w^ð1;2Þd1j

r

¹_ij^@u

ð2Þ i

@x1

r

²_ij^@u

ð1Þ i

@x1

!

njdC ð13Þ

In Eq.(13), the superscripts (1) and (2) refer to the actual and aux- iliary problems, respectively. The auxiliary problem can correspond to either of the modesIandII.w^ð1;2Þis the interaction strain energy density and is equal to

w^ð1;2Þ¼1=2

r

^ð1Þ_ij

e

^ð2Þ_ij þ

r

^ð2Þ_ij

e

^ð1Þ_ij ð14Þ It can be shown that the value of M^ð1;2Þ is equal to 2=EðK^ð1Þ_I K^ð2Þ_I þK^ð1Þ_II K^ð2Þ_II Þ. Therefore, the values of the SIFs can be computed according to the following relation:

Ks¼E

2M^ð1;sÞ; s¼I;II ð15Þ

In order to get rid of evaluation of the singular fields near the crack tip, the contour integral in Eq. (13) is transformed to an equivalent domain integral. This transformation is carried out with the help of a smoothing function qðx₁;x₂Þ and the divergence theorem:

M^ð1;2Þ¼ Z

A

r

^ð1Þ_ij ^@u

ð2Þ i

@x1 þ

r

^ð2Þ_ij ^@u

ð1Þ i

@x1 w^ð1;2Þd1j

!@q

@xj

dA ð16Þ

Eq.(16)is the domain form of the interaction integral.qðx1;x2Þin this equation should be chosen such that it equals unity on the inner most contour and vanishes on the outermost contour.

3.2. Prediction of the crack propagation path

An important concern arising during the analysis of a fracture problem is the possibility of the propagation of the crack under the applied loading. To predict whether the crack is stable or not, several criteria have been proposed, including the maximum cir- cumferential stress[51], the minimum strain energy density[52], and the maximum energy release rate [53]. In this work, the maximum circumferential stress criterion is used. According to this criterion, a crack propagates if the maximum value of the circum- ferential stress reaches a critical material constant, that is, if the equivalent stress intensity factor (Keq) reaches the fracture tough- ness of the material (KIC).Keqis written as follows:

Keq¼cos hC

2 KIcos² hC

2 3

2KIIsinðhCÞ

ð17Þ The propagation directionhCis the angle at which the circumferen- tial stress reaches its maximum value

hC¼2 tan¹ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ8ðKII=KIÞ² q

4ðKII=KIÞ 0

@

1

A ð18Þ

4. Meshless formulation of fracture mechanics

In the present work, two different approaches are adopted for numerical simulation of a cracked medium by meshless methods.

In the first approach, enriched basis functions are utilized in the formulation of the meshless method. The EFG method with enriched basis functions is well suited for this purpose. As the result, no nodal refinement is required at the crack tip. In the sec-

ond approach, conventional basis functions are used in the formu- lation of the meshless method and therefore, in order to capture the singularity of the stress field at the crack tip, a few nodes (between 6 and 12) are added to a local sub-domain in the vicinity of the crack tip. The meshless RPIM, developed by Liu and Gu[43], is a convenient choice for this purpose. In the meshless RPIM, the shape functions possess the Kronecker delta property which makes the enforcement of essential boundary conditions straightforward [54]. Herein, a concise review of the formulations of the above-mentioned meshless methods is presented.

4.1. The EFG method with enriched basis functions

The EFG method is a meshless technique which uses a set of regularly or irregularly scattered nodes on the problem domain and its boundaries to obtain a system of discretized equations. In the EFG, nodes are not connected to each other with a mesh, and only a CAD-like description of the geometry is needed to formulate the final system of equations. The EFG uses the moving least squares (MLS) method to approximate the displacement field. In the MLS, the displacement filed is written as follows:

u^hðxÞ ¼X^m

j¼1

ajðxÞpjðxÞ ¼a^TP ð19Þ

whereajare coefficients,pjare the basis functions, andm is the number of basis functions. The basis functions are usually the monomials in space coordinates, selected from the Pascal triangle.

