Reference Configurations Versus Optimal Rotations: A Derivation of Linear Elasticity from Finite Elasticity for all Traction Forces

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 $$\Gamma $$ Γ -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 $$\Gamma $$ Γ -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.