Reference configurations vs. optimal rotations: a derivation of linear elasticity from finite elasticity for all traction forces

Abstract

We rigorously derive linear elasticity as a low energy limit of pure traction nonlinear elasticity. Unlike previous results, we do not impose any restrictive assumptions on the forces, and obtain a full $\Gamma$-convergence result. The analysis relies on identifying the correct reference configuration to linearize about, and studying its relation to the rotations preferred by the forces (optimal rotations). The $\Gamma$-limit is the standard linear elasticity model, plus a term that penalizes for fluctuations of the reference configurations from the optimal rotations. However, on minimizers this additional term is zero and the limit energy reduces to standard linear elasticity.

v2: minor changes and clarification, Example 6.5 added

Cited by in corpus (4)

• Nonlinear and Linearized Models in Thermoviscoelasticity
• On the hierarchy of plate models for a singularly perturbed multi-well nonlinear elastic energy
• Stability of the Von Kármán regime for thin plates under Neumann boundary conditions
• Linearization in incompatible elasticity for general ambient spaces