Multiple holes,cracks, and inclusions in anisotropic viscoelastic solids

By using the elastic–viscoelastic correspondence principle, the problems with multiple holes, cracks, and inclusions in two-dimensional anisotropic viscoelastic solids are solved for the cases with time-invariant boundaries. Based upon this principle and the existing methods for the problems with anisotropic elastic materials, two different approaches are proposed in this paper. One is concerned with an analytical solution for certain specific cases such as two collinear cracks, collinear periodic cracks, and interaction between inclusion and crack, and the other is a boundary-based finite element method for the general cases with multiple holes, cracks, and inclusions. The former considers only specific cases in infinite domain and can be used as a reference for any other numerical methods, and the latter is applicable to any combination of holes, cracks and inclusions in finite domain, whose number, size and orientation are not restricted. Unlike the conventional finite element method or boundary element method which usually needs very fine meshes to get convergence solutions, in the proposed boundary-based finite element method no meshes are needed along the boundaries of holes, cracks and inclusions. To show the accuracy and efficiency of these two proposed approaches, several representative examples are implemented analytically and numerically, and they are compared with each other or with the solutions obtained by the finite element method.     

A line dislocation interacting with a semi-infinite crack in piezoelectric solid

Closed-form analytical solutions are presented for the physical problem of a semi-infinite crack interacting with a line dislocation under the loading of a line force and a line charge in two-dimensional infinite anisotropic piezoelectric medium. The crack can be a conventional Griffith crack or an anti-crack (a rigid line inhomogeneity). Using the extended Stroh formalism and perturbation technique, the explicit expressions of the field intensity factors and the image force on the dislocation are computed as functions of dislocation location and material constants. The results are discussed and compared with those from special cases existed in the literature. The analytical solutions obtained can be applied to studying interacting cracks and crack branching problems in piezoelectric solids.     

An Altgernating Method Based on the VNA Solution for Analysis of Damaged Solid Containing Arbitrarily Distributed Elliptical Microcracks

This paper presents the development of an alternating method for the interaction analysis of arbitrary distributed numerous elliptical microcracks. The complete analytical solutions (VNA solutions) for a single elliptical crack in an infinite solid, subject to arbitrary crack-face tractions, are implemented in the present alternating method, together with the coordinate transformations for stress tensors. First, the present method is verified by solving the problems of two interacting cracks for which accurate numerical solutions have been obtained previously. Next, the present method demonstrates obtaining efficient and accurate solutions for the problems of many interacting elliptical cracks, which cannot be solved in a practical sense by the ordinary numerical methods such as the finite element method. Furthermore, damaged solids containing periodically distributed elliptical microcracks are analyzed by the present alternating method. The effective elastic moduli are evaluated for varying microcrack density. Detailed structures of the interactions in the damaged solids are visualized and clarified. This revised version was published online in August 2006 with corrections to the Cover Date.     

An alternating method based on the VNA solution for analysis of damaged solid containing arbitrarily distributed elliptical microcracks

This paper presents the development of an alternating method for the interaction analysis of arbitrary distributed numerous elliptical microcracks. The complete analytical solutions (VNA solutions) for a single elliptical crack in an infinite solid, subject to arbitrary crack-face tractions, are implemented in the present alternating method, together with the coordinate transformations for stress tensors. First, the present method is verified by solving the problems of two interacting cracks for which accurate numerical solutions have been obtained previously. Next, the present method demonstrates obtaining efficient and accurate solutions for the problems of many interacting elliptical cracks, which cannot be solved in a practical sense by the ordinary numerical methods such as the finite element method. Furthermore, damaged solids containing periodically distributed elliptical microcracks are analyzed by the present alternating method. The effective elastic moduli are evaluated for varying microcrack density. Detailed structures of the interactions in the damaged solids are visualized and clarified. This revised version was published online in July 2006 with corrections to the Cover Date.     

Boundary collocation method for multiple defect interactions in an anisotropic finite region

The modified mapping collocation method is extended for the solution of plane problems of anisotropic elasticity in the presence of multiple defects in the form of holes, cracks, and inclusions under general loading conditions. The approach is applied to examine the stress and strain fields in an anisotropic finite region including an elliptical and a circular hole, an elliptical flexible inclusion, and a line crack. It can be readily incorporated into micro-mechanics models, capturing the relative importance of the matrix, the fiber/matrix interface, and reinforcement geometry and arrangement while estimating the effective elastic properties of composite materials. The accuracy and robustness of this method is established through comparison with results obtained from finite element analysis.     

A dislocation and a point force in an anisotropic medium with an elliptical hole

Summary The solutions of a dislocation and a point force in an infinite medium with a crack of elliptical cross-section (or an elliptical hole) are given for plane problems of anisotropic solids with one plane of symmetry.With 1 Figure     

Finite element analyses of three-dimensional crack problems in piezoelectric structures

In this paper, electromechanical fracture mechanics and finite element techniques for crack analyses are extended to three-dimensional crack configurations. Penny-shaped cracks and elliptical cracks are analyzed, subjected to combined mechanical tension and electric fields. For the penny-shaped crack, exact solutions originating from different resources are compared with numerical results. Some errors in the literature concerning the analytical solution for the elliptical crack are corrected. Numerical results of the stress-intensity factors and energy release rates for these crack configurations are presented.     

