# Behaviour of Uniform Drava River Sand in Drained Condition--A Critical State Approach
## 4 Results of Laboratory Tests
The data from Figure 9 is plotted using the power low given with Equation (5), [57]. The exponent \(\kappa\) is used as a best fit parameter, which is determined based on the data in Table 4. The data are presented in Figure 9. The upper and lower bound of critical state can easily be defined as well as the best approximation of critical state line (approximated dashed line). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Relative Density** & **Confining Pressure** & **Test** & **Young Modulus** & **Shear Modulus** & **Bulk Modulus** & **Poisson** \\ \(D_{R}\) [\%] & \(e_{\ell}^{\prime}\) [kPa] & **Type \({}^{*}\)** & \(E\) [kPa] & \(G\) [kPa] & \(K\) [kPa] & **Ratio \(v\)** \\ \hline  
\hline \hline \end{tabular} \({}^{[*]}\) C—compression; E—Extension.  
\end{table} 
Table 5: Elastic modulus of performed tests. Figure 8: Critical state of DrOS018 sand. Upper and lower bound lines, together with critical state line were then transferred back to standard \(v-\log p^{\prime}\) plane, which is plotted once again in Figure 10. From this figure it can be easily noticed that the critical state is very close to the critical state obtained by medium dense samples. The relative density \((D_{r})\) which corresponds to those value of voids \(e\) is approximately 57%. Since critical state could not be uniquely defined, authors have analysed plastic dilatancy for the DrOS018 sand. Stress dilatancy was calculated as suggested by [PERSON] [28] and [PERSON] et al. [58]. Plastic dilatancy was calculated using Equation (9). Values of elastic shear and  
Figure 10: Critical state of DrOS018 sand with upper and lower bound curves. Figure 9: Volumetric behaviour in normalised mean effective plane. bulk modulus were calculated based on the initial data reading of tests (Table 5), which is proven valid in some earlier research (for example, [28; 58; 59]). Plastic dilatancy (Equation (9)) was calculated only for compression tests for all three different initial densities and it was calculated and plotted against the stress ratio \(\eta\) (Equation (10)). \[D^{p}=\frac{d\epsilon_{v}^{p}}{d\epsilon_{q}^{p}}\approx\frac{\left(\epsilon_{ v_{j+1}}-\epsilon_{v_{j}}\right)-\left(p_{j+1}^{\prime}-p_{j}^{\prime} \right)/K}{\left(\epsilon_{q_{j+1}}-\epsilon_{q_{j}}\right)-\left(q_{j+1}-q_{ j}\right)/3G} \tag{9}\]  
where (\(de_{v}^{p}\)) is the increment of plastic volumetric deformation, (\(de_{p}^{p}\)) is the increment of plastic deviatoric deformation, (\(\epsilon_{v_{j+1}}-\epsilon_{v_{j}}\)) forward difference in total volumetric deformation, (\(\epsilon_{q_{j+1}}-\epsilon_{q_{j}}\)) forward difference in total deviatoric deformation, (\(\frac{p_{j+1}^{\prime}-p_{j}^{\prime}}{K}\)) forward difference in elastic volumetric deformation and (\(\frac{q_{j+1}^{\prime}-q_{j}^{\prime}}{3G}\)) forward difference in elastic deviatoric deformation. \[\eta=\frac{q}{p^{\prime}} \tag{10}\]  
where \(q\) is deviatoric stress and \(p^{\prime}\) mean effective stress. Figures 11-13 correspond to the initial densities of 33%, 50% and 80%, respectively. Figure 11: Plastic dilatancy for relative density \(D_{r}=33\%\). It can be identified a small negative plastic dilatancy for loose sample, indicating that the sample could exhibit small dilatative behaviour. Critical state stress can be found if the maximum stress ratio \(\eta_{max}\) is plotted against the minimum plastic dilatancy \(D_{min}\), as proposed by [PERSON] and [PERSON] (2018). Figure 12: Plastic dilatancy for relative density \(D_{r}=50\%\). Figure 13: Plastic dilatancy for relative density \(D_{r}=80\%\). [PERSON] [60] and [PERSON] [28]. Scatter plot in Figure 14 presents the relationship between the two mentioned parameters.