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

Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Nha Thanh Nguyen a, Tinh Quoc Bui b,n,1, Chuanzeng Zhang b, Thien Tich Truong a a b Department of Engineering Mechanics, Ho Chi Minh City University of Technology, Viet Nam Department of Civil Engineering, University of Siegen, Paul-Bonatz-Str 9-11, 57076 Siegen, Germany art ic l e i nf o a b s t r a c t Article history: Received 18 November 2013 Received in revised form March 2014 Accepted 25 April 2014 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-tip fields 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-tip field to extract the fracture parameters, while crack growth is controlled by utilizing the maximum circumferential stress criterion for the determination of its propagating direction 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 Keywords: Extended meshfree method Radial point interpolation method Enrichment techniques Crack propagation Stress intensity factors Fracture mechanics Introduction Advanced numerical methods have proved to be a useful tool in modeling and simulating a wide range of engineering problems The finite 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 extended finite 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 [37] n Corresponding author Tel.: ỵ 49 2717402836; fax: ỵ 49 2717404074 E-mail address: tinh.buiquoc@gmail.com (T.Q Bui) 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 An evident distinction between the mesh-based and the meshfree 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 the field 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 approximation spaces in the meshfree methods, e.g., see [5,6], to accurately describe the discontinuities and the singular field 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 approximation (MLS) shape functions, which 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 http://dx.doi.org/10.1016/j.enganabound.2014.04.021 0955-7997/& 2014 Elsevier Ltd All rights reserved Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 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 discontinuities 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 conditions in most other existing meshfree methods, e.g., the MLS based methods Additionally, as compared with lower-order finite elements that are commonly applied and can capture only the linear crackopenings, 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 parameters, we adopt the domain-form of the interaction integral in conjunction with the asymptotic crack-tip field For the quasistatic crack propagation modeling, the crack growing direction is determined from the maximum hoop-stress criterion The accuracy 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 comprehensive 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 In Section 4, two numerical examples involving single and mixed-mode cracks are investigated to illustrate the accuracy of the proposed method for evaluating the fracture parameters Two numerical examples with complex geometrical configurations for crack growth modeling are presented and discussed in details in Section Some conclusions drawn from the proposed method are reported in the last section 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 a finite element mesh The approximation of the distribution functions uðxi Þ within a subdomain Ωx D Ω 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 subdomain The well-known RPIM interpolation uh ðxi Þ; x A Ωx is defined as [4,11] ! n m a uh xị ẳ Ri rịai ỵ pj xịbj ẳ RT a ỵ PT b ẳ ẵ RT PT 1ị b iẳ1 jẳ1 T is the vector of the nodal where u ¼ uðx1 Þ uðx2 Þ … uðxn Þ displacements; Ri ðrÞ is the radial basis functions (RBFs); pj ðxÞ is the monomial in the 2D space coordinates xT ẳ ẵx; y, j ¼ 1; …; m where m is the number of polynomial basis functions The constants and bj are determined to construct the shape functions In this study, the thin plate spline function Ri rị ẳ r i , with q r i ẳ x xi ị2 ỵ y yi ị2 and the shape parameter ẳ 4:01, is used for constructing the RPIM shape functions unless otherwise stated By enforcing uh ðxÞ 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  ÃT Usd ¼ u1 u2 un ẳ R aỵ Pm b ð2Þ where the moment matrix of the RBFs R and the polynomial moment matrix Pm are given by R1 ðr Þ R2 