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

¹ Ecole Spéciale des Travaux Publics du Bâtiment et de l'Industrie (ESTP), 28 avenue du Président Wilson, 94234 Cachan Cedex, France
² AG6, Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt, Germany

^* To whom correspondence should be addressed.

	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 µL _c ²||dev sym axl ||² where axl is the axial vector of the skewsymmetric microrotation (3) and dev is the orthogonal projection on the Lie-algebra (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 µ _c > 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 µ _c > 0 remains a material parameter independent of the sample size which is impossible for stronger curvature expressions.

Key Words: Polar-materials, microstructure, parameter-identification, structured continua, solid mechanics, variational methods

First published on September 17, 2008, doi:10.1177/1081286508093581
This version was published on September 17, 2008


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