A numerical Green's function for multiple cracks in anisotropic bodies

The numerical construction of a Green's function for multiple interacting planar cracks in an anisotropic elastic space is considered. The numerical Green's function can be used to obtain a special boundary-integral method for an important class of two-dimensional elastostatic problems involving planar cracks in an anisotropic body.     

Elastic fields produced by a point source in solids of general anisotropy

Explicit expressions for three-dimensional elastostatic Green's functions in solids of general anisotropy are derived by means of an integral-representation technique and a subsequent application of the residue calculus. A direct calculation for the derivatives of the displacement Green's functions is quite complicated. However, relatively simple expressions can be obtained for the integration of the derivatives along a line or a surface. These integrals are in fact more useful, because, in most applications, not the derivatives, but their integrals along lines or surfaces are needed. Discussions regarding degenerate materials and details in evaluation of the residues at multiple poles are also given. The results presented in this paper are sufficient for the implementation of the boundary-element method for bodies of general anisotropic solids.     

Boundary integral equation formulation for Interface cracks in anisotropic materials

We present a boundary integral formulation for anisotropic interface crack problems based on an exact Green's function. The fundamental displacement and traction solutions needed for the boundary integral equations are obtained from the Green's function. The traction-free boundary conditions on the crack faces are satisfied exactly with the Green's function so no discretization of the crack surfaces is necessary. The analytic forms of the interface crack displacement and stress fields are contained in the exact Green's function thereby offering advantage over modeling strategies for the crack. The Green's function contains both the inverse square root and oscillatory singularities associated with the elastic, anisotropic interface crack problem. The integral equations for a boundary element analysis are presented and an example problem given for interface cracking in a copper-nickel bimaterial.     

Evaluation of the three-dimensional elastic Green's function in anisotropic cubic media

By using the Fourier transforms method, the three-dimensional Green's function solution for a unit force applied in an infinite cubic material is evaluated in this paper. Although the elastic behavior of a cubic material can be characterized by only three elastic constants, the explicit solutions of Green's function for a cubic material are not available in the literatures. The central problem for explicitly solving the elastic Green's function of anisotropic materials depends upon the roots of a sextic algebraic equation, which results from the inverse Fourier transforms and is composed of the material constants and position vector parameters. The close form expression of Green's function is presented here in terms of roots of the sextic equation. The sextic equation for an anisotropic cubic material is discussed thoroughly and specific results are given for possible explicit solutions.     

各向异性杆扭转问题的各向同性化域外奇点解法

本文为各向异性杆简单扭转问题提供一种各向同性化域外奇点解法。先引入坐标变换,将该问题转化成相应的各向同性杆的扭转问题。再利用后者的格林函数,按域外奇点法求解。最后,经逆变换得到所需的应力分量。它具有方法简单、不需数值积分、计算时间短和精度高等优点。     

Elastic green's function for a bimaterial composite solid containing a crack inclined to the interface

An analytical closed-form expression is derived for the elastic Green's function of a bimaterial composite solid containing a planar interface and a straight crack inclined at an arbitrary angle with the interface. The crack tip is assumed to be at the interface. Both the constituent materials of the composite are assumed to anisotropic. The Green's function satisfies the interfacial boundary conditions of continuous tractions and displacements, and zero tractions at the crack surfaces. The boundary conditions are satisfied by using the virtual force technique. The determination of the virtual forces requires solutions of a Hilbert problem which is obtained by using an orthogonal complex transform. The method is illustrated by applying it to a copper/nickel composite. The Green's function should be useful in the boundary-element method of calculating the stress and the displacement field in the solid.     

The stress intensity factors for an infinitely long transversely isotropic, thick-walled cylinder which contains a ring-shaped crack

In this study the elastostatic axisymmetric problem for a long thick-walled transversely anisotropic cylinder containing a ring-shaped internal crack is analyzed. The problem is reduced to a singular integral equation which has a simple Cauchy kernel as the dominant part by using Hankel and Fourier transform techniques. These equations are then solved numerically and the stress intensity factors are calculated.The results are given for different transversely anisotropic materials and crack geometries.     

