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SummaryThe displacements representations of the type used by Christensen and Lo (J. Mech. Phys. Solids27, 1979) are modified to allow for analytical treatment of problems involving spherical and circular material surfaces that possess constant surface tension. The modified representations are used to derive closed-form expressions for the local elastic fields and effective moduli of macroscopically isotropic composite materials containing spherical and circular inhomogeneities with the interfaces described by the complete Gurtin–Murdoch and Steigmann–Ogden models.

This paper presents the semi-analytical solution for the transverse local fields and overall transverse properties of composite materials with aligned multiple cylindrical nanofibers. The interface between each fiber and the matrix is treated as a material surface described by the Steigmann–Ogden model, which accounts for the effects of surface tension as well as for membrane and bending stiffness of the surface. Assuming a plane strain setting, the problem is formulated in the transverse plane as an infinite elastic matrix with multiple circular inhomogeneities subjected to a uniform far-field load. The expressions for all elastic fields in the composite system are obtained analytically in the form of infinite series expressions. The Maxwell methodology is used to obtain the overall transverse elastic properties. The goal of this work is twofold: (a) to study the influence of the interactions between the inhomogeneities on the local fields and overall transverse properties of the composite system, and (b) to reveal the connection of the Steigmann–Ogden model (with zero surface tension) to a specific uniform interphase layer model. The results presented in this paper demonstrate that for fiber composite materials with medium to high volume fractions, the influence of the interactions can be significant.

The boundary conditions for the [Steigmann, D.J., Ogden, R.W., 1997. Plain deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. London A 453, 853–877; Steigmann, D.J., Ogden, R.W., 1999. Elastic surface-substrate interactions. Proc. R. Soc. London A 455, 437–474.] model are re-derived for a two dimensional surface using general expression for surface energy that include surface tension. The model treats the interface as a shell of vanishing thickness possessing surface tension as well as membrane and bending stiffness. The two-dimensional plane strain problem of an infinite isotropic elastic domain subjected to the uniform far-field load and containing an isotropic elastic circular inhomogeneity whose interface is described by the Steigmann-Ogden model is solved analytically. Closed-form expressions for all elastic fields in the domain are obtained. Dimensionless parameters that govern the problem are identified. The Maxwell type approximation formula is obtained for the effective plane strain properties of the macroscopically isotropic materials containing multiple inhomogeneities with the Steigmann-Ogden interfaces. The “neutrality” conditions are analyzed. It is demonstrated that while the Steigmann-Ogden model theoretically reduces to the [Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323.; Gurtin, M.E., Murdoch, A.I., 1978. Surface stress in solids. Int. J. Solid. Struct. 14, 431–440.] model when the bending interphase effects are neglected, the two models (for the case of zero surface tension) describe two very different interphase regimes of seven regimes proposed by [Benveniste, Y., Miloh, T., 2001. Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33, 87–111.].

The paper studies the conditions under which the far-field response of a two-dimensional coated circular inhomogeneity embedded into an infinite matrix and subjected to uniform stresses at infinity is identical to that of a perfectly bonded inhomogeneity. The problem is considered in plane strain and antiplane settings. All constituents of the composite systems are assumed to be isotropic or transversely isotropic (with the axis Oz in longitudinal direction) and linearly elastic. For the plane strain problem and hydrostatic load or antiplane problem, it is shown that there always exists an equivalent inhomogeneity of the radius equal to the external radius of the coating that produces the elastic fields inside the matrix that are identical to those of the original coated inhomogeneity. For the plane strain and deviatoric load, the elastic fields in the matrix due to these two composite systems are always different, except for the equal shear moduli case. However, it is rigorously proved here that, for the deviatoric load and any combination of the material parameters, there exists the equivalent inhomogeneity of the radius equal to the external radius of the coating that induces the same dipole moments as those induced by the coated inhomogeneity. The existence of the equivalent inhomogeneities whose radius is different from the external radius of the coating is also investigated. The application of the proposed procedure to the homogenization problems leads to the new closed-form expression for the effective transverse shear modulus of transversely isotropic unidirectional composites. The findings presented here provide an insight on the influence of the interphases that could be useful in the analysis of some types of inverse problems.

