Transient analysis of multiple parallel cracks under anti-plane dynamic loading

The problem of a homogeneous linear elastic body containing multiple non-collinear cracks under anti-plane dynamic loading is considered in this work. The cracks are simulated by distributions of dislocations and an integral equation relating tractions on the crack planes and the dislocation densities is derived. The integral equation in the Laplace transform domain is solved by the Gaussian–Chebyshev integration quadrature. The dynamic stress intensity factor associated with each crack tip is calculated by a numerical inverse Laplace scheme. Numerical results are given for one crack and two or three parallel cracks under normal incidence of a plane horizontally shear stress wave.     

Stress intensity factors of an inclined elliptical crack near a bimaterial interface

In this paper the stress intensity factors are discussed for an inclined elliptical crack near a bimaterial interface. The solution utilizes the body force method and requires Green’s functions for perfectly bonded semi-infinite bodies. The formulation leads to a system of hypersingular integral equation whose unknowns are three modes of crack opening displacements. In the numerical calculation, unknown body force densities are approximated by using fundamental density functions and polynomials. The results show that the present method yields smooth variations of stress intensity factors along the crack front accurately. Distributions of stress intensity factors are presented in tables and figures with varying the shape of crack, distance from the interface, and elastic modulus ratio. It is found that the inclined crack can be evaluated by the models of vertical and parallel cracks within the error of 24% even for the cracks very close to the interface.     

Interaction of parallel dielectric cracks in functionally graded piezoelectric materials

In this paper, the problem of two interacting parallel cracks in functionally graded piezoelectric materials under in-plane electromechanical loads is studied. The formulation is based on using Fourier transforms and modeling the cracks as distributed dislocations, and the resulting singular integral equations are solved with Chebyshev polynomials. A dielectric crack model considering the crack filling effect is adopted to describe the electric boundary conditions along crack surfaces. Numerical simulations are made to show the effect of material gradient, the geometry of interacting cracks, and crack position upon fracture parameters such as stress intensity factors, electric displacement intensity factor, and COD intensity factor. By considering the effect of a dielectric medium inside the crack and crack deformation, the results obtained from the dielectric crack model are always between those from the traditional crack models with physical limitation.     

Application of an elastoplastic boundary-element method to some fracture-mechanics problems

The boundary-integral equation (BIE) method for 3D elastic fracture mechanics has been extended to the elastoplastic problem. The formulation makes use of a special elastic Green's function for the crack, thereby eliminating the need to model the crack itself. Application of the general formulation is made to problems of localized or limited plasticity. Such problems occur through the local yield of stress concentrations, together with the plastic field of the crack tip. In these problems, the elastic stress intensity factor still provides a useful characterization for cyclic-crack-growth predictions. This paper reports on an accurate and efficient calculation procedure for crack-tip stress intensity factors for cracks in welds and prestressed bolt holes, where uncracked plastic strains are important. The use of the new algorithm for crack-tip plasticity modeling is explored for small- and large-scale plasticity conditions.     

Stress intensity factors for cracks of arbitrary shape near an interfacial boundary

Modes I, II and III stress intensity factors for a crack of arbitrary planar shape near a bimaterial interface are calculated. The solution utilizes the body-force method and requires Green's functions for perfectly bonded elastic half-spaces. The formulation leads to a system of two-dimensional singular integral equations whose solutions represent the three modes of crack opening displacement. Numerical examples of a semicircular or semielliptical crack terminating at the interface and circular or elliptical cracks contained in one material are given for both internal pressure and farfield tension.     

On the dynamic behaviour of interfacial cracks between a piezoelectric layer and an elastic substrate

Surface-bonded piezoelectric layers can be used as actuators/sensors for advanced structural applications. The current paper provides a theoretical study of the dynamic behaviour of interacting cracks between a piezoelectric layer and an elastic medium under antiplane mechanical loads. The electromechanical field of a single interfacial crack is determined first using Fourier transform technique and solving the resulting integral equations. This fundamental solution is then imple- mented into a pseudo-incident wave method to account for the interaction between different cracks. The dynamic behaviour of the resulting stress field is studied with special attention being paid to the stress intensity factors at the crack tips. Typical examples are provided to show the effect of the size and position of the cracks, the material combination and the loading frequency upon the stress intensity factors.     

