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The model of an anisotropic interface in an elastic particulate composite with initial stress is developed as the first-order approximation of a transversely isotropic interphase between an isotropic matrix and spherical particles. The model involves eight independent parameters with a clear physical meaning and conventional dimensionality. This ensures its applicability at various length scales and flexibility in modeling the interfaces, characterized by the initial stress and discontinuity of the displacement and stress fields. The relevance of this model to the theory of material interfaces and its applicability in nanomechanics is discussed. The proposed imperfect interface model is incorporated in the unit cell model of a spherical particle composite with thermal stress owing to uniform temperature change. The rigorous solution to the model boundary value problem is obtained using the multipole expansion method. The reported accurate numerical data confirm the correctness of the developed theory, provide an estimate of its accuracy and applicability limits in the multiparticle environment, and reveal significant effects of the interphase or interface anisotropy and initial stress on the local fields and overall thermoelastic properties of the composite.

A circular elastic disk is coated with an elastic beam, absorbing shear and normal forces without deformation and linearly reacting to a bending moment with a change in curvature. The inexstensibility of the elastic beam introduces an isoperimetric constraint, so that the length of the initial circumference of the disk is constrained to remain fixed during the loading of the disk/coating system. The mechanical model for this system is formulated, solved for general loading, and particularized to the case of two equal and opposite traction distributions, each applied on a small boundary segment (thus modelling indentation of a coated fiber). The stress fields, obtained via complex potentials, are shown to evidence a nice correspondence with photoelastic experiments, ad hoc designed and performed. The presented results are useful for the design of coated fibers at the micro and nano scales., Comment: 35 pages, 15 figures

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

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 a study of the Bövik-Benveniste methodology for high order imperfect interface modeling of steady-state conduction problems involving coated spherical particles. Two types of imperfect interface models, that reduce the original three-phase configuration problem to the two-phase configuration problem, are discussed. In one model, the effect of the layer is accounted for via jumps in the field variables across the traces of its boundaries, while in the other via corresponding jumps across the trace of its mid-surface. Explicit expressions for the jumps are provided for both models up to the third order. The obtained higher order jump conditions are incorporated into the unit cell model of spherical particle composite. The multipole expansion method is used to derive the convergent series solutions to the corresponding boundary value problems. Numerical examples are presented to demonstrate that the use of higher order imperfect interface models allows for accurate evaluation of the local fields and overall conductivity of composites reinforced with coated spherical particles.

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.

Wells play a key role in the reduction of greenhouse gas emissions. They are not only a critical element in ensuring the permanent trapping of carbon dioxide (CO2) in deep subsurface permeable formations, but they are also being recognized as an important source of natural gas emissions. The goal of well integrity is to minimize fluid migration from permeable formations through the use of barriers. Traditionally natural and man-made barriers have been regarded as separate and independent components: once the original impermeable layers such as shale or halite have been pierced by drilling, isolation must be restored by installing steel pipe and annular cement. However this picture has been blurred by the growing recognition that a particular class of rocks can prevent and control fluid leaks through a well's life, including the hundreds of years after abandonment. Creeping formations such as halides, mudstones and possibly ice can seal uncemented sections and large defects in the cement sheath. More importantly, the radial stress they exert reduces debonding and restores integrity once the cause of a microannulus has been eliminated; this makes the geological barriers self-healing, or robust. If creeping formations are to become a fundamental element in well design and evaluation, they need to be properly understood and modeled. From an engineering point of view, the identification and characterization of geological barriers should provide four sets of constitutive properties: • The ultimate radial stress exerted by the formation, as well as its anisotropy (i.e its variation around the casing circumference). • Mechanical properties of the formation to estimate leakage rates if the closure stress has been overcome and a microannulus is formed. • A time scale over which the creep deformation can seal a given defect, or an uncemented section of the annulus. • The type and extent of geochemical reactions, if any, between the formation and the leaking fluid. This paper reviews the limited available evidence on the role and characteristic of geological barriers and adds new examples that have arisen from the study of well integrity at the basin scale. A simple model captures the essential characteristics of the formations’ behavior and allows identifying the key parameters that control the beneficial aspect of formation creep on well integrity. For a cross-section at a given depth, we model the well system (casing/cement / formation and the potential defects at the interfaces) under plane-strain condition. Modeling formation creep using viscoelasticity, we obtain the time of closure of a given set of defects between the formation and the steel casing. In the absence of defects, or when the defects have already been closed, we also obtain the time evolution of the radial stress clamping the interface. The intensity of such a normal compressive stress clamping the interfaces is the key parameter controlling the occurrence of micro-annulus induced by subsequent fluid injection, such as CO2 injection, natural gas storage or reservoir stimulation. We use both an analytical approach in the simplest case of an isotropic far-field stress and a boundary integral equations method in the Laplace domain to handle more complex configurations. Our method is notably agnostic with respect to the viscoelastic constitutive law chosen to model the geological formation time-dependent behavior. We further discuss the impact of the choice of the type of constitutive law on the obtained results. Lastly, we discuss how the controlling parameters of the problem (in-situ stress, creep formation properties) can be measured or estimated from geophysical logs, geological and geomechanical information as well as active well tests. Our analysis aims at assessing the state of the art in the design and evaluation of geological barriers from a well integrity perspective. We also highlight the remaining questions to be answered by research.

