## 3 Practical Selection of the Start Point of the Data to Be Extrapolated  
[PERSON] and [PERSON] [37] reported the experimental results obtained from multiple series of triaxial tests on Karlsruhe fine sand, which included 25 drained monotonic triaxial compression tests. The sand has almost no fine content, \(e_{\text{min}}=0.677\), and \(e_{\text{max}}=1.054\). The critical state friction angle was determined to be \(\varphi_{\text{cs}}=33.1^{\circ}\), from which it follows that \(M=6\text{sin}\varphi_{\text{cs}}/(3-\text{sin}\varphi_{\text{cs}})=1.335\). All test data are freely available from [PERSON]'s homepage on the torsten-wichtmann.de website [33]. Out of the 25 drained tests, 20 tests on medium-dense and dense samples showed strain-softening and dilatancy behavior. These 20 tests were numbered from TMD6 to TMD 25 by [37]. Table 1 of [37] presented the pre-shearing void ratios \(e_{0}\) and confining pressures \(p_{0}\) of the sand tested.  
Figure 6 shows the \(e\)-\((\eta-M)\) data obtained from the 20 tests. All the \(e\)-\((\eta-M)\) curves terminate at a stress ratio \(\eta\) greater than the critical state stress ratio \(M\). This indicates that their critical states were not attained. After the peak stress ratio is reached, the void ratio \(e\) increases and the stress ratio decreases, corresponding, respectively, to the dilatancy and strain-softening behavior. Relatively steady data fluctuations were observed in the later stage of tests TMD7 and TMD17, as shown in Figure 6a. Negligible data fluctuations were observed in the other 12 tests, as shown in Figure 6a,b. However, unsteady data fluctuations were observed in another six tests, as shown in Figure 6c. In these six tests, tests TMD11 and TMD16 showed evidently nonlinear \(e\)-\((\eta-M)\) behavior, so they were discarded from further analysis. The remaining four tests showed short random fluctuations close to the end of the tests only. For each of the four tests, an assumed end of test was specified and used for further analysis, as shown by the symbols in Figure 6c. Therefore, a total of 18 tests were considered. It can be seen from Figure 6 that the \(e\)-\((\eta-M)\) curves for these tests become approximately linear only until the stress ratio decreases to a certain value. Therefore, the entire post-peak data set cannot be used to perform linear extrapolation. The selection of a reasonable start point for the \(e\)-\((\eta-M)\) data is the key to the goodness of fit for the linear extrapolation.  
Figure 7 shows the coefficients of determination (\(R^{2}\)) of the least-squares linear fits of the \(e\)-\(\eta\) data from varying start points as defined by Equation (9) to the end of the test. The \(R^{2}\) of determination for tests TMD7, TMD12, TMD18, TMD21, and TMD 24 fluctuate relatively significantly with the change in the start point, especially when the \(s_{\eta}\) value is less than 50%. For the other 13 tests, generally speaking, the coefficient of determination increases as the \(s_{\eta}\) value decreases from 90% to 50%, and then it varies within a relatively narrow range as the \(s_{\eta}\) value further decreases from 50% to 20%. Their coefficients of determination of the least-squares linear fits of the \(e\)-\(\eta\) data from the start point at \(s_{\eta}=50\%\) to the end of the test are greater than 0.975. Therefore, \(s_{\eta}=50\%\) may be used as a reasonable start point for fitting and extrapolating the \(e\)-\(\eta\) data. This means that the latter fifty percent of the post-peak \(e\)-\(\eta\) data may be the best option for estimating the critical state void ratio with the linear extrapolation method.  
Figure 6: The \(e\)-\(\eta\) data of the drained triaxial compression tests on medium-dense and dense Karlsruhe fine sand (data reproduced from [PERSON]’s homepage on the torsten-wichtmann.de website [33]).