However, when the EFG is used for the analysis of fracture mechan- ics problems, the asymptotic fields are added to monomial terms. It can be shown that if the following basis functions are used, all the functions in the near-tip displacement field of a cracked body are spanned[9].

P^T¼ 1 x y ffiffiffi pr

cos₂^h ffiffiffi pr

sin₂^h ffiffiffi pr

sin₂^hsinh ffiffiffi pr

cos₂^hsinh

ð20Þ whererandhare measured from the crack tip. Following the stan- dard procedure of the MLS, the displacement filed can be written as:

u^hðxÞ ¼Xⁿ

i¼1

P^TðxÞA¹ðxÞBðxÞ ¼Xⁿ

i¼1

/iðxÞui ð21Þ

where/_iare the MLS shape functions. Other matrices in Eq.(21)are defined as follows:

AðxÞ ¼Xⁿ

i¼1

WiðxÞpðxiÞp^TðxiÞ;

BðxÞ ¼ ½W1ðxÞpðx1Þ;W2ðxÞpðx2Þ;. . .;WnðxÞpðxnÞ ð22Þ

/_iðxÞ ¼X^m

j¼1

p_jðxÞðA¹ðxÞBðxÞÞ_ji¼P^TðxÞðA¹BÞ_i ð23Þ In Eqs.(21)–(23),nis the number of nodes in the support domain of pointx, andWiis the weight function of thei-th node in the support domain[9].

Upon substitution of the displacement field, Eq.(21), into the modified variational principle of elasticity, the discretized system of equations can be obtained[55]:

K G G^T 0

U

K ¼ F

Q ð24Þ

In Eq.(24),U is the displacement vector, andK is the vector of Lagrange multipliers, used for enforcement of the essential bound- ary conditions. For definitions of other matrices and vectors in Eq.

(24), one may refer to[55].

4.2. The meshless radial point interpolation method

The meshless RPIM was first introduced in 2001 by Liu and Gu [43,44]and since then it has experienced a rapid development for the analysis of a wide range of problems in engineering and science. Many researchers have contributed to the further develop- ment of this method, especially the research team of Liu[55]. The formulation of the meshless RPIM is very similar to that of the EFG method, except for the interpolation technique. In the meshless RPIM, the radial basis functions (RBFs) are used to construct the shape functions[54]. Usually, polynomial basis functions are aug- mented with the RBFs. Using this interpolation technique, the dis- placement field can be written as:

u^hðxÞ ¼Xⁿ

i¼1

RiðxÞaiþX^m

j¼1

pjðxÞbj¼R^TðxÞaþp^TðxÞb ð25Þ

whereRiis a radial basis functions,p_jis a monomial,nis the num- ber of nodes in the support domain of pointx,mis the number of monomials, anda_iandb_jare constants to be determined. These con- stants are determined by enforcing the interpolated function to pass through the data points and also some specific constraint equations [55]. One major benefit of the RPIM interpolation is that its sensitiv- ity to the nodal irregularity is minimal[55], i.e., the nodes can have different densities in different parts of the domain without affecting the accuracy of the method. Different RBFs have been proposed and successfully used by researchers[56]. In this work, the thin plate spline (TPS) function is utilized in all calculations. In the RPIM, the constantsa_iandb_jare found such that the Kronecker delta func- tion property is satisfied. After some mathematical manipulations, Eq.(25)can be rewritten as:

u^hðxÞ ¼Xⁿ

i¼1

/_iðxÞui¼U^TU ð26Þ

Uis the vector of RPIM shape functions and is equal to the firstn components of the following vector:

e

U¼ ½R^T p^TG¹ ð27Þ

The vectorsRandpare defined in Eq.(25).Gis the moment matrix of the RPIM and is defined as:

G¼ R0 Pm

P^T_m 0

ð28Þ

P^T_m¼

1 1 1

x1 x2 xn

y1 y2 yn

... ...