This paper describes a boundary-element-based approach for the modeling and solution of potential problems that involve thin layers of varying curvature. On the modeling side, we consider two types of imperfect interface models that replace a perfectly bonded thin layer by a zero-thickness imperfect interface across which the field variables undergo jumps. The corresponding jump conditions are expressed via second-order surface differential operators. To quantify their accuracy with respect to the fully resolved thin layer, we use boundary element techniques, which we develop for both the imperfect interface models and the fully resolved thin layer model. Our techniques are based on the use of Green’s representation formulae and isoparametric approximations that allow for accurate representation of curvilinear geometry and second order derivatives in the jump conditions. We discuss details of the techniques with special emphasis on the evaluation of nearly singular integrals, validating them via available analytical solutions. We finally compare the two interface models using the layer problem as a benchmark.

This paper presents a comprehensive review of the far-field-based methodology of estimation of the effective properties of multi-phase composites that was pioneered by Maxwell in 1873 in the context of effective electrical conductivity of a particle-reinforced material. Maxwell suggested that a cluster of particles embedded in an infinite medium subjected to a uniform electrical field has the same far-field asymptotic as an equivalent sphere whose conductivity is equal to the effective one; this yields closed-form formula for the effective conductivity. Our review focuses on subsequent developments of Maxwell’s idea in various applications and on its range of applicability. The conclusion is that, 145 years later, the methodology is still alive and well.

The paper presents the complex variables-based approach for analytical evaluation of three-dimensional integrals involved in boundary integral representations (potentials) for the Helmholtz equation. The boundary element is assumed to be planar bounded by an arbitrary number of straight lines and/or circular arcs. The integrals are re-written in local (element) coordinates, while in-plane components of the fields are described in terms of certain complex combinations. The use of Cauchy–Pompeiu formula (a particular case of Bochner–Martinelli formula) allows for the reduction of surface integrals over the element to the line integrals over its boundary. By considering the requirement of the minimum number of elements per wavelength and using an asymptotic analysis, analytical expressions for the line integrals are obtained for various density functions. A comparative study of numerical and analytical integration for particular integrals over two types of elements is performed.

This paper examines major techniques for modeling elastostatic crack problems. The foundations of these techniques and fundamental papers that introduced, developed, and applied them are reviewed. The goal is to provide a “translation” between the different academic languages that describe the same problem.

Summary The article presents a new complex variables-based approach for analytical evaluation of threedimensional integrals involved in boundary element method (BEM) formulations. The boundary element is assumed to be planar and its boundary may contain an arbitrary number of straight lines and/or circular arcs. The idea is to use BEM integral representations written in a local coordinate system of an element, separate in-plane components of the fields involved, arrange them in certain complex combinations, and apply integral representations for complex functions. These integral representations, such as Cauchy–Pompeiu formula (a particular case of Bochner– Martinelli formula) are the corollaries of complex forms of Gauss’s theorem and Green’s identity. They reduce the integrals over the area of the domain to those over its boundary. The latter integrals can be evaluated analytically for various density functions. Analytical expressions are presented for basic integrals involved in the single- and double-layer potentials for potential (harmonic) and elasticity problems.

Maxwell’s concept of an equivalent inhomogeneity is employed for evaluating the effective elastic properties of tetragonal, fiber-reinforced, unidirectional composites with isotropic phases. The microstructure induced anisotropic effective elastic properties of the material are obtained by comparing the far-field solutions for the problem of a finite cluster of isotropic, circular cylindrical fibers embedded in an infinite isotropic matrix with that for the problem of a single, tetragonal, circular cylindrical equivalent inhomogeneity embedded in the same isotropic matrix. The former solutions precisely account for the interactions between all fibers in the cluster and for their geometrical arrangement. The solutions to several example problems that involve periodic (square arrays) composites demonstrate that the approach adequately captures microstructure induced anisotropy of the materials and provides reasonably accurate estimates of their effective elastic properties.

Maxwell’s concept of equivalent inhomogeneity is employed for evaluating the effective elastic properties of macroscopically anisotropic particulate composites with isotropic phases. The effective anisotropic elastic properties of the material are obtained by comparing the far-field solutions for the problem of a finite cluster of isotropic particles embedded in an infinite isotropic matrix with those for the problem of a single anisotropic equivalent inhomogeneity embedded in the same matrix. The former solutions precisely account for the interactions between all particles in the cluster and for their geometrical arrangement. Illustrative examples involving periodic (simple cubic) and random composites suggest that the approach provides accurate estimates of their effective elastic moduli.