ðr Þ … Rn ðr Þ R1 ðr Þ R2 ðr Þ … Rn ðr Þ 7 3ị R0 ẳ 6 ⋮ R1 ðr n Þ R2 ðr n Þ … Rn ðr n Þ ðnÂnÞ … x1 … xn y1 … yn ⋮ ⋮ ⋮ pm ðx1 Þ … pm ðxn Þ 6 T Pm ¼ 6 7 7 7 ð4Þ ðmÂnÞ The vector of the coefficients for the RBFs and the vector of the coefficients for the polynomial are defined by  ÃT aT ẳ a1 a2 an 5ị T b ¼ b1 b2 … bm ÃT ð6Þ There are n þ m variables in Eq (2), so the following m constraint conditions are used as additional equations n ∑ pj xi ịai ẳ PTm a ẳ 0; iẳ1 j ẳ 1; 2; …; m ð7Þ Combining Eqs (2) and (7) yields the following matrix form: # ! " ! R Pm a Usd ¼ ¼ G0 a0 Usd ¼ ð8Þ T P m 0 b |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} G0 Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ The vector of the coefficients a0 can be obtained by the following relation: ! a 9ị a0 ẳ ¼ G0À Usd b and substituting Eq (9) into Eq (1), we obtain h i T uh xị ẳ R T PT G0 Usd ẳ xịUsd 10ị in which, the RPIM shape functions can be expressed as h i T T PT G0 xị ẳ R h i ẳ xị xị n xị n ỵ xị n ỵ m xị ð11Þ The RPIM shape functions corresponding to the nodal displacements can be written as h i 12ị T xị ẳ ϕ1 ðxÞ ϕ2 ðxÞ … ϕn ðxÞ Finally, Eq (10) can be rewritten for the nodal displacements as n uh xị ẳ T xịUsd ẳ i ui 13ị iẳ1 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 discontinuous along the crack Finally, the displacement approximation is expressed as uh xị ẳ i xịui ỵ i A Wxị i xịHf xịịi ỵ i A W b ðxÞ ∑ i A WSðxÞ ϕi xị ! j xịij jẳ1 18ị 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 ds ¼ α c dc ð14Þ 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 functions are briefly summarized as follows More details can be found in [11,4] for instance where ϕi is 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] ( ỵ if f xị 40 Hf xịị ẳ 19ị if f xị o while the vector of the fourfold enrichment functions ψ j xị (jẳ1, 2, 3, 4) is given by p  pffiffiffi φ pffiffiffi φ pffiffiffi φ φ sin φ; r cos sin xị ẳ r sin ; r cos ; r sin 2 2 ð20Þ with r being the distance from the crack-tip xTIP to x, and φ being the angle between the tangent to the crack-line and the segment x À xTIP as depicted in Fig In Eq (18), W b denotes the set of nodes whose support contains the point x and is bisected by the crack-line (see Fig 2, the circle on the left-hand side) and W S is the set of nodes whose support contains the point x and is split by – The RPIM shape functions satisfy the Kronecker-delta function properties i xj ị ẳ ij 15ị The RPIM shape functions are of unity partition n i xị ẳ 16ị iẳ1 Fig Distance r and angle Reproducing properties n i xịxi ẳ x 17ị i¼1 – 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 Fig Definition of the sets of the nodes W b and W S , respectively Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ the crack-line and contains the crack-tip (see Fig 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 Fig The projection of a point xI belonging to Sa onto the advance vector tn 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 W b and W S 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 À be the set at the previous step n À1 Let Sa be the set of nodes for which the signed distance is updated at the current step The two vectors tn and tn À respectively are the crack-tip advance vectors at the step n and step n À The definition of the nodal sets is depicted in Fig The set Sn À is denoted by the area behind the segment 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 and not belong to the set Sn À to ensure that each point always has a unique projection onto the crack advance vector Let Stip be the set of nodes whose distance from È É the crack-tip is less than or equal to r f , Stip ¼ x : ‖x À xTIP ‖ r r f , then the set Sa is determined by the following three conditions [6]: n o À1 = Sn À ; ðx À xnTIP Þ U tn r 0; ðx À xnTIP Þ U tn À Sa ẳ x A Stip : x 21ị tip The first condition means that the nodes belong to S but not belong to Sn À , the second condition is used to pick nodes lie behind the segment and the third one ensures that these nodes lie in front of the segment Once the set Sa has 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 n ^ n ¼ nn =‖nn ‖ is its unit vector (Fig 4) The closest projection vectors of the nodes fðxI Þ belong to Sa on to the crack-tip advance vector is computed as follows [6]: ( n n 1 n^ ẵxI xnTIP ị U n^ Š if ðxI À xnTIP