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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.

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.

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.

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.

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 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.

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

Growth of pressure-induced fractures originating from a wellbore at an arbitrary angle to the direction of far field stresses is considered. A parametric study of hydraulic fracturing process from the point of view of fracture mechanics is presented in conditions of slow and fast pressurization rate. The study is based on the linear elastic fracture mechanics formulation that involves a complex hypersingular equation (CHSIE) for a plane with a circular opening and a system of arbitrary curvilinear cracks. Stepwise method is used to model fracture propagation. Dimensionless parameters influencing the fracture path are defined.

Numerical algorithm to simulate 2-D smooth crack is presented. The stepwise method based on local criteria of propagation is used. Two crack propagation criteria are employed. At the first stage of propagation, the maximum tensile stress criterion is used to take into account the abrupt change in tangent direction. At subsequent stages, the assumption that the stress intensity factor (SIF) K_2= 0 at the current crack tip is exploited. The analytical formulae for calculating SIFs are given. The displacement discontinuities (DD) involved in these formulae are found from the numerical solution of a complex hypersingular integral equation (CHSIE) for a piecewise homogeneous plane with curvilinear cracks. The new mechanism of smooth approximation of the crack path by circular arcs at each propagation stage is suggested. Numerical results are given. They confirm the efficiency of the algorithm suggested.

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.