There are principal differences between the critical state material models for soils and conventional material models of the Mohr-Coulomb type. This section describes fundamental differences and general principles of the critical state constitutive models based on the theory hypoplasticity and plasticity.
Advanced constitutive models for soils introduce the concept of state. A state is a general term to represent the current density of soil (or level of consolidation) and is typically defined by the void ratio \(e\) or the preconsolidation pressure \(p_c\). The soil properties such as stiffness and strength depend on both the current stress and current state.
The concept of critical state assumes that the soil, if continuously sheared, reaches a well-defined critical state that depends only on the stress and not on the path by which the soil arrives at the critical state. Therefore, the soil properties in the critical state depend only on the stress and are state-independent.
In elasto-plasticity the overall strain is assumed to be composed of elastic and plastic part \begin{equation} \boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}_{el} + \boldsymbol{\varepsilon}_{pl} \label{eq:epsep} \end{equation} The stress increment \(\dot{\boldsymbol{\sigma}}\) is proportional to the elastic part of strain increment according to \begin{equation} \dot{\boldsymbol{\sigma}} = \mathcal{D}:(\dot{\boldsymbol{\varepsilon}}-\dot{\boldsymbol{\varepsilon}}_{pl}) \label{eq:elastoplasticity} \end{equation} where \(\mathcal{D}\) is the instantaneous material stiffness tensor. The increment of plastic strain is provided by \begin{equation} \dot{\boldsymbol{\varepsilon}}_{pl} =\dot{\lambda}\frac{\partial g}{\partial \boldsymbol{\sigma}} \label{eq:flowrule} \end{equation} thus being normal to the plastic potential \(g\). Its magnitude is controlled by the scalar plastic multiplier \(\lambda\). The increment \(\dot{\lambda}\) follows from the consistency condition \begin{equation} \frac{\partial f}{\partial\boldsymbol{\sigma}}:\dot{\boldsymbol{\sigma}} + \frac{\partial f}{\partial\boldsymbol{\varepsilon}_{pl}}:\dot{\boldsymbol{\varepsilon}_{pl}}= 0 \label{eq:consistency} \end{equation} where \(f\) is the yield surface defined by the particular elasto-plastic material model.
The principal equation in hypoplasticity is the relation between the stress rate \(\dot{\boldsymbol{\sigma}}\) and a given strain rate \(\dot{\boldsymbol{\varepsilon}}\) expressed in the tensor form as \begin{equation} \dot{\boldsymbol{\sigma}} = (\mathcal{L}:\dot{\boldsymbol{\varepsilon}} + \boldsymbol{N}||\dot{\boldsymbol{\varepsilon}}||) \label{eq:hypo} \end{equation} where \(\mathcal{L}\) and \(\boldsymbol{N}\) are the fourth and second order operators, respectively, depending on the current stress and the current state.