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

Abstract

to SO(n). MSC (2010) 15A24, 22L30, 74A30, 74A35, 74B20, 74G05, 74G65, 74N15In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field F := ∇ϕ : → GL + (n) and the microrotation field R : → SO(n), the shear-stretch energy is necessarily of the formwhere μ > 0 is the Lamé shear modulus and μ c ≥ 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 range μ > 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 (F γ ) for simple planar shear.

Transparent anisotropy for the relaxed micromorphic model: Macroscopic consistency conditions and long wave length asymptotics

“…One of the most known generalized continuum models is the micromorphic continuum model introduced by Mindlin and Eringen [11,[16][17][18][19]54] in the early sixties of the last century. It includes many special cases among which the much older Cosserat-type models [5,12,20,34,35,40,64,70,71].…”

Grioli’s Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations

“…The optimal rotations in the weighted case rpolar µ,µc (F ) have been worked out in two and three space dimensions by the present authors in a series of papers [9,10]; cf. also [8] and [22,27] for earlier related work.…”

“…Regarding the material parameters, we proved in [9] that for any dimension n ≥ 2, it is in fact sufficient to restrict our attention to two parameter pairs: (µ, µ c ) = (1, 1), the classical case, and (µ, µ c ) = (1, 0), the non-classical case. Hence, somewhat surprisingly, the solutions for arbitrary µ > 0 and µ c ≥ 0 can be recovered from these two limit cases.…”

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

“…The above optimization problem will be further reduced with respect to the coefficients µ, µ c , see [17], to case µ = 1, µ c = 0.…”

“…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…”