Interaction effect of stress intensity factors for any number of collinear interface cracks

In this study, the stress intensity factors for any number of interface cracks are calculated for various spacings, elastic constants and number of cracks and the interaction effect of interface cracks is discussed. The problem is formulated as a system of singular integral equations on the basis of the body force method. In the numerical analysis, the unknown functions of the body force densities which satisfy the boundary conditions are expressed by the products of fundamental density functions and power series. Here, the fundamental density functions are chosen to express the stress field due to a single interface crack exactly. The accuracy of the present analysis is verified by comparing the present results with the results obtained by other researchers and examining the compliance with boundary conditions. The calculation shows that the present method gives rapidly converging numerical results for those problems as well as ordinary crack problems in homogeneous materials. The interaction effect of interface crack appears in a similar way to ordinary collinear cracks having the same geometrical condition and the maximum stress intensity factor is shown to be linearly related to the reciprocal of number of interface cracks. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dynamic fracture mechanics analysis of failure mode transitions along weakened interfaces in elastic solids

Dynamic fracture mechanics theory was employed to analyze the crack deflection behavior of dynamic mode-I cracks propagating towards inclined weak planes/interfaces in otherwise homogenous elastic solids. When the incident mode-I crack reached the weak interface, it kinked out of its original plane and continued to propagate along the weak interface. The dynamic stress intensity factors and the non-singular T-stresses of the incident cracks were fitted, and then dynamic fracture mechanics concepts were used to obtain the stress intensity factors of the kinked cracks as functions of kinking angles and crack tip speeds. The T-stress of the incident crack has a small positive value but the crack path was quite stable. In order to validate fracture mechanics predictions, the theoretical photoelasticity fringe patterns of the kinked cracks were compared with the recorded experimental fringes. Moreover, the mode mixity of the kinked crack was found to depend on the kinking angle and the crack tip speed. A weak interface will lead to a high mode-II component and a fast crack tip speed of the kinked mixed-mode crack.     

Cylindrical interface cracks in 1-3 piezocomposites

1-3 Piezocomposites are made by embedding piezoelectric fibers/rods in polymer matrix materials. Fiber–matrix interface fracture can affect the performance of piezocomposites. In this paper, axisymmetric interfacial cracks in piezocomposites are studied by considering an idealized model of a single piezoelectric fiber in a matrix material. The displacement discontinuity method is used to formulate the Mode I and II crack problems. The fundamental solutions required for DDM are derived explicitly by using the electroelastic field equations and Fourier integral transforms. The dependence of Mode I and II stress intensity factors of single and multiple interface cracks on fiber and matrix material properties, crack length and distance between cracks are investigated.     

BEM for crack‐inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξ_t and ξ_k in the transformed ξ‐plane in order to simulate SED intensity factors by the BEM. The least‐squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

Bonded elastic half-planes with an interface crack and a perpendicular intersecting crack that extends into the adjacent material—I

The solution is given for two bonded isotropic linearly elastic half-planes of different elastic properties having a crack along the interface as welt as a perpendicular crack in one of the half-planes which may intersect the interface crack. The appropriate integral equations are developed using displacement dislocations on the crack surfaces. Numerical results are presented for the stress intensity factors, strain energy release rate, stresses and displacements.     

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.     

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.     

