A monotonic bounding surface critical state model for clays

Abstract

This study investigates a simple constitutive model based on the critical state framework and bounding surface (BS) plasticity that is suitable for reconstituted clays over a wide range of overconsolidation ratios under monotonic loading. For heavily overconsolidated (OC) clays, rather than using the conventional Hvorslev line, an empirical surface is introduced into the model formulation based on two image points on the BS. The peak strength and the dilatancy of heavily OC clays can thus be predicted satisfactorily. Comparisons with triaxial test data show that the model well captures the peak strength and the dilatancy of heavily OC clays under monotonic loading.

Appendix

Atkinson [15] proposed a criterion for the peak strength of heavily overconsolidated clays as follows:

$$\frac{q}{{Mp^{\prime}}} = 1 + \frac{\beta }{\lambda - \kappa }\xi_{d}$$

(6)

where ξ _d is the state parameter measuring the vertical distance between the current stress point A(v, p′) and the CSL in the v − ln p′ space. β is the peak strength parameter. From Fig. 6, the following equation holds:

$$\xi_{d} = \varGamma - v_{\kappa } = \lambda \ln \left( {\frac{{p^{\prime}_{cr\_v} }}{{p^{\prime}}}} \right)$$

(7)

where Γ is the intercept of the CSL with the v-axis in the v − ln p′ space. v _κ is the intercept of a line, which passes through A(v, p′) and parallels with the CSL, with the v-axis in the v − ln p′ space. p ^′ _{cr_v} is the mean effective stress on the CSL at the current specific volume v, and is determined through:

$$v = \varGamma - \lambda \ln p^{\prime}_{cr\_v}$$

(8)

Fig. 6

Stress state in v − ln p′ space

Substituting Eq. (7) into (6) yields

$$\frac{q}{{Mp^{\prime}}} = 1 + \beta \frac{\lambda }{\lambda - \kappa }\ln \left( {\frac{{p^{\prime}_{cr\_v} }}{{p^{\prime}}}} \right)$$

(9)

v can also be specified through the isotropic normal compression line (NCL) as follows:

$$v = N - \lambda \ln p^{\prime}_{e}$$

(10)

where N is the intercept of the CSL with the v-axis in the v − ln p′ space. p _e ′ is the equivalent pressure, the effective pressure on the NCL at v. Combining Eq. (8) with (10) gives

$$\ln p^{\prime}_{cr\_v} = \frac{\varGamma - N}{\lambda } + \ln p^{\prime}_{e}$$

(11)

From Fig. 6, the following equation holds:

$$\kappa \ln \left( {\frac{{p^{\prime}_{c} }}{{p^{\prime}}}} \right) = \lambda \ln \left( {\frac{{p^{\prime}_{c} }}{{p^{\prime}_{e} }}} \right)$$

(12)

Combining Eq. (11) with (12) yields

$$\ln p^{\prime}_{cr\_v} = \frac{\varGamma - N}{\lambda } + \frac{\lambda - \kappa }{\lambda }\ln p^{\prime}_{c} + \frac{\kappa }{\lambda }\ln p^{\prime}$$

(13)

Substituting Eq. (13) into Eq. (9) yields

$$\frac{q}{{Mp^{\prime}}} = 1 + \frac{\beta }{\lambda - \kappa }\left( {\varGamma - N} \right) + \beta \ln \left( {\frac{{p^{\prime}_{c} }}{{p^{\prime}}}} \right)$$

(14)

(Γ − N) can be determined from the relation of the pre-consolidation pressure and the critical state pressure, and is shown as follows:

$$\varGamma - N = \left( {\kappa - \lambda } \right)\ln \frac{{2 + R_{w} }}{2}$$

(15)

Substituting Eq. (15) into Eq. (14) and noticing p _cr ′ = 2p′/(2 + R _w ) yields Eq. (3) in the main text.