A Geometrically Exact Cosserat Model Including Sharp Interfaces And Fracture

We investigate geometrically exact generalized continua of Cosserat micropolar type. The basic difference of the Cosserat model to classical models for solids is the appearance of a field of independent rotations. We introduce the Cosserat model in variational form as a minimization problem.

The model naturally incorporates an internal length scale which is characteristic of the material, e.g. the grain size and which is supposed to be a localization limiter. This length scale also introduces size effects to the extent that small samples of a material behave comparatively stiffer than large samples.

It is motivated that the traditional Cosserat couple modulus $\mu_c$ appearing in the model can and should be set to zero for macroscopic specimens liable to fracture in shear, still leading to a complete consistent Cosserat theory with independent rotations in the geometrically exact finite case in contrast to the infinitesimal, linearized Cosserat model.

Depending on material constants different mathematical existence theorems in Sobolev-spaces are given for the resulting nonlinear boundary value problems in the elastic case.

Partial focus is set to the possible regularization properties of micropolar Cosserat models compared to classical continuum models in the macroscopic case of materials failing in shear.The mathematical analysis heavily uses an extended Korn's first inequality (Neff, Proc.Roy.Soc.Edinb.A, 2002) discovered recently. The methods of choice are the direct methods of the calculus of variations.

An analytical example shows the beneficial effect of independent rotations and the possibility to describe sharp interfaces.