On probabilistic yielding of materials

Abstract

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.

Probabilistic Elastic–Plastic Cam Clay Response of Soils

“…Under the framework of probability theory, these possibilities are governed by the probability density function of yield strength (σy), which can be quantified by statistically analysing constitutive test results. Hence, to realistically simulate the probabilistic material behavior, equivalent advection and diffusion coefficients may be introduced by considering probability weights, based on probability density function of yield strength (σy), to the elastic and plastic advection and diffusion coefficients [26]. These equivalent coefficients assign probability weights to the stress response based on the probability of material being elastic or elastic-plastic.…”

“…These equivalent coefficients assign probability weights to the stress response based on the probability of material being elastic or elastic-plastic. For elastic plastic material model with uncertain yield strength (σy), the equivalent advection and diffusion coefficients (N eq (1)r and (2) ) would become [26]: (9) where 1-P[σy ≤ σ] represents the probability of material being elastic and P[σy ≤ σ]represents the probability of material being elastic-plastic. Hence, given the statistical description of Young's modulus, Poisson's ratio, and yield strength of any material, the governing FPK equation (Eq.…”

“…strength parameter p0 (related to the pre-consolidation pressure) equals the diameter of the ellipse in the direction of the axis of hydrostatic pressure p [27]. For Cam Clay elastic-perfectly plastic material model with uncertain (M), the equivalent advection and diffusion coefficients take the following forms: (17) For a elastic-perfectly plastic model, Therefore (18) where represents the probability of material being elastic and represents the probability of material being elastic-plastic. In this study, the material constant for the Cam Clay model, M, was assumed to have a Weibull distribution with a mean of 1.2 MPa and a COV of 10%.…”

“…In 2009, Jeremi´c and Sett introduced the concept of probabilistic yielding. They showed that by considering the uncertainty in yield stress, the average (mean) constitutive response and the most probable (mode) constitutive response show nonlinear behavior from the very beginning, even for the simplest elastic-perfectly plastic material model [18]. To make this method more general, Sett and Jeremi´c also discussed the effect of probabilistic yielding on constitutive simulation under cyclic loading [19] .…”

An Efficient Meshless Solution for the Fokker-Planck-Kolmogorov Equations of Probabilistic Elastoplasticity

“…The first attempt to propagate the uncertainties through elastoplastic constitutive equations was made by Anders and Hori, [3] who relied on a bounding media analysis. Later, Jeremić et al, [4,5] developed a second-order exact E-L form of the Fokker-Planck-Kolmogorov (FPK) equation for the general elastoplastic constitutive rate equation. Finally, Rosić and Matthies, [6] presented the variational theory behind the mixed-hardening stochastic plasticity problem along with stochastic versions of the relevant established computational algorithms.…”