Thermo-elastoplastic constitutive model for unsaturated soils

Abstract

This study presents a simple approach to modelling the effect of temperature on the deformation and strength of unsaturated/saturated soils by using the average skeleton stress and degree of saturation. The concept of thermo-induced equivalent stress is introduced to consider the influence of temperature on the pre-consolidated stress. A skeleton stress–saturation framework is applied to enable the model to describe the thermo-elastoplastic behaviour of both unsaturated and saturated soils, as the skeleton stress can smoothly shift to Terzaghi’s effective stress if saturation changes from the unsaturated to the saturated condition. The new model only employs seven parameters, of which five parameters are the same as those used in the Cam-Clay model. The other two parameters can be easily determined by oedometer tests and simple thermo-mechanical tests. Numerical simulations of isotropic loading tests and triaxial shear tests under different conditions are conducted to illustrate the performance of the proposed model. By comparing with experimental temperature controlled oedometer tests and triaxial tests, it is confirmed that the proposed model is able to capture the thermo-mechanical behaviour of unsaturated/saturated normally and over-consolidated soils with a set of unified parameters.

Appendix

In the SWCC model proposed by Zhang and Ikariya [50], the incremental relationship between suction and the degree of saturation is given as:

$$dS_{r} = k_{s}^{ - 1} ds$$

(34)

where k _s is the tangential stiffness of the suction-saturation relationship.

The skeleton curves of the SWCC model are given with tangential and arc-tangential functions in three different ways according to the state of the moisture as,

• (i) Primary drying curve from slurry:

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

  (35)

  or

  $$s = \frac{1}{{c_{1} }}\ln \left[ {1 + e^{{c_{1} s_{d} }} \tan \left(\frac{\pi }{2}\frac{{S_{r}^{s0} - S_{r}^{{}} }}{{S_{r}^{s0} - S_{r}^{r} }}\right)} \right]$$

  (36)

• (ii) Secondary drying curve experienced in the drying-wetting process:

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

  (37)

  or

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

  (38)

• (iii) Wetting curve:

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

  (39)

  or

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

  (40)

where S _d is a parameter corresponding to the drying AEV and S _w is a parameter corresponding to the wetting AEV, as shown in Fig. 21. c ₁ and c ₂ are scaling factors that control the shape of the curves. S ^s0 _r is the degree of saturation of a slurry under a fully saturated condition and is equal to 1.0.

Fig. 21

SWCC of unsaturated soil

With reference to the scanning curve in the drying-wetting process between the skeleton curves, the incremental relationship between suction and saturation is expressed as:

$$k_{s}^{ - 1} = k_{s0}^{ - 1} + k_{s1}^{ - 1}$$

(41)

k _s0 is the gradient of the suction-saturation relationship under the condition that the inner variable r equals 0. k _s1 is expressed as:

$$k_{s1}^{{}} = k_{s1}^{s} \left(1 + c_{3} \frac{1 - r}{r}\right)$$

(42)

where, c ₃ is a scaling factor which controls the curvature of the scanning curve. k ^s _s1 is the gradient of the corresponding skeleton curve on which the moisture state, (S _r , s) is located under the condition that r equals 1, as shown in Fig. 21. According to the illustration in Fig. 21, the inner variable, r is defined as:

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

(43)

Equation (41) means that the stiffness of k _s consists of two parts, k _s0 and k _s1 in a way that its value resembles the value of two springs. It can be clearly understood from Eqs. (41) and (42) that if r = 0, k _s1 will be infinite and k _s  = k _s0. If r = 1 and k _s0 ≫ k ^s _s1 , k _s will equal to k ^s _s1 , which coincides with the gradient of the skeleton curve. This explanation can also be illustrated as shown in Fig. 21.

Eight parameters are involved in the proposed SWCC, of which three parameters, c ₁ , c ₂ and c ₃ are determined, by the curve fitting method while the other five parameters, k _s0, S ^s _r , S ^r _r , S _d and S _w have definite physical meanings and can be easily determined by moisture testing.

Figure 22 shows a theoretical prediction of the SWCC of a theoretical unsaturated silt. It is very clear that all the main features of the moisture characteristics can be properly described.

Fig. 22

Simulated SWCC of unsaturated fictional silt