## 4 Kepes Element for Effective Stress Behaviour
### Formulation  
Kepes model describes materials which dissipate energy. It is a special case of a plastic material, in which the set of the admissible stress depends on the strain state. The process of energy dissipation for the Kepes material as well as the plastic material is rate independent. The phenomenon of rate independency makes Kepes and plastic elements different from the classic viscous element (dashpot). The assumption is made that for an undeformed body stresses are zero. The one-dimensional constitutive relationship for the Kepes model is similar to the relationship of a classical slider element which describes ideal plasticity but with the plastic limit not constant but dependent on the strain: \(\sigma_{0}(\varepsilon)\). For the one-dimensional case the continuous set of admissible stress for the Kepes material is as follows [16]  
\[\Theta_{kp}:=[-\sigma_{0};\sigma_{0}] \tag{6}\]  
where \(\sigma_{0}=\kappa|\varepsilon|\) and \(\kappa\) is a material constant. The Kepes element can be effectively used to model the behaviour of soil in effective stress as observed in oedometer or isotropic consolidation tests. The typical \(e-\log\sigma^{\prime}\) curve obtained from these tests is shown in Figure 5. While we are going to formulate the relationship between vertical strain (equivalent to the volumetric strain for the assumed conditions) and stress, the void ratio \(e\) can be calculated, for a given soil's initial void ratio \(e_{0}\), using the well-known formula:  
\[e=\varepsilon(1+e_{0})+e_{0} \tag{7}\]  
The first model we will analyse is a connection in series of the Kepes element and a spring (see Figure 6a). To formulate the constitutive relationship in effective stress for such a scheme, let us first denote the logarithm of the effective stress as  
\[\sigma^{\prime}_{L}:=\left\{ \begin{array}{ll}-\log|\sigma^{\prime}|&\text{ for}\quad\sigma^{\prime}\leq 0\\ \log|\sigma^{\prime}|&\text{for}\quad\sigma^{\prime}>0\end{array} \right. \tag{8}\]  
Figure 4: Comparison of the solution with [PERSON] model with long-term incremental loading (IL) oedometer test results. In the definition (8) natural logarithm can also be used. The choice depends on the way results of oedometer or isotropic consolidation tests are presented. Here, as the common logarithm is used, the slopes of the virgin compression line and the unloading-reloading line will be related to the compression index (\(C_{c}\)) and to the swelling index (\(C_{s}\)), respectively. Now, for the Kepes element we can define \(\sigma_{L}^{\prime}=\beta\sigma_{0}^{\prime}\) where \(\sigma_{0}^{\prime}=\kappa|\varepsilon_{p}|\), so:  
\[\sigma_{L}^{\prime}=\beta\kappa|\varepsilon_{p}| \tag{9}\]  
and the derivative with respect to time is:  
\[\hat{\sigma}_{L}^{\prime}=\dot{\beta}\kappa|\varepsilon_{p}|+\beta\kappa\frac {\varepsilon_{p}}{|\varepsilon_{p}|}\dot{\varepsilon}_{p} \tag{10}\]  
For elements connected in a series, as shown in Figure 6a, the stress in the spring is  
\[\sigma_{L}^{\prime}=M\big{(}\varepsilon-\varepsilon_{p}\big{)} \tag{11}\]  
where \(M\) is the oedometric (constrained) modulus. Thus, the strain in the Kepes element is  
\[\varepsilon_{p}=\frac{1}{M}\big{(}M\varepsilon-\sigma_{L}^{\prime}\big{)} \tag{12}\]  
hence  
\[\dot{\varepsilon}_{p}=\dot{\varepsilon}-\frac{\hat{\sigma}_{L}^{\prime}}{M} \tag{13}\]  
Substituting Equation (10) to Equation (13) we get  
\[\dot{\varepsilon}_{p}=\dot{\varepsilon}-\frac{1}{M}\bigg{(}\dot{\beta}\kappa| \varepsilon_{p}|+\beta\kappa\frac{\varepsilon_{p}}{|\varepsilon_{p}|}\dot{ \varepsilon}_{p}\bigg{)} \tag{14}\]  
Figure 5: Typical \(e-\log\sigma^{\prime}\) relationship. Figure 6: Rheological schemes proposed for drained conditions: (**a**) Spring and Kepes element connected in series, (**b**) Spring and Kepes element connected in parallel with dashpot.