On the Theory of Relaxation in Nonlinear Elasticity with Constraints on the Determinant

By Sergio Conti, Georg Dolzmann. 2014

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We consider vectorial variational problems in nonlinear elasticity of the form $${I[u]=\int W(Du)\,{\rm d}x}$$I[u]=∫W(Du)dx, where W is continuous on matrices with a positive determinant and diverges to infinity along sequences of matrices whose determinant is positive and tends to zero. We show that, under suitable growth assumptions, the functional $${\int W^{\rm qc}(Du)\,{\rm d}x}$$∫Wqc(Du)dx is an upper bound on the relaxation of I, and coincides with the relaxation if the quasiconvex envelope Wqc of W is polyconvex and has p-growth from below with $${p\geqq n}$$p≧n. This includes several physically relevant examples. We also show how a constraint of incompressibility can be incorporated in our results.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/b80232416767c47a2e23419670ce71ac4dff4e4b