Higher order tip enrichment of eXtended Finite Element Method in thermoelasticity

An eXtended Finite Element Method (XFEM) is presented that can accurately predict the stress intensity factors (SIFs) for thermoelastic cracks. The method uses higher order terms of the thermoelastic asymptotic crack tip fields to enrich the approximation space of the temperature and displacement fields in the vicinity of crack tips—away from the crack tip the step function is used. It is shown that improved accuracy is obtained by using the higher order crack tip enrichments and that the benefit of including such terms is greater for thermoelastic problems than for either purely elastic or steady state heat transfer problems. The computation of SIFs directly from the XFEM degrees of freedom and using the interaction integral is studied. Directly computed SIFs are shown to be significantly less accurate than those computed using the interaction integral. Furthermore, the numerical examples suggest that the directly computed SIFs do not converge to the exact SIFs values, but converge roughly to values near the exact result. Numerical simulations of straight cracks show that with the higher order enrichment scheme, the energy norm converges monotonically with increasing number of asymptotic enrichment terms and with decreasing element size. For curved crack there is no further increase in accuracy when more than four asymptotic enrichment terms are used and the numerical simulations indicate that the SIFs obtained directly from the XFEM degrees of freedom are inaccurate, while those obtained using the interaction integral remain accurate for small integration domains. It is recommended in general that at least four higher order terms of the asymptotic solution be used to enrich the temperature and displacement fields near the crack tips and that the J- or interaction integral should always be used to compute the SIFs.     

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.     

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.     

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.     

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.     

曲线裂纹裂尖SIFs等效分析的广义参数Williams单元确定方法

基于Williams级数建立的含直线裂纹平面弹性体裂尖应力强度因子（Stress Intensity Factors，简记SIFs）分析的广义参数Williams单元（简记W单元），因裂纹面必须满足σ_θ=0、τ_ρθ=0（θ=±π）的边界条件，而曲线裂纹面无法满足该边界条件使得W单元不能直接适用。该文针对此问题，将含曲线裂纹平面弹性体裂尖局部区域进行等效处理，以裂尖为圆心，选取合适的半径建立等效区，即截取斜率呈单调变化的曲线微段，在微段两端引切线交叉成折线段，以该折线段近似代替曲线微段。算例分析表明:当等效区的尺寸取建议值时，该文方法的计算结果精度较高，证明了该文改进方法在分析曲线裂纹尖端SIFs的合理性，同时给出了等效区尺寸的建议值，即ψ_min=5π/6，且等效区截取曲线长度宜大于原曲线长度的1/5。     

A unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A new unified and integrated method for the numerical‐analytical calculation of J_k‐integrals of an in‐plane traction free interfacial crack in homogeneous orthotropic and isotropic bimaterials is presented. The numerical algorithm, based on the boundary element crack shape sensitivities, is generic and flexible. It applies to both straight and curved interfacial cracks in anisotropic and/or isotropic bimaterials. The shape functions of semidiscontinuous quadratic quarter point crack tip elements are correctly scaled to adapt the singular oscillatory near tip field of tractions. The length of crack is designated as the design variable to compute the strain energy release rate precisely. Although an analytical equation relating J₁ and stress intensity factors (SIFs) exists, a similar relation for J₂ in debonded anisotropic solids for decoupling SIFs is not available. An analytical expression was recently derived by this author for J₂ in aligned orthotropic/orthotropic bimaterials with a straight interface crack. Using this new relation and the present computed J_k values, the SIFs can be decoupled without the need for an auxiliary equation. Here, the aforementioned analytical relation is reconstructed for cubic symmetry/isotropic bimaterials and used with the present numerical algorithm. An example with known analytical SIFs is presented. The numerical and analytical magnitudes of J_k for an interface crack in orthotropic/orthotropic and cubic symmetry/isotropic bimaterials show an excellent agreement.     

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.     

An over‐deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis

An over‐deterministic method has been employed for calculating the stress intensity factors (SIFs) as well as the coefficients of the higher‐order terms in the Williams series expansions in cracked bodies, using the conventional finite element analysis. For a large number of nodes around the crack tip, an over‐determined set of simultaneous linear equations is obtained, and using the fundamental concepts of the least‐squares method, the coefficients of the Williams expansion can be calculated for pure mode I, pure mode II and mixed mode I/II conditions. A convergence study has been conducted to examine the effects of the number of nodes used, the number of terms in Williams expansion and the distance of the selected nodes from the crack tip, on the accuracy of the results. It is shown that the simple method presented in this paper, yields accurate results even for coarse finite element meshes or in the absence of singular elements. The accuracy of SIFs and the coefficients of higher‐order terms are validated by using the available results in the literature.     

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.     

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.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.     

Hypersingular quarter-point boundary elements for crack problems

The present paper deals with the study and effective implementation for Stress Intensity Factor computation of a mixed boundary element approach based on the standard displacement integral equation and the hypersingular traction integral equation. Expressions for the evaluation of the hypersingular integrals along general curved quadratic line elements are presented. The integration is carried out by transformation of the hypersingular integrals into regular integrals, which are evaluated by standard quadratures, and simple singular integrals, which are integrated analytically. The generality of the method allows for the modelling of curved cracks and the use of straight line quarter-point elements. The Stress Intensity Factors can be computed very accurately from the Crack Opening Displacement at collocation points extremely close to the crack tip. Several examples with different crack geometries are analyzed. The computed results show that the proposed approach for Stress Intensity Factors evaluation is simple, produces very accurate solutions and has little dependence on the size of the elements near the crack tip.     

