Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

Abstract: This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of equations of motion, viewing motion as a curve in an infinite-dimensional Banach space of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies; residual stress results from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two- dimensional curved annulus embedded in a sphere.