求解双材料界面裂纹应力强度因子的扩展有限元法

基于双材料界面裂纹尖端的基本解,构造扩展有限元法(eXtended Finite Element Methods, XFEM)裂尖单元结点的改进函数。有限元网格剖分不遵从材料界面,考虑3种类型的结点改进函数:弱不连续改进函数、Heaviside改进函数和裂尖改进函数,建立XFEM的位移模式,给出计算双材料界面裂纹应力强度因子(Stress Intensity Factors, SIFs)的相互作用积分方法。数值结果表明:XFEM无需遵从材料界面剖分网格,该文的方法能够准确评价双材料界面裂纹尖端的SIFs。     

Cohesive and non-cohesive fracture by higher-order enrichment of XFEM

A comprehensive study is performed on the use of higher-order terms of the crack tip asymptotic fields as enriching functions for the eXtended FEM (XFEM) for both cohesive and traction-free cracks. For traction-free cracks, the Williams asymptotic field is used to obtain highly accurate stress intensity factors (SIFs), directly from the enriched degrees of freedom without any post-processing. The low accuracy of the results of the original research on this subject by Liu et al. [Int. J. Numer. Meth. Engng., 2004; 59:1103–1118] is remedied here by appropriate modifications of the enrichment scheme. The modifications are simple and can be easily included into an XFEM computer code. For cohesive cracks, the relevant asymptotic field is used, and two widely used criteria including the SIFs criterion and the stress criterion are examined for the crack growth simulation. Both linear and nonlinear cohesive laws are used. For the stress criterion, averaging is avoided due to the highly accurate crack tip approximation because of the higher-order enrichment. Then, a modified stress criterion is proposed, which is shown to be applicable to a wider class of problems. Several numerical examples, including straight and curved cracks, stationary and growing cracks, single and multiple cracks, and traction-free and cohesive cracks, are studied to investigate in detail the robustness and efficiency of the proposed enrichment scheme. Copyright © 2012 John Wiley & Sons, Ltd.     

Direct evaluation of stress intensity factors for curved cracks using Irwin's integral and XFEM with high‐order enrichment functions

This paper presents a comprehensive study on the use of Irwin's crack closure integral for direct evaluation of mixed‐mode stress intensity factors (SIFs) in curved crack problems, within the extended finite element method. The approach employs high‐order enrichment functions derived from the standard Williams asymptotic solution, and SIFs are computed in closed form without any special post‐processing requirements. Linear triangular elements are used to discretize the domain, and the crack curvature within an element is represented explicitly. An improved quadrature scheme using high‐order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. Furthermore, because the Williams asymptotic solution is derived for straight cracks, an appropriate definition of the angle in the enrichment functions is presented and discussed. This contribution is an important extension of our previous work on straight cracks and illustrates the applicability of the SIF extraction method to curved cracks. The performance of the method is studied on several circular and parabolic arc crack benchmark examples. With two layers of elements enriched in the vicinity of the crack tip, striking accuracy, even on relatively coarse meshes, is obtained, and the method converges to the reference SIFs for the circular arc crack problem with mesh refinement. Furthermore, while the popular interaction integral (a variant of the J‐integral method) requires special auxiliary fields for curved cracks and also needs cracks to be sufficiently apart from each other in multicracks systems, the proposed approach shows none of those limitations. Copyright © 2017 John Wiley & Sons, Ltd.     

Direct evaluation of accurate coefficients of the linear elastic crack tip asymptotic field

An improvement to the extended finite element method (XFEM) and generalised finite element method (GFEM) is introduced. It enriches the finite element approximation of the crack tip node as well as its surrounding nodes with not only the first term but also the higher order terms of the linear elastic crack tip asymptotic field using a partition of unity method (PUM). Numerical results show that together with a reduced quadrature rule to the enriched elements, this approach predicts accurate stress intensity factors (SIFs) directly (i.e. without extra post‐processing) after constraining the enriched nodes properly. However, it does not predict accurately the coefficients of the higher order terms. For that a hybrid crack element (HCE) is introduced which is powerful and convenient not only for directly determining the SIF but also the coefficients of higher order terms in the plane linear elastic crack tip asymptotic field. Finally, the general expressions for the coefficients of the second to fifth terms of the linear elastic crack tip asymptotic field of three‐point bend single edge notched beams (TPBs) with span to depth ratios widely used in testing are extended to very deep cracks with the use of the HCE.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi‐materials

