Sofia G. Mogilevskaya

A complex boundary integral method for multiple circular holes in an infinite plane

Abstract 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.

A Galerkin boundary integral method for multiple circular elastic inclusions with homogeneously imperfect interfaces

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

The universal algorithm based on complex hypersingular integral equation to solve plane elasticity problems

The effective numerical algorithm to solve a wide range of plane elasticity problems is presented. The method is based on the use of the complex hypersingular boundary integral equation (CHBIE) for blockyThe terminology refers to multiregions of interacting elastic bodies with generalized interfaces ranging from fixed to having displacement discontinuities. The terminology derives from Linkov (1983) and was associated with the hypersingular formulation in Linkov and Mogilevskaya (1991). systems and bodies with cracks and holes. The BEM technique is employed to solve this equation. The unknown functions (displacement discontinuities (DD) or tractions) are approximated by Lagrange polynomials of the arbitrary degree. For the tip elements the asymptotics for the DD are taken into account. The boundaries of the blocks, cracks and holes are approximated by the arcs of the circles and the straight elements. In this case all the integrals (hypersingular, singular and regular) involved in this equation are evaluated in a closed form. Numerical results are given and compared either to the ones obtained by the other authors or to analytical solutions.

Evaluation of effective transverse mechanical properties of transversely isotropic viscoelastic 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.

Benchmark Results for the Problem of Interaction Between a Crack and a Circular Inclusion

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

Growth of Pressure-Induced Fractures in the Vicinity of a Wellbore

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.

On ‘strange’ properties of some symmetric inhomogeneities

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.

Evaluation of some approximate estimates for the effective tetragonal elastic moduli of two-phase fiber-reinforced composites

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.

On convergence of the generalized Maxwell scheme: conductivity of composites containing cubic arrays of spherical particles

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.

Geomechanical Simulation of Fluid-Driven Fractures

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.