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In any geometrically nonlinear, isotropic and quadratic Cosserat micropolarextended continuum model formulated in the deformation gradient field $F =\nabla\varphi : \Omega \to GL^+(n)$ and the microrotation field $R: \Omega \toSO(n)$, the shear-stretch energy is necessarily of the form  $$W_{\mu,\mu_c}(R;F) = \mu\, \| sym(R^T F - 1) \|^2 + \mu_c\, \| skew(R^T F -1) \|^2 .$$  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 indimension $n = 3$, as a function of $F \in GL^+(n)$. In a previous contribution(Part I), we have first shown that, for all $n \geq 2$, the full range ofweights $\mu > 0$ and $\mu_c \geq 0$ can be reduced to either a classical or anon-classical limit case. We have then derived the associated closed formexpressions for the optimal planar rotations in $SO(2)$ and proved their globaloptimality. In the present contribution (Part II), we characterize thenon-classical optimal rotations in dimension n = 3. After a lift of theminimization problem to the unit quaternions, the Euler-Lagrange equations canbe symbolically solved by the computer algebra system Mathematica. Among thesymbolic expressions for the critical points, we single out two candidates$rpolar^{\pm}_{\mu,\mu_c}(F) \in SO(3)$ which we analyze and for which we cancomputationally validate their global optimality by Monte Carlo statisticalsampling of $SO(3)$. Geometrically, our proposed optimal Cosserat rotations$rpolar^{\pm}_{\mu,\mu_c}(F)$ act in the "plane of maximal strain" and ourpreviously obtained explicit formulae for planar optimal Cosserat rotations in$SO(2)$ reveal themselves as a simple special case. Further, we derive theassociated reduced energy levels of the Cosserat shear--stretch energy andcriteria for the existence of non-classical optimal rotations.

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