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

© 2017, Springer International Publishing AG. The rotation polar (F) ∈ SO (3) arises as the unique orthogonal factor of the right polar decomposition F=polar(F)U of a given invertible matrix F∈ GL+(3). In the context of nonlinear elasticity Grioli (Boll Un Math Ital 2:252–255, 1940) discovered a geometric variational characterization of 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) μ> 0 and μc≥ 0 (Grioli: μ= μc). 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: = ∇ φ: Ω → GL+(3) and the microrotation field R: Ω → SO (3). The corresponding set of non-classical energy-minimizing rotations rpolarμ,μc±(F):=arg minR∈SO(3){Wμ,μc(R;F):=μ||sym(RTF-1)||2+μc||skew(RTF-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 φnanowhich models an idealized nanoindentation and compare the corresponding optimal rotation patterns rpolar1,0±(Fnano) 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.

Jaar van publicatie:2017