The Degenerate Scales for Plane Elasticity Problems in Piecewise Homogeneous Media Under General Boundary Conditions

Abstract

The degenerate scale issue for 2D-boundary integral equations and boundary element methods has been already investigated for Laplace equation, antiplane and plane elasticity, bending plate for Dirichlet boundary condition. Recently, the problems of Robin and mixed boundary conditions and of piecewise heterogeneous domains have been considered for the case of Laplace equation. We investigate similar questions for plane elasticity for more general boundary conditions. For interior problems, it is shown that the degenerate scales do not depend on the boundary condition. For exterior problems, the two degenerate scales (homogeneous medium) or two of them (heterogeneous medium) are tightly linked with the behavior at infinity of the solutions. The dependence of this behavior on the boundary conditions is investigated. We give sufficient conditions for the uniqueness of the solution. Numerical applications are provided and validate the set of theoretical results.

Degenerate scales for thin elastic plates with Dirichlet boundary conditions

“…Lower bound From [16], we deduce: ρ 0 e −3/2 ≤ ρ i , ρ 0 is the degenerate scale factor for Laplace equation and ρ i , i = 1, 2 are the degenerate scale factors for the biharmonic equation.…”

“…3.2 Obtention of the discriminant matrix from an exterior boundary value problem Now, we consider an exterior problem, following the ideas of [27] for Laplace equation, later applied to Lamé's Eq. [16]. The domain that is exterior to the bounded boundary is denoted by + (Fig.…”