The extended finite element method (XFEM) is improved to directly evaluate mixed mode stress intensity factors (SIFs) without extra post‐processing, for homogeneous materials as well as for bimaterials. This is achieved by enriching the finite element (FE) approximation of the nodes surrounding the crack tip with not only the first term but also the higher order terms of the crack tip asymptotic field using a partition of unity method (PUM). The crack faces behind the tip(s) are modelled independently of the mesh by displacement jump functions. The additional coefficients corresponding to the enrichments at the nodes of the elements surrounding the crack tip are forced to be equal by a penalty function method, thus ensuring that the displacement approximations reduce to the actual asymptotic fields adjacent to the crack tip. The numerical results so obtained are in excellent agreement with analytical and numerical results available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical simulation of bi-material interfacial cracks using EFGM and XFEM

In this paper, bi-material interfacial cracks have been simulated using element free Galerkin method (EFGM) and extended finite element method (XFEM) under mode-I and mixed mode loading conditions. Few crack interaction problems of dissimilar layered materials are also simulated using extrinsic partition of unity enriched approach. Material discontinuity has been modeled by a signed distance function whereas strong discontinuity has been modeled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The results obtained by EFGM and XFEM for bi-material edge and center cracks are compared with those available in literature. In order to check the validity of simulations, the results have been obtained for two different ratio of Young’s modulus.     

Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method

New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near‐tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack‐tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed‐mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M‐integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors. Copyright © 2006 John Wiley & Sons, Ltd.     

Interaction integrals for fracture analysis of functionally graded magnetoelectroelastic materials

This paper presents the domain form of interaction integrals based on three independent formulations for computation of stress intensity factors, electric displacement intensity factors and magnetic induction intensity factors for cracks in functionally graded magnetoelectroelastic materials. Conservation integrals of J-type are derived based on the governing equations for magnetoelectroelastic media and the crack tip asymptotic fields of homogeneous magnetoelectroelastic medium as auxiliary fields. Each of the formulations differs in the way auxiliary fields are imposed in the evaluation of interaction integrals and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded magnetoelectroelastic medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.     

Partition of unity enrichment for bimaterial interface cracks

Partition of unity enrichment techniques are developed for bimaterial interface cracks. A discontinuous function and the two‐dimensional near‐tip asymptotic displacement functions are added to the finite element approximation using the framework of partition of unity. This enables the domain to be modelled by finite elements without explicitly meshing the crack surfaces. The crack‐tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The concept of partition of unity facilitates the incorporation of the oscillatory nature of the singularity within a conforming finite element approximation. The mixed‐mode (complex) stress intensity factors for bimaterial interfacial cracks are numerically evaluated using the domain form of the interaction integral. Good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized. Copyright © 2004 John Wiley & Sons, Ltd.     

An optimally convergent discontinuous Galerkin‐based extended finite element method for fracture mechanics

The extended finite element method (XFEM) enables the representation of cracks in arbitrary locations of a mesh. We introduce here a variant of the XFEM rendering an optimally convergent scheme. Its distinguishing features are as follows: (a) the introduction of singular asymptotic crack tip fields with support on only a small region around the crack tip (the enrichment region), (b) only one and two enrichment functions are added for anti‐plane shear and planar problems, respectively and (c) the relaxation of the continuity between the enrichment region and the rest of the domain, and the adoption of a discontinuous Galerkin (DG) method therein. The method is provably stable for any positive value of a stabilization parameter, and by weakly enforcing the continuity between the two regions it eliminates ‘blending elements’ partly responsible for the suboptimal convergence of some early XFEMs. Moreover, the particular choice of enrichment functions results in a surprisingly sparse stiffness matrix that remains reasonably conditioned as the mesh is refined. More importantly, the stress intensity factors can be extracted with a satisfactory accuracy as primary unknowns. Quadrature strategies required for the optimal convergence are also discussed. Finally, the DG method was modified to retain stability based on an inf‐sup condition. Copyright © 2009 John Wiley & Sons, Ltd.     

