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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.

Modern advances in material science and surface chemistry lead to creation of composite materials with enhanced mechanical, thermal, and other properties. It is now widely accepted that the enhancements are achieved due to drastic reduction in sizes of some phases of composite structures. This leads to increase in surface to volume ratios, which makes surface- or interface-related effects to be more significant. For better understanding of these phenomena, the investigators turned their attention to various theories of material surfaces. This paper is a review of two most prominent theories of that kind, the Gurtin–Murdoch and Steigmann–Ogden theories. Here, we provide comprehensive review of relevant literature, summarize the current state of knowledge, and present several new results.

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.

The Maxwell concept of equivalent inhomogeneity generalized to account for the interactions between the particles in the cluster and combined with recently reported results on the polarizability of a cube is used to evaluate the effective conductivities of the materials reinforced by cubic arrays of spherical particles. New numerical results demonstrate that the estimates of the effective properties based on the generalized Maxwell scheme with the equivalent inhomogeneity of cubic shape converge to the accurate periodic benchmark solutions, unlike spherical shape-based estimates.

The problem of an infinite isotropic elastic space subjected to uniform far-field load and containing an isotropic elastic spherical inhomogeneity with Steigmann–Ogden interface is considered. The interface is treated as a shell of vanishing thickness possessing surface tension as well as membrane and bending stiffnesses. The constitutive and equilibrium equations of the Steigmann–Ogden theory for a spherical surface are written in explicit forms. Closed-form analytical solutions are derived for two cases of loading conditions—the hydrostatic loading and deviatoric loading with vanishing surface tension. The single inhomogeneity-based estimates of the effective properties of macroscopically isotropic materials containing spherical inhomogeneities with Steigmann–Ogden interfaces are presented. It is demonstrated that, in the case of vanishing surface tension, the Steigmann–Ogden model describes a special case of thin and stiff uniform interphase layer.

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.

This paper examines three sets of approximate formulae for the overall tetragonal effective elastic properties of two-phase fiber-reinforced unidirectional composites with isotropic phases. The fibers are of circular cross-sections and periodically distributed in a matrix in a square pattern. The formulae by Kantor and Bergman, Luciano and Barbero, and estimates based on non-interacting Maxwell’s type approximations are rewritten in unified notations. The latter approximations coincide with the most of well-known estimates of the effective medium theories (composite cylinder model, generalized self-consistent model and the Mori–Tanaka method), as well as with one of the Hashin–Shtrikman variational bounds. The approximate estimates are compared with the exact periodic solutions to determine the range of their applicability. The simplest and most accurate formulae are identified and suggested as a set of approximate expressions for accurate estimates of the effective elastic properties of composite materials with a square symmetry.

A simple analytical study of a single-atom-thick sheet of graphene under biaxial tension is presented. It is based on the combination of the approaches of continuum and molecular mechanics. On the molecular level the Tersoff-Brenner potential with a modified cut-off function is used as an example. Transition to a continuum description is achieved by employing the Cauchy–Born rule. In this analysis the graphene sheet is considered as a crystal composed of two simple Bravais lattices and the mutual atomic relaxation between these lattices is taken into account. Following this approach a critical failure surface is produced for strains in biaxial tension. The adopted methodology is discussed in the context of the alternative approaches.

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 concept of Maxwell's equivalent inhomogeneity is re-evaluated in the context of the effective elastic properties of unidirectional multi-phase composite materials with the fibers of circular cross-section. The following key questions are addressed in this paper. Is the concept of equivalent inhomogeneity valid only for the materials with low volume fractions, as originally suggested by Maxwell? When do the interactions among the fibers matter? Could Maxwell's concept be generalized to allow for accurate estimates of the effective properties? The approach advocated in the paper provides an unique opportunity to study, in the framework of one model, the effects of the interactions among fibers on the effective transversely isotropic elastic properties of multi-phase composite materials.

A new computational approach for calculation of the effective transverse mechanical properties of unidirectional fiber-reinforced composites with linear viscoelastic matrix and elastic fibers is presented. The approach requires the knowledge of stresses outside a cluster representing the structure of composite in question. The effective properties are found from the assumption that the viscoelastic stresses at the distances far away from the cluster are the same as those from a single equivalent inhomogeneity. The approach directly takes into account the interactions between the inhomogeneities. The comparison of the results with several benchmark solutions reveals the advantages of the developed approach.

A complete solution has been obtained for the problem of multiple interacting spherical inhomogeneities with a Gurtin–Murdoch interface model that includes both surface tension and surface stiffness effects. For this purpose, a vectorial spherical harmonics-based analytical technique is developed. This technique enables solution of a wide class of elasticity problems in domains with spherical boundaries/interfaces and makes fulfilling the vectorial boundary or interface conditions a routine procedure. A general displacement solution of the single-inhomogeneity problem is sought in a form of a series of the vectorial solutions of the Lame equation. This solution is valid for any non-uniform far-field load and it has a closed form for polynomial loads. The superposition principle and re-expansion formulas for the vectorial solutions of the Lame equation extend this theory to problems involving multiple inhomogeneities. The developed semi-analytical technique precisely accounts for the interactions between the nanoinhomogeneities and constitutes an efficient computational tool for modeling nanocomposites. Numerical results demonstrate the accuracy and numerical efficiency of the approach and show the nature and extent to which the elastic interactions between the nanoinhomogeneities with interface stress affect the elastic fields around them.

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.

The interaction between a straight crack and a circular inhomogeneity with interface stiffness and energy is considered. The Gurtin and Murdoch model is adopted, wherein the interface between the inhomogeneity and the matrix is regarded as a material surface that possesses its own mechanical properties and surface tension. The elastostatics problem is decomposed into two complimentary problems for (1) a circular disk with unknown distributions of traction and displacements along its boundary and (2) a loaded isotropic plane containing a circular hole with unknown distributions of traction and displacements along its boundary. The unknown distributions are determined through the application of the constitutive relations at the material surface. For selected values of the dimensionless parameters that quantify the geometry, material properties and applied loading, the stress field, stress intensity factors and energy release rates are calculated using a complex boundary integral equation approach. Particular attention is paid to crack-tip shielding and anti-shielding that develops as a result of the stresses introduced by the material surface. An illustrative example involving a perforated plate loaded in tension suggests that the material surface produces a modest level of expected effective toughening.

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 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.

A two-dimensional problem of multiple interacting circular nano-inhomogeneities or/and nano-pores is considered. The analysis is based on the Gurtin and Murdoch model [Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323.] in which the interfaces between the nano-inhomogeneities and the matrix are regarded as material surfaces that possess their own mechanical properties and surface tension. The precise component forms of Gurtin and Murdoch's three-dimensional equations are derived for interfaces of arbitrary shape to provide a basis for critical review of various modifications used in the literature. The two-dimensional specification of these equations is considered and their representation in terms of complex variables is provided. A semi-analytical method is proposed to solve the problem. Solutions to several example problems are presented to: (i) examine the difference between the results obtained with the original and modified Gurtin and Murdoch's equations, (ii) compare the results obtained using Gurtin and Murdoch's model and those for a problem of nano-inhomogeneities with thin membrane-type interphase layers, and (iii) demonstrate the effectiveness of the approach in solving problems with multiple nano-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 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.