Creep characteristics and unified macro–meso creep model for saturated frozen soil under constant/variable temperature conditions

Abstract

In permafrost regions, under the interference of climate warming and human activities, the deformation stability of geological engineering with frozen soil as the medium is directly affected, and the typical ones are the creep characteristics under the influence of temperature and stress level. In order to reveal the internal mechanism of such characteristics, this paper carried out the creep tests of saturated frozen soil under constant/variable temperature conditions, and the following meaningful conclusions are drawn: an increase in temperature and deviatoric stress level leads to an increase in the effect of weakening mechanism on the meso-scale, which cause the creep mechanical behavior transition from a stable state to an unsteady state, and eventually reach a failure state; The creep strain and meso-scale broken law caused by temperature history are important features that affect the creep characteristics at subsequent temperature condition. Furthermore, with the breakage mechanics for geological materials and meso-mechanics theories, the strengthening and weakening mechanisms on the meso-scale under the influence of temperature and stress level (deviatoric stress and confining pressure) were defined and quantitatively described in the creep process, and then, an unified macro–meso creep constitutive model was proposed, which includes stress concentration tensor and breakage ratio. Finally, the creep deformation of saturated frozen soil under constant temperature and variable temperature condition was well predicted. The conclusion of the article can be used to evaluate and predict the long-term deformation stability of geological engineering in cold regions.

Appendix

Unified creep model in cryogenic triaxial conditions.

According to Eqs. (18)–(21), the stress concentration tensor needs to be obtained first: $A_{1}^{{b\left( {n + 1} \right)}}$ and $A_{1}^{{f\left( {n + 1} \right)}}$ at $t_{n + 1}$.

In Sect. 3.3.2, there are two methods of the stress concentration tensor that are given. For saturated frozen soil, it can choose the second solution method. Thus, the expression of the stress concentration tensor coefficient for the bonded elements by Eq. (35-1) is

$$A_{1}^{{b\left( {n + 1} \right)}} = 1$$

(38)

Meanwhile, the frictional elements are uniformly distributed on the meso-scale in an elliptical shape, so that $R_{ijmn}^{M} \left( {t_{n + 1} } \right)$ can be obtained as

$$R_{ijmn}^{M} \left( {t_{n + 1} } \right) = \frac{{\beta^{{t_{n + 1} }} }}{{2G^{{t_{n + 1} }} }}{\mathbb{J}} + \frac{{\alpha^{{t_{n + 1} }} }}{{3K^{b} }}{\mathbb{K}}$$

(39)

with

$$\beta^{{t_{n + 1} }} = \frac{{6\left( {2G^{{t_{n + 1} }} + K^{b} } \right)}}{{5\left( {4G^{{t_{n + 1} }} + 3K^{b} } \right)}};\alpha^{{t_{n + 1} }} = \frac{{3K^{b} }}{{4G^{{t_{n + 1} }} + 3K^{b} }}$$

(40)

Submitting Eqs. (38), (39), (24) and (26) to Eq. (35-2), we can get

$$A_{1}^{n + 1} = \left( {\frac{1}{{2\sqrt 2 G^{{t_{n + 1} }} B^{{t_{n + 1} }} \left( {1 - \beta^{{t_{n + 1} }} } \right) + \beta^{{t_{n + 1} }} }}} \right)$$

(41)

$$A_{2}^{n + 1} = \left( {\frac{{I_{1}^{{t_{n} }} }}{{9K^{b} B^{{t_{n + 1} }} \alpha \left( {1 - \alpha^{{t_{n + 1} }} } \right) + \alpha^{{t_{n + 1} }} I_{1}^{{t_{n} }} }}} \right)$$

(42)

And, the evolution equation of the breakage ratio at $t_{n + 1}$ is

$$\lambda_{v}^{n + 1} = 1 - \exp \left[ { - \omega \left( {\varepsilon_{s}^{{f\left( {n + 1} \right)}} } \right)^{\vartheta } } \right]$$

(43)

In the time increment linearization method proposed in this article, the time increment $\Delta t$ must satisfy

$$\Delta t = t_{n + 1} - t_{n} \ll \frac{T}{\Pr }$$

(44)

where T is the total prediction time, and N guarantees the prediction accuracy. In this paper, N may be 240 for predicting the experimental time within 24 h, which is also the main research time for many scholars to carry out creep research for frozen soil.

Finally, by submitting Eqs. (38) to (42) into Eqs. (20) and (21), the calculation equations under conventional triaxial compression conditions can be obtained:

$$\varepsilon_{s} \left( {t_{n + 1} } \right) = \left( {1 - \lambda_{v}^{n + 1} } \right)\varepsilon_{s}^{b} \left( {t_{n + 1} } \right) + \lambda_{v}^{n + 1} \varepsilon_{s}^{f} \left( {t_{n + 1} } \right)$$

(45)

with

$$\varepsilon_{s}^{b} \left( {t_{n + 1} } \right) = \frac{Q}{{3G^{b} }} + \frac{2Q}{{3\eta^{1} }}t_{n + 1} + \frac{Q}{{3G^{b1} }}\left[ {1 - {\text{exp}}\left( { - \frac{{G^{b1} }}{{\eta^{2} }}t_{n + 1} } \right)} \right]$$

(46)

$$\varepsilon_{s}^{f} \left( {t_{n + 1} } \right) = \frac{{2B^{n + 1} \sqrt 2 }}{3}t_{n + 1} A_{1}^{{f\left( {n + 1} \right)}} Q$$

(47)

All the calculation processes above are carried out under constant temperature conditions, and the schematic diagram of the calculation process is shown in Fig. 

Fig. 13

Schematic diagram of the calculation process of the unified macro–meso creep constitutive model

13. For variable temperature conditions, it only needs to change the model parameters in the target constant temperature. But the temperature changes approximately linearly and is not constant in the time interval [$t_{a}$, $t_{b}$] and [$t_{c}$, $t_{d}$]. Therefore, the model parameters of the bonded elements and the breakage parameters should also be regarded as approximately linear change laws. Taking $G^{b}$ in the time interval [$t_{a}$, $t_{b}$] as an example, the other parameters ($K^{b} , G^{b1}$, $\eta^{1}$, $\eta^{2}$, $\omega$, $\vartheta$) can also be determined with a similar methods, and the solution is.

$$G^{b} \left( t \right) = G^{b} \left( { - 4.0\,^{ \circ } {\text{C}}} \right) - \frac{{G^{b} \left( { - 4.0\,^{ \circ } {\text{C}}} \right) - G^{b} \left( { - 1.5\,^{ \circ } {\text{C}}} \right)}}{{\left( {t_{b} - t_{a} } \right)}}t,t \in \left[ {t_{a} , t_{b} } \right]$$

(48)

Finally, Eq. (48) is taken into consideration in Eqs. (7) and (8), the shear strain at variable temperature conditions can be calculated.