There Is Neither Flattening nor Self-contact or Contact with an Obstacle. Non Smooth Evolution

Abstract

In this chapter the predictive theory of the motion of a solid with large deformation is described. The novel and main point of the theory is the choice of the quantities which describe the shape changes. Observation shows that the stretch matrix and the rotation matrix of the classical polar decomposition are adapted to account for part of the shape change. Moreover, an external action can be applied on the surface of the solid by a curvilinear beam. This observation and mathematics show that a third order gradient theory is mandatory (a deformation involving third order space derivatives is needed). The velocities of deformation involve the angular velocity and its gradient, the gradient of the velocity and the third gradient velocity. The free energy depends on the stretch matrix and it may be a convex function of this matrix, giving properties we are accustomed to in the small deformation theory. The volume impenetrability condition is that the three eigenvalues of the stretch matrix are larger than \(\alpha > 0, \alpha \) quantifies the resistance to crushing of the material. The theory is completely developed for a viscoelastic solid including the evolution of the temperature. The theory coherent in terms of mechanics is also coherent in terms of mathematics. Incompressibility and plasticity are taken into account within this framework. This long chapter has a classical structure: description of the shape change and of the shape change velocities, equations of motion, choice of the state quantities, choice of the quantities which describe the evolution, the laws of thermodynamics, the free energy and the pseudo-potential of dissipation, solution of the equations predicting the evolution.