Sharp conditions for the linearization of finite elasticity

TL;DR

This paper investigates the conditions under which finite elasticity problems can be accurately approximated by linear elasticity, revealing that external load compatibility critically affects the limiting energy.

Contribution

It characterizes the variational limit of rescaled nonlinear elastic energies and identifies conditions where the linear approximation fails to capture the true minimal energy.

Findings

The limiting minimal value can be lower than the linear elastic energy when compatibility conditions are not met.

Results apply to both compressible and incompressible elasticity.

Strict load compatibility is necessary for the linearization to be valid.

Abstract

We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.