This paper presents a new boundary element-based approach for solving three-dimensional problems of an elastic medium containing multiple cracks of arbitrary shapes. The medium could be loaded by far-field stress (for infinite domains), surface tractions (including those at the cracks surfaces), or point loads. Constant body forces are also allowed. The elastic fields outside of the cracks are represented by integral identities. Triangular elements are employed to discretize the boundaries. Integration over each element is performed analytically. In-plane components of the fields are combined in various complex combinations to simplify the integration. No singular integrals are involved since the limit, as the field point approaches the boundary, is taken after the integration. The collocation method is used to set up the system of linear algebraic equations to find the unknown boundary displacements and tractions. No special procedure is required to evaluate the fields outside of the boundaries, as the integration is performed before the limit is taken. Several numerical examples are presented to demonstrate the capacity of the method.

A complete solution has been obtained for periodic particulate nanocomposite with the unit cell containing a finite number of spherical particles with the Gurtin–Murdoch interfaces. For this purpose, the multipole expansion approach by Kushch et al. [Kushch, V.I., Mogilevskaya, S.G., Stolarski, H.K., Crouch, S.L., 2011. Elastic interaction of spherical nanoinhomogeneities with Gurtin–Murdoch type interfaces. J. Mech. Phys. Solids 59, 1702–1716] has been further developed and implemented in an efficient numerical algorithm. The method provides accurate evaluation of local fields and effective stiffness tensor with the interaction effects fully taken into account. The displacement vector within the matrix domain is found as a superposition of the vector periodic solutions of Lame equation. By using local expansion of the total displacement and stress fields in terms of vector spherical harmonics associated with each particle, the interface conditions are fulfilled precisely. Analytical averaging of the local strain and stress fields in matrix domain yields an exact, closed form formula (in terms of expansion coefficients) for the effective elastic stiffness tensor of nanocomposite. Numerical results demonstrate that elastic stiffness and, especially, brittle strength of nanoheterogeneous materials can be substantially improved by an appropriate surface modification.

Maxwell's methodology is combined with the boundary element method (BEM) for evaluating the two-dimensional effective elastic properties of composite, porous, and microcracked isotropic materials with periodic or random structure. The approach is based on the idea that the effective properties of the material can be deduced from the effects that a cluster of fibers, pores, or cracks embedded in an infinite matrix has on the far-fields. The fibers, pores, or cracks can have arbitrary shapes, sizes, and elastic properties, provided that the overall behavior is isotropic, and their effects are assumed to be the same as those of an equivalent circular inhomogeneity. The key aspect of the approach is to precisely account for the interactions between all the constituents in the cluster that represent the material in question. This is done by using the complex-variables version of the BEM to solve the problem of a finite cluster of fibers, pores or cracks embedded an infinite isotropic, linearly elastic matrix. The effective properties of the material are evaluated by comparing the far-field solutions for the cluster with that of the equivalent inhomogeneity. It is shown that the model adequately captures the influence of the micro-structure of the material on its overall properties.

The problem of calculating the effective conductivity of isotropic composite materials with periodic or random arrangements of spherical particles is revisited by using the equivalent inhomogenitymethod. The approach can be viewed as an extension of classical Maxwell’s methodology. It is based on the idea that the effective conductivity of the composite material can be deduced from the effect of the cluster embedded in an infinite space on the far-fields. The key point of the approach is to precisely account for the interactions between all the particles in the cluster that represent the composite material in question. It is done by using a complete, multipole-type analytical solution for the problem of an infinite isotropic matrix containing a finite cluster of isotropic spherical particles, regarded as the Finite Cluster Model of particulate composite. The effective conductivity of the composite is evaluated by applying the “singularto-singular” re-expansion formulae and comparing the far-field asymptotic behavior with the equivalent inhomogeneity solution. The model allows to adequately capture the influence of the micro-structure of composite material on its overall properties. Numerical realization of the method is simple and straightforward. Comparison of the numerical results obtained by the proposed approach with those available in literature (both for periodic and random arrangements) demonstrate its accuracy and numerical efficiency.

