## 2 The Most Appropriate Data Set for Extrapolation  
[PERSON] and [PERSON] [32] proposed a fundamental model for simulating triaxial compression tests on cohesionless soils. In this model, the concept of state-dependent dilatancy was introduced, and a dilatancy function was proposed as given by  
\[D=D_{0}(\text{e}^{m\psi}-\frac{\eta}{M}) \tag{2}\]  
where e is Euler's number; \(D_{0}\) and \(m\) are two positive model parameters; and \(\psi=e-e_{\text{cs}}\) is the state parameter defined by [PERSON] and [PERSON] [4], which measures how far the soil state under the current void ratio \(e\) and confining pressure \(p\) is from the critical state. Such defined dilatation ratio makes it possible to model the dilatancy characteristics dependent on different void ratios and pressures. Because the \(e\)-log \(p\) plots of critical-state lines for cohesionless soil are in general not straight, [PERSON] and [PERSON] [34] proposed a more appropriate empirical formula to depict the critical state line as  
\[e_{\text{cs}}=e_{\Gamma}-\lambda_{c}(p_{\text{cs}}/p_{a})^{\xi} \tag{3}\]  
where \(e_{\Gamma}\), \(\lambda_{c}\), and \(\xi\) are the material constants and \(p_{a}\) is the atmospheric pressure (101.325 kPa), serving as a scaling factor for normalization or dimensional homogeneity. To model the elastic behavior, shear modulus was defined as a function of not only \(p\) but also of \(e\), as given by  
\[G=G_{0}\frac{(2.97-e)^{2}}{1+e}\sqrt{pp_{a}} \tag{4}\]  
where \(e\) is used instead of the initial void ratio \(e_{0}\) used in [35] and \(G_{0}\) is a dimensionless material constant. The unit of \(G\) is reflected by \((pp_{a})^{0.5}\). Within elasticity theory, the elastic bulk modulus \(K\) is equal to  
\[K=G\frac{2(1+\
u)}{3(1-2\
u)} \tag{5}\]  
where \(\
u\) is [PERSON]'s ratio, which was considered to be a material constant. The yield criterion was assumed to be \(f=q-\eta p=0\). A state-dependent plastic modulus \(K_{\rm p}\) was proposed as given by  
\[K_{p}=(h_{1}-h_{2}e)G\bigg{(}\frac{M}{\eta}-\mathrm{e}^{\eta\psi}\bigg{)} \tag{6}\]  
where \(h_{1}\) and \(h_{2}\) are two material constants and \(n\) is a positive model parameter. Such defined plastic modulus makes it possible to model the peak stress ratio response and strain softening of dense soils, the strain hardening of loose soils, and the failure at the critical state stress ratio \(M\). A systematic procedure for the calibration of model parameters was also provided. Equations (7) and (8) show the relationship between the stress and strain increments of Li and [PERSON] [32] model. \[\mathrm{d}\varepsilon_{q}=\mathrm{d}\varepsilon_{q}^{e}+\mathrm{d}\varepsilon_ {q}^{p}=\bigg{(}\frac{1}{3G}+\frac{1}{K_{p}}\bigg{)}\mathrm{d}q-\frac{\eta}{K _{p}}\mathrm{d}p \tag{7}\]  
\[\mathrm{d}\varepsilon_{v}=\mathrm{d}\varepsilon_{v}^{e}+\mathrm{d}\varepsilon _{v}^{p}=D\frac{1}{K_{p}}\mathrm{d}q+(\frac{1}{K}-\frac{D\eta}{K_{p}})\mathrm{ d}p \tag{8}\]  
Based on the triaxial test data reported by [PERSON] and [PERSON] [36], [PERSON] and [PERSON] [32] calibrated the model parameters for Toyoura sand as follows: \(G_{0}=125\), \(\
u=0.05\), \(M=1.25\), \(e_{\rm T}=0.934\), \(\lambda_{\rm c}=0.019\), \(\zeta=0.7\), \(D_{0}=0.88\), \(m=3.5\), \(h_{1}=3.15\), \(h_{2}=3.05\), and \(n=1.1\). In the present paper, [PERSON] and [PERSON] [32] model and parameters are used to perform a series of simulations of drained triaxial compression tests on Toyoura sand with a range of initial ('pre-shearing') confining pressures \(p_{0}\) and void ratios \(e_{0}\). Figure 1 shows the critical state line of Toyoura sand under confining pressures not greater than 500 kPa. A total of 20 numerical simulations of drained shear tests on dilative sand specimens were carried out. The pre-shearing states of the specimens are shown by the abscissa and ordinate values of the solid circles in Figure 1. Each circle is numbered for ease of identification.