An effective numerical method for an interfacial crack in a finite bi-material plate

In this paper an effective numerical method is presented for analyzing the stress intensity factors associated with the stress field near a partially debonded interface in a finite bi-material plate. The strees functions are assumed such that they can represent the stress singularity at the crack tips, satisfying not only the equilibrium equations in the domain, but also the stress and displacement conditions on the crack surfaces and across the interface. Therefore, only the boundary conditions of the plate need be considered, and they can be satisfied approximately by the Boundary Collocation Method. Numerical examples demonstrated that the proposed method gives satisfactory results and has many advantages compared to other methods.     

A general method for multiple crack problems in a finite plate

A novel method for the multiple crack problems in a finite plate is proposed in this paper. The basic stress functions of the solution consist of two parts. One is the Fredholm integral equation solution for the crack problem in an infinite plate, and the other is that of the weighted residual method for general plane problems. The combined stress functions are used in the analysis and the boundary conditions on the crack surfaces and the boundary are considered. After the coefficients of the functions have been determined, the stress intensity factors (SIF) at the crack tips can be calculated. Some numerical examples are given and it was observed that when the cracks are very short, the results compare very favorably with the existing results for an infinite plate. Furthermore, the influence of the boundary can be considered. This method can be used for arbitrary multiple crack problems in a finite plate.     

A definition and evaluation procedure of generalized stress intensity factors at cracks and multi-material wedges

This paper proposes a definition of generalized stress intensity factors that includes classical definitions for crack problems as special cases. Based on the semi-analytical solution obtained from the scaled boundary finite-element method, the singular stress field is expressed as a matrix power function with its dimension equal to the number of singular terms. Not only real and complex power singularities but also power-logarithmic singularities are represented in a unified expression without explicitly determining the type of singularity. The generalized stress intensity factors are evaluated directly from the scaled boundary finite-element solution for the singular stress field by following standard stress recovery procedures in the finite element method. The definition and evaluation procedure are valid to multi-material wedges composed of any number of isotropic and anisotropic materials. Numerical examples, including a cracked homogeneous plate, a bimaterial plate with an interfacial crack, a V-notched bimaterial plate and a crack terminating at a material interface, are analyzed. Features of this unified definition are discussed.     

Edge function analysis of stress intensity factors in cracked anisotropic plates

The edge function method, which involves the use of analytic solutions to model field behavior in the various parts of an elastic region, is applied to the analysis of a finite anisotropic plate with a single crack. Analytical solutions for the stress singularities at each crack tip facilitate the inexpensive calculation of accurate values of the stress intensity factors. A boundary Galerkin variational principle is used to match the boundary conditions. The method is applicable to isotropic and anisotropic materials and is demonstrated for a number of fracture problems involving variation of the crack position, crack orientation and material orientation. For the range of geometries examined in this paper, the calculated values of the stress intensity factors do not show a major dependence on the material anisotropy. The formulation of the method makes it easily applicable to the study of the interaction of several cracks and also to a limited study of crack propagation or damage development in a composite laminate.     

Boundary element analysis of fracture mechanics in anisotropic bimaterials

This paper presents a boundary element formulation for the analysis of linear elastic fracture mechanics problems involving anisotropic bimaterials. The most important feature associated with the present formulation is that it is a single domain method, and yet it is accurate, efficient and versatile. In this formulation, the displacement integral equation is collocated on the uncracked boundary only, and the traction integral equation is collocated on one side of the crack surface only. The complete Green's functions for anisotropic bimaterials are also derived and implemented into the boundary integral formulation so that discretization along the interface can be avoided except for the interfacial crack part. A special crack-tip element is introduced to capture exactly the crack-tip behavior.Numerical examples are presented for the calculations of stress intensity factors for a straight crack with various locations in infinite bimaterials. It is found that very accurate results can be obtained by the proposed method even with relatively coarse discretization. Numerical results also show that material anisotropy can greatly affect the stress intensity factor.     

