a twice-interpolation finite element method (tfem) for ...

2nd Reading December 28, 2012 17:33 WSPC/0219-8762

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International Journal of Computational Methods Vol. 9, No. 4 (2012) 1250055 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S0219876212500557

Int. J. Comput. Methods 2012.09. Downloaded from www.worldscientific.com by HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY on 02/04/13. For personal use only.

A TWICE-INTERPOLATION FINITE ELEMENT METHOD (TFEM) FOR CRACK PROPAGATION PROBLEMS

S. C. WU∗,†,§ , W. H. ZHANG∗ , X. PENG‡ and B. R. MIAO∗ ∗State

Key Lab of Traction Power, Southwest Jiaotong University Chengdu 230009, China †State Key Lab of Advanced Design and Manufacturing for Vehicle Body Changsha 410082, China ‡School of Engineering, Institute of Mechanics & Advanced Materials Cardiff University, Wales, UK §[email protected] Received 1 June 2010 Accepted 21 May 2011 Published 31 December 2012 The twice-interpolation finite element method (TFEM) constructs the trial function for the Galerkin weak form through two stages of sequential interpolation without additional degrees of freedom and achieves better accuracy and convergence compared to the conventional finite element method (FEM). The TFEM has been shown to be insensitive to the quality of the elemental mesh and thus has the potential to simulate fracture problems. Intense examples and issues with cracking problems are investigated in this study. It is observed that the stress intense factor (SIF) of the crack tip can be evaluated with satisfactory accuracy. Since the present TFEM can produce more continuous nodal gradients, better stress fields can be reproduced, especially around the crack, without requiring more nodes. It is also shown that crack propagation can be reproduced readily and the cracking path agrees well with the reference solution and experimental results. Keywords: Crack propagation; twice-interpolation; finite element method; ALOF software.

1. Introduction The numerical simulation of crack initiation and its progress is essential for predicting the service life of engineering structures. In the standard finite element method (FEM), nodal relaxation techniques are frequently used for modeling cracks, where each node along the free surface is split into two nodes [Swenson and Ingrafea (1996)]. The partition of unity method [Melenk and Babuˇska (1996)], which does not require frequent meshing and remeshing processes, has also emerged as a powerful tool for crack propagation analysis. Moreover, mesh-free methods, such as the § Corresponding

author. 1250055-1


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element-free Galerkin method (EFG) [Belytschko et al. (1995)] and the meshless local Petrov–Galerkin method (MLPG) [Atluri and Zhu (1998)], which are more flexible than the standard FEM, have been developed to solve fracture problems with arbitrary node distributions. In addition, some other mesh-free approaches [Rabczuk and Belytschko (2004); Zheng et al. (2008); Liu and Zhang (2009); Chen et al. (2010); Liu et al. (2010); Liu et al. (2011)] have also been proposed for progressive cracking problems. However, these well-known reduced mesh methods often result in high computation costs, and thus, many robust solutions and other excellent properties developed for the FEM are not utilized. As a special example of the partition of unity method, the extended finite element method (XFEM) [Sukumar et al. (2000); Belytschko et al. (2009)], which offers advantages over other numerical methods for discontinuous problems of dynamic fractures, was developed and has been successfully employed in both commercial ALOF [Wu and Wu (2011)] to solve complex three-dimensional (3D) fracture problems. If only the XFEM is used to simulate the cracking process, the numerical accuracy cannot always be assured. The XFEM also requires a finer mesh, especially in the moving crack tip. By integrating the XFEM and the virtual node method (VNM) [Tang et al. (2009)] into the ALOF, satisfactory accuracy and lower calculation costs can be achieved for progressive cracking problems. Recently, a novel twice-interpolation finite element method (TFEM) [Zheng et al. (2010)] was proposed for solid mechanics problems. In the TFEM, the trial function for the Galerkin weak form is constructed through two stages of sequential interpolation. Similar to the standard FEM, the TFEM is based on an elemental mesh, with the unknowns being the nodal displacements. In the standard FEM, the trial functions are obtained by interpolation according to nodal displacement and are represented as C 0 on the element edges. The first stage of TFEM is the same as that of the standard FEM, but the average nodal gradients must be computed for the second stage of the interpolation, unlike with the second stage of the interpolation for the Hermite FEM. Then, the approximation functions can be further rebuilt, in which both the nodal displacement and average gradients are selected as the interpolation conditions. Thus, the shape functions constructed by the twice-interpolation step, exhibit more continuous nodal gradients and higherorder polynomial contrast compared to the standard FEM when analyzing the same mesh. Intensive numerical benchmark problems have shown that the TFEM exhibits better accuracy and convergence properties than standard FEM. In the present work, the TFEM is further developed for fracture problems into ALOF. In order to maintain the standard FEM computer procedure to the extent possible, the nodal relaxation technique is utilized to model the crack growth. In addition, a node projection technique is formulated to improve the modeling of the crack evolution. The outline of this paper is as follows. Section 2 briefly reviews the twice-interpolation strategy, and Sec. 3 details the evaluation of the stress intensity factor and simulation of the crack growth. An intensive numerical study is conducted in Sec. 4, and the conclusions are presented in Sec. 5. 1250055-2