The stress intensity factors of slightly undulating interface cracks of bimaterials

A modified interface crack with slightly undulating profile, which has a good agreement with reality and retains the simplicity of a mathematical model, is presented in this paper. This model is utilized to reveal some of the properties of uneven cracks, especially the stress intensity factors. As we know, many failures occurring in the interface are induced by crucial lateral stresses which are parallel to the interface. Hence, when the lateral stresses are much stronger than others, the corresponding solution is also derived for understanding how the lateral stresses affect the stress intensity factors as the crack is uneven. In the present paper, the Hilbert's problem enables different perturbed-interface cracks to be solved in an unified manner. Muskhelishvili's potential formulation is used to derive, by means of a perturbation analysis technique, an homogeneous and general Hilbert's problem.     

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.     

Numerical analysis of the inclusion-crack interactions using an integral equation

 A general-purpose integral formulation is proposed for the analysis of the interaction between inclusions and cracks embedded in an elastic isotropic homogeneous infinite medium subjected to a remote loading. This formulation is tailored for the inclusions of arbitrary shapes with the presence of cracks. The discretization is limited to the inclusions (with continuous quadratic triangular and quadrilateral elements) and the cracks (using discontinuous quadratic elements). For the calculation of the stress intensity factors at the crack tips, special crack tip elements are used to model the variation of the displacements near the crack tips. Maximum circumferential stress criterion is adopted to determine the crack propagating direction. Numerical results of benchmark examples are compared with other available methods. Received: 8 January 2002 / Accepted: 24 September 2002     

A Method to Extract Interface Stress Intensity Factors for Bimaterial Problems using Interlayers

A numerical method is presented here to determine stress intensity factors for interface cracks in plane, isotropic, elastic bimaterial fracture problems. The method relies on considering a companion problem wherein a very thin elastic interlayer with a crack, is artificially inserted between the two material regions of the original bimaterial problem. Modes I and II stress intensity factors are obtained for the companion problem using the modified virtual crack closure method. These stress intensity factors for the companion problem are then converted to the stress intensity factors for the original interface crack problem with the help of a universal relation. This universal relation between the stress intensity factors of the two problems is established by considering an asymptotic problem where the thickness of the interlayer is small compared with all other length scales. Two benchmark problems are considered to demonstrate the effectiveness of the interlayer approach for determining interface stress intensity factors.     

Stress intensity factors for an interface crack along an elliptical inclusion

The problem of a crack along the interface of an elliptical elastic inclusion embedded in an infinite plate subjected to uniform stresses at infinity is analyzed by the body force method. The crack tip stress intensity factors are calculated for various inclusion geometries and material combinations. Based on numerical results, the effect of the inclusion geometry on the stress intensity factors is investigated. It is found that for small interface cracks the stress intensity factors are mainly determined by the stresses, occurring at the crack center point before the crack initiation, and interface curvature radius alone.     

Anti-plane problem of periodic interface cracks in a functionally graded coating-substrate structure

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.     

含界面裂纹圆形夹杂的十二次对称二维准晶热应力分析

针对点热源作用下,无限大十二次对称二维准晶基体和圆形弹性夹杂界面之间含多条裂纹的问题进行了研究。基于复变函数分区全纯理论、留数定理、广义 Liouville 定理、Riemann-Schwarz 解析延拓定理及复应力函数奇性主部分析方法,获得了集中热源作用于准晶基体内任意一点时,准晶基体和圆形弹性夹杂内外温度场、声子场热应力的一般复势解。由此获得了含一条界面裂纹和两条界面裂纹时温度场以及声子场热应力的封闭形式解答,将所得结果与已有结果进行了对比,验证了该方法的有效性。最后通过数值算例分析了夹杂半径、点热源强度及裂纹角度对热应力和裂纹尖端热应力强度因子的影响规律。结果表明：随着热源强度的增大,裂纹尖端的声子场热应力也逐渐增大;随着裂纹角度的增大,裂纹尖端的声子场热应力强度因子变大;随着半径的增大,热应力强度因子的变化趋势越来越明显,并且取得的峰值越高,即裂纹角度和夹杂半径的增加,促进了裂纹的扩展。这些结论为准晶材料的结构设计和使用提供了科学依据。