Mathematical and numerical modeling of the non-associated plasticity of soils—Part 1: The boundary value problem

Abstract

The soil is characterized by the influence of the hydrostatic stress, which leads to a yield surface with a shape of a pyramid for Mohr–Coulomb criteria and a shape of a cone for Drucker–Prager one. These materials are also characterized by a non-associated plasticity where the plastic yielding rule does not follow the normality rule. The usual mechanical models use two independent functions to describe this particular collapse. Unfortunately, this manner broke the model formulation. The purpose of this work is to present a consistent formulation of the non-associated plasticity of soil. The frame of the mathematical analysis is the concept of the implicit standard material. The cornerstone of this new idea is the construction of a single function called the bipotential playing in the same time the roles of the yield surface and the plastic potential. The bipotential concept is then intended to involve the constitutive law, cover the normality rule even for the non-associated soil and the proof of the solution existence. The formulation was initially performed for the case of a regular point out of the cone apex and in present, it is extended to the irregular point located at the apex. The paper presents firstly the implicit standard material method. Then, the methodology to build a full model for the boundary value problem is detailed. Particular expressions and relations are sufficiently explained and discussed. Attention is made to the evolution problem and the variational principles related to the elastic–plastic behavior.