Þ Utn Z fxI ị ẳ 22ị n1 n1 if xI xTIP Þ Utn o À ðxI À xTIP Þ and the Heaviside function value of a node xI A Sa is determined by À1 Þ U tn Š Hðf xI ịị ẳ signẵxI xnTIP 23ị 2.5 Weak-form and discrete equations We consider a 2D, small strain, and linear elastic problem in the domain Ω bounded by Γ, and subjected to the body force vector b in the domain and traction t on Γ t The weak-form of the equilibrium equations can be expressed as [22] Z Z Z ∇s δu : r dΩ À δuT b dΩ uT t d ẳ 24ị t where ∇s is the symmetric gradient operator and r is 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 Kd ẳ f 25ị T where d ¼ fu; α; βg is the vector of unknown In Eq (25), the stiffness matrix K is dependent on either enriched or nonenriched nodes in the domain For enriched nodes, the stiffness matrix is determined as uu Kuα Kuβ Kij ij ij αu 7 K K Kij ẳ Kij 26ị ij ij βu βα ββ Kij Kij Kij where R R u T u u T α αu T Ku Kuu ij ẳ Bi ị DBj d; ij ¼ Ω ðBi Þ DBj dΩ ¼ ðKji Þ ; Z Z T Bui ịT DBj d ẳ Ku K Bi ịT DBj d; Ku ij ẳ ij ẳ ji Þ ; Kαβ ij Z Ω ¼ Ω ðBαi ÞT DBj d ẳ T K ji ị ; K ij Z ẳ Bi ịT DBj d 27ị while for non-enriched ones Kij ẳ Kuu ij 28ị The external force vector f in Eq (25) for enriched nodes is explicitly given by Fig Geometrical description for the set Sa u α β β β β f i ¼ ff i ; f i ; f i1 ; f i2 ; f i3 ; f i4 gT ð29Þ Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ whereas for non-enriched nodes u fi ¼ fi 30ị with Z Z u f i ẳ i b d ỵ i t d fi ẳ f ij ẳ 31ị t Z Z i Hb d ỵ Z t i Ht d 32ị Z i j b dỵ t i j t d; with j ẳ 4ị 33ị Additionally, the displacement gradient matrices Bui ; Bαi ; Bβi and the matrix of the elastic constants D dependent on Young's modulus E and Poisson's ratio ν are given respectively by 3 ðϕi HÞ;x ϕi;x i Hị;y Bui ẳ i;y 5; Bi ẳ 5; i Hị;y i Hị;x ϕi;y ϕi;x ðϕi ψ j Þ ;x i j ị;y 34ị B ẳ 5; j ẳ 4ị i i j ị;y i j ị;x Dẳ E ỵ ị1 2ị 1Àν 0 ðplane strainÞ ð1 À 2νÞ=2 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 W b and W S 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 strain fields g Calculate the SIFs using the interaction integral method h Compute the crack propagation angle based on the information of the computed SIFs to determine the crack growth direction i Specify a given size of crack growth, update the new crackline including the crack path and tips using the vector level set method as described in Section 2.4 (7) End the loop over the propagation steps (8) Visualization of the results Crack growth simulation and the SIFs implementation In the simulation of the crack propagation problems an appropriate 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 angle θc is calculated based on the mixed-mode stress intensity factor as follows [17,1]: θc ¼ tan À 1@ q1 ỵ 8K II =K I ị2 A 4ðK II =K I Þ ð36Þ The SIFs are computed using the domain-form of the interaction integral [18,19] The coordinates are assumed here to be the local crack-tip coordinates with the x-axis parallel to the crackfaces The path-independent J-integral is expressed as  Z  ∂u wδ1j À sij i nj d 37ị Jẳ x1 where w ẳ 1=2sij εij Þ is the strain energy density and nj is the jth component of the outward unit vector normal to an arbitrary contour Γ enclosing the crack-tip For general mixed-mode crack problems, the relationship between the value of the J-integral and the SIFs is J ẳ K 2I ỵ K 2II ị E~ 38ị where E~ ẳ E is for plane-stress and E~ ẳ E=1 ị for plane-strain problems, and K I and K II are 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 (superscript 1) is the one corresponding to the actual state and state (superscript 2) is the auxiliary state which is often chosen as the known asymptotic crack-tip field for mode-I or mode-II The interaction integral can be then expressed as # Z " ð1Þ ∂uð2Þ ∂q ð2Þ ∂ui ð1;2Þ i I 1;2ị ẳ s1ị ỵ s W dA 39ị 1j ij ∂x ij ∂x ∂xj 1 A Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ where W ð1;2Þ is the interaction strain energy determined by W 1;2ị 2ị ẳ 12 1ị ij ij ỵ s s 2ị 1ị ij ij ị and the smoothing weight function     2y 2jxj 1À q ¼ 1À c c 4.1 Mode-I: an edge-cracked plate under a tensile loading ð40Þ 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-tip eld as follows: E~ K I ẳ I 1;2ị Let us consider a rectangular plate with an edge-crack subjected to a uniformed tensile loading s ¼ 1:0 as depicted in Fig 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]  