2D motion with large deformations

We study the motion of a solid with large deformations. The solid may be loaded on its surface by needles, rods, beams, plates,... It turns out that it is wise to choose a third gradient theory for the body. The stretch matrix of the polar decomposition has to be symmetric. This is an internal constraint which introduces a reaction stress in the Piola–Kirchhoff–Boussinesq stress. By use of a Galerkin approximation, combined with suitable a priori estimates and a passage to the limit, we prove that there exists a motion which solves a variational formulation of the complete equations of mechanics, at least locally in time. This motion may be interrupted by crushing resulting in a discontinuity of velocity with respect to time, i.e., an internal collision.