A thermodynamically consistent framework for visco-elasto-plastic creep and anisotropic damage in saturated frozen soils

Abstract

This paper presents a poro-visco-elastic-plastic damageable model for saturated frozen soils within a rigorous theoretical framework. The effective fluid pressure, obtained considering the interfacial energy, is combined with the total Cauchy stress in order to formulate the hydro-mechanical effective stress applied to a soil skeleton. On the other hand, a two-stress variable constitutive relationship is adopted for saturated frozen soils to describe the essential features of frozen and unfrozen behaviour. Based on the continuum damage theory, the cross-anisotropic damage variables for saturated frozen soils are deduced. The proposed damage criterion and the new nonlinear damage surface for saturated frozen soils are all governed by the second invariants of the “double effective stress”, which combines the damaged effective stress with the effective hydro-mechanical stress. The validity of the visco-elasto-plastic model with no damage is verified by comparing its modelling results with experimental results obtained from uniaxial creep tests.

Appendices

Appendix I: Deriving the equation linking $\varepsilon _{v}$ and $\hat{{p}}_{\alpha }$

We assume that the pores are fully saturated by liquid water before the porous material is exposed to low temperatures, and the current total mass of liquid water and ice crystals per unit of initial volume $d\Omega _{0}$ is

$$\begin{aligned} m_{w} =\rho _{i} \phi _{i} +\rho _{w} \phi _{w} =\rho _{i} (\phi _{0} S_{i} +\varphi _{i} )+\rho _{w} (\phi _{0} S_{w} +\varphi _{w} ). \end{aligned}$$

(I.1)

The first term of Eq. (I.1) can be rewritten as follows:

$$\begin{aligned} \rho _{i} (\phi _{0} S_{i} +\varphi _{i} )=(\rho _{i} -\rho _{w} +\rho _{w} )(\phi _{0} S_{i} +\varphi _{i} )=(\rho _{i} -\rho _{w} )(\phi _{0} S_{i} +\varphi _{i} )+\rho _{w} \phi _{0} S_{i} +\rho _{w} \varphi _{i}. \end{aligned}$$

(I.2)

Substitution of Eq. (I.2) in (I.1) provides:

$$\begin{aligned} m_{w} =\rho _{w} \phi _{0} +(\rho _{i} -\rho _{w} )(\phi _{0} S_{i} +\varphi _{i} )+\rho _{w} \varphi _{i} +\rho _{w} \varphi _{w} =\rho _{w} \phi _{0} +(\rho _{i} -\rho _{w} )\phi _{0} S_{i} +\rho _{w} \varphi _{w} +\rho _{i} \varphi _{i}. \end{aligned}$$

(I.3)

Since the water is not allowed to escape from the soil sample upon loading, the total mass $m_{w}$ remains equal to the initial water mass $\rho _{w} \phi _{0}$. Therefore, we have:

$$\begin{aligned} (\rho _{i} -\rho _{w} )\phi _{0} S_{i} +\rho _{w} \varphi _{w} +\rho _{i} \varphi _{i} =\rho _{w} \left[ {(\frac{\rho _{i} }{\rho _{w} }-\text{1 })\phi _{0} S_{i} \text{+ }(\varphi _{w} +\frac{\rho _{i} }{\rho _{w} }\varphi _{i} )} \right] \text{=0 }. \end{aligned}$$

(I.4)

Finally, using the relation of $\frac{\rho _{i} }{\rho _{w} }\approx 0.09\approx 1-\frac{\rho _{i} }{\rho _{w}}\le 1$ , we get the following expression:

$$\begin{aligned} {0.09}\phi _{0} S_{i} \text{+ }b\varepsilon _{v} {+}\left( \frac{1}{N_{ii} }+\frac{1}{N_{iw} }\right) \hat{{p}}_{i} +\left( \frac{1}{N_{ww} }+\frac{1}{N_{iw} }\right) \hat{{p}}_{w} -3\left( \alpha _{i} +\alpha _{w}\right) (T-T_{m} )\text{=0 }. \end{aligned}$$

(I.5)

Appendix II: Deriving the visco-elastic dissipation inequality

The derivation of Eq. (29) uses the following relation:

$$\begin{aligned} \frac{\partial ^{2}\Psi _{s} }{\partial \varphi _{i} \partial \varphi _{i} }=N_{ii} ,\frac{\partial ^{2}\Psi _{s} }{\partial \varphi _{i} \partial \varphi _{w} }=N_{iw} ,\frac{\partial ^{2}\Psi _{s} }{\partial \varphi _{w} \partial \varphi _{i} }=N_{wi} ,\frac{\partial ^{2}\Psi _{s} }{\partial \varphi _{w} \partial \varphi _{w} }=N_{ww}. \end{aligned}$$

(II.1)

For the sake of simplicity, the following assumptions are used:

$$\begin{aligned} A_{1}= & {} [(\varphi _{i} -\beta _{i} \varepsilon _{v}^{ve} )+3\alpha _{i} (T-T_{m} )-b_{i} (\varepsilon _{v} -\varepsilon _{v}^{ve} )]^{2} \end{aligned}$$

(II.2a)

$$\begin{aligned} B_{1}= & {} [(\varphi _{w} -\beta _{w} \varepsilon _{v}^{ve} )+3\alpha _{w} (T-T_{m} )-b_{w} (\varepsilon _{v} -\varepsilon _{v}^{ve} )]^{2} \end{aligned}$$

(II.2b)

$$\begin{aligned} C_{1}= & {} [(\varphi _{w} -\beta _{w} \varepsilon _{v}^{ve} )+3\alpha _{i} (T-T_{m} )-b_{i} (\varepsilon _{v} -\varepsilon _{v}^{ve} )] \end{aligned}$$

(II.2c)

$$\begin{aligned} D_{1}= & {} [(\varphi _{i} -\beta _{i} \varepsilon _{v}^{ve} )+3\alpha _{w} (T-T_{m} )-b_{w} (\varepsilon _{v} -\varepsilon _{v}^{ve} )]. \end{aligned}$$

(II.2d)

Therefore, the visco-elastic state equations can be derived:

$$\begin{aligned} \sigma _{m}= & {} \frac{\partial W_{s} }{\partial \varepsilon _{v} }=K(\varepsilon _{v} -\varepsilon _{v}^{ve} )-N_{ii} b_{i} A_{\text{1 }} -N_{ww} b_{w} B_{\text{1 }} -N_{iw} (b_{w} C_{1} \text{+ }b_{i} D_{\text{1 }} )-3\alpha K(T-T_{m}) \end{aligned}$$

(II.3a)

$$\begin{aligned} s_{ij}= & {} \frac{\partial W_{s} }{\partial e_{ij} }=2G(e_{ij} -e_{ij}^{ve} )\end{aligned}$$

(II.3b)

$$\begin{aligned} \hat{{p}}_{i}= & {} \frac{\partial W}{\partial \varphi _{i} }=N_{ii} A_{1} +N_{iw} C_{1} \end{aligned}$$

(II.3c)

$$\begin{aligned} \hat{{p}}_{w}= & {} \frac{\partial W}{\partial \varphi _{w} }=N_{ww} B_{1} +N_{iw} D_{1}. \end{aligned}$$

(II.3d)

Substitution of (II-3) and (31) in (6) , we can deduce the Inequality (32).