a 2  a 3  a 4 ! pffiffiffiffiffiffi a K exact ¼ s πa 1:12 À 0:23 þ 10:55 À 21:72 þ 30:39 I W W W W ð34Þ ð42Þ The stress intensity factor for a mode-II crack can also be obtained in the same way 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 quadrature are used for evaluating the stiffness matrix and the force vector Generally, a  quadrature is adequate but except in the elements/cells surrounding the crack-tip where a  Gaussian quadrature is used instead First, the finite size effect on the SIFs is investigated numerically 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 K I factor for various ratios a=W are presented in Table in Table Comparison of the K I factor between the analytical and the proposed X-RPIM solutions a=W 0.2 0.3 0.4 0.5 0.6 Present (15  30) Exact K I 3.07 4.56 6.67 10.02 15.64 Present (24  40) KI Error (%) KI Error (%) 3.15 4.64 6.75 10.30 16.27 2.35 1.86 1.25 2.82 4.01 2.98 4.32 6.71 10.21 16.15 3.16 5.31 0.65 1.89 3.26 16 14 12 10 0 Fig (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) Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ comparison with the exact solution [20] The distribution of the 20  40 scattered nodes of the cracked plate is depicted in Fig 5b The percentage errors of the SIFs compared to the analytical solution are also estimated As seen in Table the K I results derived from the X-RPIM are in good agreement with the analytical 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 in Fig 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 a¼ 3.5 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 numerical 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 XRPIM 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 in Fig is considered The geometrical and material parameters used for this example are L ¼ 16; W ¼ 7; 18 16 14 12 10 −2 −2 10 Fig The deformed shape of the cracked plate with 20  40 nodes, a ¼3.5, enlarged by a factor of 50 Fig Schematic configuration of a mixed-mode edge-crack in a rectangular plate under a uniform shear loading Table Comparison of the SIFs for an edge-cracked plate under a shear loading obtained by the FEM and the X-RPIM Table Influence of the support domain size on the SIFs a/W Methods KI FEM Exact MLS-EFG X-RPIM (MQ) X-RPIM (TPS) K II Scaling factor αc 1.6 1.7 1.8 1.9 2.0 7.74 7.75 8.03 8.06 7.74 8.01 7.36 7.45 7.74 8.08 7.84 7.86 7.74 8.19 7.80 7.83 7.74 8.65 7.85 7.88 0.3 0.4 0.5 19.82 25.80 34.09 Present X-RPIM FEM 15  30 20  40 25  50 19.82 24.64 33.87 19.33 24.47 34.57 19.50 25.42 34.55 2.46 3.49 4.54 Present X-RPIM 15  30 20  40 25  50 2.63 3.65 4.66 2.37 3.53 4.69 2.59 3.55 4.59 Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 20 20 18 18 16 16 14 14 12 12 10 10 8 6 4 2 0 10 15 0 10 15 Fig 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 a ¼ 3:5; E ¼ 1000; ν ¼ 0:25; and τ ¼ The analytical SIFs results K I ¼ 34 and K II ¼ 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, the finite 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 in 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 in Fig for two different sets of the scattered nodes, respectively The deformations look reasonable as well 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 the first 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 a fillet 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 Fig Schematic configuration of a fillet with a crack Type-1 (a) and type-2 (b) boundary conditions 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 in Fig representing two types of the boundary conditions In the first type (type-1) boundary conditions, as shown in Fig 9a, the displacements along the entire bottom edge are fully constrained, whereas only both ends of the bottom one are fixed 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 is fixed at the right end corner for both types of the boundary conditions The problems are considered in the plane strain condition and linear elasticity with Young's modulus E ¼  1011 Pa and Poisson's ratio ν ¼ 0:3 The initial crack-length is set to be a ¼ mm and the applied load is Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 150 150 100 100 50 50 0 50 100 150 200 250 300 350 400 Fig 10 Distributed nodes and evolution of the crack path from a fillet: type-1 boundary condition taken as P ¼ N The specimen for