Research Article

Injective weak solutions in second-gradient nonlinear elasticity

Healey, Timothy J.^a1 and Krömer, Stefan^a2

^a1 Cornell University, Ithaca, NY 14853, USA.

^a2 Universität Augsburg, 86135 Augsburg, Germany. stefan.kroemer@math.uni-augsburg.de

Abstract

We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question here. In particular, we demonstrate that the determinant of the gradient of any admissible deformation with finite energy is strictly positive on the closure of the domain. With this in hand, Gâteaux differentiability of the potential energy at a minimizer is automatic, yielding the existence of a weak solution. We indicate how our results hold for a general class of boundary value problems, including “mixed” boundary conditions. For each of the two possible pure displacement formulations (in second-gradient problems), we show that the resulting deformation is an injective mapping, whenever the imposed placement on the boundary is itself the trace of an injective map.

(Received December 27 2007)

(Online publication July 19 2008)

Key Words:

• Gradient estimate;
• injective deformations;
• Euler-Lagrange equation;
• nonlinear elasticity

Mathematics Subject Classification:

• 74B20;
• 49K20