A Hidden Convexity of Nonlinear Elasticity

Abstract

AbstractA technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive ‘Hadamard instability;’ this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response.