both types of loading is discretized using the same set of 1115 irregularly scattered nodes as depicted in Fig 10 or Fig 11 The problem is solved incrementally by increasing the crack size Δa ¼ mm in each step In all numerical simulations, a total number of 14 steps are performed Figs 10 and 11 visualize the crack growth paths from the fillet 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 the fillet is additionally depicted in Fig 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 downward and propagates toward the bottom of the structure (see Figs 11 and 12) In contrast, the crack grows almost directly toward the opposite fillet 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 a fixed bottom edge and subjected to a uniform tensile load P at the top edge as 0 50 100 150 200 250 300 350 400 Fig 11 Distributed nodes and evolution of the crack path from a fillet: type-2 boundary condition type−1 boundary condition type−2 boundary condition Fig 12 Close-up of the crack paths for both types of the boundary conditions at the vicinity of the fillet sketched in Fig 13 We take this example because of the availability 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  107 kN=m2 and Poisson's ratio ν ¼ 0:2 The geometrical parameters are the same as used in [23] with H ¼ m; H ¼ 1:2 m; H ¼ 1:5 m; D ¼ 0:4 m; W ¼ 0:7 m; Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i 10 N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ W ¼ 0:3 m and the initial crack length is a ¼ 0:1 m The applied tensile load is to P ¼ kN=m2 in this model The problem domain is discretized with a set of 1986 irregularly scattered nodes as shown in Fig 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 in Fig 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 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 and finally collapses with the hole Fig 15 shows a close-up view at the crack path in the perforated panel with 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 Fig 15 Close-up of the crack paths of a perforated panel in a circular hole subjected to a uniform tensile loading 2.5 1.5 0.5 0 0.5 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 Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i N.T Nguyen et al / Engineering Analysis with Boundary Elements ∎ (∎∎∎∎) ∎∎∎–∎∎∎ Conclusions In this paper, we present a new approach based on the partition of unity extended meshfree Galerkin method using the radial point interpolation method (X-RPIM) associated with the vector level set method for modeling the crack growths in 2D elastic solids An extrinsic enrichment technique is applied to the standard basis function to capture the singular fields at the crack-tips and a Heaviside jump function is employed to describe the discontinuous displacements on the crack-faces far away from the crack-tips The domain form of the interaction integral is used to calculate the stress intensity factors while the maximum circumferential stress criterion is taken to determine the crack growth direction The accuracy of the proposed X-RPIM is demonstrated by interpreting the numerical results of the SIFs for mode-I and mixed-mode crack problems A good agreement of the SIFs results with that by other methods is obtained Two numerical examples for crack growth problems with complex geometrical configurations are studied The predicted crack growth paths show a good agreement among methods with that by other numerical methods The essential conclusions drawn from the present study can be summarized as follows:  The present X-RPIM formulation can be developed straightfor     wardly from an existing meshfree or FEM computer code with little efforts By using the enrichment technique and the level set method without solving the partial differential equations, the proposed X-RPIM is accurate and convenient in modeling crack growth problems By automatically satisfying the Kronecker-delta function using the RPIM shape functions, the present formulation is easy in treating the essential boundary conditions, and no special technique is thus required A complete independence of the discretization of the problem domain is obtained by the vector representation and updating techniques with the aid of the vector level set technique through a signed distance function Compared with the reference solutions, it is