The direct formulation of the complex variables boundary element method is generalized to allow for solving problems with constant body forces. The hypersingular integral equation for two-dimensional piecewise homogeneous medium is presented and the numerical solution is described. The technique can be used to solve a wide variety of problems in engineering. Several examples are presented to verify the approach and to demonstrate its key features. The results of calculations performed with the proposed approach are compared with available analytical and numerical benchmark solutions.

A computational technique based on Maxwell's methodology is presented for evaluating the effective thermal conductivity of isotropic materials with periodic or random arrangement of spherical pores. The basic idea of the approach is to construct an equivalent sphere in an infinite space whose effects on the temperature at distant points are the same as those of a finite cluster of spherical pores arranged in a pattern representative of the material in question. The thermal properties of the equivalent sphere then define the effective thermal properties of the material. This procedure is based on a semi-analytical solution of a problem of an infinite space containing a cluster of non-overlapping spherical pores under prescribed temperature gradient at infinity. The method works equally well for periodic and random arrays of spherical pores.

SUMMARY A two-dimensional transient heat conduction problem of multiple interacting circular inhomogeneities, cavities and point sources is considered. In general, a non-perfect contact at the matrix/inhomogeneity interfaces is assumed, with the heat flux through the interface proportional to the temperature jump. The approach is based on the use of the general solutions to the problems of a single cavity and an inhomogeneity and superposition. Application of the Laplace transform and the so-called addition theorem results in an analytical transformed solution. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. Several numerical examples are given to demonstrate the accuracy and the efficiency of the method. The approximation error decreases exponentially with the number of the degrees of freedom in the problem. A comparison of the companion two- and threedimensional problems demonstrates the effect of the dimensionality. Copyright q 2009 John Wiley & Sons, Ltd.

This paper is concerned with the problem of an isotropic, linear viscoelastic half-plane containing multiple, isotropic, circular elastic inhomogeneities. Three types of loading conditions are allowed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, and a force uniformly distributed over the whole boundary of the half-plane. The half-plane is subjected to farfield stress that acts parallel to its boundary. The inhomogeneities are perfectly bonded to the material matrix. An inhomogeneity with zero elastic properties is treated as a hole; its boundary can be either traction free or subjected to uniform pressure. The analysis is based on the use of the elastic-viscoelastic correspondence principle. The problem in the Laplace space is reduced to the complementary problems for the bulk material of the perforated half-plane and the bulk material of each circular disc. Each problem is described by the transformed complex Somigliana’s traction identity. The transformed complex boundary parameters at each circular boundary are approximated by a truncated complex Fourier series. Numerical inversion of the Laplace transform is used to obtain the time domain solutions everywhere in the half-plane and inside the inhomogeneities. The method allows one to adopt a variety of viscoelastic models. A number of numerical examples demonstrate the accuracy and efficiency of the method.

This paper considers a transient heat conduction problem for an infinite medium with multiple nonoverlapping spherical cavities. Suddenly applied, steady Dirichlet-, Neumannor Robin-type boundary conditions are assumed. The approach is based on the use of the general solution to the problem of a single cavity and superposition. Application of the Laplace transform and the so-called addition theorem results in a semi-analytical transformed solution for the case of multiple cavities. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. A large-time asymptotic series for the temperature is obtained. The limiting case of infinitely large time results in the solution for the corresponding steady-state problem. Several numerical examples that demonstrate the accuracy and the efficiency of the method are presented. Copyright 2008 John Wiley & Sons, Ltd.

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.