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A Twice-Interpolation Finite Element Method (TFEM ) for Crack Propagation Problems

2. Twice-Interpolation Strategy The implementation of and variable definitions for the TFEM were detailed [Zheng et al. (2010)]. Because the numerical problems in this work are solved based on the triangular element, the detailed expression of the two-dimensional (2D) triangular element is shown briefly.


2.1. The first stage of interpolation In the standard FEM, the approximation of interested point x can be obtained by u(x) = Ni u[i] + Nj u[j] + Nm u[m] , Ni = L i ,

Nj = L j ,

Nm = L m ,

(1) (2)

in which L is the area coordinate and N is the shape function. Si , Sj , and Sm are set of element related to nodes i, j, and m, respectively, as shown in [Zheng et al. (2010)]. The supporting nodes for point x include all the nodes in the element set Si , Sj , and Sm . We define qs as the nodal displacement vector of supporting node: qs = {q1 , q2 , . . . , qns }T ,

(3)

in which ns is the total number of supporting nodes. For any point in Si , Sj , and Sm , the classical interpolation can be written in the form, u(x) =

ns

Nl (x)ql .

(4)

l=1

The averaged derivatives of node i can be computed as u ¯[i] ,x =

ns

¯ [i] ql , N l,x

(5)

l=1

¯ [i] = N l,x

[i][e]

(ωe · Nl,x ),

(6)

e∈Si

in which ωe is the weight function of element e and can be computed as follows: ∆e e ∈ Si , (7) ωe = ∆e e∈Si

where ∆e is the area of element e. 2.2. The second stage of interpolation In second stage of interpolation, the averaged derivatives, rather than the nodal derivatives, are adopted to reproduce the interpolated value at point x, [i] [i] u ˆ(x) = ϕi u[i] + ϕix u¯,x + ϕiy u ¯,y + ϕj u[j] + ϕjx u ¯[j] ¯[j] ,x + ϕjy u ,y

+ ϕm u[m] + ϕmx u ¯[m] ¯[m] ,x + ϕmy u ,y , 1250055-3

(8)


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in which ϕi , ϕix are ϕiy are constructed to satisfy the relationship of ϕi (xl ) = δil , ϕix,x (xl ) = δil . When i is equated to l, δil = 1; otherwise, δil = 0. ϕi = Li + L2i Lj + L2i Lm − Li L2j − Li L2m , 1 1 2 2 ϕix = −cj Lm Li + Li Lj Lm + cm Li Lj + Li Lj Lm , 2 2 1 1 ϕiy = bj Lm L2i + Li Lj Lm − bm L2i Lj + Li Lj Lm . 2 2

(9) (10) (11)

Similarly, ϕj , ϕjx , ϕjy , ϕm , ϕmx , and ϕmy can also be obtained by a cyclic permu[i] [i] tation of the indices of i, j, and m. Substituting u ¯,x , u¯,y and ϕ into Eq. (8) provides the final trial functions of the twice-interpolation process: u ˆ(x) =

ns

ˆl (x)ql , N

(12)

l=1

ˆl = ϕi N [i] + ϕix N ¯ [i] + ϕiy N ¯ [i] + ϕj N [j] + ϕjx N ¯ [j] + ϕjy N ¯ [j] N l l,x l,y l l,x l,y [m]

+ ϕm Nl

¯ [m] + ϕmy N ¯ [m] . + ϕmx N l,x l,y

(13)

2.3. Modification when C 0 node is required It has been found that the nodal derivatives obtained by the twice-interpolation process are continuous. However, in solving practical problems, some nodes are required to be recovered to C 0 , that is, the nodal property of the classical FEM, such as the nodes located on the essential boundary or material boundary lines. Therefore, these specified nodes should be modified. Suppose that node i is a C 0 node and that the point of interest x is in element e. Then, the average nodal derivatives u ¯i,x can be obtained by [e]

u ¯i,x = ui,x .

