Simple shear in nonlinear Cosserat elasticity: bifurcation and induced microstructure

We discuss the simple shear problem for a geometrically exact Cosserat model. In contrast to linear Cosserat elasticity, where the unique solution is available in closed form we exhibit a multitude of solutions to the nonlinear problem, even if the two fields of deformations phi and microrotations (R) over bar remain homogeneous. This motivates a search for new conditions on the microrotations (R) over bar which single out a unique, physically reasonable, response. The influence of material parameters, notably the Cosserat couple modulus mu(c) and the internal length scale L-c on the response is also studied. For small Cosserat couple modulus mu(c) > 0 we observe a pitchfork bifurcation of the homogeneous response and for vanishing internal length L-c = 0 and zero Cosserat couple modulus mu(c) = 0 the Cosserat model may show highly oscillating microstructure solutions which are energetically better than the homogeneous response. Thus, the large scale nonlinear Cosserat limit is not necessarily a classical limit.