The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part I: A general parameter reduction formula and energy-minimizing microrotations in 2D

Abstract

In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field $F:=\nabla \phi :\Omega \to {\mathrm{GL}}^{+}(n)$ and the microrotation field $R:\Omega \to \mathrm{SO}(n)$, the shear-stretch energy is necessarily of the form \begin{equation*} W_{μ,μ_c}(R\,;F) := μ\,\left\lVert{\mathrm{sym}(R^T F - \boldsymbol{1})}\right\rVert^2 + μ_c\,\left\lVert{\mathrm{skew}(R^T F - \boldsymbol{1})}\right\rVert^2\;, \end{equation*} where $\mu >0$ is the Lamé shear modulus and ${\mu }_c\ge 0$ is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations ${\mathrm{argmin}}_{R\in \mathrm{SO}(n)}{W}_{\mu ,{\mu }_c}(R;F)$ as a function of $F\in {\mathrm{GL}}^{+}(n)$ and weights $\mu >0$ and ${\mu }_c\ge 0$. For $n\ge 2$, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: $(\mu ,{\mu }_c)=(1,1)$ and $(\mu ,{\mu }_c)=(1,0)$. In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range $\mu >{\mu }_c\ge 0$ in dimension $n=2$. Currently, optimality results for $n\ge 3$ are out of reach for us, but we contribute explicit representations for $n=2$ which we name ${\mathrm{rpolar}}_{\mu ,{\mu }_c}^{\pm }(F)\in \mathrm{SO}(2)$ and which arise for $n=3$ by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations ${\mathrm{rpolar}}_{\mu ,{\mu }_c}^{\pm }(F_{\gamma })$ for simple planar shear.

17 pages, 3 figures