Calculus of Variations and Geometric Measure Theory

C. Maor - M. G. Mora

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

created by mora on 28 Aug 2020
modified on 17 May 2021


Accepted Paper

Inserted: 28 aug 2020
Last Updated: 17 may 2021

Journal: Journal of Nonlinear Science
Year: 2021
Links: Link to the published paper (open access)


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.

Keywords: Gamma-convergence, nonlinear elasticity, linear elasticity, Neumann boundary conditions, pure traction