A Galerkin boundary integral method for multiple circular elastic inclusions

The problem of an infinite, isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered. The analysis procedure is based on the use of a complex singular integral equation. The unknown tractions at each circular boundary are approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using the classical Galerkin method and the Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the effectiveness of the approach. Copyright © 2001 John Wiley & Sons, Ltd.     

A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions

This paper considers the problem of an infinite, isotropic viscoelastic plane containing an arbitrary number of randomly distributed, non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size. All inclusions are assumed to be perfectly bonded to the material matrix but the elastic properties of the inclusions can be different from one another. The Kelvin model is employed to simulate the viscoelastic plane. The numerical approach combines a direct boundary integral method for a similar problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described in [Crouch SL, Mogilevskaya SG. On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries. Int J Numer Methods Eng 2003;58:537–578], and a time-marching strategy for viscoelastic material analysis described in [Mesquita AD, Coda HB, Boundary integral equation method for general viscoelastic analysis. Int J Solids Struct 2002;39:2643–2664]. Several numerical examples are given to verify the approach. For benchmark problems with one inclusion, results are compared with the analytical solution obtained using the correspondence principle and analytical Laplace transform inversion. For an example with two holes and two inclusions, results are compared with numerical solutions obtained by commercial finite element software—ANSYS. Benchmark results for a more complicated example with 25 inclusions are also given.     

Complex variable boundary integral method for linear viscoelasticity: Part II—application to problems involving circular boundaries

Complex variable integral equations for linear viscoelasticity derived in Part I [Huang Y, Mogilevskaya SG, Crouch SL. Complex variable boundary integral method for linear viscoelasticity. Part I—basic formulations. Eng Anal Bound Elem 2006; in press, doi:10.1016/j.enganabound.2005.12.007.] are employed to solve the problem of an infinite viscoelastic plane containing a circular hole. The viscoelastic material behaves as a Boltzmann model in shear and its bulk response is elastic. Constant or time-dependent stresses are applied at the boundary of the hole, or, if desired, at infinity. Time-dependent variables on the circular boundary (displacements or tractions in the direct formulation of the complex variable boundary integral method or unknown complex density functions in the indirect formulations) are represented by truncated complex Fourier series with time-dependent coefficients and all the space integrals involved are evaluated analytically. Analytical Laplace transform and its inversion are adopted to accomplish the evaluation of the associated time convolutions. Several examples are given to demonstrate the validity and reliability of the method. Generalization of the approach to the problems with multiple holes is discussed.     

A boundary integral method for multiple circular holes in an elastic half-plane

This paper presents a semi-analytical method for solving the problem of an isotropic elastic half-plane containing a large number of randomly distributed, non-overlapping, circular holes of arbitrary sizes. The boundary of the half-plane is assumed to be traction-free and a uniform far-field stress acts parallel to that boundary. The boundaries of the holes are assumed to be either traction-free or subjected to constant normal pressure. The analysis is based on solution of complex hypersingular integral equation with the unknown displacements at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-plane. Several examples available in the literature are re-examined and corrected, and new benchmark examples with multiple holes are included to demonstrate the effectiveness of the approach.     

A semi-analytical solution for multiple circular inhomogeneities in one of two joined isotropic elastic half-planes

The paper presents a semi-analytical method for solving the problem of two joined, dissimilar isotropic elastic half-planes, one of which contains a large number of arbitrary located, non-overlapping, perfectly bonded circular elastic inhomogeneities. In general, the inhomogeneities may have different elastic properties and sizes. The analysis is based on a solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. Apart from round-off, the only errors introduced into the solution are due to truncation of the Fourier series. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the inhomogeneities. Numerical examples are included to demonstrate the effectiveness of the approach.     