(14)

Note that u ¯i,y can be computed similar to u ¯i,x . It is found that if all the nodes in the problem domain are recovered to be C 0 , the TFEM will degenerate to the standard FEM. Thus, by coupling novel TFEM and standard FEM, a satisfactory ratio of accuracy-versus-cost can be obtained.

3. Stress Intensity Factor Evaluation 3.1. Stress intensity factor In this section, we briefly discuss the criterion used to specify the direction of the cracking. Furthermore, we also review the domain form of the interaction integral used to determine the mixed-mode stress intensity factors (SIF). 1250055-4


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Under general mixed-mode loadings, the asymptotic near-tip circumferential and shear stresses take the following form: σθθ KI 1 3 cos(θ/2) + cos(3θ/2) = √ sin(θ/2) + sin(3θ/2) 2πr 4 σrθ KII 1 −3 sin(θ/2) − 3 sin(3θ/2) . (15) +√ 2πr 4 cos(θ/2) + 3 cos(3θ/2) In this paper, we use the maximum circumferential stress criterion, which states that the crack propagates from the tip in a direction θc so that the circumferential stress σθθ , obtains the maximum. The circumferential stress in the direction of the crack propagation is the principal stress. Therefore, the critical angle θc , which defines the radial direction of propagation, can be determined by setting the shear stress in Eq. (15) equal to zero. The following equation is thus obtained: θ 1 1 1 √ KI sin(θ) + KII (3 cos(θ) − 1) = 0. (16) cos 2 2 2 2πr This leads to the equation defining the angle of the crack propagation, θc , in the tip coordinate system. KI sin(θc ) + KII (3 cos(θc ) − 1) = 0,

(17)

where KI and KII are the pure Mode I and pure Mode II SIF, respectively. Solving the equation for the angle of crack propagation gives the equation,   2 KI 1  KI ± + 8. (18) θc = 2 arctan 4 KII KII To obtain the SIF, the domain forms of the interaction integrals are used. For completeness, these are discussed here. The coordinates are the crack tip coordinates with the x-axis parallel to the crack faces. For general mixed-mode problems, we have the following relationship between the value of the J integral and the SIFs: J=

2 KII KI2 + , E∗ E∗

(19)

where E ∗ is defined in terms of Young’s modulus of E and Poisson’s ratio of ν:   plane stress E E E∗ = . (20)   1 − ν 2 plane strain (1)

(1)

Two states of a crack body are considered in the computation. State 1 (σij , εij , (1)

(2)

(2)

(2)

ui ) corresponds to the actual state, and State 2 (σij , εij , ui ) corresponds to 1250055-5


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an auxiliary state, which will be chosen as the asymptotic fields for Modes I and II. The J-integral for the sum of the two states is defined as: J (1+2) = J (1) + J (2) + M (1,2) ,

(21)


where M (1,2) is called the interaction integral for States 1 and 2 2 1 (1) ∂ui (2) ∂ui M (1,2) = − σij W (1,2) δ1j − σij nj dΓ ∂x1 ∂x1 Γ

(22)

and W (1,2) is the interaction strain energy (1) (2)

(2) (1)

W (1,2) = σij εij = σij εij .

(23)

After rearranging terms, expressing Eq. (21) for the combined states gives 2 (1) (2) (1) (2) J (1+2) = J (1) + J (2) + ∗ (KI KI + KII KII ). (24) E Combining Eqs. (21) and (24) yields 2 (1) (2) (1) (2) M (1,2) = ∗ (KI KI + KII KII ). (25) E Making the judicious choice of State 2 as the pure Mode I asymptotic field with (2) (2) KI = 1 and KII = 0, give the Mode I stress intensity factor for State 1 in terms of the interaction integral E ∗ (1,Mode I) (1) M . (26) KI = 2 (2)

(2)