Many interface formulations, e.g. based on asymptotic thin interphase models or material surface theories, involve higher-order differential operators and discontinuous solution fields. In this article, we are taking first steps towards a variationally consistent discretization framework that naturally accommodates these two challenges by synergistically combining recent developments in isogeometric analysis and cut-cell finite element methods. Its basis is the mixed variational formulation of the elastic interface problem that provides access to jumps in displacements and stresses for incorporating general interface conditions. Upon discretization with smooth splines, derivatives of arbitrary order can be consistently evaluated, while cut-cell meshes enable discontinuous solutions at potentially complex interfaces. We demonstrate via numerical tests for three specific nontrivial interfaces (two regimes of the Benveniste–Miloh classification of thin layers and the Gurtin–Murdoch material surface model) that our framework is geometrically flexible and provides optimal higher-order accuracy in the bulk and at the interface.

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.

This paper explores the possibility of using the Gurtin-Murdoch theory of material surface for modeling mechanical processes in nanomaterials reinforced by two-dimensional flexible and extensible nanoplatelets. In accordance with the theory, reinforcement is modeled by a vanishing thickness prestressed membrane embedded in an isotropic elastic matrix material. The governing equations for the model are reviewed with a detailed discussion of the conditions at the membrane tips. Plane strain assumption is made and with the purpose of representing the displacements in the bulk material, a single layer elastic potential is employed, with the density representing the jump in tractions across the membrane. Expressions for the remaining elastic fields are provided in terms of complex integral representations. The case of a rectilinear membrane of finite length is considered in detail. Numerical solution for this case is based on the use of approximations that automatically incorporate the membrane-tip conditions into the resulting system of boundary integral equations. Numerical examples illustrate the influence of governing dimensionless parameters and present simulations of the local elastic fields in the materials under study. Additionally, it is shown that, in absence of surface tension, the problem considered here is related to that of a membrane type elastic inhomogeneity embedded into a homogeneous elastic medium.

This paper reports new developments on the complex variables boundary element approach for solving three-dimensional problems of cracks in elastic media. These developments include implementation of higher order polynomial approximations for the boundary displacement discontinuities and more efficient analytical techniques for evaluation of integrals. The approach employs planar triangular boundary elements and is based on the integral representations written in a local coordinate system of an element. In-plane components of the fields involved in the representations are separated and arranged in certain complex combinations. The Cauchy–Pompeiu formula is used to reduce the integrals over the element to those over its contour and evaluate the latter integrals analytically. The system of linear algebraic equations to find the unknown boundary displacement discontinuities is set up via collocation. Several illustrative numerical examples involving a single (penny-shaped) crack and multiple (semi-cylindrical) cracks are presented.

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.

The effects of surface elasticity and surface tension on the transverse overall behavior of unidirectional nano-scale fiber-reinforced composites are studied. The interfaces between the nano-fibers and the matrix are regarded as material surfaces described by the Gurtin and Murdoch model. The analysis is based on the equivalent inhomogeneity technique. In this technique, the effective elastic properties of the material are deduced from the analysis of a small cluster of fibers embedded into an infinite plane. All interactions between the inhomogeneities in the cluster are precisely accounted for. The results related to the effects of surface elasticity are compared with those provided by the modified generalized self-consistent method, which only indirectly accounts for the interactions between the inhomogeneities. New results related to the effects of surface tension are presented. Although the approach employed is applicable to all transversely isotropic composites, in this paper we consider only a hexagonal arrangement of circular cylindrical fibers.

A new approach is proposed for the experimental study of the effective shear modulus of porous elastic materials using the uniaxial tension test. The idea is to measure strains at a few points surrounding a cluster of holes that represents the structure of the material. The representative cluster is placed in the material with the same elastic properties as those of the matrix. The measured strains lead to the properties of the equivalent circular inhomogeneity, which would produce the same elastic fields as from the cluster. An aluminum plate containing a cluster of seven circular or hexagonal holes was used. The effective shear modulus obtained from the strain data was compared with theoretical predictions and various bounds, and it was shown that the laboratory estimated values are quite accurate. The experimental technique can be used for the determination of the effective Poisson’s ratio of porous media and/or cellular solids if more detailed strain data are obtained.

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.

