A numerical procedure for multiple circular holes and elastic inclusions in a finite domain with a circular boundary

Abstract

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 GaussSeidel iterative algorithm.

Semi-analytical solution for a viscoelastic plane containing multiple circular holes

“…As an attempt in this direction, we describe here a semi-analytical solution for the problem of an infinite viscoelastic plane containing multiple holes. The time-independent analog of this approach has been presented earlier in the series of papers [Mogilevskaya and Crouch 2001;2002;Wang et al 2003a;2003b;Legros et al 2004]. The technique presented in those papers was based on the use of complex or real versions of the two-dimensional Somigliana's formula.…”

A fast and accurate algorithm for a Galerkin boundary integral method

“…Based on the above global representations of the unknown boundary parameters, all of the integrals involved in (2) can be evaluated analytically (see Wang et al [22]; Wang [23]). As a result, explicit expressions are obtained for the potentials, which are written as a superposition of the influences from the individual elements and from the conditions at infinity.…”

“…Taking into account the particular shapes of the boundaries considered in this paper, we express the unknown boundary parameters involved in (2) in terms of series expansions of orthogonal functions [12,[20][21][22][23].…”

“…Mogilevskaya and Crouch [12]; Wang et al [20]- [22]; Wang [23]) for solving two-dimensional elastostatics problems involving numerous straight cracks and circular inhomogeneities. In the method, the boundary parameters are represented globally in terms of series expansions of orthogonal functions and all integrations are performed analytically.…”

“…Unlike in the superblock approach, the coefficients in all series expansions for the unknown functions, including in the asymptotic and Taylor expansions, are calculated analytically whenever possible. As a result, the approach retains the advantages of the Galerkin boundary integral method [12,[20][21][22][23], which requires no boundary discretization and does not rely on collocation. Accordingly, the results are not artificially distorted in the vicinity of a boundary, and the individual physical features (inclusions, holes, and cracks) can be arbitrarily close together.…”