Choosing State 2 as Mode II asymptotic fields with KI = 0 and KII = 1 gives the Mode II stress intensity factor for State 1 in terms of the interaction integral E ∗ (1,Mode II) (1) M . (27) KII = 2 The contour integral in Eq. (22) is not flexible for the FEM. Therefore, we must recast the integral into an equivalent domain form by multiplying the integrand by a bounded weighting function, (q(x) = 1), on an open set containing the crack tip and vanishing on an outer prescribed contour C0 . Then, for each contour Γ, as shown in Fig. 1, the crack faces are assumed to be free of stresses and straight in the region A, which is bounded by contour C0 . The interaction integral can then be written as: 2 1 (1) ∂ui (2) ∂ui (1,2) (1,2) = δ1j − σij − σij (28) M W qmj dΓ, ∂x1 ∂x1 C where C = Γ + C+ + C− + C0 and m is the outward unit normal to the contour C. Assuming that the divergence theorem holds true and passes the limit as contour Γ is shrunk to the crack tip, which is justified by the Dominated Convergence theorem, the following interaction integral in domain form can be obtained 2 1 ∂q (1) ∂ui (2) ∂ui − σij − W (1,2) δ1j dA, (29) σij M (1,2) = ∂x1 ∂x1 ∂xj A where mj = −nj on Γ and mj = nj on C0 , C+ and C− . 1250055-6


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Fig. 1. Convention at the crack tip. Domain A is enclosed by Γ, C+, C− and C0 .

Fig. 2. Elements selected around the crack tip for the integral.

To numerically evaluate the integral above, domain A is defined in terms of the length of the elements in contact with the crack tip, with this quantity being designated as h. Then, domain A is set to be all elements that have a node within a square. Figure 2 shows a typical set of elements for domain A with rq = 4h. The weighting function is represented as q(x, y) = (rq − x)(rq + x)(rq − y)(rq + y).

(30)

Note that the coordinates are actually crack tip coordinates with the x1 -axis being parallel to the crack faces. 3.2. Nodal projection and relaxation For triangular elements, the mesh irregularity has little effect on the accuracy of the TFEM [Zheng et al. (2010)]. Therefore, if the node along the crack moves, acceptable solutions can still be obtained. To model the accurate path of the cracks and modify the mesh as little as possible, the nodes along the crack path are projected onto the crack path. In nodal projection, the total number of elements and nodes does not change, and the nodal coordinates along cracks only require a simple modification. 1250055-7


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Fig. 3. Nodal projection.

As shown in Fig. 3, the nodal projection is divided into three cases. In Case 1, the crack intersects two edges of the triangle, and the common node of these two edges is moved onto the crack. In Case 2, the crack intersects only one edge, and thus, the node near the crack is moved onto the crack. In Case 3, the triangle contains the crack tip, and the nearest node is moved to the crack tip. After nodal projection, the crack path is represented by a set of element edges. The traditional method, known as nodal relaxation, is employed to model the discontinuity. In nodal relaxation, each node on the crack path is split into two nodes, and the elements on two sides of the crack are set to two different nodes, causing the displacement to be discontinuous.

4. Benchmarks and Experiment The formulated TFEM is then incorporated into the ALOF and used for solving some experimental fracture problems, benchmarked in terms of accuracy and robustness. Numerical results and stress plots are obtained with the ALOF software. The variable units used are the international standard unit system, unless specially denoted. To examine the accuracy and convergence, the relative L2 error in the displacement norm and energy norm, respectively, are given by exact − unumerical )2 dΩ Ω (u , (31) ed = (uexact )2 dΩ Ω 1 exact − εnumerical )T D(εexact − εnumerical )dΩ 2 Ω (ε ee = , (32) 1 exact )T Dεexact dΩ 2 Ω (ε in which the superscript “exact” and “numerical” stand for the exact solution and a numerical result, such as the present TFEM. For the element used in the TFEM, a four-point Hammer integration is utilized to evaluate the stiffness matrix. The nodes located on the essential boundary and material dividing lines are all set to be C 0 . 1250055-8


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4.1. Edge crack plate in tension As the first example, a plate with a crack under the action of uniform tension on the top edge is analyzed. A sketch of the problem is shown in Zheng et al. [2008], with the following parameters: H = 1, L = 1 and σ = 1. The material used exhibits linear elastic behavior with a Young’s modulus of E = 1 × 1010 and ν = 0.25, and the plane strain conditions are assumed. The value of the Mode I stress intensity KI factors, as a function of the crack lengths, are computed as follows: r = a/H,

(33)

F = 1.120 − 0.231r + 10.550r − 21.710r + 30.382r , √ KI = F σ πa. 2

3

4

(34) (35)

The mesh used by the TFEM contains 806 uniform nodes. The SIF obtained by numerical methods is shown in Fig. 4. The computed SIFs agree well with the benchmark value for all values of the crack length and for similar meshes; the solutions are better than for the standard FEM. The stress fields in the loading direction, obtained by both the TFEM and the standard FEM, are plotted in Fig. 5. It is observed that the stress field obtained by the TFEM is more continuous and smooth.