Computation of Dyadic Green's Functions for Electrodynamics in Quasi-Static Approximation with Tensor Conductivity

Homogeneous non-dispersive anisotropic materials, characterized by a positive constant permeability and a symmetric positive definite conductivity tensor, are considered in the paper. In these anisotropic materials, the electric and magnetic dyadic Green's functions are defined as electric and magnetic fields arising from impulsive current dipoles and satisfying the time-dependent Maxwell's equations in quasi-static approximation. A new method of deriving these dyadic Green's functions is suggested in the paper. This method consists of several steps: equations for electric and magnetic dyadic Green's functions are written in terms of the Fourier modes; explicit formulae for the Fourier modes of dyadic Green's functions are derived using the matrix transformations and solutions of some ordinary differential equations depending on the Fourier parameters; the inverse Fourier transform is applied to obtained formulae to find explicit formulae for dyadic Green's functions.     

Steady-state response of a Cosserat medium with a spherical inclusion

Summary Two concepts of asymmetric eigenstrain and eigentorsion are employed to derive a general steady-state theory of inhomogeneous anisotropic micropolar media containing defects with the help of Green's function technique. In particular, a dynamic inclusion problem for homogeneous isotropic centrosymmetric micropolar elasticity is investigated. By means of Green's functions an exact closed-form solution is presented for the case of a spherical inclusion embedded in an infinitely extended Cosserat medium. With this result, the micropolar dynamic Eshelby tensors for the inside and outside elastic fields of the spherical inclusion are defined and determined. It is confirmed that the classical dynamic and static Eshelby tensors are obtained as two special cases of the micropolar dynamic Eshelby tensors, respectively.     

Crack growth in an orthotropic medium

Abstract

The elastostatic problem of an orthotropic body having a central inclined crack and subjected at infinity to a uniform biaxial load is considered. It is assumed that the crack line does not coincide with an axis of elastic symmetry of the body. The problem must be considered as one of general orthotropy, due in particular to the fact that the elastic coefficients of the material change with rotation of the reference system. The stress fields at the crack tip are reported and the presence of the non‐singular terms underlined. The Strain Energy Density Theory is extended to orthotropic materials. It is assumed that the Critical Strain Energy Density Factor has a polar variation. The crack initiation is determined via minimization of the ratio of the strain energy density over the material critical strain energy density, pointing out the effects of orthotropy and load biaxiality. The effects of the non‐singular terms on crack growth for different orthotropic materials is also studied, underling the relation between orthotropy and non‐singular terms.     

Interfaces Between two Dissimilar Elastic Materials

In this paper the near tip solutions for interface corners written in terms of the stress intensity factors are presented in a unified expression. This single expression is applicable for any kinds of interface corners including corners and cracks in homogeneous materials as well as interface corners and interface cracks lying between two dissimilar materials, in which the materials can be any kinds of linear elastic anisotropic materials or piezoelectric materials. Through this unified expression of near tip solutions, the singular orders of stresses and their associated stress/electric intensity factors for different kinds of interface problems can be determined through the same formulae and solution techniques. This unified feature of solving interface problems is then implemented numerically through several different interface problems. Moreover, in order to improve the accuracy and efficiency of numerical computation, a special boundary element based upon the Green's function of bimaterials is introduced in this paper.     

Fracture problems of rubbers: J-integral estimation based upon η factors and an investigation on the strain energy density distribution as a local criterion

The plane elasticity problem studied is of a circular inclusion having a circular arc-crack along the interface and a crack of arbitrary shape in an infinite matrix of different material subjected to uniform stresses at infinity. The solution of the problem is given using Muskhelishvili's complex variable method with sectionally holomorphic functions. First, the solution to the (auxiliary) problem of a dislocation (or force) applied at a point in the matrix with the circular inclusion partially bonded is derived fully in its general form by solving the appropriate Rieman-Hilbert problem. It is subsequently used as the Green's function for the initial problem by introducing an unknown density function associated with a distribution of dislocations along the crack in the matrix. The initial problem is then reduced to a singular integral equation (SIE) over the crack in the matrix only. The SIE is solved numerically by appropriate quadratures and the stress intensity factors reported for the arc-cut and a straight crack in the matrix for a range of values of the geometrical parameters.     

Effect of microcrack array on stress intensity factor of main crack

An improved technique for solving crack-microcrack interaction problems is developed, based on the Green's function for a dislocation dipole placed in the vicinity of a main crack. The microcrack array is characterized by the microcrack density, orientation and length distributions. The stress field near the main crack tip is taken as a sum of a singular term of an anisotropic problem with stress intensity factor K, and a regular term. K and the regular term are determined from a system of equations obtained by self-consistency arguments. The proposed technique is illustrated by numerical solutions of several interaction problems for particular configurations of microcrack array.     

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.     

