Quasiconvex envelopes in nonlinear elasticity

Annie Raoult (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We give several examples of modeling in nonlinear elasticity where a quasiconvexification procedure is needed. We first recall that the three-dimensional Saint Venant-Kirchhoff energy fails to be quasiconvex and that its quasiconvex envelope can be obtained by means of careful computations. Second, we turn to the mathematical derivation of slender structure models: an asymptotic procedure using Gamma-convergence tools leads to models whose energy is quasiconvex by construction. Third, we construct an homogenized quasiconvex energy for square lattices.

Mots clés: nonlinear elasticity; membranes; lattices; quasiconvexity; relaxation; homogenization

Domaines: Equations aux dérivées partielles [math.AP]; Mécanique [physics.med-ph]

Annie Raoult. Quasiconvex envelopes in nonlinear elasticity. J. Schröder, P. Neff. Poly-,Quasi- and Rank-one Convexity in Applied Mechanics, Springer, pp.17-51, 2010, CISM Courses and Lectures, vol. 516. ⟨hal-00365910⟩