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

Abstract

Let F ∈ GL + (3) and consider the right polar decomposition F = R p (F ) · U into an orthogonal factor R p (F ) ∈ SO(3) and a symmetric, positive definite factor U =This variational characterization of the orthogonal factor R p (F ) ∈ SO(n) holds in any dimension n ≥ 2 (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations rpolar µ,µc (F ) := arg minfor given weights µ > 0 and µ c ≥ 0. We identify a classical parameter range µ c ≥ µ > 0 for which Grioli's Theorem is recovered and a non-classical parameter range µ > µ c ≥ 0 giving rise to a new type of globally energy-minimizing rotations which can substantially deviate from R p (F ). In mechanics, the weighted energy subject to minimization appears as the shear-stretch contribution in any geometrically nonlinear, quadratic, and isotropic Cosserat theory.

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

“…Due to the parameter reduction, it suffices to consider the Cosserat shear‐stretch energy in the limit case given by A Cosserat strain energy is called isotropic, if it satisfies the invariance for all . Exploiting the isotropy of the Cosserat shear‐stretch energy, it can be equivalently expressed in terms of a rotation acting relative to the polar factor , see the introduction to [] for details. After this second reduction step, we obtain the equivalent energy where is a positive definite matrix.…”

“…Hence, on the basis of the previous works [] and [], it suffices to solve Problem in order to characterize the global minimizers for the quadratic Cosserat shear‐stretch energy in the entire non‐classical parameter range. For a short overview of the previous results, see [].…”

“…The corresponding minimal energy levels were also provided. In dimension three, the following explicit formulae for the solutions to Problem were obtained using computer algebra [, Corollary 2.7]: Corollary Let such that . Then the solutions to Problem are given by the energy‐minimizing relative rotations where is an optimal rotation angle satisfying In particular, for , we have .…”

Soliton solutions in geometrically nonlinear Cosserat micropolar elasticity with large deformations

“…Spinor methods were used in [17] to simplify the Euler-Lagrange equation and subsequent works appeared in [18,19], with an intrinsically two-dimensional model studied in [20]. An investigation in optimisation of the Cosserat shear-stretch energy in searching for the optimal Cosserat rotation is made in [21,22]. In [23], the polarity of ferromagnets gave rise to the description of the defects in order parameters as the solitary waves under the external magnetic stimuli, followed by the study in the elastic crystals as a micropolar continuum in [24], again with the description of soliton solution for the topological defects.…”

Geometrically nonlinear Cosserat elasticity in the plane: applications to chirality

“…Therefore, in addition to the standard deformation field ϕ there is an independent rotation field R, which means that R is an orthogonal matrix. Many models in continuum mechanics were motivated by this idea which has resulted in many interesting research lines, sometime with varying names [2,3,4,5,6,7,8,9,10,11,12,13] The three-dimensional static nonlinear Cosserat model has seen a tremendous increase of interest in recent years [14,15,16,17,18,19,20]. This is connected to its possibility to model uncommon effects like for instance lattice rotations.…”