2015/07/20 by Andreas Fischle, Fischle, Andreas, Patrizio Neff +1
Materials Science · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Black Holes and Theoretical Physics #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlocal and gradient elasticity in micro/nano structures #math.AP
paper · pdf · doi:10.48550/arxiv.1507.05480
17 pages, 3 figures
arxiv created 2015/07/20 · openalex publication_date 2015/07/20 · arxiv updated 2015/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field F := ∇φ: Ω→ GL+(n) and the microrotation field R: Ω→ SO(n), the shear-stretch energy is necessarily of the form Wμ,μc(R ;F) := μ ‖sym(RT F - \boldsymbol1)‖2 + μc ‖skew(RT F - \boldsymbol1)‖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 argminR ∈ SO(n)Wμ,μc(R ;F) as a function of F ∈ GL+(n) and weights μ> 0 and μc ≥ 0. For n ≥ 2, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: (μ, μc) = (1,1) and (μ,μc) = (1,0). In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range μ> μc ≥ 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.