Efficient solution algorithms for multiaxial probabilistic elasto-plastic constitutive simulations of soils

A Fokker-Planck-Kolmogorov (FPK) equation approach has recently been developed to probabilistically solve any elastic-plastic constitutive equation with uncertain material parameters by transforming the nonlinear, stochastic constitutive rate equation into a linear, deterministic partial differential equation (PDE) and thereby simplifying the numerical solution process. For an uniaxial problem, conventional numerical techniques, such as the finite difference or finite element methods, may be used to solve the resulting univariate FPK PDE. However, for a multiaxial problem, an efficient algorithm is necessary for tractability of the numerical solution of the multivariate FPK PDE. In this paper, computationally efficient algorithms, based on a Fourier spectral approach, are presented for solving FPK PDEs in (stress) space and (pseudo) time, having space-independent but time-dependent coefficients and both space- and time-dependent coefficients, that commonly arise in probabilistic elasto-plasticity. The algorithms are illustrated by probabilistically simulating 2 common laboratory constitutive experiments in geotechnical engineering, namely, the unconfined compression test and the unconsolidated undrained triaxial compression test.