On the computation of two-dimensional stress intensity factors using the boundary element method

General two-dimensional linear elastic fracture problems are investigated using the boundary element method. The √r displacement and 1/√r traction behaviour near a crack tip are incorporated in special crack elements. Stress intensity factors of both modes I and II are obtained directly from crack-tip nodal values for a variety of crack problems, including straight and curved cracks in finite and infinite bodies. A multidomain approach is adopted to treat cracks in an infinite body. The body is subdivided into two regions: an infinite part with a finite hole and a finite inclusion. Numerical results, compared with exact solution whenever possible, are accurate even with a coarse discretization.     

A non-singular boundary integral formula for frequency domain analysis of the dynamic <Emphasis Type="Italic">T</Emphasis>-stress

A non-singular 2-D boundary integral equation (BIE) in the Fourier-space frequency domain for determining the dynamic T-stress (DTS) is presented in this paper. This formulation, based upon the Fourier transform of the asymptotic expansion for the stress field in the vicinity of a crack tip, can be conveniently implemented as a post-processing step in a frequency-domain boundary element analysis of cracks. The proposed BIE is accurate as it can be directly collocated at the crack tip in question. The technique is also computationally effective as it simply requires a similar computing effort as that used in determining the dynamic stress components at an interior point of a domain. Five numerical examples involving both straight and curved cracks are studied to validate the proposed technique. For the frequency domain analysis of the DTS in these examples, the exponential window method is employed to obtain its time history.     

A Straightforward Direct Traction Boundary Integral Method for Two-Dimensional Crack Problems Simulation of Linear Elastic Materials

This paper presents a direct traction boundary integral equation method (DTBIEM) for two-dimensional crack problems of materials. The traction boundary integral equation was collocated on both the external boundary and either side of the crack surfaces. The displacements and tractions were used as unknowns on the external boundary, while the relative crack opening displacement (RCOD) was chosen as unknowns on either side of crack surfaces to keep the single-domain merit. Only one side of the crack surfaces was concerned and needed to be discretized, thus the proposed method resulted in a smaller system of algebraic equations compared with the dual boundary element method (DBEM). A new set of crack-tip shape functions was constructed to represent the strain field singularity exactly, and the SIFs were evaluated by the extrapolation of the RCOD. Numerical examples for both straight and curved cracks are given to validate the accuracy and efficiency of the presented method.     

Modeling of interface cracks in fiber-reinforced composites with the presence of interphases using the boundary element method

An advanced boundary element method (BEM) with thin-body capabilities was developed recently for the study of interphases in fiber-reinforced composite materials (Y.J. Liu, N. Xu and J.F. Luo, Modeling of interphases in fiber-reinforced composites under transverse loading using the boundary element method, ASME J. Appl. Mech. 67 (2000) 41–49). In this BEM approach, the interphases are modeled as thin elastic layers based on the elasticity theory, as opposed to spring-like models in the previous BEM and some FEM work. In the present paper, this advanced BEM is extended to study the interface cracks at the interphases in the fiber-reinforced composites. These interface cracks are curved cracks between the fiber and matrix, with the presence of the interphases. Stress intensity factors (SIFs) for these interface cracks are evaluated based on the developed models. The BEM approach is validated first using available analytical and other numerical results for curved cracks in a single material and straight interface cracks between two materials. Then, the interface cracks at the interphases of fiber-reinforced composites are studied and the effects of the interphases (such as the thickness and materials) on the SIFs are investigated. As a special case, results of the SIFs for sub-interface cracks are also presented. It is shown that the developed BEM is very accurate and efficient for the interface crack analyses, and that the properties of the interphases have significant influences on the SIFs for interface cracks in fiber-reinforced composites.     

An enriched element‐failure method (REFM) for delamination analysis of composite structures

This paper develops an enriched element‐failure method for delamination analysis of composite structures. This method combines discontinuous enrichments in the extended finite element method and element‐failure concepts in the element‐failure method within the finite element framework. An improved discontinuous enrichment function is presented to effectively model the kinked discontinuities; and, based on fracture mechanics, a general near‐tip enrichment function is also derived from the asymptotic displacement fields to represent the discontinuity and local stress intensification around the crack‐tip. The delamination is treated as a crack problem that is represented by the discontinuous enrichment functions and then the enrichments are transformed to external nodal forces applied to nodes around the crack. The crack and its propagation are modeled by the ‘failed elements’ that are applied to the external nodal forces. Delamination and crack kinking problems can be solved simultaneously without remeshing the model or re‐assembling the stiffness matrix with this method. Examples are used to demonstrate the application of the proposed method to delamination analysis. The validity of the proposed method is verified and the simulation results show that both interlaminar delamination and crack kinking (intralaminar crack) occur in the cross‐ply laminated plate, which is observed in the experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional simulation of crack with curved front with direct estimation of stress intensity factors

This paper consists of an extension of simulation with direct estimation of stress intensity factors to the three‐dimensional case. Here, it combines X‐FEM with localized multigrids and direct estimation of quantities of interest along the crack front (SIF, T‐stress, etc.) based on crack tip asymptotic series expansion. In practice, a three‐dimensional patch is introduced locally with a truncated basis of Williams series expansion and is linked in a weak sense with the X‐FEM localized multigrids. Some examples (with available analytical solutions) illustrate the efficiency and the robustness of the method. These examples consider planar cracks with curved front, but the proposed method aims to apply to any continuously curved crack. Copyright © 2014 John Wiley & Sons, Ltd.