Direct boundary integral procedure for a Boltzmann viscoelastic plane with circular holes and elastic inclusions

A direct boundary integral method in the time domain is presented to solve the problem of an infinite, isotropic Boltzmann viscoelastic plane containing a large number of randomly distributed, non-overlapping circular holes and perfectly bonded elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, in general, be different. The method is based on a direct boundary integral approach for the problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described by Crouch and Mogilevskaya [1], and a time marching strategy for viscoelastic analysis described by Mesquita and Coda [2–8]. Benchmark problems and numerical examples are included to demonstrate the accuracy and efficiency of the method.     

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

Summary This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.     

Null-field integral equation approach for free vibration analysis of circular plates with multiple circular holes

In this paper, a semi-analytical approach for the eigenproblem of circular plates with multiple circular holes is presented. Natural frequencies and modes are determined by employing the null-field integral formulation in conjunction with degenerate kernels, tensor rotation and Fourier series. In the proposed approach, all kernel functions are expanded into degenerate (separable) forms and all boundary densities are represented by using Fourier series. By uniformly collocating points on the real boundary and taking finite terms of Fourier series, a linear algebraic system can be constructed. The direct searching approach is adopted to determine the natural frequency through the singular value decomposition (SVD). After determining the unknown Fourier coefficients, the corresponding mode shape is obtained by using the boundary integral equations for domain points. The result of the annular plate, as a special case, is compared with the analytical solution to verify the validity of the present method. For the cases of circular plates with an eccentric hole or multiple circular holes, eigensolutions obtained by the present method are compared well with those of the existing approximate analytical method or finite element method (ABAQUS). Besides, the effect of eccentricity of the hole on the natural frequency and mode is also considered. Moreover, the inherent problem of spurious eigenvalue using the integral formulation is investigated and the SVD updating technique is adopted to suppress the occurrence of spurious eigenvalues. Excellent accuracy, fast rate of convergence and high computational efficiency are the main features of the present method thanks to the semi-analytical procedure.     

A circular inclusion in a finite domain I. The Dirichlet-Eshelby problem

Summary This is the first paper in a series concerned with the precise characterization of the elastic fields due to inclusions embedded in a finite elastic medium. A novel solution procedure has been developed to systematically solve a type of Fredholm integral equations based on symmetry, self-similarity, and invariant group arguments. In this paper, we consider a two-dimensional (2D) circular inclusion within a finite, circular representative volume element (RVE). The RVE is considered isotropic, linear elastic and is subjected to a displacement (Dirichlet) boundary condition. Starting from the 2D plane strain Navier equation and by using our new solution technique, we obtain the exact disturbance displacement and strain fields due to a prescribed constant eigenstrain field within the inclusion. The solution is characterized by the so-called Dirichlet-Eshelby tensor, which is provided in closed form for both the exterior and interior region of the inclusion. Some immediate applications of the Dirichlet-Eshelby tensor are discussed briefly.     

Two circular inclusions with inhomogeneous interfaces interacting with a circular Eshelby inclusion in anti-plane shear

Summary An analytical solution in infinite series form for two circular cylindrical elastic inclusions embedded in an infinite matrix with two circumferentially inhomogeneous imperfect interfaces interacting with a circular Eshelby inclusion in anti-plane shear is derived by employing complex variable techniques. All of those coefficients in the series can be uniquely determined in a simple and transparent way. Numerical examples are given to illustrate the effect of imperfection and circumferential inhomogeneity of the two interfaces as well as the size, location and elastic properties of the two circular inclusions on the stress fields induced within the two circular inclusions and the Eshelby inclusion.     

Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes

The paper considers the problem of multiple interacting circular nano-inhomogeneities or/and nano-pores located in one of two joined, dissimilar isotropic elastic half-planes. The analysis is based on the solutions of the elastostatic problems for (i) the bulk material of two bonded, dissimilar elastic half-planes and (ii) the bulk material of a circular disc. These solutions are coupled with the Gurtin and Murdoch model of material surfaces [Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57:291–323; Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct 1978;14:431–40.]. Each elastostatic problem is solved with the use of complex Somigliana traction identity [Mogilevskaya SG, Linkov AM. Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 1998;22:88–92]. The complex boundary displacements and tractions at each circular boundary are approximated by a truncated complex Fourier series, and the unknown Fourier coefficients are found from a system of linear algebraic equations obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the nano-inhomogeneities. Numerical examples demonstrate that (i) the method is effective in solving the problems with multiple nano-inhomogeneities, and (ii) the elastic response of a composite system is profoundly influenced by the sizes of the nano-features.     

Torsional rigidity of a circular bar with multiple circular inclusions using the null-field integral approach

In this article, a systematic approach is proposed to calculate the torsional rigidity and stress of a circular bar containing multiple circular inclusions. To fully capture the circular geometries, the kernel function is expanded to the degenerate form and the boundary density is expressed into Fourier series. The approach is seen as a semi-analytical manner since error purely attributes to the truncation of Fourier series. By collocating the null-field point exactly on the real boundary and matching the boundary condition, a linear algebraic system is obtained. Convergence study shows that only a few number of Fourier series terms can yield acceptable results. Finally, torsion problems are revisited to check the validity of our method. Not only the torsional rigidities but also the stresses of multiple inclusions are also obtained by using the present approach.     

A novel method for solving the displacement and stress fields of an infinite domain with circular holes and/or inclusions subject to a screw dislocation

In this paper, the degenerate kernel and superposition technique are employed to solve the screw dislocation problems with circular holes or inclusions. The problem is decomposed into the screw dislocation problem with several holes and the interior Laplace problems for several circular inclusions. Following the success of the null-field integral equation approach, the typical boundary value problems can be solved easily. The kernel functions and unknown boundary densities are expanded by using the degenerate kernel and Fourier series, respectively. To the authors?? best knowledge, the angle-type fundamental solution is first derived in terms of degenerate kernel in this paper. Finally, four examples are demonstrated to verify the validity of the present approach.     

A circular inclusion in a finite domain II. The Neumann-Eshelby problem

Summary This is the second paper in a series concerned with the precise characterization of the elastic fields due to inclusions embedded in a finite elastic medium. In this part, an exact and closed form solution is obtained for the elastic fields of a circular inclusion embedded in a finite circular representative volume element (RVE), which is subjected to the traction (Neumann) boundary condition. The disturbance strain field due to the presence of an inclusion is related to the uniform eigenstrain field inside the inclusion by the so-called Neumann-Eshelby tensor. Remarkably, an elementary, closed form expression for the Neumann-Eshelby tensor of a circular RVE is obtained in terms of the volume fraction of the inclusion. The newly derived Neumann-Eshelby tensor is complementary to the Dirichlet-Eshelby tensor obtained in the first part of this work. Applications of the Neumann-Eshelby tensor are discussed briefly.     

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.     

An inverse method for determining elastic material properties and a material interface

A numerical procedure which integrates optimization, finite element analysis and automatic finite element mesh generation is developed for solving a two-dimensional inverse/parameter estimation problem in solid mechanics. The problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix. Traction and displacement boundary conditions sufficient for solving a direct problem are applied to the boundary of the domain. In addition, displacements are measured at discrete points on the part of the boundary where the tractions are prescribed. The inverse problem is solved using a modified Levenberg-Marquardt method to match the measured displacements to a finite element model solution which depends on the unknown parameters. Numerical experiments are presented to show how different factors in the problem and the solution procedure influence the accuracy of the estimated parameters.     

