Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements

TL;DR

This paper proves the existence of solutions for a nonlinear Cosserat micropolar elasticity model using calculus of variations, emphasizing convex energy and dislocation density tensor as a curvature measure.

Contribution

It provides a rigorous existence proof for the nonlinear Cosserat model under convexity assumptions, utilizing the dislocation density tensor as a curvature measure.

Findings

Existence of minimizers established for the nonlinear Cosserat model.

Use of dislocation density tensor as a curvature measure.

Proof based on coercivity and weak lower semi-continuity.

Abstract

We reconsider the geometrically nonlinear Cosserat model for a uniformly convex elastic energy and write the equilibrium problem as a minimization problem. Applying the direct methods of the calculus of variations we show the existence of minimizers. We present a clear proof based on the coercivity of the elastically stored energy density and on the weak lower semi-continuity of the total energy functional. Use is made of the dislocation density tensor $\bar{\boldsymbol{K}}=\bar{\boldsymbol{R}}^T\,\mathrm{Curl}\,\bar{\boldsymbol{R}}$ as a suitable Cosserat curvature measure.