4.2. Center-oblique crack in an infinite plate In this section, a finite plate with an oblique central crack is studied under axial loading conditions, as shown in Zheng et al. [2008]. The material used is linear elastic in behavior, with E = 1 × 1010 and ν = 0.25. The plate is subjected to tensile loading of σ1 and σ2 in the x- and y-directions, respectively. The exact stress intensity factors are a function of the angle of the crack, β, with the x axial

(a) SIF for difference length of crack.

(b) Compare of relative error of SIF.

Fig. 4. SIF of the edge crack problem. 1250055-9


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(a) TFEM solution.

(b) FEM solution.

Fig. 5. Stresses in the loading direction.

being given by √ KI = P sin2 β πa, √ KII = P sin β cos β πa.

(36) (37)

A finite plate with a crack is modeled by a square plate with a width of L = 10, a crack length of a = 1.414 and an axial loading of P = 1. Since the crack length is much smaller than the specimen’s dimensions, the numerical results are compared to the reference solution of a crack in an infinite plate. The mesh used by the TFEM contained 2392 uniform nodes. To improve the accuracy, the mesh is fine around the crack tip in the stress plot shown in Fig. 6. Modes I and II SIFs as a function of the angle β are presented in Fig. 7; the result agrees well with the reference solution result.

Fig. 6. Stress field of σx and σy . 1250055-10


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Fig. 7. SIF of the center-oblique oblique problem.

4.3. Plate with two holes This example concerns the growth of cracks emanating from holes in a square plate, loaded in tension as shown in Fig. 8. The material used exhibits linear elastic properties with E = 1 × 1010 and ν = 0.25, and the plane strain conditions are assumed. The crack path, after 28 steps of crack growth, is plotted in Fig. 8, together with a reference crack path.

Reference solu. The TFEM solu.

Path comparison of TFEM and the Reference

Fig. 8. Computational paths using the TFEM with a fine mesh for two holes of the plate. 1250055-11


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Fig. 9. Stress field of σx and σxy after the 28th growth step.

The reference solution is presented for a very fine mesh (12,174 nodes), using the standard FEM. The numerical solution agrees well with the reference solution, as shown in Fig. 8. Figure 9 shows the stress distribution of σx and σxy at the 28th growth step under the ∆a = 0.025. 4.4. Beam under four-point loading We consider the condition of precracks of a = 0.2 and c = 0.4, with the beam being subjected to four-point shear loading. The boundary conditions are shown in Fig. 10, with a beam length of L = 4, a width of b = 1 and a load of P = 1. The mesh used for the TFEM contains 144 uniform nodes. Together with the experimental crack path, the computed crack path at step 18 is shown in Fig. 11. It is clearly observed that using the fracture theory and the TFEM, the crack propagation path can be simulated with satisfactory accuracy. Additional research also shows that TFEM approaches the experimental values more closely by increasing the node density and element order. Figure 12 shows the stress distribution of σx and σy at the 18th growth step with a cracking increment of ∆a = 0.05.

Fig. 10. Stretch of the beam under four-point loading. 1250055-12


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Fig. 11. Computational cracking paths at the 18th step and experimental solution.

Fig. 12. Stress distributions of σx and σy after the 18th growth step.

5. Conclusions and Discussions The TFEM is advantageous compared to the standard FEM, as demonstrated by the example fracture problems in this paper. Example problems are studied to examine the accuracy and robustness of the TFEM. The following conclusions can be drawn: (a) Using a similar mesh, the TFEM can achieve better accuracy than the traditional FEM for both Mode I cracks and mixed-mode cracks, in terms of the accuracy of the SIF of the crack tip. (b) TFEM can achieve more smoothing stress field of cracking because of its two stages of interpolation. (c) The TFEM can simulate crack propagation readily and accurately by incorporating the TFEM into the ALOF platform. It must be noted that when compared to the standard FEM, TFEM does not increase the total degrees of freedom of a model, which can be easily misunderstood based on the above formulations. In the second stage of interpolation, each node possesses three interpolation conditions, including displacements and its derivatives in x- and y-directions. As a result, linear trial functions can be constructed with only one node, and incomplete 1250055-13