.. . ... p_mðx1Þ p_mðx2Þ p_mðxnÞ 2

66 66 66 64

3 77 77 77 75

ð29Þ

R0¼

R1ðr1Þ R2ðr1Þ Rnðr1Þ R1ðr2Þ R2ðr2Þ Rnðr2Þ

... ...

... R1ðrnÞ R2ðrnÞ RnðrnÞ 2

66 66 4

3 77 77

5 ð30Þ

where rk in the formulation of RiðrkÞ is the Euclidean distance between nodesiandk.

Since the RPIM shape functions possess the Kronecker delta property, the enforcement of essential boundary conditions is straightforward. After substituting the approximate displacement field, Eq.(26), into the Galerkin weak-form of the problem, the final discretized system of equations is obtained.

5. Numerical results and discussions

In this section, four representative numerical examples are pre- sented. Two of these examples deal with the evaluation of the SIFs in stationary cracks. The other two examples demonstrate the applicability of the proposed techniques for prediction of the crack growth path. In the first example problem, both the EFG method with enriched basis functions and the meshless RPIM with nodal refinement are utilized for obtaining the SIF. However, the rest of the examples are solved only by the meshless RPIM. All of the results are compared with those found in literature or with the results obtained by the well-established FEM package ANSYS.

5.1. Example 1: SIF evaluation of a single-edge-crack in a rectangular domain under uniform tension

In this example, a rectangle containing an edge crack is ana- lyzed by the developed techniques. The problem geometry and loading is shown in Fig. 6(a). For analysis of this problem, both the enriched EFG method and the meshless RPIM with nodal refinement are used. Due to the symmetry of the problem domain and the loading, only half of the domain is modeled. InFig. 6(b), a typical nodal distribution of the meshless RPIM is shown for a crack of length a¼3:2. The nodal refinement around the crack tip is also depicted in this figure. The nodal distribution of the EFG method for the same crack length is depicted inFig. 6(c). A structured nodal distribution without any refinement at the crack tip is used for the EFG. A closed-form solution for the stress inten- sity factor of this problem is given by Gdoutos[2]:

K_I¼

r

p^{ffiffiffiffiffiffi}

p

a

1:120:23 a

b þ10:55 a b

2

21:72 a b

3

þ30:39 a b

4

; a

b<0:6 ð31Þ

For the numerical analysis of the problem with the mentioned meshless methods, three different nodal arrangements are used.

In the EFG, three structured 77, 1111, and 1616 nodal dis- tributions are used, while in the RPIM, nodal points are scattered irregularly. The number of nodes in the three different arrange- ments of the RPIM is 42, 123, and 210. For easy reference, these nodal arrangements are referred to as arrangement 1, 2, and 3, respectively. The domain integrals in both the EFG method and the RPIM are evaluated by the BDM.Fig. 7depicts the distribution of the BDM integration points, for the nodal arrangements shown inFig. 6. This figure shows that in case of a refined nodal arrange- ment, the distribution of the integration points conform to that of the nodal points. Also, it can be seen that in case of the EFG with enriched basis functions, the density of the integration points near the crack tip is more than other parts of the region. In all of the analyses, plane strain condition is assumed and the material prop- erties areE¼1000 Pa and

t

¼0:3.

For different ratios of the crack length to the rectangle width (a/b), the SIFs are obtained with the EFG and RPIM and with the three different nodal arrangements. InTables 1 and 2the results obtained by the RPIM and EFG, are compared with the analytical solutions given by Eq.(31). These tables clearly show the excellent agreement of the results obtained by the proposed methods.

Although the results of both the EFG and the RPIM methods reveal an acceptable accuracy, these tables suggest that the results obtained by RPIM with a nodal refinement are more stable and reliable than those obtained by enriched EFG method. The most interesting point is that the maximum error of the results obtained by the proposed RPIM with only 123 nodes is no more than 2%. To achieve this level of accuracy with other methods the number of nodes should be far more than this. For instance, Nguyen et al.

[10]have used 960 nodes to obtain the same level of accuracy.

