General Stress–Dilatancy Relation for Granular Soils

The stress–dilatancy relation is a significant component in modeling the stress-strain behavior of soils, and reflects the relationship between the excess friction angle ${\phi }_{\mathrm{ex}}$ (i.e., the difference between peak and critical state angles ${\phi }_{\mathrm{ex}}={\phi }_{\mathrm{ps}}-{\phi }_{\mathrm{cs}}$) and the maximum dilatancy angle ${\psi }_{\max }$. The stress–dilatancy relation can be expressed as ${\phi }_{\mathrm{ex}}=\alpha {\psi }_{\max }$, in which $\alpha$ is a coefficient. Bolton (1986) found $\alpha =0.48$ and 0.8 for clean sands under the conventional triaxial condition [i.e., $b=0$, where $b=({\sigma }_2^{\prime }-{\sigma }_3^{\prime })/({\sigma }_1^{\prime }-{\sigma }_3^{\prime })$] and plane-strain (PS) condition ($0.2<b<0.4$), respectively, which indicates that $\alpha$ is dependent on the $b$-value. However, the stress–dilatancy relation of soils (especially coarse granular soils) for the whole range of the $b$-values (i.e., $0\le b\le 1$) has not been fully investigated. In addition, a general formulation is needed for this relationship.

Three categories of true triaxial tests were conducted for dense coarse granular soils (CGSs) with the same initial gradation and the same maximum diameter of 10 mm. The C1 tests were conducted for angular particle soils sheared under constant-$b$ and constant-$p^{\prime }$ conditions ($p^{\prime }=200–800\mathrm{kPa}$). The C2 tests were conducted for angular particle soils sheared under constant-$b$ and constant-${\sigma }_3^{\prime }$ conditions (${\sigma }_3^{\prime }=100–400\mathrm{kPa}$). The C3 tests were conducted for rounded particle soils sheared under constant-$b$ and constant-${\sigma }_3^{\prime }$ conditions (${\sigma }_3^{\prime }=100–400\mathrm{kPa}$). Five different $b$-values ($b=0$, 0.25, 0.5, 0.75, and 1) were applied. Detailed descriptions of the true triaxial apparatus, specimen preparation, and test procedures were reported by Xiao et al. (2014).

Fig. 1 shows that the value of $\alpha$ is related to the $b$-value, indicating that a $b$-dependent stress–dilatancy relation is observed for CGSs. A general stress–dilatancy relation for CGSs can be written aswhere material constants ${\alpha }_0$ (= 1.7), $\chi$ (= 1.36), and $\mu$ (= 0.72) can be obtained from the relationship between $\alpha$ and $b$. Fig. 2 shows that Eq. (1) can accurately predict the test data for soils with different particle angularity and shearing conditions.