The Cosserat couple modulus for continuous solids is zero viz the linearized Cauchy-stress tensor is symmetric

Abstract

We investigate weaker than usual constitutive assumptions in linear Cosserat theory still providing for existence and uniqueness and show a continuous dependence result for Cosserat couple modulus µc → 0. This result is needed when using Cosserat elasticity not as a physical model but as a numerical regularization device. Thereafter it is shown that the usually adopted material restrictions of uniform positivity for a linear Cosserat model cannot be consistent with experimental findings for continuous solids: the analytical solutions for both the torsion and the bending problem in general predict an unbounded stiffness for ever thinner samples. This unphysical behaviour can only be avoided for specific choices of parameters in the curvature energy expression. However, these choices do not satisfy the usual constitutive restrictions. We show that the possibly remaining linear elastic Cosserat problem is nevertheless well-posed but that it is impossible to determine the appearing curvature modulus independent of boundary conditions. This puts a doubt on the use of the linear elastic Cosserat model (or the geometrically exact model with µc > 0) for the physically consistent description of continuous solids like polycrystals in the framework of elasto-plasticity. The problem is avoided in geometrically exact Cosserat models if the Cosserat couple modulus µc is set to zero.