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 a transient heat conduction problem for an infinite medium with two non-overlapping circular cavities. Suddenly applied, steady Dirichlet type boundary conditions are assumed. The approach is based on superposition and the use of the general solution to the problem of a single cavity. Application of the Laplace transform results in a semi-analytical solution for the temperature in the form of a truncated Fourier series. The large-time asymptotic formulae for the solution are obtained by using the analytical solution in the Laplace domain. The method can be extended to problems with multiple cavities and inhomogeneities.

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 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.

In this paper we describe a two-dimensional numerical algorithm to calculate overall elastic properties of rocks. We assume that rocks can be viewed as an assemblage of blocks divided by discontinuity surfaces such as large-scale fractures. For each block we use the concept of representative volume element. We model a representative volume element as a finite body containing multiple defects in the form of inhomogeneities, pores, and microcracks and use simplified assumptions regarding the shapes of the defects. Effective moduli for all blocks can be used as input data for a boundary element (or finite element) computer code to model behavior of rock masses at a different length scale as an assemblage of homogeneous blocks of different shapes and elastic properties. The approach is based on a numerical solution of a hypersingular integral equation for a representative volume element. The use of global approximations and a specially tailored fast multipole technique allows one to efficiently solve large-scale problems and accurately calculate the elastic properties of an equivalent homogeneous material. An additional benefit of the approach is that it allows for an accurate representation of the displacement and stress fields anywhere within the material. A series of computer experiments is conducted to demonstrate the efficiency and accuracy of the algorithm. The possible extension of the approach for three-dimensional problems is discussed.

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.

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.

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.

The project supported graduate students working on experimental and numerical modeling of rock fracture, with the following objectives: (a) perform laboratory testing of fluid-saturated rock; (b) develop predictive models for simulation of fracture; and (c) establish educational frameworks for geologic sequestration issues related to rock fracture. These objectives were achieved through (i) using a novel apparatus to produce faulting in a fluid-saturated rock; (ii) modeling fracture with a boundary element method; and (iii) developing curricula for training geoengineers in experimental mechanics, numerical modeling of fracture, and poroelasticity.

Complex hypersingular (finite-part) integrals and integral equations are considered in the functional class of N. Muskhelishvili. The appropriate definition is given. Three regularization (equivalence) formulae follow from this definition. They reduce hypersingular integrals to singular ones and allow to derive hypersingular analogues for Sokhotsky-Plemelj's formulae and for conditions that are necessary and sufficient for the function to be piecewise holomorphic. Two approaches to get and investigate complex hypersingular equations follow from these results: one of them is based on the equivalence formulae; as to the other, it is based on above-mentioned conditions. As an example, authors' equation for plane elasticity is studied. The existence of a unique solution is stated and some advantages over singular equations are outlined. To solve hypersingular equations the quadrature rules are presented. The accuracy of different quadrature formulae is compared, the examples being used. They confirm the need to take into account asymptotics and to carry out a thorough analytical investigation to get safe numerical results.

The paper presents an analysis of elasticity problems involving a single inhomogeneity which possesses certain types of symmetries. As observed earlier, isotropic problems of that kind exhibit some ‘strange’ and remarkable properties. Under the action of uniform far-field stresses, the averages of the fields inside the inhomogeneities preserve the structure of the far-field loads. Here, it is shown that these properties are exhibited for a wider class of problems, which include anisotropic and non-uniform materials subjected to either far-field loads or constant transformational strains within the inhomogeneity. The proposed modified Eshelby technique facilitates a straightforward analysis of these problems, which is based entirely on the assumed symmetry. It is also shown that some remarkable properties of symmetric inhomogeneities discovered here are related to the so-called ‘strange’ properties of the Eshelby inclusions extensively covered in the literature. Some implications of these findings are discussed.

The importance of accounting for multiple cracks and the blocky structure of rock is emphasized and a method to solve plane problems for such structures is developed. The method makes use of (i) hypersingular equations to deal with discontinuities combined with (ii) complex variables to facilitate evaluation of singular and hypersingular integrals. New approximations for the accurate representation of displacement discontinuities are suggested; they include linear combination of common and conjugated polynomials for ordinary elements together with a special form of asymptotic function for crack tip elements. This results in simple analytical recurrence formulae for the “crucial” (singular and hypersingular) integrals over arbitrary curvilinear elements (both ordinary and tip). The remaining non-singular integrals over elements are efficiently evaluated by common methods. Moreover, in the important particular cases of a straight element and a circular are element they are also evaluated in a simple analytical form. All the significant details of the method are presented. A computer program, worked out on this basis, provides an opportunity do deal with blocky systems, defined by multiple straight and curvilinear cracks as well as a basis for the simulation of propagating curvilinear cracks. An example illustrating the efficiency of the method is given.

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.