The relaxed-polar mechanism of locally optimal Cosserat rotations for an idealized nanoindentation and comparison with 3D-EBSD experiments

Abstract

The rotation ${\rm polar}(F) \in {\rm SO}(3)$ arises as the unique orthogonal factor of the right polar decomposition $F = {\rm polar}(F) \cdot U$ of a given invertible matrix $F \in {\rm GL}^+(3)$. In the context of nonlinear elasticity Grioli (1940) discovered a geometric variational characterization of ${\rm polar}(F)$ as a unique energy-minimizing rotation. In preceding works, we have analyzed a generalization of Grioli's variational approach with weights (material parameters) $\mu > 0$ and $\mu_c \geq 0$ (Grioli: $\mu = \mu_c$). The energy subject to minimization coincides with the Cosserat shear-stretch contribution arising in any geometrically nonlinear, isotropic and quadratic Cosserat continuum model formulated in the deformation gradient field $F := \nabla\varphi: \Omega \to {\rm GL}^+(3)$ and the microrotation field $R: \Omega \to {\rm SO}(3)$. The corresponding set of non-classical energy-minimizing rotations $$ {\rm rpolar}^\pm_{\mu,\mu_c}(F) := \substack{{\rm argmin}\\ R\,\in\,{\rm SO(3)}} \Big\{ W_{\mu, \mu_c}(R\,;F) := \mu\, || {\rm sym}(R^TF - 1)||^2 + \mu_c\, ||{\rm skew}(R^TF - 1)||^2 \Big\} $$ represents a new relaxed-polar mechanism. Our goal is to motivate this mechanism by presenting it in a relevant setting. To this end, we explicitly construct a deformation mapping $\varphi_{\rm nano}$ which models an idealized nanoindentation and compare the corresponding optimal rotation patterns ${\rm rpolar}^\pm_{1,0}(F_{\rm nano})$ with experimentally obtained 3D-EBSD measurements of the disorientation angle of lattice rotations due to a nanoindentation in solid copper. We observe that the non-classical relaxed-polar mechanism can produce interesting counter-rotations. A possible link between Cosserat theory and finite multiplicative plasticity theory on small scales is also explored.