ALEX 2018 Workshop: Abstracts

Stress-space relaxation

Sergio Conti${}^{(1)}$, Stefan Müller${}^{\mathrm{(}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{)}}$ , and Michael Ortiz${}^{\mathrm{(}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{3}\mathrm{)}}$

(1) Institut für Angewandte Mathematik, Universität Bonn (Germany)

(2) Hausdorff Center for Mathematics, Universität Bonn (Germany)

(3) Division of Engineering and Applied Science, California Institute of Technology (USA)

The theory of relaxation, based on the concept of quasiconvexity, has been very successful in the study of microstructure in nonlinear elasticity. There are situations, however, in which a formulation with the elastic deformation as the only independent variable is not appropriate, even after minimizing out some internal variables. We consider here a setting in which the natural independent variable is the stress and not the strain field, such as critical-state theory of plasticity. We give a general relaxation framework involving building upon the general tools of $𝒜$-quasiconvexity [1] and discuss its application to the relaxation of isotropic models in which the yield surface depends on the first two invariants only. Our results can be used to interpret numerical results on fused silica glass [2]

Acknowledgments: This work was partially supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 1060 “The mathematics of emergent effects”, project A5.