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

摘要

We reconsider the geometrically nonlinear Cosserat model for a uniformlyconvex elastic energy and write the equilibrium problem as a minimizationproblem. Applying the direct methods of the calculus of variations we show theexistence of minimizers. We present a clear proof based on the coercivity ofthe elastically stored energy density and on the weak lower semi-continuity ofthe 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.