Radial Deformations and Cavitation in Riemannian Manifolds with Applications to Membrane Shells

Abstract

This study is a geometric version of Ball's work, Philos. Trans. Roy. Soc. London Ser. A 306 (1982), no. 1496, 557-611. Radial deformations in Riemannian manifolds are singular solutions to some nonlinear equations given by constitutive functions and radial curvatures. A geodesic spherical cavity forms at the center of a geodesic ball in tension by means of given surface tractions or displacements. The existence of such solutions depends on the growth properties of the constitutive functions and the radial curvatures. Some close relationships are shown among radial curvature, the constitutive functions, and the behavior of bifurcation of a singular solution from a trivial solution. In the incompressible case the bifurcation depends on the local properties of the radial curvature near the geodesic ball center but the bifurcation in compressible case is determined by the global properties of the radial curvatures. A cavity forms at the center of a membrane shell of isotropic material placed in tension by means of given boundary tractions or displacements when the Riemannian manifold under question is a surface of $\R^3$ with the induced metric. In addition, cavitation at the center of ellipsoids of $\R^n$ is also described if the Riemannian manifold under question is $(\R^n g)$ where $g(x)$ are symmetric, positive matrices for $x\in\R^n.$