## 2 The Most Appropriate Data Set for Extrapolation
The density index \(I_{\rm D}=(e_{\rm max}-e_{0})/(e_{\rm max}-e_{\rm min})\) is used as a proxy for \(e_{0}\) (Toyoura sand having the maximum void ratio \(e_{\rm max}=0.977\) and the minimum void ratio \(e_{\rm min}=0.597\)[36]). Figure 2 shows typical results obtained from the simulations of drained shear tests on Toyoura sand with a pre-shearing confining pressure \(p_{0}\) of 200 kPa and different density indices. The critical state stress ratio \(M\) obtained is identical to the input value of 1.25. The critical void ratio obtained, 0.8894, is also identical to the theoretical value that would be derived from traditional calculations with \(q_{\rm cs}=Mp_{\rm cs}=3(p_{\rm cs}-p_{0})\) and \(e_{\rm cs}=e_{\rm T}-\lambda_{\rm c}(p_{\rm cs}/p_{\rm a})^{\zeta}\). Figure 2a clearly shows the peak stress ratio response and the post-peak strain-softening behavior, and Figure 2b shows the dilating behavior. It is noted that none of the simulated tests attain the critical state at an axial strain \(\varepsilon_{a}\) of 25%, which is approximately the highest strain level attainable in triaxial tests. For all the simulated tests, a conservative value of the axial strain for attaining the critical state is approximately 80%. As expected, the higher the density index, the higher the axial strain necessary to attain the critical state. The \(e\)-\(\eta\) data set of Toyoura sand with a pre-shearing confining pressure \(p_{0}\) of 200 kPa and a density index \(I_{\rm D}\) of 2/3 (for which \(e_{0}=0.724\)) is plotted as \(e\) versus \((\eta-M)\), as shown by the solid grey curve in Figure 3. For practical consideration, \(\varepsilon_{a}=25\%\) is assumed to be the end value of the experiment. A subset of the post-peak data from a specified start point to an assumed end of test at \(\varepsilon_{a}=25\%\) was fitted and extrapolated, while the simulation values ranging from 25% to 80% are used for comparison with the fitted and extrapolated curve for validation. The location of the start point is defined as  
\[s_{\eta}=\frac{\eta-\eta_{\rm EOT}}{\eta_{\rm max}-\eta_{\rm EOT}} \tag{9}\]  
where \(\eta_{\rm EOT}\) is the stress ratio at the end of the test; \(s_{\eta}\) is expressed in percentage and indicates approximately what the percentage of the post-peak data used for extrapolation is. A best-fit line of the \(e\)-\(\eta\) data from the start point at \(s_{\eta}=90\%\) up to the end point at \(\varepsilon_{a}=25\%\) is shown by the dashed line in \[e_{\text{cs}}=e_{\Gamma}-\lambda_{c}(p_{\text{cs}}/p_{a})^{\xi} \tag{3}\]0 This line agrees excellently with the fitted data, which covers approximately ninety percent of the post-peak data. Extrapolating this line to \(\eta-M=0\) leads to an estimated critical state void ratio \(\varepsilon_{\rm cses}=0.8894\), which is identical to the theoretical value of the critical state void ratio. The colored curves in Figure 4 represent the simulation results of the \(e\)-\(D\) and \(\varepsilon_{v}\)-\(D\) data sets of Toyoura sand under the conditions \(p_{0}=200\) kPa and \(I_{\rm D}=2/3\). The relationship between \(e\) and \(\varepsilon_{v}\) is given by \(\varepsilon_{v}=\ln[(1+e_{0})/(1+e)]\). During the dilation process, \(e\) increases and \(D\) varies from a negative minimum to zero, which corresponds to the critical state condition. Best-fit lines for the data sets from the start points at \(s_{\eta}=90\%\), 70%, and 50% up to the end point at \(\varepsilon_{a}=25\%\) are shown by the dashed lines in Figure 4a-c, respectively. Extrapolating these lines to \(D=0\) leads to the estimated critical state void ratios \(\varepsilon_{\rm cses}\) and volumetric strains \(\varepsilon_{\rm cses}\). It can be seen that the ordinate values of the dashed lines at \(D=0\) are more or less higher than those of the corresponding colored curves. This indicates that the critical state void ratio and volumetric strain are overestimated by the linear extrapolation of the \(e\)-\(D\) and \(\varepsilon_{v}\)-\(D\) data sets, respectively.