The role of nonlinear hardening/softening in probabilistic elasto-plasticity

The elastic-plastic modelling and simulations have been studied extensively in the last century. However, one crucial area of material modelling has received very little attention. The uncertainties in material properties probably have the largest influence on many aspects of structural and solids behaviour. Despite its importance, effects of uncertainties of material properties on overall response of structures and solids have rarely been studied. Most of the small number of studies on effects of material variabilities have used repetitive deterministic models through Monte-Carlo-type simulations. While this approach might appear sound, it cannot be both computationally efficient and statistically accurate (have statistically appropriate number of data points). Recently, we have developed a methodology to solve the probabilistic elastic-plastic differential equations. The methodology is based on Eulerian-Lagrangian form of the Fokker-Planck-Kolmogorov equation and provides for full description of the probability density function (PDF) of stress response for a given strain. In this paper we describe our development in some details. In particular, we investigate the effects of nonlinear hardening/softening on predicted PDF of stress. As it will be shown, the nonlinear hardening/ softening will create a discrepancy between the most likely stress solution and the deterministic solution. This discrepancy, in fact, means that the deterministic solution is not the most likely outcome of the corresponding probabilistic solution if material parameters are uncertain (and they always are, we just tend to simplify that fact and use, for example, mean values for deterministic simulations). A number of examples will be presented, illustrating methodology and main results, some of which are quite surprising as mentioned above.