This paper presents a technique for developing convex dual variational principles for the governing PDEs of nonlinear elastostatics and elastodynamics. The method allows the definition of variational dual solutions even when the PDEs arise from non-convex energy functionals whose minimizers do not exist. The approach involves transforming the primal PDE problem into a dual variational problem using dual Lagrange multiplier fields. The key insight is to treat the primal PDE as constraints and introduce a strictly convex auxiliary potential to be optimized. This leads to a dual variational principle that is convex, with its Euler-Lagrange equations corresponding to the primal PDE system. The method is demonstrated for the Saint-Venant-Kirchhoff material in all dimensions and the incompressible neo-Hookean material in 2D. It is also applied to compute stressed and unstressed elastostatic and elastodynamic solutions for a non-convex, double-well energy. The stability of a dual elastodynamic equilibrium solution is shown, even in regions with negative elastic stiffness (non-hyperbolic regions), where the primal problem is ill-posed and exhibits a Hadamard instability. The dual methodology is shown to be effective in modeling physically observed softening behavior in macroscopic mechanical response. The paper also discusses the extension of the dual methodology to time-dependent problems, leading to well-defined boundary-value problems in space-time domains. The dual functional is shown to be convex and lower semi-continuous, allowing the application of the direct method of the Calculus of Variations to prove the existence of minimizers. The concept of a variational dual solution is introduced, which is applicable to standard models such as the Saint-Venant-Kirchhoff and incompressible neo-Hookean materials. The paper concludes with the development of computational results and the implications of the dual methodology for the modeling of nonlinear elasticity.This paper presents a technique for developing convex dual variational principles for the governing PDEs of nonlinear elastostatics and elastodynamics. The method allows the definition of variational dual solutions even when the PDEs arise from non-convex energy functionals whose minimizers do not exist. The approach involves transforming the primal PDE problem into a dual variational problem using dual Lagrange multiplier fields. The key insight is to treat the primal PDE as constraints and introduce a strictly convex auxiliary potential to be optimized. This leads to a dual variational principle that is convex, with its Euler-Lagrange equations corresponding to the primal PDE system. The method is demonstrated for the Saint-Venant-Kirchhoff material in all dimensions and the incompressible neo-Hookean material in 2D. It is also applied to compute stressed and unstressed elastostatic and elastodynamic solutions for a non-convex, double-well energy. The stability of a dual elastodynamic equilibrium solution is shown, even in regions with negative elastic stiffness (non-hyperbolic regions), where the primal problem is ill-posed and exhibits a Hadamard instability. The dual methodology is shown to be effective in modeling physically observed softening behavior in macroscopic mechanical response. The paper also discusses the extension of the dual methodology to time-dependent problems, leading to well-defined boundary-value problems in space-time domains. The dual functional is shown to be convex and lower semi-continuous, allowing the application of the direct method of the Calculus of Variations to prove the existence of minimizers. The concept of a variational dual solution is introduced, which is applicable to standard models such as the Saint-Venant-Kirchhoff and incompressible neo-Hookean materials. The paper concludes with the development of computational results and the implications of the dual methodology for the modeling of nonlinear elasticity.