The influence of uncertain material parameters on stress-strain response

In this paper we present an approach to account for uncertainty in inelastic material properties. The methodology is based on numerical solution of forward Kolmogorov (also known as Fokker-Planck) equation corresponding to the stochastic elastic-plastic constitutive differential equation. The advantage of the forward Kolmogorov (Fokker-Planck) approach is evident as it transforms original non-linear stochastic ordinary differential equation into linear deterministic partial differential equation. The developed methodology is capable of fully describing the uncertainties of the response variable (e.g stress for displacement-controlled simulation) to exact second order for given probabilistic behavior of (elastic and elastic-plastic) material properties. When the material parameters are modeled as random fields, this method could be used to upscale the point-location scale constitutive equation taking into account the material inhomogeneity. Point-location is used here to describe constitutive, differentially small element behavior, which in the sense of material variability is tied to the single coordinate point. The method can also be used for sensitivity analysis of elastic-plastic material models when material properties are modeled as random variables. The developments are general in nature and applicable to any incremental elastic-plastic material model. It will be shown that the most probable (mode) stress response does not need to coincide with deterministic result if material parameters have probabilistic distribution. This implies that by that by neglecting probabilistic distributions of material properties and using means, results in numerical simulations that do not represent the most likely results. Copyright ASCE 2006.