# Dilatant Nature of Sand Shear Strength
## 2 Stress-Dilatancy Relationships for Sand
\begin{array}{l}\eta=q/p^{\prime}_{j},\\ D^{p}=\delta e^{p}_{v}/\delta e^{p}_{q},\\ Q=M^{\circ}-\alpha A^{\circ},\\ A=\beta A^{\circ},\\ M^{\circ}=g(\theta)M^{\circ}_{c},\\ M^{\circ}_{c}=6 sin\phi^{\circ}/(3-sin\phi^{\circ}),\\ g(\theta)=(3-sin\phi^{\circ})/\left\{2(\sqrt{3}cos\theta-sin\phi^{\circ}sin \theta)\right\},\\ \theta=\frac{1}{3}sin^{-1}\left\{-\left(3\sqrt{3}J_{3}\right)/2J_{2}^{3/2} \right\},\\ J_{2}=\frac{1}{2}s_{ij}s_{ij},\\ J_{3}=\frac{1}{3}s_{ij}s_{k}s_{kl},\\ s_{ij}=\sigma^{\prime}_{ij}-p^{\prime}\delta_{ij},\\ p^{\prime}=\frac{1}{3}\sigma^{\prime}_{kk},\\ q=\sqrt{3}J_{2},\\ \theta_{c}=\frac{1}{3}sin^{-1}\left\{-\left(3\sqrt{3}J_{\epsilon 3}\right)/ \left(2J_{\epsilon 2}\right)\right\},\\ I_{\epsilon 2}=\frac{2}{2}\delta e^{p}_{ij}\delta e^{p}_{ij},\\ I_{\epsilon 3}=\frac{1}{3}\delta e^{p}_{ij}\delta e^{p}_{ik}\delta e^{p}_{kl},\\ \delta e_{ij}=\delta e^{p}_{ij}-\frac{1}{3}\delta e^{p}_{ij},\\ \delta e^{p}_{v}=\delta e^{p}_{kk},\\ \delta e^{p}_{q}=\sqrt{4J_{\epsilon 2}/3},\\ \delta e^{p}_{ij}=\delta e_{ij}-\delta e^{\epsilon}_{ij},\\ \end{array} \)  
For conventional drained compression tests at constant, \(\sigma^{\prime}_{3}\):  
\[A^{\circ}=\ \left\{1/cos(\theta-\theta_{\epsilon})\right\}\left\{1-\frac{2}{3} M^{\circ}sin\left(\theta+\frac{2}{3}\pi\right)\right\} \tag{14}\]  
which represents the slope of the stress-plastic dilatancy line in the \(\eta-D^{p}\) plane for a purely frictional state (\(\alpha=0\), \(\beta=1\)). The stress ratio can be calculated from the following equation:  
\[\sigma^{\prime}_{1}/\sigma^{\prime}_{3}=\ \left\{3-2\eta sin\left(\theta- \frac{2}{3}\pi\right)\right\}\bigg{/}\left\{3-2\eta sin\left(\theta+\frac{2}{ 3}\pi\right)\right\} \tag{15}\]  
and the secant angle of friction, which is expressed in degrees, can be calculated from the equation:  
\[\phi^{\prime}=2\arctan\sqrt{\sigma^{\prime}_{1}/\sigma^{\prime}_{3}}-90^{\circ} \tag{16}\]  
Neglecting the elastic strain increments, the behavior of the stress-dilatancy of Toyoura sand for TXC at different initial densities and confining pressures (\(\sigma_{c}=p_{0}=100\) kPa) is shown in Figure 2[32]. For the pre-failure stage, the lines, \(\eta-D\), have the slopes, \(A\), and intersect the vertical axis at Q, with the parameters, \(\alpha>0\) and \(\beta>1\). More complex stress-plastic dilatancy relationships can also be observed in sand shear [26]. Usually, Points F, which represent failure states (Figure 2), lie almost on a straight line with a slope, \(A_{f}=\beta_{f}A^{\circ}\), and intersect the vertical axis at the point representing the critical frictional state (\(Q=M^{\circ}_{c}\)). \(\alpha\) and \(\beta\) are new soil parameters that express the combined effect of the anisotropy, the grain breakage, the non-coaxiality of the stress, and the strain increment tensors, as well as other undefined effects. The values of these parameters for various stages of shearing should be determined experimentally [26; 27; 28; 29]. For TXC and TXE at failure, \(\alpha_{f}=0\) and \(\beta_{f}=1\), whereas for BXC,\(\alpha_{f}=0\) and \(\beta_{f}=1.4\), for sands without breakage effects [22; 23; 24; 25; 26]. The FSC assumes that for sands, \(\phi^{{}^{\circ}}=\phi^{\prime}_{cv}\)[22]. It is also assumed that the critical state angle is independent of the deformation mode [8; 23; 33; 34]. For most natural sands, the values, \(\phi^{{}^{\circ}}=\phi^{\prime}_{cv}\), range from \(30^{\circ}\) to \(34^{\circ}\) (Table 1). The characteristic values of the stress-dilatancy relationship for the drained TXC, BXC, and TXE tests when \(\sigma^{\prime}_{3}\) is a constant stress path are presented in Table 2[22; 23; 24; 25]. In this study, for clarity, the characteristic values at failure for TXC, BXC, and TXE are marked with the subscripts, \(c\), \(b\), and \(e\), respectively (e.g., \(\phi^{\prime}_{c}\), \(\phi^{\prime}_{b}\), and \(\phi^{\prime}_{e}\) and \((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{e}\), \((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{b}\), and \((\delta\varepsilon_{v}/\delta\varepsilon_{a})_{e}\)).