On the coupling of large deformations and elastic-plasticity in the mechanics of a simple system

Abstract

The buckling and the post buckling of a simple model involving both large deformations and elastic-plasticity are studied. Plasticity is described by the classical incremental constitutive law involving in particular non regularized transition between plastic loading and elastic unloading. The kinematics is chosen in order to take into account any rotation, however large, without having to resort to approximations or asymptotic expansions. The work focusses essentially on the specificity of the coupling between these two types of nonlinearity. At small strains the global behavior is correctly given by Hutchinson’s simple model involving the same elastic-plastic constitutive law and linearized deformations. But at very large strains, the response notably differs from that of Hutchinson’s model. Indeed plasticity seems to have no effect and the behavior is that of a system involving the same nonlinear deformations but with a constant modulus. From the point of view of bifurcation analysis this result is qualitatively interesting due to the fact that the elastic-plastic constitutive law can be regarded as a strong, or non-smooth, nonlinearity, whereas large deformations is a smooth, we could say weaker, non linearity, so that the result contradicts the intuition that the strongest non linearity should be associated with the strongest bifurcation effect. The analysis is carried out using two different geometrical nonlinearities.