Discontinuous mappings and the limit load in boundary value problems of nonlinear elasticity

The article discusses a boundary value problem of nonlinear elasticity for mapping (deformation) in two weak formulations: in the form of a variational equilibrium equation and in the form of minimization of a multidimensional integral functional. Some questions of a mathematical correctness of the nonlinear elasticity BVP are discussed. Using methods of calculus of variations, by the example of two simple problems, it is proved that for some nonlinear elastic models in the corresponding boundary value problems there can be mappings that have discontinuities of the slip type, as well as there is the limit load - such a finite value of external forces, above which the boundary value problem has no solution at all.