Integrability conditions between the first and second Cosserat deformation tensor in geometrically nonlinear micropolar models and existence of minimizers

摘要

In this note we extend integrability conditions for the symmetric stretchtensor $U$ in the polar decomposition of the deformation gradient$\nabla\varphi=F=R\,U$ to the non-symmetric case. In doing so we recoverintegrability conditions for the first Cosserat deformation tensor. Let $F=\barR\,\bar U$ with $\bar R:\Omega\subset\mathbb{R}^3\longrightarrow\mathrm{SO}(3)$and $\bar U:\Omega\subset\mathbb{R}^3\longrightarrow \mathrm{GL}(3)$. Then$\mathfrak{K}:={\bar R}^T\mathrm{Grad}\,{\bar R}=\mathrm{Anti}\Big(\frac{1}{\mathrm{det} \bar U}\Big[\bar U(\mathrm{Curl} \bar U)^T-\frac{1}{2}\mathrm{tr}(\bar U(\mathrm{Curl} \bar U)^T) 1\!\!1 \Big]\bar U\Big),$ giving aconnection between the first Cosserat deformation tensor $\bar U$ and thesecond Cosserat tensor ${\mathfrak{K}}$. (Here, Anti denotes an isomorphismbetween $\mathbb{R}^{3\times 3}$ and$\mathfrak{So}(3):=\{\,\mathfrak{A}\in\mathbb{R}^{3\times 3\times3}\,|\,\mathfrak{A}.u\in\mathfrak{so}(3)\;\forall u\in \mathbb{R}^3\}$.) Theformula shows that it is not possible to prescribe $\bar U$ and $\mathfrak{K}$independent from each other. We also propose a new energy formulation ofgeometrically nonlinear Cosserat models which completely separate the effectsof nonsymmetric straining and curvature. For very weak constitutive assumptions(no direct boundary condition on rotations, zero Cosserat couple modulus,quadratic curvature energy) we show existence of minimizers in Sobolev-spaces.