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

Andreas Fischle
Patrizio Neff
Journal / Anthology

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
Year: 2017
Volume: 97
Issue: 7
Page range: 843-871

In any geometrically nonlinear, isotropic and quadratic Cosserat micropolar extended continuum model formulated in the deformation gradient field F := ∇ϕ :  → GL+(n) and the microrotation field R :  → SO(n), the shear–stretch energy is necessarily of the form Wμ,μc (R ; F) := μ  sym(RT F − 1)  2 + μc  skew(RT F − 1)  2 . We aim at the derivation of closed form expressions for the minimizers ofWμ,μc (R ; F) in SO(3), i.e., for the set of optimal Cosserat microrotations in dimension n=3, as a function of F ∈ GL+(3). In a previous contribution (Part I), we have first shown that, for all n ≥ 2, the full range of weightsμ > 0 and μc ≥ 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 rpolar± μ,μc (F) ∈ 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 rpolar± μ,μc (F) act in the plane of maximal stretch. 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.

*Applied Mathematics