demonstrated that the obtained SIFs results from the X-RPIM are accurate Crack paths predicted by the X-RPIM method match well with those from the reference works Consequently, the applicability and the accuracy of the proposed method in modeling complex crack growths are verified The advantages of the present X-RPIM highlight its great potential in dealing with complex crack propagation problems in other advanced materials and complicated structures in engineering applications 11  Nevertheless, some other interesting issues including the convergence, error estimation, adaptivity, etc pertaining to the method should be studied in our further works and they will be reported in the future References [1] Belytschko T, Black T Elastic crack growth in finite elements with minimal remeshing Int J Numer Methods Eng 1999;45:601–20 [2] Aliabadi HM Boundary element formulations in fracture mechanics Appl Mech Rev 1997;50:83–96 [3] Bui QT, Nguyen NM, Zhang Ch An efficient meshfree method for vibration analysis of laminated composite plates Comput Mech 2011;48:175–93 [4] Bui QT, Khosravifard, Zhang Ch, Hematiyan MR, Golub MV Dynamic analysis of sandwich beams with functionally graded core using a truly meshfree radial point interpolation method Eng Struct 2013;47:90–104 [5] Fleming M, Chu YA, Moran B, Belytschko T Enriched element-free Galerkin methods for crack tip fields Int J Numer Methods Eng 1997;40:1483–504 [6] Ventura G, Xu XJ, Belytschko T A vector level set method and new discontinuity approximations for crack growth by EFG Int J Numer Methods Eng 2002;54:923–44 [7] Guiamatsia I, Falzon GB, Davies OAG, Iannucci L Element-free Galerkin modeling of composite damage Compos Sci Technol 2009;69:2640–8 [8] Gingold RA, Monaghan JJ Smoothed particle hydrodynamics: theory and application to non-spherical stars Mon Not R Astron Soc 1977;181:375–89 [9] Belytschko T, Lu YY, Gu L Element-free Galerkin method Int J Numer Methods Eng 1994;37:229–56 [10] Atluri SN, Zhu T A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics Comput Mech 1998;22:117–27 [11] Wang JG, Liu GR A point interpolation meshless method based on radial basis functions Int J Numer Methods Eng 2002;54:1623–48 [12] Liu KW, Jun S, Zhang Y Reproducing kernel particle methods Int J Numer Methods Eng 1995;20:1081–106 [13] Gu YT, Wang W, Zhang LC, Feng XQ An enriched radial point interpolation method (e-RPIM) for analysis of crack tip fields Eng Fract Mech 2011;78: 175–190 [14] Gu YT An enriched radial point interpolation method based on weak-form and strong-form Mech Adv Mater Struct 2011;18:578–84 [15] Ventura G, Budyn E, Belytschko T Vector level sets for description of propagating cracks in finite elements Int J Numer Methods Eng 2003;58:1571–92 [16] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P Meshless methods: An overview and recent developments Comput Methods Appl Mech Eng 1996;139:3–47 [17] Erdogan F, Sih G On the crack extension in plates under plane loading and transverse shear J Basic Eng 1963;85:519–27 [18] Yau J, Wang S, Corten H A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity J Appl Mech 1980;47:335–41 [19] Shih C, Asaro R Elastic-plastic analysis of cracks on biomaterial interfaces: Part I—small scale yielding J Appl Mech 1988;55:299–316 [20] Gdoutos EE Fracture mechanics – an introduction (solid mechanics and its applications), 2nd ed The Netherlands; 2005 [21] Sumi Y, Yang C, Wang ZN Morphological aspects of fatigue crack propagation Part II-effects of stress biaxiality and welding residual stress Int J Fract 1996;82:221–35 [22] Wang S, Zhang H Partition of unity-based thermomechanical meshfree method for two-dimensional crack problems Arch Appl Mech 2011;81: 1351–63 [23] Leonel DE, Venturini SW Multiple random crack propagation using a boundary element formulation Eng Fract Mech 2011;78:1077–90 Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i ... 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... Please cite this article as: Nguyen NT, et al Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method Eng Anal Boundary Elem (2014), http://dx.doi.org/10.1016/j.enganabound.2014.04.021i... Another key factor in the meshfree methods is the in uence domain, which is used to determine the number of field nodes within the interpolation domain of interest Often, the following relation