A second-order exact expression for the evolution of probability density function of stress is derived for general, one-dimensional (1-D) elastic-plastic constitutive rate equations with uncertain material parameters. The Eulerian-Lagrangian (EL) form of Fokker-Planck-Kolmogorov (FPK) equation is used for this purpose. It is also shown that by using EL form of FPK, the so called ''closure problem'' associated with regular perturbation methods used so far, is resolved too. The use of EL form of FPK also replaces repetitive and computationally expensive deterministic elastic-plastic computations associated with Monte Carlo technique. The derived general expressions are specialized to the particular cases of point location scale linear elastic and elastic-plastic constitutive equations, related to associated Drucker-Prager with linear hardening. In a companion paper, the solution of FPK equations for 1D is presented, discussed and illustrated through a number of examples.

In this paper, a solution is presented for evolution of probability density function (PDF) of elasticplastic stress-strain relationship for material models with uncertain parameters. Developments in this paper are based on already derived general formulation presented in the companion paper. The solution presented is then specialized to a specific Drucker-Prager elastic-plastic material model. Three numerical problems are used to illustrate the developed solution. The stress-strain response (1D) is given as a PDF of stress as a function of strain. The presentation of the stress-strain response through the PDF differs significantly from the traditional presentation of such results, which are represented by a single, unique curve in stress-strain space. In addition to that the numerical solutions are verified against closed form solutions where available (elastic). In cases where the closed form solution does not exist (elastic-plastic), Monte Carlo simulations are used for verification.

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SUMMARYThe 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.

SUMMARYUncertainty in material properties can have large effect on numerical modeling of solids and structures. This is particularly true as all natural and man-made materials exhibit spatial non-uniformity and point-wise uncertainty in their behaviors.A methodology that accounts for the probabilistic yielding of elastic-plastic materials is presented. The recently developed Eulerian-Lagrangian form of the Fokker-Planck-Kolmogorov equation is used to obtain a second-order exact solution to elastic-plastic constitutive differential equations. In this paper that solution is used in deriving the weighted probabilities of elastic, elastic-plastic behavior and yielding. A number of examples for two commonly used material models, von Mises and Drucker-Prager, illustrated the findings.