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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 (pq)-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.

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Acknowledgements

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.

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Correspondence to M. Ortiz.

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Conti, S., Müller, S. & Ortiz, M. Symmetric Div-Quasiconvexity and the Relaxation of Static Problems. Arch Rational Mech Anal 235, 841–880 (2020). https://doi.org/10.1007/s00205-019-01433-1

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