As expected, the ratio of the crack length to the rectangle width (a/b) greatly alter the values of SIFs. The SIFs increase as the ratio a/bincreases. In addition, from the results presented in Tables 1 and 2, one can clearly observe a good convergence of the SIFs with respect to the nodal density. By comparing the SIFs obtained by the EFG and RPIM it is seen that both methods have yielded very accu-

rate results, but RPIM has a marginal superiority over the EFG.

Since the results of the RPIM are obtained by ordinary basis func- tions while the EFG results are obtained by the enriched basis func- tions, it is concluded that not only the RPIM is slightly more accurate than the EFG, but also its implementation can be carried out more easily for fracture mechanics problems.

The stress field ahead of the crack tip is also computed using the meshless RPIM and depicted inFig. 8. This figure corresponds to the crack length ofa¼3:2, and the nodal arrangement 3, i.e., the arrangement with 210 nodes. The results of the proposed tech- nique are compared with those obtained by the FEM (ANSYS) with a very fine mesh. It is seen that by using the BDM, not only the SIFs, but also the distribution of the stress field can be obtained with a high accuracy. Considering that a small number of nodal points is used to obtain the RPIM results, the proposed technique is found to be promising for the fast and efficient analysis of fracture mechanics problems.

5.2. Example 2: SIFs evaluation in a rectangle with an inclined edge crack

In this problem, a rectangle with an inclined edge crack is ana- lyzed by the proposed meshless RPIM. The problem geometry and Fig. 6.A rectangle with an edge-crack, (a) the geometry and loading, (b) nodal distribution of the meshless RPIM, (c) nodal distribution of the EFG method.

Fig. 7.Distribution of the BDM integration points for the (a) RPIM, and (b) EFG.

Table 1

Stress intensity factors obtained by the RPIM and BDM.

a/b AnalyticalKI Arrangement 1 Arrangement 2 Arrangement 3

KI Error (%) KI Error (%) KI Error (%)

0.2 3.07 3.10 1.0 3.06 0.3 3.07 0.0

0.3 4.56 4.29 5.9 4.48 1.7 4.53 0.6

0.4 6.67 6.48 2.8 6.61 0.9 6.65 0.3

0.5 10.02 9.82 2.0 9.93 0.9 10.01 0.1

Table 2

Stress intensity factors obtained by the EFG and BDM.

a/b AnalyticalKI 77 nodes 1111 nodes 1616 nodes

KI Error (%) KI Error (%) KI Error (%)

0.2 3.07 2.60 15.3 3.04 1.0 3.12 1.6

0.3 4.56 4.20 7.9 4.63 1.5 4.56 0.0

0.4 6.67 6.00 10.0 6.54 1.9 6.62 0.7

0.5 10.02 7.80 22.2 9.92 1.0 9.99 0.3

the applied boundary conditions are depicted inFig. 9(a). The prob- lem is analyzed with different angles of the crack,h, from75 to +75 degrees with respect to the horizon. In this example problem, it is assumed that plane strain condition prevails and the material

properties areE¼200 GPa and

t

¼0:3. InFig. 9(b) a typical nodal distribution of the meshless method is shown.

In Fig. 10 the results obtained by the proposed meshless method are compared with those obtained by ANSYS with a fine mesh. It is demonstrated that the SIFs obtained by the present method are in a good agreement with the reference results of ANSYS. This figure clearly shows the excellent accuracy of the pro- posed technique for prediction of the SIFs in mixed-mode condi- tions. Similar to the previous example, these accurate results are obtained with a small number of nodes. For instance, there are only 249 nodes in the nodal arrangement of Fig. 9(b). The deformed shape of the domain is also obtained by the proposed meshless RPIM and depicted in Fig. 11. The deformation induced in the domain is scaled by a factor of 610⁹.

5.3. Example 3: crack propagation from a fillet in a structural member As the third example, a crack propagation problem from a fillet is investigated. This problem is based on an experimental set-up which was first designed and performed by Sumi et al. [57]. In the original set-up, cracked structural members which were welded to I-beams of variable depth were considered. In the pre- sent work, the computational domain of that experiment is mod- eled and simulated by the proposed meshless RPIM with the BDM. Depending on the depth of the I-beam, two different types of boundary conditions are defined and applied in the models. In Fig. 8.Comparison of the (a) radial, and (b) circumferential stresses ahead of the crack tip ath= 0°fora= 3.2 obtained by the proposed RPIM with BDM and the FEM (ANSYS).

