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

Abstract

The rotation $\mathrm{polar}(F)\in \mathrm{SO}(3)$ arises as the unique orthogonal factor of the right polar decomposition $F=\mathrm{polar}(F)\cdot U$ of a given invertible matrix $F\in {\mathrm{GL}}^{+}(3)$. In the context of nonlinear elasticity Grioli (1940) discovered a geometric variational characterization of $\mathrm{polar}(F)$ as a unique energy-minimizing rotation. In preceding works, we have analyzed a generalization of Grioli's variational approach with weights (material parameters) $\hat{I}¼>0$ and $\hat{I}{¼}_c\ge 0$ (Grioli: $\hat{I}¼=\hat{I}{¼}_c$). 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 $F:=\nabla \ddot{I}†:\hat{I}\text{c◯}\to {\mathrm{GL}}^{+}(3)$ and the microrotation field $R:\hat{I}\text{c◯}\to \mathrm{SO}(3)$. The corresponding set of non-classical energy-minimizing rotations $${\mathrm{rpolar}}_{\hat{I}¼,\hat{I}{¼}_c}^{\pm }(F):=\begin{array}{c}\mathrm{argmin}\\ R\in \mathrm{SO}(3)\end{array}\{W_{\hat{I}¼,\hat{I}{¼}_c}(R;F):=\hat{I}¼||\mathrm{sym}(R^{T}F-1)|{|}^2+\hat{I}{¼}_c||\mathrm{skew}(R^{T}F-1)|{|}^2\}$$ 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 $\ddot{I}{†}_{\mathrm{nano}}$ which models an idealized nanoindentation and compare the corresponding optimal rotation patterns ${\mathrm{rpolar}}_{1,0}^{\pm }(F_{\mathrm{nano}})$ 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.

28 pages, 11 figures