Symmetric Div-Quasiconvexity and the Relaxation of Static Problems
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 divdiv-quasiconvexity; a special case of Fonseca and Müller's A-quasiconvexity with A=div acting on R^(n×n)_(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 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.
© 2019 Springer-Verlag GmbH Germany, part of Springer Nature. First Online: 05 August 2019. This work was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 1060 "The mathematics of emergent effects", project A5, and through the Hausdorff Center for Mathematics, GZ 2047/1, project-ID 390685813.
Accepted Version - 1907.04549.pdf