A multisurface elastoplastic model for frozen soil

Abstract

Many geotechnical engineering problems in the cold region, including cracking of pavements, damage to the foundations of structures, and fracture of pipelines are blamed due to the deformation and failure of frozen soils. In this study, cryogenic suction is used as one of the constitutive variables together with the solid-phase stress to model the joint influence of temperature and confining pressure on the constitutive behaviour of frozen soils. A nonlinear relationship is proposed to link cryogenic cohesion with cryogenic suction, to consider the strength increase due to the lowering of temperature. In the space of the solid-phase stress and cryogenic suction, loading-collapse yield surface, subloading surface, and unified hardening parameter are then integrated to produce a novel multisurface constitutive model for frozen soil. The proposed model can predict the mechanical behaviours such as softening/hardening and dilation/compression of frozen soil under various temperatures and stresses. Some special characteristics of frozen soil, including the deformation induced by ice segregation and softening related to pressure melting, can also be well explained by this model. The developed model is validated by using several triaxial compression test results of different frozen soils in the literature.

Appendices

Appendix 1: Derivation of elastoplastic constitutive flexibility matrix C

The partial differentials of the yield function to the stress in Eqs. (30), (31) and (32) are presented are as follows:

$$\begin{array}{*{20}c} {\frac{{\partial f_{1} }}{\partial p} = \frac{{M^{2} \left( {p + p_{{\text{s}}} } \right)^{2} - q^{2} }}{{\left( {p + p_{{\text{s}}} } \right)\left[ {M^{2} \left( {p + p_{{\text{s}}} } \right)^{2} + q^{2} } \right]}}} \\ \end{array}$$

(40)

$$\begin{array}{*{20}c} {\frac{{\partial f_{1} }}{\partial q} = \frac{2q}{{M^{2} \left( {p + p_{{\text{s}}} } \right)^{2} + q^{2} }}} \\ \end{array}$$

(41)

$$\begin{aligned} \frac{{\partial f_{1} }}{\partial S} & = - \left[ {\frac{{k_{t} }}{{k_{t} + S}} - \frac{{k_{t} S}}{{\left( {k_{t} + S} \right)^{2} }}} \right]\frac{{\left[ {M^{2} \left( {p + p_{{\text{s}}} } \right)^{2} \left( {p - p_{x} } \right) + q^{2} \left( {p + p_{x} + 2p_{{\text{s}}} } \right)} \right]}}{{\left( {p + p_{{\text{s}}} } \right)\left( {p_{x} + p_{{\text{s}}} } \right)\left[ {M^{2} \left( {p + p_{{\text{s}}} } \right)^{2} + q^{2} } \right]}} \\ & \quad - \frac{{\ln \left( {\frac{{p_{x0} }}{{p_{c} }}} \right)\lambda_{0} \beta \left( {\lambda_{0} - \kappa } \right)\left( {1 - r} \right)p_{c} \left( {\frac{{p_{x0} }}{{p_{c} }}} \right)^{{X_{1} }} \exp \left( { - \beta S} \right)}}{{\left( {p_{x} + p_{{\text{s}}} } \right)\left( {\lambda - \kappa } \right)^{2} }} \\ \end{aligned}$$

(42)

$$\begin{array}{*{20}c} {\frac{{\partial f_{1} }}{{\partial p_{x} }} = - \frac{1}{{p_{x} + p_{s} }}} \\ \end{array}$$

(43)

where $X_{1} = \left( {\lambda_{0} - \lambda } \right)/\left( {\lambda - \kappa } \right)$.

The movement of the yield surface is controlled by the increment of $\varepsilon_{v}^{mp}$ and $\varepsilon_{v}^{sp}$, i.e. $p_{x} = F\left( {\varepsilon_{v}^{mp} , \varepsilon_{v}^{sp} } \right)$. Therefore, by using the chain rule, the partial differential of $p_{x}$ can be obtained as:

$$\begin{array}{*{20}c} {\frac{{\partial p_{x} }}{{\partial \varepsilon_{v}^{mp} }} = \frac{{\partial p_{x} }}{{\partial p_{x0} }}\frac{{\partial p_{x0} }}{\partial H}\frac{\partial H}{{\partial \varepsilon_{v}^{mp} }} } \\ \end{array}$$

(44)

$$\begin{array}{*{20}c} {\frac{{\partial p_{x} }}{{\partial \varepsilon_{v}^{sp} }} = \frac{{\partial p_{x} }}{{\partial p_{x0} }}\frac{{\partial p_{x0} }}{\partial H}\frac{\partial H}{{\partial \varepsilon_{v}^{sp} }} } \\ \end{array}$$

(45)

where

$$\begin{array}{*{20}c} {\frac{{\partial p_{x} }}{{\partial p_{x0} }} = X_{1} \left( {\frac{{p_{x0} }}{{p_{c} }}} \right)^{{X_{1} - 1}} } \\ \end{array}$$

(46)

$$\begin{array}{*{20}c} {\frac{{\partial p_{x0} }}{\partial H} = c_{p} p_{x0} } \\ \end{array}$$