Boundary element analysis for viscoelastic solids containing interfaces/holes/cracks/inclusions

With the aid of the elastic–viscoelastic correspondence principle, the boundary element developed for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. Green's functions for the problems of two-dimensional linear anisotropic elastic solids containing holes, cracks, inclusions, or interfaces have been obtained analytically using Stroh's complex variable formalism. Through the use of these Green's functions and the correspondence principle, special boundary elements in the Laplace domain for viscoelastic solids containing holes, cracks, inclusions, or interfaces are developed in this paper. Subregion technique is employed when multiple holes, cracks, inclusions, and interfaces exist simultaneously. After obtaining the physical responses in Laplace domain, their associated values in time domain are calculated by the numerical inversion of Laplace transform. The main feature of this proposed boundary element is that no meshes are needed along the boundary of holes, cracks, inclusions and interfaces whose boundary conditions are satisfied exactly. To show this special feature by comparison with the other numerical methods, several examples are solved for the linear isotropic viscoelastic materials under plane strain condition. The results show that the present BEM is really more efficient and accurate for the problems of viscoelastic solids containing interfaces, holes, cracks, and/or inclusions.     

Interactions of cracks and inclusions in homogeneous elastic media

The work is devoted to static problems of elasticity for an infinite homogeneous medium containing planar parallel cracks and heterogeneous inclusions of arbitrary shapes. Cracks and inclusions occupy a finite region of the medium that is subjected to arbitrary external forces. The problem is reduced to a system of surface integral equations for crack opening vectors and volume integral equations for the stress tensor in the region. Gaussian approximating functions are used for discretization and efficient numerical solution of this system. Such functions are centered at the nodes of a regular node grid that covers all the inclusions and the crack surfaces. For Gaussian functions, the elements of the matrix of the discretized system have forms of standard integrals that can be tabulated and calculated fast. The matrix of the discretized system is not sparse but it has Teoplitz’s structure, and the number of independent matrix elements is much smaller than the total number of the elements. In addition, fast Fourier transform technique can be used for calculation matrix-vector products with such matrices. It accelerates substantially the process of iterative solutions of the discretized system. The method is mesh free. Examples of numerical solutions of the problems for planar circular cracks and spherical inclusions are presented and compared with analytical and numerical solutions available in the literature.     

Determination of the Stress Intensity Factor of an elliptical breathing crack in a rotating shaft

The failures due to the propagation of fatigue cracks are one of the most frequent problems in rotating machines. Those failures sometimes are catastrophic and are sufficient to provoke the loss of the complete machine with high risks for people and other equipments. When a cracked shaft rotates, the breathing mechanism appears. The crack passes from an open state to a close state with a transition in which a partial opening or closing of the crack is produced. In this work, a new general expression that gives the Stress Intensity Factor (SIF) along the crack front of an elliptical crack in a rotating shaft in terms of the crack depth ratio, the crack aspect ratio, the relative position on the front and the angle of rotation has been developed for linear elastic materials. By the moment, no expressions of the SIF in term of these variables have been found in the literature. To this end, a quasi-static 3D numerical model of a cracked shaft with straight and elliptical cracks subjected to rotary bending using the Finite Element Method (FEM) has been made. To simulate the rotation of the shaft, different angular positions have been considered. The SIF in mode I along the crack front has been calculated for each angular position of the cracked shaft and for different crack geometries. The expression results have been compared with solutions obtained from the literature. It has been found that they are in good agreement. The model has been applied to other crack geometries with good results. The obtained SIF expression allows studying the dynamic behavior of cracked shafts and can be used to analyze the crack propagation.     

A new hybrid finite element approach for plane piezoelectricity with defects

In this paper, a new type of hybrid fundamental solution-based finite element method (HFS-FEM) is developed for analyzing plane piezoelectric problems with defects by employing fundamental solutions (or Green’s functions) as internal interpolation functions. The hybrid method is formulated based on two independent assumptions: an intra-element field covering the element domain and an inter-element frame field along the element boundary. Both general elements and a special element with a central elliptical hole or crack are developed in this work. The fundamental solutions of piezoelectricity derived from the elegant Stroh formalism are employed to approximate the intra-element displacement field of the elements, while the polynomial shape functions used in traditional FEM are utilized to interpolate the frame field. By using Stroh formalism, the computation and implementation of the method are considerably simplified in comparison with methods using Lekhnitskii’s formalism. The special-purpose hole element developed in this work can be used efficiently to model defects such as voids or cracks embedded in piezoelectric materials. Numerical examples are presented to assess the performance of the new method by comparing it with analytical or numerical results from the literature.     

