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

Abstract

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 (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 rpolar ± µ,µc (F ) := arg min R ∈ SO(3) Wµ,µ c (R ; F ) := µ sym(R T F − 1) 2 + µc skew(R T F − 1) 2represents 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 ϕnano which models an idealized nanoindentation and compare the corresponding optimal rotation patterns rpolar ± 1,0 (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.

Characterization of the critical points for the shear-stretch strain energy of a Cosserat problem

“…Following an ingenious observation from [7] and [17], which discusses a reduction of the parameters µ, µ c , we have the following straightforward computation. Making the notation 14) in the case µ = µ c we obtain the relation between the free energy W µ,µc (·; F ) and the reduced free…”

“…If f > 1, then we have the family of critical points q (17) ± = ± λ+1 2λ , q 1 , q 2 , q 3 where q 2 1 + q 2 2 + q 2 3 = λ−1 2λ . This family contains the critical points q (7) ±;± , q…”

“…In our paper we search for "small" rotations that minimize the free energy of a linear model for a Cosserat body, energy that is obtained in the hypotheses of isochoric homogeneous deformation and homogeneous microrotations. Details regarding the physical model are presented in Section 2, following the papers [2][3][4][5][6][7]. There is a rich literature on nonlinear models of a Cosserat body, see [8,9].…”

“…8. If λ 1 + λ 2 > 2, then we have four critical points q (7) ±;± = ± 1 2 + 1 λ1+λ2 , 0, 0, ± 1 2 − 1 λ1+λ2 .…”

“…(ii) If λ 1 +λ 2 > 2, then the critical points q (7) ±;± are the only global minima for G 1,0 and consequently, the corresponding rotations R q (7) +;…”