The geometrically nonlinear Cosserat micropolar shear-stretch energy. Part I: A general parameter reduction formula and energy-minimizing microrotations in 2D

摘要

In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field F := ?_? : ? → GL+(n) and the microrotation field R : ? → SO(n), the shear-stretch energy is necessarily of the form W_(μ,μ_c) (R ; F) := μ||sym(RT F - 1)||~2 + μ_c||skew(R~T F - 1)||~2 ,where μ > 0 is the Lamé shear modulus and μc ≥ 0 is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations argmin+(R ∈ SO(n)) W_(μ,μc) (R ; F) as a function of F ∈ GL~+(n) and weights μ > 0 and μ_c ≥ 0. For n ≥ 2, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: (μ,μ_c) = (1, 1) and (μ,μ_c) = (1, 0). In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter rangeμ > μ_c ≥ 0 in dimension n=2. Currently, optimality results for n ≥ 3 are out of reach for us, but we contribute explicit representations for n=2 which we name rpolar _(μ,μc)~± (F) ∈ SO(2) and which arise for n=3 by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations rpolar _(μ,μc)~± (Fγ ) for simple planar shear.