二维压电材料动强度因子的扩展有限元计算

采用扩展有限元求解二维弹性压电材料动断裂问题。扩展有限元的网格独立于裂纹,因此网格生成可大大地简化,且裂纹扩展时不需重构网格。采用相互作用积分技术计算动强度因子。比较了标准的力裂尖加强函数和力-电裂尖加强函数对动强度因子的影响,结果表明标准的力裂尖加强函数能有效地分析压电材料动断裂问题。分析了极化方向对动强度因子的影响。数值分析表明采用扩展有限元获得的动强度因子与其他数值方法解吻合得很好。     

Extended finite element method for three‐dimensional crack modelling

An extended finite element method (X‐FEM) for three‐dimensional crack modelling is described. A discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three‐dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Representation of singular fields without asymptotic enrichment in the extended finite element method

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi‐analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix, and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A Heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended SBFEM. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code. Copyright © 2013 John Wiley & Sons, Ltd.     

Coefficients of the crack tip asymptotic field for wedge splitting specimens

The stress intensity factor (SIF) and the coefficients of higher order terms of the crack tip asymptotic field of typical wedge splitting specimens with two different loading arrangements are directly computed using a hybrid crack element. Accurate analytical expressions for the first five terms are obtained by fitting the computed data. Numerical results show that the coefficients of terms higher than three are negligibly small, this may explain that the wedge splitting specimen is more stable than other geometries. The first five terms are not sensitive to support conditions. However, for short cracks coefficients of terms, except the SIF, are quite sensitive to the loading arrangement even when the loads are statically equivalent.     

Interaction integrals for thermal fracture of functionally graded piezoelectric materials

This paper presents domain form of the interaction integrals based on three independent formulations for computation of the stress intensity factors and electric displacement intensity factor for cracks in functionally graded piezoelectric materials subjected to steady-state thermal loading. Each of the formulation differs in the way auxiliary fields are imposed in the evaluation of interaction integral and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded piezoelectric medium.     

Mixed-mode fracture analysis of composite bonded joints considering adhesives of different ductility

We propose a method for simulating linear elastic crack growth through an isogeometric boundary element method directly from a CAD model and without any mesh generation. To capture the stress singularity around the crack tip, two methods are compared: (1) a graded knot insertion near crack tip; (2) partition of unity enrichment. A well-established CAD algorithm is adopted to generate smooth crack surfaces as the crack grows. The M integral and \(J_k\) integral methods are used for the extraction of stress intensity factors (SIFs). The obtained SIFs and crack paths are compared with other numerical methods.     

An asymptotic analysis of dynamic crack growth in rubber

Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks are important for several reasons. For example, they facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. The present study focuses on the asymptotic fields in front of a crack that propagates dynamically in rubber. Static analyses of this type of problem have been made in previous studies. In order to be able to compare the present results with these earlier studies, the constitutive model from Knowles and Sternberg (J. Elast. 3:67–107, 1973) was adopted. It is assumed that viscoelastic stresses become negligible compared with the singular elastic stresses close to the crack tip. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. However, for a neoHookean type of model, inertia comes into play and causes a maximum theoretical crack speed that equals the shear wave speed.     

Higher order terms of the crack tip asymptotic field for a wedge-splitting specimen

The coefficients of the crack tip asymptotic field of a typical wedge-splitting specimen are computed using a hybrid crack element (HCE), which has the potential to directly calculate not only the stress intensity factor (SIF) but also the coefficients of the higher order terms of the crack tip asymptotic field. The approximate closed-form expression for SIF proposed by Guinea et al. (1996) is calibrated by the results of the HCE. Approximate expressions for the second and third order terms for the wedge-splitting specimen are obtained by fitting the computed data. Numerical results show that the coefficients for terms higher than three are negligibly small, thus the wedge-splitting specimen is more stable than other geometries.     

XFEM analysis of the effects of voids,inclusions and other cracks on the dynamic stress intensity factor of a major crack

This paper is devoted to the extraction of the dynamic stress intensity factor (DSIF) for structures containing multiple discontinuities (cracks, voids and inclusions) by developing the extended finite element method (XFEM). In this method, four types of enrichment functions are used in the framework of the partition of unity to model interface discontinuity within the classical finite element method. In this procedure, elements that include a crack segment, the boundary of a void or the boundary of an inclusion are not required to conform to discontinuous edges. The DSIF is evaluated by the interaction integral. After the effectiveness of the implemented XFEM program is verified, the effects of voids, inclusions and other cracks on the DSIF of a stationary major crack are investigated by using XFEM. The results show that the dynamic effects have an influence on the path independence of the interaction integral, and these voids, inclusions and other cracks have a significant effect on the DSIF of the major crack.