Probabilistic elasto-plasticity: formulation in 1D

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.