The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part II: Non-classical energy-minimizing microrotations in 3D and their computational validation

Abstract

In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field $F=\nabla \ddot{I}†:\hat{I}\text{c◯}\to G{L}^{+}(n)$ and the microrotation field $R:\hat{I}\text{c◯}\to SO(n)$, the shear-stretch energy is necessarily of the form $$W_{\hat{I}¼,\hat{I}{¼}_c}(R;F)=\hat{I}¼\Vert sym(R^{T}F-1){\Vert }^2+\hat{I}{¼}_c\Vert skew(R^{T}F-1){\Vert }^2.$$ We aim at the derivation of closed form expressions for the minimizers of $W(R;F)$ in $SO(3)$, i.e., for the set of optimal Cosserat microrotations in dimension $n=3$, as a function of $F\in G{L}^{+}(n)$. In a previous contribution (Part I), we have first shown that, for all $n\ge 2$, the full range of weights $\hat{I}¼>0$ and $\hat{I}{¼}_c\ge 0$ can be reduced to either a classical or a non-classical limit case. We have then derived the associated closed form expressions for the optimal planar rotations in $SO(2)$ and proved their global optimality. In the present contribution (Part II), we characterize the non-classical optimal rotations in dimension n = 3. After a lift of the minimization problem to the unit quaternions, the Euler-Lagrange equations can be symbolically solved by the computer algebra system Mathematica. Among the symbolic expressions for the critical points, we single out two candidates $rpola{r}_{\hat{I}¼,\hat{I}{¼}_c}^{\pm }(F)\in SO(3)$ which we analyze and for which we can computationally validate their global optimality by Monte Carlo statistical sampling of $SO(3)$. Geometrically, our proposed optimal Cosserat rotations $rpola{r}_{\hat{I}¼,\hat{I}{¼}_c}^{\pm }(F)$ act in the "plane of maximal strain" and our previously obtained explicit formulae for planar optimal Cosserat rotations in $SO(2)$ reveal themselves as a simple special case. Further, we derive the associated reduced energy levels of the Cosserat shear--stretch energy and criteria for the existence of non-classical optimal rotations.

30 pages, 8 figures