压电螺型位错与含界面裂纹椭圆夹杂的干涉效应

运用求解复杂多连通域问题的复变函数方法，获得了压电螺型位错与含界面裂纹椭圆夹杂的干涉问题，复势函数的精确级数形式解。利用广义Peach-Koehler公式导出作用于螺型位错的位错力公式。分析结果表明，当裂纹的曲率或长度达到临界值，界面裂纹的存在会改变压电螺型位错与椭圆夹杂的干涉性质。     

A new integral equation formulation for the analysis of crack-inclusion interactions

A new integral equation method for the analysis of the interactions between cracks and elastic inclusions embedded in a two-dimensional, linearly elastic, isotropic infinite medium subjected to in-plane force is presented. By distributing dislocations along the crack lines and forces along the matrix-inclusion interfaces, a set of coupled integral equations is obtained. The discretization procedure of the integrals involved is discussed and the relations between the stress intensity factors and the values of the dislocation functions at the respective crack tips are derived. Several sample problems are presented in order to determine the versatility and the accuracy of this approach.     

Some elasticity problems of a general anisotropic solid subjected to anti-plane loadings

Plane elasticity problems of a general anisotropic material subjected to both in-plane and anti-plane loadings are formulated based on Lekhnitskii’s complex variable approach. Some useful solutions to half-plane problems and elliptical cavity problems are derived for cases involving anti-plane loadings. Asymptotical crack-tip elastic fields are reviewed and comparisons are made for crack problems and slender elliptical cavity problems. Specifically, it is shown that the stress state at a rounded crack tip can be characterized by a stress rounding factor and the stress intensity factor of the associated slit crack. An important conclusion is that, for a fixed loading condition and a fixed shape of the cavity, the influence of the shape of the cavity on the stress concentration factor can be separated from the influence of the material properties.     

Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

This paper considers a 2‐D fracture analysis of anisotropic piezoelectric solids by a boundary element‐free method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least‐squares (MLS) approximation. The numerical scheme of the boundary element‐free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended ‘stress intensity factors’ (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary element modeling of cracks in piezoelectric solids

This study focuses on the application of boundary element methods for linear fracture mechanics of two-dimensional piezoelectric solids. A complete set of piezoelectric Green's functions, based on the extended Lekhnitskii's formalism and distributed dislocation modeling, are presented. Special Green's functions are obtained for an infinite medium containing a conducting crack or an impermeable crack. The numerical solution of the boundary integral equation and the computation of fracture parameters are discussed. The concept of crack closure integral is utilized to calculate energy release rates. Accuracy of the boundary element solutions is confirmed by comparing with analytical solutions reported in the literature. The present scheme can be applied to study complex cracks such as branched cracks, forked cracks and microcrack clusters.     

Numerical studies on crack problems in three-layered elastic media using an image method

Two-dimensional crack problems in a three-layered material are analyzed numerically under the conditions of plane strain. An image method is adopted to obtain fundamental solutions for dislocation dipoles in trilayered media. The governing equations for equilibrium cracks can be constructed by distributed dislocation technique and their solutions are sought in terms of the displacement discontinuity method (DDM). Comparisons are made with available analytical or reference solutions for several examples at various contrasts of material constants, and good agreements are found. A crack within a brittle adhesive layer joining two semi-infinite blocks can propagate in a variety of ways. In particular, crack paths in the form of sigmoidal waves within the adhesive layer are revisited to reveal the sensitivities of cracking paths to initial crack locations and directions and residual stresses. In addition, Z-shape and H-shape cracks alternating from interface to interface are re-examined to highlight the transition of failure modes and the role of the interlayer thickness.     

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 combined finite element/strip element method for analyzing elastic wave scattering by cracks and inclusions in laminates

 A new numerical technique combining the finite element method and strip element method is presented to study the scattering of elastic waves by a crack and/or inclusion in an anisotropic laminate. Two-dimensional problems in the frequency domain are studied. The interior part of the plate containing cracks or inclusions is modeled by the conventional finite element method. The exterior parts of the plate are modeled by the strip element method that can deal problems of infinite domain in a rigorous and efficient manner. Numerical examples are presented to validate the proposed technique and demonstrate the efficiency of the proposed method. It is found that, by combining the finite element method and the strip element method, the shortcomings of both methods are avoided and their advantages are maintained. This technique is efficient for wave scattering in anisotropic laminates containing inclusions and/or cracks of arbitrary shape. Received 2 February 2001     

Volume integral equation method for multiple circular and elliptical inclusion problems in antiplane elastostatics

A Volume Integral Equation Method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple isotropic and anisotropic circular/elliptical inclusions subject to remote antiplane shear. This method is applied to two-dimensional problems involving long parallel cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of isotropic and anisotropic inclusions. The effects of the number of isotropic and anisotropic inclusions and various fiber volume fractions on the stress field at the interface between the matrix and the central circular/elliptical inclusion are also investigated in detail. The accuracy of the method is validated by solving single isotropic and orthotropic circular/elliptical inclusion problems and multiple isotropic circular and elliptical inclusion problems for which solutions are available in the literature.