Solution of three-dimensional, anisotropic, nonlinear problems of magnetostatics using two scalar potentials, finite and infinite multipolar elements and automatic mesh generation

The two-scalar potentials idea has been used with success for the computation of static magnetic fields in the presence of nonlinear isotropic magnetic materials by the finite element method. In this communication we formulate the two-scalar-potentials method for anisotropic materials and present a computer program and the solution of an example problem. The use of infinite multipolar elements is also discussed. Several advanced methods and ideas are employed by the program: scalar potentials, rather than vector potentials, giving only one unknown quantity; the finite element method, in which the solution is approximated by a continuous function; the Galerkin method to solve the differential equations; accurate infinite elements, which avoid the introduction of an artificial boundary for unbounded problems; automatic mesh generation, which means that the user can construct a large mesh and represent a complicated geometry with little effort; automatic elimination of nodes outside the iron, which restricts the iterations to the nonlinear anisotropic region with economy of computer time; use of sparse matrix technology, which represents a further economy in computer time when assembling the linear equations and solving them by either Gauss elimination or iterative techniques such as the conjugated gradient method, etc. The combination of these techniques is very convenient.     

Nonplanar Interfacial Cracks in Anisotropic Bimaterials

This paper presents a general approach for the two-dimensional elastic problem of a crack lying along an elliptical interface seperating two dissimilar anisotropic materials. The analysis is based upon the use of the Eshelby–Stroh formalism of anisotropic elasticity theory and a special conformal mapping technique devised by Lekhniskii. The resulting elastic fields are fully described by a pair of function vectors whose components are holomorphic functions. These function vectors define the two-phase potentials of the bi-material. The associated expressions are universal in the sense of being applicable to any applied load. As in the case of a planar interface crack, the crack tip stress field is free of oscillation if the bimaterial matrix H is real. The general results are applied to specific examples and explicit forms of solutions are obtained. This revised version was published online in July 2006 with corrections to the Cover Date.     

Crack analysis in unidirectionally and bidirectionally functionally graded materials

Mixed-mode crack analysis in unidirectionally and bidirectionally functionally graded materials is performed by using a boundary integral equation method. To make the analysis tractable, the Young's modulus of the functionally graded materials is assumed to be exponentially dependent on spatial variables, while the Poisson's ratio is assumed to be constant. The corresponding boundary value problem is formulated as a set of hypersingular traction boundary integral equations, which are solved numerically by using a Galerkin method. The present method is especially suited for straight cracks in infinite FGMs. Numerical results for the elastostatic stress intensity factors are presented and discussed. Special attention of the analysis is devoted to investigate the effects of the material gradients and the crack orientation on the elastostatic stress intensity factors.     

Elastic analysis of a half-plane containing an inclusion and a void using a mixed volume and boundary integral equation method

A mixed volume and boundary integral equation method is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing an isotropic or anisotropic inclusion and a void subject to remote loading parallel to a traction-free boundary. A detailed analysis of the stress field is carried out for three different geometries of the problem. It is demonstrated that the method is very accurate and effective for investigating local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions and multiple voids.     

BEM for crack-hole problems in thermopiezoelectric materials

The boundary element formulation for analysing interaction between a hole and multiple cracks in piezoelectric materials is presented. Using Green's function for hole problems and variational principle, a boundary element model (BEM) for a 2-D thermopiezoelectric solid with cracks and holes has been developed and used to calculate stress intensity factors of the crack-hole problem. In BEM, the boundary condition on the hole circumference is satisfied a priori by Green's function, and is not involved in the boundary element equations. The method is applicable to multiple-crack problems in both finite and infinite solids. Numerical results for stress and electric displacement intensity factors at a particular crack tip in a crack-hole system of piezoelectric materials are presented to illustrate the application of the proposed formulation.     

Inverse formulation for geometrically exact stress resultant shells

The inverse elastostatic method deals with a class of problems in which a deformed configuration of an elastic body is known while the initial stress‐free configuration or the stress in the deformed state is to be determined. The method is imperative for certain problems in engineering applications. Computational methods of inverse elastostatics have been established for elastic continua. In this paper, we present an inverse method for thin‐wall structures modeled as geometrically exact stress resultant shells. The theoretical basis and the details of implementation are discussed. Numerical examples involving both isotopic and anisotropic materials are presented. The practical utility of the method is demonstrated using an example of human aneurysm stress analysis. Copyright © 2007 John Wiley & Sons, Ltd.