Bifurcations at a spherical cavity in a compressible solid with spherical isotropy

Abstract

AbstractThis paper investigates bifurcations, including surface mode and localized mode of deformations, at a spherical cavity in an infinite compressible solid with spherical isotropy. Two types of surface modes are found satisfying the zero traction rate conditions on the surface of the hole: (1) the torsional modes, corresponding to bifurcations with zero radial velocity and zero rate of dilatation; and (2) the spheroidal modes, corresponding to bifurcations with zero rate of spin on the tangential planes. Both of these bifurcations can be written in terms of surface harmonics of degree n. Three‐dimensional pictures of some of the bifurcation modes are devised for both torsional and spheroidal modes, showing both the displaced nodes and nodal lines. Torsional bifurcation occurs only if the ratio of the radial shear modulus to the tangential shear modulus (Gr/Gt) vanishes. Three types of spheroidal modes are possible depending on the combination of the five material parameters; their eigenvalue equations are obtained analytically. As expected, numerical results show that the conditions of bifurcations are independent of the size of the spherical hole. Radial bifurcation modes (n = 0) are always possible at the peak applied stress; and pre‐peak bifurcations are also possible for mode shapes composed of surface harmonics of higher degree (n > 0). Spherically isotropic solids (i.e. Gt/Gr < 1) are more conducive to bifurcations than isotropic solids if n > 2. Surface mode with large degree (n → ∞ ) is not necessarily the lowest possible bifurcation; this finding differs from the conclusion by Bassani et al.1 for incompressible isotropic solids.