This paper considers the problem of a semi-infinite, isotropic, linear viscoelastic half-plane containing multiple, non-overlapping circular holes. The sizes and the locations of the holes are arbitrary. Constant or time dependent far-field stress acts parallel to the boundary of the half-plane and the boundaries of the holes are subjected to uniform pressure. Three types of loading conditions are assumed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, a force uniformly distributed over the whole boundary of the half-plane. The solution of the problem is based on the use of the correspondence principle. The direct boundary integral method is applied to obtain the governing equation in the Laplace domain. The unknown transformed displacements on the boundaries of the holes are approximated by a truncated complex Fourier series. A system of linear equations is obtained by using a Taylor series expansion. The viscoelastic stresses and displacements at any point of the half-plane are found by using the viscoelastic analogs of Kolosov–Muskhelishvili’s potentials. The solution in the time domain is obtained by the application of the inverse Laplace transform. All the operations of space integration, the Laplace transform and its inversion are performed analytically. The method described in the paper allows one to adopt a variety of viscoelastic models. For the sake of illustration only one model in which the material responds as the standard solid in shear and elastically in bulk is considered. The accuracy and efficiency of the method are demonstrated by the comparison of selected results with the solutions obtained by using finite element software ANSYS.

The paper presents a numerical study of the three-dimensional problem of cracks interacting with a cylindrical uniformly pressurized borehole. The theoretical developments describe general case in which the axis of the borehole can be inclined to the vertical direction, the cracks are either located outside of the borehole or emanate from it, and the in-situ stresses are uniform with major principal stress acting in vertical direction. The tractions are prescribed at the cracks surfaces that includes two limiting cases of traction-free cracks (“fast pressurization”) or cracks subjected to uniform load equal to that applied at the surface of the borehole (“slow pressurization”). The study is based on the complex integral representations for the three-dimensional fields around the borehole-crack system. The boundary surfaces are approximated using triangular mesh and quadratic polynomials are employed for approximating the boundary unknowns. The prescribed boundary conditions are met using “limit after discretization” procedure. The linear algebraic system to find the unknowns is set up by the collocation method. Two numerical benchmarks are presented.

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.

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.

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.

This paper discusses applications of Somiglina's identities to the solutions of elasticity problems with spherical boundaries. The components of the boundary displacements and tractions involved in the identities are represented as truncated series of surface spherical harmonics, and all of the integrals involved in the formulae are evaluated analytically. The classical problems of a solid sphere, a spherical cavity, and a perfectly bonded spherical inhomogeneity (an inclusion with the elastic properties different from those of the surrounding material) are solved with the use of Somiglina's identities. Extensions of the new solutions to more complicated three-dimensional problems with spherical boundaries are discussed.

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.

The basic formulations (direct and indirect) of the complex variable boundary integral method for linear viscoelasticity are presented. Complex variable temporal integral equations for the formulations are obtained for viscoelastic solids whose behavior in shear is governed by a Boltzmann model while the bulk behavior is purely elastic. The functions involved in the integral equations are the time-dependent complex boundary tractions and displacements for the direct approach and the unknown time-dependent complex density functions for the indirect approaches. The temporal integral equations give the displacements and stresses at a point inside a viscoelastic region in terms of time convolution and space integrals over the boundary of this region. The equations are valid for the boundaries of arbitrary shapes provided that these boundaries are sufficiently smooth. Complex variable temporal boundary equations are obtained by taking the inner point to the boundary. Numerical treatment of spatial and time convolution integrals involved in the boundary equations is discussed.

A numerical method is presented for simulating the occurrence of localized slip and separation along the interfaces of multiple, randomly distributed, circular elastic inclusions in an infinite elastic plane. The method is an extension of a direct boundary integral approach previously described elsewhere for solving problems involving perfectly bonded circular inclusions. Here, we allow displacement discontinuities to develop along the inclusion/matrix interfaces in accordance with a linear Mohr–Coulomb yield condition combined with a tensile strength cut-off. The displacements, tractions, and displacement discontinuities on the inclusion boundaries are all represented by truncated Fourier series, and an explicit iterative algorithm is adopted to determine zones of slip and separation under the prevailing loading conditions. Several examples are given to demonstrate the accuracy and generality of the approach.

This paper presents a numerical method for modeling the micromechanical behavior and macroscopic properties of fiber-reinforced composites and perforated materials. The material is modeled by a finite rectangular domain containing multiple circular holes and elastic inclusions. The rectangular domain is assumed to be embedded within a larger circular domain with fictitious boundary loading represented by truncated Fourier series. The analytical solution for the complementary problem of a circular domain containing holes and inclusions is obtained by using a combination of the series expansion technique with a direct boundary integral method. The boundary conditions on the physical external boundary are satisfied by adopting an overspecification technique based on a least squares approximation. All of the integrals arising in the method can be evaluated analytically. As a result, the elastic fields and effective properties can be expressed explicitly in terms of the coefficients in the series expansions. Several numerical experiments are conducted to verify the accuracy and efficiency of the numerical method and to demonstrate its application in determination of the macroscopic properties of composite materials.

