This paper presents a technique for developing convex dual variational principles for the governing PDEs of nonlinear elastostatics and elastodynamics. This allows the definition of variational dual solutions and dual solutions corresponding to the PDEs of nonlinear elasticity, even when the energy functionals are non-quasiconvex and their energy minimizers do not exist. The existence of variational dual solutions is proven rigorously for elastostatics in the Saint-Venant Kirchhoff material (in all dimensions) and for the incompressible neo-Hookean material in 2D. The methodology is demonstrated through computational results, including the computation of stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy. A key finding is the stability of a dual elastodynamic equilibrium solution with non-vanishing length regions of negative elastic stiffness, which demonstrates an explosive 'Hadamard instability' in the primal problem, highlighting implications for modeling softening behavior in macroscopic mechanical responses. The approach is based on transforming the primal PDE problem into a dual variational problem involving a dual-to-primal mapping, which ensures that the Euler-Lagrange equations of the dual problem are the primal PDE system. The dual functional is shown to be convex, and its existence of minimizers is established under certain conditions, providing a robust framework for solving nonlinear elasticity problems.This paper presents a technique for developing convex dual variational principles for the governing PDEs of nonlinear elastostatics and elastodynamics. This allows the definition of variational dual solutions and dual solutions corresponding to the PDEs of nonlinear elasticity, even when the energy functionals are non-quasiconvex and their energy minimizers do not exist. The existence of variational dual solutions is proven rigorously for elastostatics in the Saint-Venant Kirchhoff material (in all dimensions) and for the incompressible neo-Hookean material in 2D. The methodology is demonstrated through computational results, including the computation of stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy. A key finding is the stability of a dual elastodynamic equilibrium solution with non-vanishing length regions of negative elastic stiffness, which demonstrates an explosive 'Hadamard instability' in the primal problem, highlighting implications for modeling softening behavior in macroscopic mechanical responses. The approach is based on transforming the primal PDE problem into a dual variational problem involving a dual-to-primal mapping, which ensures that the Euler-Lagrange equations of the dual problem are the primal PDE system. The dual functional is shown to be convex, and its existence of minimizers is established under certain conditions, providing a robust framework for solving nonlinear elasticity problems.