Fig. 9.A rectangular domain with an inclined crack, (a) problem geometry and loading, (b) nodal points of the meshless method forh¼30.

Fig. 10.The SIFs of (a) mode I and (b) mode II for a rectangle with an inclined crack.

the first case, referred here to as case 1, it is assumed that a very thick beam is attached to the bottom of the cracked body, while in the other case, case 2, the depth of the beam is very small.

The problem geometry and boundary conditions for case 1, and 2

are shown inFig. 12(a) and (b) respectively. All the dimensions in this figure are in millimeters.

Since, there is no connectivity between the nodes in a meshless method the procedure of crack growth modeling can be carried out easily. Nodes can be added to and removed from the domain with- out any difficulties. In each step of the crack propagation, initially, the nodes are scattered uniformly over the domain. A few nodes are then added near the crack tip. In Fig. 13, a typical nodal arrangement used in one of the analyses of this example is shown.

A uniform pressure is applied to the upper most boundary of the specimens. The initial length of the crack in both cases is 5 mm, and the increment of the crack growth is also 5 mm. it is assumed that the cracked body is in state of plane strain, and the material properties areE¼200 GPa and

t

¼0:3. The crack trajec- tories for both cases are determined using the proposed techniques and depicted inFig. 14. The propagation paths predicted in the pre- sent work are in close agreement with the results found in litera- ture, for instance, see[9]. In case 1, the crack path bends sharply downward, while in case 2, the crack propagates almost directly.

The SIFs of the cracked body in each step of crack evolution for both cases are plotted inFig. 15. Normally, as the crack propagates, the value of the SIFs corresponding to the opening mode grows.

This growth in value of K_I becomes steep when the crack approaches the other boundary of the body.

5.4. Example 4: crack propagation in a three-point bending specimen with circular holes

The last example deals with crack growth in a three-point bend- ing beam with three circular holes. This example is based on the experiments performed by Ingraffea and Grigoriu[58]at Cornell University. They have performed several experiments with differ- ent configurations of the holes and notches. In this work, two con- figurations of the original test are simulated by the proposed meshless RPIM. The problem geometry and loading conditions are shown inFig. 16. In order to comply with the original test con- ditions, all the dimensions of this figure are in inches. For analysis of this example problem, two geometrical configurations are selected. In the first configuration, case 1,a¼1 andd¼4, while in the second configuration, case 2,a¼1:5 andd¼5. It is assumed that the specimen is in state of plane strain and the following material properties are used in all analyses: E¼200 GPa and

t

¼0:3.

Fig. 11.The deformed shape of a cracked domain using the meshless RPIM with the BDM.

Fig. 12.The problem geometry and boundary conditions of example 3, for (a) case 1, and (b) case 2.

Fig. 13.A typical nodal arrangement of example 3, with 648 nodes.

Fig. 14.The predicted crack trajectories in example 3.

For case 1, a constant crack propagation step of 0.3 in. is selected and the crack trajectory is simulated by the proposed meshless RPIM. For case 2, the crack propagation step varies between 0.3 and 0.1 in. As reported by Ingraffea and Grigoriu [58], in case 2 the crack path intersects the middle hole. In other published works, when the minimum crack growth increment of 0.1 in. is selected, the simulated path does not intersect the middle hole[50,59]. In this paper, the same result is obtained. Therefore, crack growth increments that vary from 0.3 to 0.05 in. are also selected, and it is observed that the simulated path intersects the middle hole. As mentioned in the previous examples, one major benefit of the proposed method is that the number of nodes that is required to acquire an accurate solution is less than other mesh- less and finite element methods and X-FEM. For instance, Geniaut and Galenne[59]have used 4440 linear triangular elements, for modeling of this problem. Cai et al. [39] have solved the same problem with a meshless method using 2278 nodes. The numbers

of nodes that are used in the different steps of analyses of this example are around 700. Fig. 17shows a typical nodal arrange- ment used in the analysis of this example. The number of nodal points in this figure is 677.

