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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.

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The unitary polar factor Q = U p in the polar decomposition of Z = U p H is the minimizer over unitary matrices Q for both Log(Q * Z) 2 and its Hermitian part sym * (Log(Q * Z)) 2 over both R and C for any given invertible matrix Z ∈ C n×n and any matrix logarithm Log, not necessarily the principal logarithm log. We prove this for the spectral matrix norm for any n and for the Frobenius matrix norm in two and three dimensions. The result shows that the unitary polar factor is the nearest orthogonal matrix to Z not only in the normwise sense, but also in a geodesic distance. The derivation is based on Bhatia's generalization of Bernstein's trace inequality for the matrix exponential and a new sum of squared logarithms inequality. Our result generalizes the fact for scalars that for any complex logarithm and for all z ∈ C \ {0} min ϑ∈(−π,π] | Log C (e −iϑ z)| 2 = | log |z|| 2 , min ϑ∈(−π,π] | Re Log C (e −iϑ z)| 2 = | log |z|| 2 .

In any geometrically nonlinear, isotropic and 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 formWe aim at the derivation of closed form expressions for the minimizers of W μ,μc (R ; F ) in SO(3), i.e., for the set of optimal Cosserat microrotations in dimension n = 3, as a function of F ∈ GL + (3). In a previous contribution (Part I), we have first shown that, for all n ≥ 2, the full range of weights μ > 0 and μ c ≥ 0 can be reduced to either a classical or a non-classical limit case. We have then derived the associated closed form expressions for the optimal planar rotations in SO(2) and proved their global optimality. In the present contribution (Part II), we characterize the non-classical optimal rotations in dimension n = 3. After a lift of the minimization problem to the unit quaternions, the Euler-Lagrange equations can be symbolically solved by the computer algebra system Mathematica. Among the symbolic expressions for the critical points, we single out two candidates rpolar ± μ,μc (F ) ∈ SO(3) which we analyze and for which we can computationally validate their global optimality by Monte Carlo statistical sampling of SO(3). Geometrically, our proposed optimal Cosserat rotations rpolar ± μ,μc (F ) act in the plane of maximal stretch. Our previously obtained explicit formulae for planar optimal Cosserat rotations in SO(2) reveal themselves as a simple special case. Further, we derive the associated reduced energy levels of the Cosserat shear-stretch energy and criteria for the existence of non-classical optimal rotations. 1 The Cosserat brothers never proposed any specific expression for the local energy W = W (U ). The chosen quadratic ansatz for W = W (U ) is motivated by a direct extension of the quadratic energy in the linear theory of Cosserat models, see, e.g. [41,65,66]. We consider a true volumetric-isochoric split in Sect. 3.4.Let us abbreviate R(q) := π S 3 (q). It is precisely the restriction of the lifted energy to the unit quaternions for which the Cosserat shear-stretch energy of the relative rotation is well-defined W 1,0 S 3 (q ; D) = W 1,0 ( R(q) ; D).

We consider the problem to determine the optimal rotations ∈ SO( ) which minimizeThe objective function is the reduced form of the Cosserat shear-stretch energy, which, in its general form, is a contribution in any geometrically nonlinear, isotropic, and quadratic Cosserat micropolar (extended) continuum model. We characterize the critical points of the energy ( ; ), determine the global minimizers and compute the global minimum. This proves the correctness of previously obtained formulae for the optimal Cosserat rotations in dimensions two and three. The key to the proof is the result that every real matrix whose square is symmetric can be written in some orthonormal basis as a block-diagonal matrix with blocks of size at most two. K E Y W O R D SCosserat theory, Grioli's theorem, (non-symmetric) matrix square root, micropolar media, polar decomposition, relaxed-polar decomposition, rotations, special orthogonal group, symmetric square A M S 2 0 1 0 S

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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.