A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy

Abstract

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 μ L 2 c dev sym ∇ axl A 2 , where axl A is the axial vector of the skewsymmetric microrotation A ∈ so (3), dev is the orthogonal projection on the Lie-algebra 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 μ and λ only one Cosserat coupling constant μc > 0 and one length scale parameter Lc > 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.

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“…One of the most known generalized continuum models is the micromorphic continuum model introduced by Mindlin and Eringen [11,[16][17][18][19]54] in the early sixties of the last century. It includes many special cases among which the much older Cosserat-type models [5,12,20,34,35,40,64,70,71].…”

Semicontinuity theorem in the micropolar elasticity

“…A new approach has been recently taken by Jeong and Neff in [9] in considering the weakest possible, conformally invariant curvature expression (see [10,22]). In this way, physical inconsistencies present for uniform positive curvature assumptions can be avoided (see [16]).…”

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“…An infinitesimal conformal mapping [33,36] preserves (to first order) angles and shapes of infinitesimal figures. The included inhomogeneity is therefore only a global feature of the mapping.…”