Hybrid stress finite element analysis of bending of a plate with a through flaw

A hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through-the-thickness crack is presented. Reissner's sixth-order plate theory including the effects of transverse shear deformation is used. The dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements. The stress intensity factors can be calculated directly from the crack tip singular stress solution functions. The effects of the plate thickness, the ratio between the crack size and the inplane dimension of the plate, and the singular element size on the stress intensity factor solution are investigated. The effects of the explicit enforcement of traction-free conditions along crack surfaces, which are the natural boundary conditions in the present hybrid stress finite element model, are also investigated. The numerical results of bending of a plate with a straight central crack compare favourably with analytical solutions. It is also found that the explicit enforcement of traction-free conditions along crack surfaces is mandatory to obtain meaningful results for the Mode I type of bending stress intensity factor.     

A new boundary integral equation method for analysis of cracked linear elastic bodies

Abstract

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.     

Kirchhoff型裂纹板的边界元法

本文用的复变函数理论,导出了含裂板弯曲问题的基本解。该基本解满足自由裂纹的边界条件。将其引入直接或间接积分方程中,只要对板的外边界进行离散,就可计算有限尺寸裂纹板的弯曲问题。算例表明,本文所得到的基本解用以求解裂纹板弯曲问题划分的单元较少,精度较高。本文的方法还可用以求解含有形状比较复杂的裂纹或孔洞板弯曲问题的基本解。     

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.     

Numerical simulations of cracked plate using XIGA under different loads and boundary conditions

This article deals with the numerical simulation of cracked plate using extended isogeometric analysis (XIGA) under different loads and boundary conditions. The plate formulation is done using first-order shear deformation theory. The crack faces are modeled by the Heaviside function, whereas the singularity in stress field at the crack tip is modeled by crack tip enrichment functions. The stress intensity factors for the cracked plate are numerically computed using a domain-based interaction integral. The results obtained by XIGA for the center and edge crack plate are compared with extended finite element method and/or literature results for different types of loads and boundary conditions.     

各向异性板应力强度因子的变分解法

本文在文献的基础上给出了含边缘裂纹各向异性板的应力与位移展开式, 并应用广义变分原理求解含对称中心裂纹、对称边缘裂纹与对称孔边裂纹正交各向异性板的应力强度因子。所得结果的收敛性是令人满意的。同时在各向同性的情况下, 所得结果与文献之结果非常符合。      

Analysis of cracked plates using an iterative hybrid technique of boundary element method and distributed dislocation method

An iterative hybrid technique of boundary element method (BEM) and distributed dislocation method (DDM) is introduced for solving two dimensional crack problems. The technique decomposes the problem into (n + 1) subsidiary problems where n is the number of crack branches. The required solution will be the sum of these (n + 1) solutions. The first subsidiary problem is to find the stress distribution induced in the plate in the absence of the crack using BEM. All of the remaining subsidiary problems, are stress disturbance ones that will be solved using DDM. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the plate boundaries and crack faces until all of the boundary conditions are satisfied.     

On higher order parameters in cracked composite plates under far‐field pure shear

This paper presents the analytical solution of the crack tip fields as well as the crack parameters in an infinitely large composite plate with a central crack subjected to pure shear loading. To this end, the complex variable method is employed to formulate an asymptotic solution for the crack tip fields in an anisotropic plane. Using a stress‐based definition of the crack tip modes of loading, only the mode II crack parameters are found to be non‐zero under pure shear load. Special focus is given to the determination of the higher order parameters of the crack tip asymptotic field, particularly the first non‐singular term, ie, the T‐stress. Unlike the isotropic materials, in which the T‐stress is zero under pure shear, it is found that the T‐stress is non‐zero for the case of anisotropic materials, being the only material‐dependent crack tip stress parameter. The veracity of our exact crack tip fields is assessed and verified through a comparison made with respect to the finite element (FE) solution. Finally, we demonstrate the significance of the T‐stress on stresses near the crack tip in composite plates under pure shear loads.     