(47)

$$\begin{array}{*{20}c} {\frac{\partial H}{{\partial \varepsilon_{v}^{mp} }} = \frac{1}{{\Omega }}} \\ \end{array}$$

(48)

$$\begin{array}{*{20}c} {\frac{\partial H}{{\partial \varepsilon_{v}^{sp} }} = \frac{1}{{\Omega }}} \\ \end{array}$$

(49)

Appendix 2: The elements in the matrix ${\varvec{C}}$

The elements in the matrix are as follows:

$$\begin{array}{*{20}c} {C_{11} = A_{0} \left( {A_{1} + A_{4} } \right) + \frac{1}{{K_{m} }}} \\ \end{array}$$

(50)

$$\begin{array}{*{20}c} {C_{12} = A_{0} \left( {A_{2} + A_{5} } \right)} \\ \end{array}$$

(51)

$$\begin{array}{*{20}c} {C_{13} = A_{0} \left( {A_{3} + A_{6} } \right) + \frac{1}{{K_{s} }}} \\ \end{array}$$

(52)

$$\begin{array}{*{20}c} {C_{21} = \chi A_{0} A_{1} } \\ \end{array}$$

(53)

$$\begin{array}{*{20}c} {C_{22} = \chi A_{0} A_{2} + \frac{1}{3G}} \\ \end{array}$$

(54)

$$\begin{array}{*{20}c} {C_{23} = \chi A_{0} A_{3} } \\ \end{array}$$

(55)

where

$$\begin{array}{*{20}c} {A_{0} = \frac{1}{{\frac{{\partial f_{1} }}{{\partial p_{x} }}\frac{{\partial p_{x} }}{{\partial p_{x0} }}\frac{{\partial p_{x0} }}{\partial H} \left( {\frac{\partial H}{{\partial \varepsilon_{v}^{sp} }}\frac{{\partial S_{{{\text{seg}}}} }}{{\partial \varepsilon_{v}^{mp} }} - \frac{\partial H}{{\partial \varepsilon_{v}^{mp} }}\frac{{\partial S_{{{\text{seg}}}} }}{{\partial \varepsilon_{v}^{sp} }} } \right)}}} \\ \end{array}$$

(56)

$$\begin{array}{*{20}c} {A_{1} = \frac{{\partial f_{1} }}{\partial p}\frac{{\partial S_{{{\text{seg}}}} }}{{\partial \varepsilon_{v}^{sp} }}} \\ \end{array}$$

(57)

$$\begin{array}{*{20}c} {A_{2} = \frac{{\partial f_{1} }}{\partial q}\frac{{\partial S_{{{\text{seg}}}} }}{{\partial \varepsilon_{v}^{sp} }}} \\ \end{array}$$

(58)

$$\begin{array}{*{20}c} {A_{3} = - \frac{{\frac{{\partial f_{2} }}{\partial S}\frac{{\partial f_{1} }}{{\partial p_{x} }}\frac{{\partial p_{x} }}{{\partial p_{x0} }}\frac{{\partial p_{x0} }}{\partial H}\frac{\partial H}{{\partial \varepsilon_{v}^{sp} }} - \frac{{\partial f_{1} }}{\partial S}\frac{{\partial f_{2} }}{{\partial S_{{{\text{seg}}}} }}\frac{{\partial S_{{{\text{seg}}}} }}{{\partial \varepsilon_{v}^{sp} }}}}{{\frac{{\partial f_{2} }}{{\partial S_{{{\text{seg}}}} }}}}} \\ \end{array}$$

(59)

$$\begin{array}{*{20}c} {A_{4} = - \frac{{\partial f_{1} }}{\partial p}\frac{{\partial S_{{{\text{seg}}}} }}{{\partial \varepsilon_{v}^{mp} }}} \\ \end{array}$$

(60)

$$\begin{array}{*{20}c} {A_{5} = - \frac{{\partial f_{1} }}{\partial q}\frac{{\partial S_{{{\text{seg}}}} }}{{\partial \varepsilon_{v}^{mp} }}} \\ \end{array}$$

(61)

$$\begin{array}{*{20}c} {A_{6} = \frac{{\frac{{\partial f_{2} }}{\partial S}\frac{{\partial f_{1} }}{{\partial p_{x} }}\frac{{\partial p_{x} }}{{\partial p_{x0} }}\frac{{\partial p_{x0} }}{\partial H}\frac{\partial H}{{\partial \varepsilon_{v}^{mp} }} - \frac{{\partial f_{1} }}{\partial S}\frac{{\partial f_{2} }}{{\partial S_{{{\text{seg}}}} }}\frac{{\partial S_{{{\text{seg}}}} }}{{\partial \varepsilon_{v}^{mp} }}}}{{\frac{{\partial f_{2} }}{{\partial S_{{{\text{seg}}}} }}}}} \\ \end{array}$$

(62)

$$ \begin{array}{*{20}c} {\chi = \frac{{\frac{{\partial f_{1} }}{\partial q}}}{{\frac{{\partial f_{1} }}{\partial p}}}} \\ \end{array}, K_{m}=K, K_{s}=\frac{3(1+e_{0})(S+p_{at})}{\kappa_s}$$

(63)