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cubic trial functions can be constructed with two nodes. Thus, using the visible criterion, discontinuity can be easily embedded in an element. However, discontinuity in an element will cause problems in integration and postprocesses. The nodal relaxation method can be easily utilized with little modification to the FEM code, which is always preferred by engineers. Thus, the technique of embedding discontinuity into elements is not investigated in this paper, even though such as technique is useful for the TFEM. Acknowledgments Financial support from the National Nature Science Foundation of China (Grant No. 51005068), the Major State Basic Research Development program of China (973 Program, Grant No. 2011CB711105-1), the Independent Research Project of the State Key Lab of Traction Power of Southwest Jiaotong University (Grant No. 2012TPL T18) and the Open Research Fund Program of the State Key Lab of Advanced Design and Manufacturing for Vehicle Body (Grant No. 31115030) is gratefully acknowledged. Authors also give sincere thanks to Dr. Zheng C and Dr. Tang XH for their help, constructive discussion and collaboration. References Atluri, S. N. and Zhu, T. [1998] “A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics,” Comput. Mech. 2(2), 117–127. Belytschko, T., Lu, Y. Y. and Gu, L. [1995] “Crack propagation by element-free Galerkin methods,” Eng. Fract. Mech. 51(2), 295–315. Belytschko, T., Gracie, R. and Ventura, G. [2009] “A review of extended/generalized finite element methods for material model,” Model. Simul. Mater. Sci. Eng. 17(4), 1–24. Chen, L., Liu, G. R., Nourbakhsh, N. and Zeng, K. Y. [2010] “A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks,” Comput. Mech. 45(2–3), 109–125. Liu, G. R., Jiang, Y., Chen, L., Zhang, G. Y. and Zhang, Y. W. [2011] “A singular cellbased smoothed radial point interpolation method for fracture problems,” Comput. Struct. 89(13–14), 1378–1396. Liu, G. R., Nourbakhshnia, N., Chen, L. and Zhang, Y. W. [2010] “A novel general formulation for singular stress field using the ES-FEM method for the analysis of mixed-mode cracks,” Int. J. Comput. Meth. 7(1), 191–214. Liu, G. R. and Zhang, G. Y. [2009] “A normed G-space and weakened weak (W2 ) formulation of a cell-based smoothed point interpolation method,” Int. J. Comput. Meth. 6(1), 147–179. Melenk, J. M. and Babuˇska, I. [1996] “The partition of unity finite element method: Basic theory and applications,” Comput. Meth. Appl. Mech. Eng. 139(1–4), 289–314. Rabczuk, T. and Belytschko, T. [2004] “Cracking particles: A simplified meshfree method for arbitrary evolving cracks,” Int. J. Numer. Meth. Eng. 61(13), 2316–2343. Sukumar, N., Moes, N., Moran, B. and Belytschko, T. [2000] “Extended finite element method for three-dimensional crack modeling,” Int. J. Numer. Meth. Eng. 48(11), 1549–1570. Swenson, D. and Ingrafea, A. [1996] “Modeling mixed-mode dynamic crack propagation using finite elements: Theory and applications,” Comput. Mech. 3(6), 381–397. 1250055-14


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Tang, X. H., Wu, S. C., Zheng, C. and Zhang, J. H. [2009] “A novel virtual node method for polygonal elements,” Appl. Math. Mech. Engl. Ed. 30(10), 1233–1246. Wu, S. C. and Wu, Y. C. [2011] “ALOF: New 3D fatigue crack propagation analysis software,” Comput. Aided Eng. 20(1), 136–140 (in Chinese). Zheng, C., Wu, S. C., Tang, X. H. and Zhang, J. H. [2008] “A mesh-free poly-cell Galerkin (MPG) approach for problems of elasticity and fracture,” CMES 38(2), 149–178. Zheng, C., Wu, S. C., Tang, X. H. and Zhang, J. H. [2010] “A novel twice-interpolation finite element method for solid mechanics problems,” Acta Mech. Sin. 26(2), 265–278.

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a twice-interpolation finite element method (tfem) for ...

a twice-interpolation finite element method (tfem) for ...

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