Symmetric Div-Quasiconvexity and the Relaxation of Static Problems

Abstract

We consider problems of static equilibrium in which the primary unknown is the stress field and the solutions maximize a complementary energy subject to equilibrium constraints. A necessary and sufficient condition for the sequential lower-semicontinuity of such functionals is symmetric $\mathrm{div}$-quasiconvexity; a special case of Fonseca and Müller’s $\mathcal {A}$-quasiconvexity with $\mathcal {A}= \mathrm{div}$ acting on $\mathbb {R}^{n\times n}_\mathrm {sym}$. We specifically consider the example of the static problem of plastic limit analysis and seek to characterize its relaxation in the non-standard case of a non-convex elastic domain. We show that the symmetric $\mathrm{div}$-quasiconvex envelope of the elastic domain can be characterized explicitly for isotropic materials whose elastic domain depends on pressure p and Mises effective shear stress q. The envelope then follows from a rank-2 hull construction in the (p, q)-plane. Remarkably, owing to the equilibrium constraint, the relaxed elastic domain can still be strongly non-convex, which shows that convexity of the elastic domain is not a requirement for existence in plasticity.