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.     

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.     

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.     

Numerical analysis of doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic medium using the boundary integral equation method

In this paper, the boundary integral equation approaches are used to study the doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic plane medium. For the doubly periodic rigid-line inclusion problems, the special integral equation containing the axial and shear forces within the rigid-line inclusion is used. The doubly periodic crack problems are dealt with using the displacement discontinuous integral equation approach. Stress intensity factors, effective elastic properties for doubly periodic array of cracks/rigid-line inclusions are calculated and compared with the available numerical solutions.     

Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.     

Elastic analysis of a half-plane with multiple inclusions using volume integral equation method

A volume integral equation method (VIEM) is used to calculate the elastostatic field in an isotropic elastic half-plane containing circular inclusions subject to remote loading parallel to the traction-free boundary. The material of the inclusions may be either isotropic or anisotropic and they are assumed to be distributed in square or hexagonal array. A detailed analysis of the stress field at the interface between the matrix and one of the inclusions is carried out for different distances between the inclusion and the surface of the half-plane. The results of the calculations are compared with available results. The VIEM is shown to be very accurate and effective for investigating the local stresses in the presence of multiple inclusions. The method can be applied to multiple inclusions of arbitrary geometry and elastic properties embedded in extended isotropic elastic media.     

Multiple elliptical inclusions of arbitrary orientation in composites

A volume integral equation method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing multiple elliptical inclusions of arbitrary orientation subjected to uniform tensile stress at infinity. The inclusions are assumed to be long parallel elliptical cylinders composed of isotropic and anisotropic elastic material perfectly bonded to the isotropic matrix. The solid is assumed to be under plane strain on the plane normal to the cylinders. A detailed analysis of the stress field at the matrix–inclusion interface for square and hexagonal packing arrays is carried out, taking into account different values for the number, orientation angles and concentration of the elliptical inclusions. The accuracy and efficiency of the method are examined in comparison with results available in the literature.     

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.     

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.     

The numerical solution of boundary-value problems for an elastic body with an elliptic hole and linear cracks

Using the boundary-element method which is a combination of a fictitious load and a displacement discontinuity, numerical solutions are obtained for two-dimensional (plane deformation) boundary-value problems for the elastic equilibrium of infinite and finite homogeneous isotropic bodies having elliptic holes with cracks and cuts of finite length. Using the method of separation of variables, the boundary-value problem is solved in the case of an infinite domain containing an elliptic hole with a linear cut on whose contour the symmetry conditions are fulfilled.     

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values. Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori–Tanaka schemes.     

A boundary spectral method for elastostatic problems with multiple spherical cavities and inclusions

The problem of an infinite solid containing an arbitrary number of non-overlapping spherical cavities and inclusions with arbitrary sizes and locations is considered. The infinite solid and the spherical inclusions are made of different isotropic, linearly elastic materials. The spherical cavities are assumed to carry arbitrary tractions, and the spherical inclusions are assumed to be perfectly bonded to the infinite solid. The boundary and interfacial displacements and tractions are represented by truncated series of surface spherical harmonics. The problem involving multiple spherical features is replaced by a sequence of problems involving a single spherical feature via Schwarz's alternating method which accounts for the interactions in the course of an iterative process. Problems involving a single spherical feature are solved by employing the Papkovich–Neuber functions, and the interactions are evaluated by applying a least squares method. A robust scheme is introduced to control the total errors on the spherical boundaries and interfaces and to choose the number of terms in the series expansions. Several numerical examples are given to address the efficiency and the accuracy of the proposed method.     

Dynamic stress concentration studies by boundary integrals and Laplace transform

The dynamic stress field and its concentrations around holes of arbitrary shape in infinitely extended bodies under plane stress or plane strain conditions are numerically determined. The material may be linear elastic or viscoelastic, while the dynamic load consists of plane compressional waves of harmonic or general transient nature. The method consists of applying the Laplace transform with respect to time to the governing equations of motion and formulating and solving the problem numerically in the transfomed domain by the boundary integral equation method. The stress field can then be obtaind by a numerical inversion of the trasformed solution. The correspondence principle is invoked for the case of viscoelastic material behavious. The method is simplified for the case of harmonic waves where no numerical inversion is involved.     

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.     

Mixed mode fatigue growth of curved cracks emanating from fastener holes in aircraft lap joints

The finite element alternating method is extended further for analyzing multiple arbitrarily curved cracks in an isotropic plate under plane stress loading. The required analytical solution for an arbitrarily curved crack in an infinite isotropic plate is obtained by solving the integral equations formulated by Cheung and Chen (1987a, b). With the proposed method several example problems are solved in order to check the accuracy and efficiency of the method. Curved cracks emanating from loaded fastener holes, due to mixed mode fatigue crack growth, are also analyzed. Uniform far field plane stress loading on the plate and sinusoidally distributed pin loading on the fastener hole periphery are assumed to be applied. Small cracks emanating from fastener holes are assumed as initial cracks, and the subsequent fatigue crack growth behavior is examined until long arbitrarily curved cracks are formed near the fastener holes under mixed mode loading conditions.     

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.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

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.     

Application of the boundary-domain integral equation in elastic inclusion problems

A boundary-domain integral equation is used to calculate the elastic stress and strain field in a finite or infinite body of isotropic, orthotropic or anisotropic materials characterized with inclusions of arbitrary shapes. Based on the Betti–Rayleigh reciprocal work theorem between the unknown state and a known fundamental solution, the equilibrium of the body with inclusions is formulated in terms of boundary-domain integral equations. The resulting equation involves only the fundamental solution of isotropic medium, and hence the use of complicated fundamental solution for anisotropic materials could be avoided. Numerical examples are given to ascertain the correctness and effectiveness of the boundary-domain integral equation technique for the inclusion problems.     

Kirchhoff Approximation Revisited—Some New Results for Scattering in Isotropic and Anisotropic Elastic Solids

Through a series of numerical studies that compare the Kirchhoff approximation to more exact scattering theories, it is demonstrated that the Kirchhoff approximation can accurately predict the pulse–echo peak-to-peak responses of spherical pores and circular cracks in isotropic media over a very wide range of cases that extend well beyond the limits normally associated with this approximation. The reason for this good agreement is shown to lie in the ability of the Kirchhoff approximation to model accurately the very early time response of the flaw. It is also shown that in the Kirchhoff approximation the pulse–echo response of an arbitrary traction-free scatterer in an isotropic elastic solid is identical to the same response obtained using a scalar (fluid) scattering model. This leads to simple analytical expressions for the pulse–echo far-field scattering amplitude of some canonical geometries (circular cracks, spherical voids, cylindrical holes) and to simplified numerical expressions for more general scatterers. For general anisotropic volumetric flaws in a anisotropic elastic solid, it is shown that a high-frequency asymptotic evaluation of the Kirchhoff approximation yields an explicit analytical expression for the pulse–echo leading-edge response of the flaw. Explicit expressions are also given for the pitch–catch response of an elliptical-shaped flat crack in a general anisotropic solid.