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We consider problems of static equilibrium in which the primary unknown isthe stress field and the solutions maximize a complementary energy subject toequilibrium constraints. A necessary and sufficient condition for thesequential lower-semicontinuity of such functionals is symmetric ${\rmdiv}$-quasiconvexity, a special case of Fonseca and M\"uller's$A$-quasiconvexity with $A = {\rm div}$ acting on $R^{n\times n}_{sym}$. Wespecifically consider the example of the static problem of plastic limitanalysis and seek to characterize its relaxation in the non-standard case of anon-convex elastic domain. We show that the symmetric ${\rm div}$-quasiconvexenvelope of the elastic domain can be characterized explicitly for isotropicmaterials whose elastic domain depends on pressure $p$ and Mises effectiveshear stress $q$. The envelope then follows from a rank-$2$ hull constructionin the $(p,q)$-plane. Remarkably, owing to the equilibrium constraint therelaxed elastic domain can still be strongly non-convex, which shows thatconvexity of the elastic domain is not a requirement for existence inplasticity.

Symmetric div-quasiconvexity and the relaxation of static problems