Stress boundedness and existence of radial minimizers in constrained nonlinear elasticity

For simple elastic bodies in small strain regime, the convexity of the energy allows one to discuss equilibrium problems under stress constraints (the specification of an admissible convex region in the stress space) in terms of the complementary energy. In the presence of large strains, the necessary lack of energy convexity does not allow one to retrace the same path. A significant concept of complementary energy in large strain regime rests on the Legendre transform of the energy with respect to the deformation gradient, its cofactor and determinant. The related minimum problem necessarily requires that constraints be assigned to the derivatives of the energy density with respect to the variables already listed (these derivatives are required to be in a convex subset of an appropriate linear space). A problem is to characterize the related stresses in terms of a constrained energy. We tackle this problem for radially symmetric simple bodies under radial deformations and show how the resulting stress is bounded. We also prove the existence of radially symmetric minimizers for the constrained elastic energy under Dirichlet boundary conditions.