混合夹杂问题的边界元法

该文提出了一种求解含固体、流体和孔隙等多类型夹杂的混合夹杂问题的边界元法。混合夹杂问题实质也是多连通域问题,但内边界的位移和面力都是未知量,导致该问题因定解条件不足而无法直接求解。根据不同类型夹杂的本构关系建立了各夹杂与基体界面面力与位移之间的关联矩阵,从而形成除给定边界条件以外的补充定解条件,使问题得以解决。以平面问题为例,分别对只含固体夹杂、流体夹杂以及同时含有孔隙、固体和流体夹杂的情况进行了计算,模拟了含100个随机分布夹杂的板材的弹性模量,验证了该方法的有效性、程序的正确性和可靠性。     

Eigensolutions of a circular flexural plate with multiple circular holes by using the direct BIEM and addition theorem

The purpose of this paper is to present an analytical formulation to describe the free vibration of a circular flexural plate with multiple circular holes by using the null field integral formulation, the addition theorem and complex Fourier series. Owing to the addition theorem, all kernel functions are represented in the degenerate form and further transformed into the same polar coordinates centered at one of circles, where the boundary conditions are specified. Thus, not only the computation of the principal value for integrals is avoided but also the calculation of higher-order derivatives in the flexural plate problem can be easily determined. By matching the specified boundary conditions, a coupled infinite system of simultaneous linear algebraic equations is derived as an analytical model for the title problem. According to the direct searching approach, natural frequencies are numerically determined through the singular value decomposition (SVD) in the truncated finite system. After determining the unknown Fourier coefficients, the corresponding mode shapes are obtained by using the direct boundary integral formulations for the domain points. Several numerical results are presented. In addition, the inherent problem of spurious eigenvalue using the integral formulation is investigated and the SVD updating technique is adopted to suppress the occurrence of spurious eigenvalues. Excellent accuracy, fast rate of convergence and high computational efficiency are advantages of the present method thanks to its analytical features.     

Stress analysis of multilayered plates around circular holes

In this paper, a technique to study the 3-dimensional stress state around a circular hole in laminated plates is developed. First, the 3-dimensional elasticity problem for a thick plate with a circular hole is formulated in a systematic fashion by using the z-component of the Galerkin vector and that of Muki's harmonic vector function. This problem was originally solved by Alblas[1]. The reasons for reconsidering it are to introduce a technique which may be used in solving the elasticity problem for a multilayered plate and to verify and extend the results given by Alblas. Among the additional results of particular interest, one may mention the significant effect of the Poisson's ratio on the behavior and the magnitude of the stresses. Secondly, the elasticity problem for a laminated thick plate, which consists of two bonded dissimilar layers and which contains a circular hole, is considered. The problem is formulated for arbitrary axisymmetric tractions on the hole surface. Through the expansion of the boundary conditions into Fourier series, the problem is reduced to an infinite system of algebraic equations which is solved by the method of reduction. Of particular interest in the problem are the stresses along the interface as they relate to the question of delamination failure of the composite plate. These stresses are calculated and are observed to become unbounded at the hole boundary. An approximate treatment of the singular behavior of the stress state is presented, and the stress intensity factors are calculated. It is also observed that, the results compare rather well with those obtained from the finite element method.     

Scale and boundary conditions effects in elastic properties of random composites

Summary We study elastic anti-plane responses of unidirectional fiber-matrix composites. The fibers are of circular cylinder shape, aligned in the axial direction, and arranged randomly, with no overlap, in the transverse plane. We assume that both fibers and matrix are linear elastic and isotropic. In particular, we focus on the effects of scale of observation and boundary conditions on the overall anti-plane (axial shear) elastic moduli. We conduct this analysis numerically, using a two-dimensional square spring net-work, at the mesoscale level. More specifically, we consider finite windows of observation, which we increase in size. We subject these regions to several different boundary conditions: displacement-controlled, traction-controlled, periodic, and mixed (combination of any of the first three) to evaluate the mesoscale moduli. The first two boundary conditions give us scale-dependent bounds on the anti-plane elastic moduli. For each boundary condition case we consider many realizations of the random composite to obtain statistics. In this parametric study we cover a very wide range of stiffness ratios ranging from composites with very soft inclusions (approximating holes) to those with very stiff inclusions (approaching rigid fibers), all at several volume fractions.