Existence, Uniqueness and Stability in Linear Cosserat Elasticity for Weakest Curvature Conditions

Abstract

We investigate the weakest possible constitutive assumptions on the curvature energy in linear Cosserat models still providing for existence, uniqueness and stability. The assumed curvature energy is μL2 c ∥dev sym ∇axl A∥2 where axl A is the axial vector of the skewsymmetric microrotation A ∈ so (3 ) and dev is the orthogonal projection on the Lie-algebra sl (3 ) of trace free matrices. The proposed Cosserat parameter values coincide with values adopted in the experimental literature by R. S. Lakes. It is observed that unphysical 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. One Cosserat coupling parameter µc > 0 and only one length scale parameter L c > 0. Use is made of a new coercive inequality for conformal Killing vectorfields. An interesting point is that no (controversial) essential boundary conditions on the microrotations need to be specified; thus avoiding boundary layer effects. Since the curvature energy is the weakest possible consistent with non-negativity of the energy, it seems that the Cosserat couple modulus µc > 0 remains a material parameter independent of the sample size which is impossible for stronger curvature expressions.