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Abstract A 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. 

Abstract We address a problem that extends a fundamental classical result of continuum mechanics from the time of its inception, as well as answers a fundamental question in the recent, modern nonlinear elastic theory of dislocations. Interestingly, the implication of our result in the latter case is qualitatively different from its well-established analog in the linear elastic theory of dislocations. It is a classical result that if $$u\in C^2({\mathbb {R}}^n;{\mathbb {R}}^n)$$ u ∈ C 2 ( R n ; R n ) and $$\nabla u \in SO(n)$$ ∇ u ∈ S O ( n ) , it follows that u is rigid. In this article this result is generalized to matrix fields with non-vanishing $${\text {curl }}$$ curl . It is shown that every matrix field $$R\in C^2(\varOmega ;SO(3))$$ R ∈ C 2 ( Ω ; S O ( 3 ) ) such that $${\text {curl }}R = constant$$ curl R = c o n s t a n t is necessarily constant. Moreover, it is proved in arbitrary dimensions that a measurable rotation field is as regular as its distributional $${\text {curl }}$$ curl allows. In particular, a measurable matrix field $$R: \varOmega \rightarrow SO(n)$$ R : Ω → S O ( n ) , whose $${\text {curl }}$$ curl in the sense of distributions is smooth, is also smooth.