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 φ and microrotations remain homogeneous. This motivates a search for new conditions on the microrotations which single out a unique, physically reasonable, response. The influence of material parameters, notably the Cosserat couple modulus μ _c and the internal length scale L _c on the response is also studied. For small Cosserat couple modulus μ _c  > 0 we observe a pitchfork bifurcation of the homogeneous response and for vanishing internal length L _c  = 0 and zero Cosserat couple modulus μ _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.