A fast and accurate algorithm is presented to increase the computational efficiency of a Galerkin boundary integral method for solving two-dimensional elastostatics problems involving numerous straight cracks and circular inhomogeneities. The efficiency is improved by computing the combined influences of groups, or blocks, of elements—with each element being an inclusion, a hole, or a crack—using asymptotic expansions, multiple shifts, and Taylor series expansions. The coefficients in the asymptotic and Taylor series expansions are computed analytically. Implementation of this algorithm involves a single- or multi-level grid, a clustering technique, and a tree data structure. An iterative procedure is adopted to solve the coefficients in the series expansions of boundary unknowns block by block. The elastic fields in each block are calculated by superposition of the direct influences from the nearby elements and the grouped far-field influences from all the other elements. This fast multipole algorithm is considerably more efficient for large-scale practical problems than the conventional approach.

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.

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.

The paper presents a numerical method for solving the problem of an infinite, isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with uniform interphase layers. The bonds between the inclusions and the interphases as well as between the interphases and the matrix are assumed to be perfect. In general, the inclusions may have different elastic properties and sizes; the thicknesses of the interphases and their elastic properties are arbitrary. The analysis is based on a numerical solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex Fourier series. The Galerkin technique is used to obtain a system of linear algebraic equations. The resulting numerical method allows one to calculate the elastic fields everywhere in the matrix and inside the inclusions and the interphases. Using the assumption of macro-isotropic behavior in a plane section one can find the effective elastic moduli for an equivalent homogeneous material. The method can be viewed as an extension of our previous work (Int. J. Solids Struct. 39 (2002) 4723) where simpler spring-like interface conditions were modeled. The problem of overlapping of the fibers and matrix inherent to spring type interface is discussed in the context of the present model. Numerical examples are included to demonstrate the effectiveness of the new approach.

This paper describes a numerical procedure for solving two-dimensional elastostatics problems with multiple circular holes and elastic inclusions in a finite domain with a circular boundary. The inclusions may have arbitrary elastic properties, different from those of the matrix, and the holes may be traction free or loaded with uniform normal pressure. The loading can be applied on all or part of the finite external boundary. Complex potentials are expressed in the form of integrals of the tractions and displacements on the boundaries. The unknown boundary tractions and displacements are approximated by truncated complex Fourier series. A linear algebraic system is obtained by using Taylor series expansion without boundary discretization. The matrix of the linear system has diagonal submatrices on its diagonal, which allows the system to be effectively solved by using a block Gauss-Seidel iterative algorithm.

A complex boundary integral equation method, combined with series expansion technique, is presented for the problem of an infinite, isotropic elastic plane containing multiple circular holes. Loading is applied at infinity or on the boundaries of the holes. The sizes and locations of the holes are arbitrary provided they do not overlap. The analysis procedure is based on the use of a complex hypersingular integral equation that expresses a direct relationship between all the boundary tractions and displacements. The unknown displacements on each circular boundary are represented by truncated complex Fourier series, and all of the integrals involved in the complex integral equation are evaluated analytically. A system of linear algebraic equations is obtained by using a Taylor series expansion, and the block Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the accuracy, versatility, and efficiency of the approach.

This paper considers the problem of an infinite, isotropic elastic plane containing an arbitrary number of non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, if desired, be different. The analysis is based on the two-dimensional version of Somigliana's formula, which gives the displacements at a point inside a region V in terms of integrals of the tractions and displacements over the boundary S of this region. We take V to be the infinite plane, and S to be an arbitrary number of circular holes within this plane. Any (or all) of the holes can contain an elastic inclusion, and we assume for simplicity that all inclusions are perfectly bonded to the material matrix. The displacements and tractions on each circular boundary are represented as truncated Fourier series, and all of the integrals involved in Somigliana's formula are evaluated analytically. An iterative solution algorithm is used to solve the resulting system of linear algebraic equations. Several examples are given to demonstrate the accuracy and efficiency of the numerical method. Copyright © 2003 John Wiley & Sons, Ltd.

