Andreas Fischle

A Logarithmic Minimization Property of the Unitary Polar Factor in the Spectral and Frobenius Norms

The unitary polar factor

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

Q=U_p

Stable identification of linear isotropic Cosserat parameters: bounded stiffness in bending and torsion implies conformal invariance of curvature

in the polar decomposition of

A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm

Z=U_p \, H

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part II: Non‐classical energy‐minimizing microrotations in 3D and their computational validation***

is the minimizer over unitary matrices

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

Q

Symmetric Cauchy stresses do not imply symmetric Biot strains in weak formulations of isotropic hyperelasticity with rotational degrees of freedom.

for both

Optimality of the relaxed polar factors by a characterization of the set of real square roots of real symmetric matrices

\|{\rm Log}(Q^* Z)\|^2

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

and its Hermitian part

The unitary polar factor Q=U minimizes norm{Log(Q^* Z)}^2 and norm{sym Log(Q^* Z)}^2 in the spectral norm in any dimension and the Frobenius matrix norm in three dimensions

\|{{\rm sym}{_{_*}}\!}({\rm Log}(Q^* Z))\|^2