A NUMERICAL SOLUTION METHOD FOR AN INFINITESIMAL ELASTO-PLASTIC COSSERAT MODEL

We present a finite element implementation of a Cosserat elasto-plastic model and provide a rigorous numerical analysis of the introduced time-incremental algorithm. The model allows the use of standard tools from convex analysis as known from classical Prandtl–Reuss plasticity. We derive the dual stress formulation and prove that for vanishing Cosserat couple modulus μc → 0 the classical problem is approximated. Our numerical results show the robustness of the approximation. Notably, for positive couple modulus μc > 0 there is no need for a safe-load assumption. For small μc the response is numerically indistinguishable from the classical response.

A GEOMETRICALLY EXACT PLANAR COSSERAT SHELL-MODEL WITH MICROSTRUCTURE: EXISTENCE OF MINIMIZERS FOR ZERO COSSERAT COUPLE MODULUS

The existence of minimizers to a geometrically exact Cosserat planar shell model with microstructure is proven. The membrane energy is a quadratic, uniformly Legendre–Hadamard elliptic energy in contrast to traditional membrane energies. The bending contribution is augmented by a curvature term representing the interaction of the rotational microstructure in the Cosserat theory. The model includes non-classical size effects, transverse shear resistance, drilling degrees of freedom and accounts implicitly for thickness extension and asymmetric shift of the midsurface. Upon linearization with zero Cosserat couple modulus μc = 0, one recovers the infinitesimal-displacement Reissner–Mindlin model. It is shown that the Cosserat shell formulation admits minimizers even for μc = 0, in which case the drill-energy is absent. The midsurface deformation m is found in H1(ω, ℝ3). Since the existence of energy minimizers rather than equilibrium solutions is established, the proposed analysis includes the large deformation/large rotation buckling behaviour of thin shells.