A Galerkin boundary integral method is presented to solve the problem of an infinite, isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with homogeneously imperfect interfaces. Problems of interest might involve thousands of inclusions with no restrictions on their locations (except that the inclusions may not overlap), sizes, and elastic properties. The tractions are assumed to be continuous across the interfaces and proportional to the corresponding displacement discontinuities. The analysis is based on a numerical solution of a complex hypersingular integral equation with the unknown tractions and displacement discontinuities at each circular boundary approximated by truncated complex Fourier series. The method allows one to calculate the stress and displacement fields everywhere in the matrix and inside the inclusions. Numerical examples are included to demonstrate the effectiveness of the approach.

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.

This paper is a reply to the challenge by Helsing and Jonsson (2002, ASME J. Appl. Mech., 69, pp. 88–90) for other investigators to confirm or disprove their new numerical results for the stress intensity factors for a crack in the neighborhood of a circular inclusion. We examined the same problem as Helsing and Jonsson using two different approaches—a Galerkin boundary integral method (Wang et al., 2001, in Rock Mechanics in the National Interest, pp. 1453–1460) (Mogilevskaya and Crouch, 2001, Int. J. Numer. Meth. Eng., 52, pp. 1069–1106) and a complex variables boundary element method (Mogilevskaya, 1996, Comput. Mech., 18, pp. 127–138). Our results agree with Helsing and Jonsson’s in all cases considered.

An effective method is proposed for solving the boundary integral equation (BIE) for the problem of a crack along a curvilinear surface in an elastic space on the basis of the transformation of the initial integrodifferential equation into an equation without derivatives. This is achieved by using the concept of the finite-part integral (FPI). Quadrature formulas are presented for such integrals over arbitrary convex polygons by approximating displacement discontinuities on the boundary by polynomials. The well-known BIE for three-dimensional cracks contain either derivatives of the unknown functions or derivatives of a surface integral /1–7/. In both cases the presence of the derivatives significantly complicates the solution. However, as is shown in /8/, these difficulties are reduced in the case of a plane crack of normal discontinuity if the FPI concept is utilized /9, 10/. In this connection, it is useful to investigate the possibility of applying such an approach to the more general problem of a crack of arbitrary discontinuity and to develop the numerical side of its utilization. Both aims are pursued in this paper: the extension of this idea to the general case of three-dimensional cracks is given and methods are indicated for evaluating the integrals that originate by presenting quadrature formulas convenient for the numerical realization of the BIE method on a computer.

A new technique is presented for evaluating the effective properties of linearly elastic, multi-phase unidirectional composites. Various effects on the fiber/matrix interfaces (perfect bond, homogeneously imperfect interfaces, uniform interphase layers) are allowed. The analysis of nano-composite materials based on the Gurtin and Murdoch model of material surface is also included. The basic idea of the approach is to construct a circular inhomogeneity in an infinite plane whose effects on the displacements and stresses at distant points are the same as those of a finite cluster of inhomogeneities (fibers of circular cross-section) arranged in a pattern representative of the composite material in question. The elastic properties of the equivalent inhomogeneity then define the effective elastic properties of the material. The volume ratio of the composite material is found after the size of the equivalent circular inhomogeneity is defined in the course of the solution procedure. This procedure is based on a semi-analytical solution of a problem of an infinite plane containing a cluster of non-overlapping circular inhomogeneities subjected to loading at infinity. The method works equally well for periodic and random composites and – importantly – eliminates the necessity for averaging either stresses or strains. New results for nano-composite materials are presented.

A two-dimensional, transient, uncoupled thermoelastic problem of an infinite medium with a circular nano-scale cavity is considered. The analysis is based on the generalized Gurtin and Murdoch model [Murdoch, A.I., 2005. Some fundamental aspects of surface modelling. Journal of Elasticity 80, 33–52.] where the surface of the cavity possesses its own surface tension and thermomechanical properties. A semi-analytical solution for the problem is obtained using a complex variable boundary integral equation method and the Laplace transform. Several examples are presented to study the significance of surface thermomechanical properties and surface tension, and to compare the results obtained using the generalized Gurtin and Murdoch model and a thin interphase layer model.