Variational derivation of the Flamant solution for a nonlinear elastic wedge

TL;DR

This paper derives the classical Flamant solution for a nonlinear elastic wedge as a leading order asymptotic response under small boundary loads, extending linear elasticity results to hyperelastic materials.

Contribution

It provides a variational derivation of the Flamant solution from nonlinear elasticity, including general hyperelastic energies with super-quadratic growth.

Findings

Proves the Flamant solution as the leading order response for nonlinear elastic wedges.

Establishes a geometric rigidity inequality in $L^p$ for truncated wedge domains.

Derives an asymptotic variational principle characterizing the Flamant solution.

Abstract

Concentrated forces acting at the tip of a two-dimensional wedge give rise to the classical Flamant solution to linear elasticity, whose displacement and strain are singular at the tip of the wedge. Starting from nonlinear elasticity, we prove that the Flamant solution gives the leading order response of a slightly truncated wedge to small boundary displacements or loads. This asymptotic result holds for general hyperelastic energies with super-quadratic growth at infinity; it also holds in the borderline case of quadratic growth at infinity, so long as the tip of the wedge is subjected to small enough displacements or loads. A main point of the proof is to restore compactness to low-energy sequences. We do so by applying a logarithmic change of variables sufficiently far from the tip. To justify this change of variables, we prove a geometric rigidity inequality in $L^p$ for truncated wedge domains with a constant that is uniform in the truncation length. This follows from the bi-Lipschitz invariance of the constant in the $L^p$ Friesecke--James--M\"uller inequality. Using this change of variables, we derive an asymptotic variational principle characterizing the Flamant solution in the singular limit of an ideal wedge.