The principle of virtual work and a symplectic reduction of non-local continuum mechanics

Abstract

We study a symplectic formulation of continuum mechanics on the Fréchet manifold of (smooth) embeddings J : M → N. From a virtual work principle we derive a generalized Hamiltonian dynamics, which allows to include the description non-hyperelastic media as well as a non-linear and non-local material behaviour. Dividing out the (rigid) translation group, the deformation gradient and the stress tensor appear—via Marsden-Weinstein reduction—as natural geometric objects. The balance law on the reduced phase space becomes a weak equation for exact differential forms, which relates the dynamics of the deformation gradient to the stress. This equation is subjected to an interesting gauge freedom and yields the classical continuum dynamics as well as boundary conditions for the stress.