Fig. 15.Stress intensity factors of the crack corresponding to the (a) opening and (b) sliding modes.

Fig. 16.Configuration of the problem geometry and loading in a three-point bending beam with three circular holes.

Fig. 17.A typical nodal distribution of a three-point bending beam by the proposed meshless RPIM.

Fig. 18.Comparison of the predicted crack trajectory of example 4 – case 1, with the experimental results.

The crack propagation path obtained by the present method and the one observed in experiments of Ingraffea and Grigoriu [58]are plotted inFig. 18. The crack path shown in this figure cor- responds to the configuration of case 1. It is observed that the crack path computed by the proposed meshless method matches per- fectly with the one obtained from the experiments [58]. These results are obtained by a crack propagation increment of 0.3 in.

Compared to case 1, the trajectory of the crack in the configura- tion of case 2 is more complicated. In this case, the effect of the holes on the distribution of the stress field is more evident. There- fore, a variable crack growth increment is chosen for this configu- ration. At first, the increment of crack evolution is selected between 0.3 and 0.1.Fig. 19shows the predicted crack path by the selected increments. Here again, a very close agreement between the experimental and simulated results is observed. In addition to the obtained results by the present method, the path obtained by Khoei et al. [50] with the polygonal FEM is also depicted.

This problem is solved again with minimum crack increment of 0.05 in. The minimum crack increment is only used in the last four stages of crack growth. As compared to the previous case with minimum increment of 0.1 in., a more accurate path is predicted at the final stage of propagation. This time, the predicted crack path intersects with the middle hole of the beam. The predicted and observed paths are almost identical. The crack path evolution of this final analysis is presented inFig. 20.

6. Conclusions

In this paper, new numerical strategies for accurate and effi- cient computation of fracture parameters of stationary as well as propagating cracks were presented. Both the element free Galerkin method and the meshless radial point interpolation method were adopted for numerical analysis of fracture mechanics problems.

In the EFG method, the basis functions were enriched using the near field asymptotic field, therefore, no nodal refinement was per- formed at the crack tip. In contrast, in the meshless RPIM ordinary basis functions were used. As a result, a refined nodal distribution was considered in a local sub-domain in the vicinity of the crack tip. For both of the methods the numerical integrations need to receive a special treatment. The reason for this requirement is two- fold: the shape functions and their derivative are discontinuous across the crack face, and also they have a severe variation near the crack tip. In the present work, the background decomposition method was proposed for accurate evaluation of such integrals with minimum computational labor. Through the numerical exam- ples, it was shown that when the BDM is used for evaluation of domain integrals of the meshless method, a simple yet powerful tool is obtained for accurate determination of fracture parameters of cracks. The following conclusions were drawn in the present work:

By using the BDM for evaluation of the integrals of fracture mechanics problems, an optimum distribution of integration points is obtained, i.e., each part of the domain receives as much integration point as is required for accurate evaluation of the integrals. Therefore, the integrals with near singular or discon- tinuous integrands can be evaluated accurately and with mini- mum effort.

In the BDM formulation, the domain on the two sides of a crack receive different integration points, therefore, the discontinuity of the integrand is handled accurately.

We have numerically shown that the BDM is suitable for both the meshless methods that use enrichment techniques and for methods that utilize a refined nodal distribution at the crack tip. Acceptable results were obtained by the two strategies in the present work. However, it was observed that when ordinary basis functions are used and only a few nodes (around 6–12) are added to the region near the crack tip, very accurate and stable results can be obtained without introducing any additional complexity to the problem formulation.

Fig. 19.Crack trajectory of a three-point bending beam with three circular holes, example 4 – case 2, minimum crack increment is 0.1 in.

Fig. 20.Crack trajectory of a three-point bending beam with three circular holes, example 4 – case 2, minimum crack increment is 0.05 in.