A unified constitutive model for unsaturated soil under monotonic and cyclic loading

Abstract

This study presents a sophisticated elastoplastic constitutive model for unsaturated soil using Bishop-type skeleton stress and degree of saturation as state variables in the framework of critical state soil mechanism. The model is proposed in order to describe the coupled hydromechanical behavior of unsaturated soil irrespective of what kind of the loadings or the drainage conditions may be. At the same time, a water retention characteristic curve considering the influence of deformation on degree of saturation is also proposed. In the model, the superloading and subloading concepts are introduced to consider the influences of overconsolidation and structure on deformation and strength of soils. The proposed model only employs nine parameters, among which five parameters are the same as those used in Cam-Clay model. The other four parameters have the clear physical meanings and can be easily determined by conventional soil tests. The capability and accuracy of the proposed model have been validated carefully through a series of laboratory tests such as isotropic loading tests and triaxial monotonic and cyclic compression tests under different mechanical and hydraulic conditions.

Appendix moisture characteristics curve [40]

In order to properly consider the influence of the degree of saturation on stress–strain–dilatancy relationships of unsaturated soils, a suitable MCC for suction–saturation relation is proposed by Zhang and Ikariya [40]. The MCC includes skeleton curves and scanning curves so that at any moisture state (S_r, s), it is possible to obtain an incremental relation between suction and the degree of saturation as

$$ {\text{d}}S_{r} = k_{s}^{ - 1} {\text{d}}s $$

(A-1)

where k_s is the tangential stiffness of suction–saturation relation.

Skeleton curves for the moisture characteristics with tangential and arc-tangential functions are given in three different ways according to the state of the moisture as

• (1) Primary drying curve from slurry:

  $$ S_{r}^{{}} = S_{r}^{s0} - \frac{2}{\pi }(S_{r}^{s0} - S_{r}^{r} )\tan^{ - 1} ((e^{{c_{1} s}} - 1)/e^{{c_{1} s_{d} }} ) $$

  (A-2)

  or

  $$ s = \frac{1}{{c_{1} }}\ln \left[ {1 + e^{{c_{1} s_{d} }} \tan \left( {\frac{\pi }{2}\frac{{S_{\text{r}}^{{{\text{s}}0}} - S_{\text{r}}^{{}} }}{{S_{\text{r}}^{{{\text{s}}0}} - S_{\text{r}}^{\text{r}} }}} \right)} \right] $$

  (A-3)

• (2) Secondary drying curve experienced drying–wetting process:

  $$ S_{\text{r}}^{{}} = S_{\text{r}}^{\text{s}} - \frac{2}{\pi }(S_{\text{r}}^{\text{s}} - S_{\text{r}}^{\text{r}} )\tan^{ - 1} ((e^{{c_{1} s}} - 1)/e^{{c_{1} s_{d} }} ) $$

  (A-4)

  or

  $$ s = \frac{1}{{c_{1} }}\ln \left[ {1 + e^{{c_{1} s_{d} }} \tan \left( {\frac{\pi }{2}\frac{{S_{\text{r}}^{\text{s}} - S_{\text{r}}^{{}} }}{{S_{\text{r}}^{\text{s}} - S_{\text{r}}^{\text{r}} }}} \right)} \right] $$

  (A-5)

• (3) (iii) Wetting curve:

  $$ S_{\text{r}}^{{}} = S_{\text{r}}^{\text{s}} - \frac{2}{\pi }(S_{\text{r}}^{\text{s}} - S_{\text{r}}^{\text{r}} )\tan^{ - 1} ((e^{{c_{2} s}} - 1)/e^{{c_{2} s_{w} }} ) $$

  (A-6)

  or

  $$ s = \frac{1}{{c_{2} }}\ln \left[ {1 + e^{{c_{2} s_{w} }} \tan \left( {\frac{\pi }{2}\frac{{S_{\text{r}}^{\text{s}} - S_{\text{r}}^{{}} }}{{S_{\text{r}}^{\text{s}} - S_{\text{r}}^{\text{r}} }}} \right)} \right] $$

  (A-7)

where S_d is a parameter corresponding to drying AEV and S_w is a parameter corresponding to WEV, as shown in Fig. 14. c₁ and c₂ are scaling factors that control the shape of the curves. $S_{r}^{s0}$ is the degree of saturation of a slurry under fully saturated condition and is equal to 1.0.

Fig. 14

Image of moisture characteristic curve of unsaturated soil

As to the scanning curve in the drying–wetting process between the skeleton curves, the incremental relation between suction and saturation is expressed as

$$ k_{\text{s}}^{ - 1} = k_{{{\text{s}}0}}^{ - 1} + k_{{{\text{s}}1}}^{ - 1} $$

(A-8)

$k_{{{\text{s}}0}}$ is the gradient of suction–saturation relation under the condition that inner variable r equals to 0. $k_{s1}$ is expressed as:

$$ k_{{{\text{s}}1}}^{{}} = k_{{{\text{s}}1}}^{\text{s}} \left( {1 + c_{3} \frac{1 - r}{r}} \right) $$

(A-9)

where c₃ is a scaling factor controlling the curvature of the scanning curve. $k_{s1}^{s}$ is the gradient of the corresponding skeleton curve on which the moisture state (S_r, s) is locating under the condition that r equals to 1, as shown in Fig. 14. According to the illustration shown in Fig. 14, the inner variable r is defined as

$$ r = \left\{ {\begin{array}{*{20}c} {\delta_{2} /\delta \quad {\text{d}}s > 0} \\ {\delta_{1} /\delta \quad ds \le 0} \\ \end{array} } \right\} $$

(A-10)

Equation (A-8) means that the stiffness of $k_{s}$ consists of two parts, $k_{s0}$ and $k_{s1}$, in a way that its value looks like the value of a spring consisted from two series of springs. It is easily understood from Equations (A-8) and (A-9) that if r = 0, $k_{s1}$ will be infinite and $k_{s}^{{}} = k_{s0}^{{}}$. While if r = 1 and $k_{s0} > > k_{s1}^{s}$, $k_{s}$ will be equal to $k_{s1}^{s}$, which coincides with the gradient of the skeleton curve. This explanation can also be easily understood by means of the illustration shown in Fig. 14.

Eight parameters are involved in the proposed moisture characteristics curve, among which three parameters c₁, c₂ and c₃ are determined with curve fitting method, while other five parameters, $k_{s0}$, $S_{\text{r}}^{\text{s}}$, $S_{\text{r}}^{\text{r}}$, S_d and S_w, have definite physical meaning and can be determined by the test of moisture characteristics easily.

In the area between the primary drying curve and the secondary drying curve, the upper bound for the scanning curve is changed from the secondary drying curve to the primary drying curve; and the lower bound for the scanning curve is changed from the wetting curve to a combined curve made from the wetting curve in the region $S_{{_{r} }} \le S_{r}^{s}$ and the line s = 0 in the region $S_{{_{r} }} > S_{r}^{s}$. The scanning rule, however, is totally the same as those for the secondary drying–wetting process discussed in Equations (A-8)–(A-10).