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A new paradigm: The linear isotropic Cosserat model with conformally invariant curvature energy. (English) Zbl 1157.74002

Summary: We discuss a linear Cosserat model with weakest possible constitutive assumptions on the curvature energy still providing for existence, uniqueness, and stability. The assumed curvature energy is the conformally invariant expression \(\mu L_c^2\| \text{dev sym}\nabla \text{axl}\overline A \| ^2\), where axl\(\overline A\) is the axial vector of the skewsymmetric microrotation \(\overline A \in\mathfrak{so}(3)\), dev is the orthogonal projection on the Lie algebra \(\mathfrak{sl}(3)\) of trace free matrices, and sym is the orthogonal projection onto symmetric matrices. It is observed that unphysical singular stiffening for small samples is avoided in torsion and bending while size effects are still present. The number of Cosserat parameters is reduced from six to four: in addition to the (size-independent) classical linear elastic Lamé moduli \(\mu\) and \(\lambda\), only one Cosserat coupling constant \(\mu_c > 0\) and one length scale parameter \(L_c > 0\) need to be determined. We investigate those deformations not leading to moment stresses for different curvature assumptions, and thereby hypothesize a novel invariance principle of linear isotropic Cauchy elasticity which is extended to the Cosserat and couple-stress (Koiter-Mindlin) model with conformal curvature.

MSC:

74A35 Polar materials
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