Stability of rigid solids embedded in an elastic medium

Abstract

The stability of a two-dimensional continuum consisting of rigid solids embedded in an elastic homogeneous medium with a fixed boundary is investigated. With each solid is associated the scalar \(\sigma = \frac{1}{S}\sum\limits_\alpha {\vec F_\alpha } \user1{ }.\user1{ }\overrightarrow {0M} _\alpha\) where \(\vec F_\alpha\) are balanced dead loads acting at points Mα of the solid, and where S is the area of the surface of the solid. If μ is the shear modulus of the elastic medium, it is shown that (i) the inequality σ+4μ>0, when it applies to each solid, is a sufficient condition for stability of the continuum; (ii) the inequality σ+4μ≧0 is a necessary condition for stability of a single circular solid embedded in an infinite elastic medium.