Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion

Abstract

We study the problem of minimizing the functional $I(\phi ) = \int\limits_\Omega {W(x,D\phi )dx}$ on a new class of mappings. We relax summability conditions for admissible deformations to φ ∈ W ¹ _n (Ω) and growth conditions on the integrand W(x, F). To compensate for that, we require the condition $\frac{{\left| {D\phi (x)} \right|^n }} {{J(x,\phi )}} \leqslant M(x) \in L_s (\Omega )$, s > n − 1, on the characteristic of distortion. On assuming that the integrand W(x, F) is polyconvex and coercive, we obtain an existence theorem for the problem of minimizing the functional I(φ) on a new family of admissible deformations A.