Application of the body force method to the calculation of stress intensity factors for a crack in the arbitrarily shaped plate

The body force method was applied to the stress analysis of the arbitrarily shaped plate with an inner or edge crack. Stress intensity factors were obtained for various problems and its accuracy was discussed.The resultant forces along the boundary were satisfied as boundary condition. Moreover, it was made clear that in order to obtain the solution, the equilibrium conditions for the total body forces distributed along the boundary must be satisfied. The basic stress functions used for the analysis not only express the stress field of a point force but also satisfy the free surface condition of the crack.     

The anisotropic R-criterion for crack initiation

The anisotropic nature of mixed modes I-II crack tip plastic core region and crack initiation is investigated in this study using an angled crack plate problem under various loading conditions. Hill’s anisotropic yield criterion along with singular elastic stress field at the crack tip is employed to obtain the non-dimensional variable-radius crack tip plastic core region. In addition, the R-criterion for crack initiation proposed by the authors for isotropic materials is also extended to include anisotropy. The effect of Hill’s anisotropic constants on the shape and size of the crack tip plastic core region and crack initiation angle is presented for both plane stress and plane strain conditions at the crack tip. The study shows a significant effect of anisotropy on the crack tip core region and crack initiation angle and calls for further development of anisotropic crack initiation theory.     

求解含裂纹各向异性板与单向复合材料板应力强度因子的变分解法

本文推导了含边缘裂纹各向异性板与单向复合材料板的应力与位移的函数项级数表达式。该级数每一项均满足平衡方程、协调方程与裂纹表面静力边界条件。通过最小势能原理的变分方程来满足其余静力边界条件以确定级数中的待定系数,从而求得应力强度因子。最后,本文给出了长条板拉伸试件、紧凑拉伸试件与三点弯曲试件的计算结果。收敛是迅速的,几乎所有算例都能满足两位收敛。     

不同孔口形状对含孔复合材料板孔边应力状态影响的研究

对复合材料结构而言,孔口边界条件的建立和处理比金属材料要复杂得多。针对含不同孔形的复合材料板,根据非均质各向异性弹性理论和复变函数理论,通过保角映射方法建立精确的边界条件,解决了某些复杂孔形的边值问题。得到了含圆、矩形和六边形孔复合材料板孔边应力的解析解。并针对不同孔形在受外荷载作用的情况下的应力状态,以及它们对孔边应力集中系数的影响进行了探讨。     

Effect of width and length on stress intensity factors of internally cracked plates under various boundary conditions

This paper is concerned with the analysis of stress intensity factors of a strip with a longitudinal crack subject to tension and bending along its edges, and the tension of rectangular plates with a central crack. For both problems three types of boundary conditions, that is, stress conditions, displacement conditions and their combinations are considered.Analysis is based on Laurent expansions of the complex potentials satisfying the stress free relations along the crack.The expansion coefficients are determined from boundary conditions along outer edges, by using a perturbation technique in the first problem and a boundary collocation procedure based on resultant forces and mean displacements in the second problem. Numerical calculations are performed for various plate configurations, and the results are summarized in forms ready for practical use. The accuracy of numerical results are also examined, and they are regarded as correct up to four figures.     

Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method

A new formulation of the boundary element method (BEM) is proposed in this paper to calculate stress intensity factors for cracked 2-D anisotropic materials. The most outstanding feature of this new approach is that the displacement and traction integral equations are collocated on the outside boundary of the problem (no-crack boundary) only and on one side of the crack surfaces only, respectively. Since the new BEM formulation uses displacements or tractions as unknowns on the outside boundary and displacement differences as unknowns on the crack surfaces, the formulation combines the best attributes of the traditional displacement BEM as well as the displacement discontinuity method (DDM). Compared with the recently proposed dual BEM, the present approach doesn't require dua elements and nodes on the crack surfaces, and further, it can be used for anisotropic media with cracks of any geometric shapes. Numerical examples of calculation of stress intensity factors were conducted, and excellent agreement with previously published results was obtained. The authors believe that the new BEM formulation presented in this paper will provide an alternative and yet efficient numerical technique for the study of cracked 2-D anisotropic media, and for the simulation of quasi-static crack propagation.