Constitutive model with double yield surfaces of freeze-thaw soil considering moisture migration

Abstract

In cold regions, the effects of moisture migration will destroy the original structure of the soil and cause the redistribution of moisture inside the soil. In order to research the effects of moisture migration on soil’s mechanics behaviors under freeze-thaw cycles, an elastoplastic constitutive model with double yield surfaces considering moisture migration was proposed based on the internal state variables (ISV) theory. The amount of moisture migration was obtained by the segregation potential model and used as an intrinsic variable to obtain the Helmholtz free energy function coupled with the amount of moisture migration. Two kinds of plastic deformation mechanisms, including plastic volumetric compression and plastic shear, were considered to develop the coupled model based on the ISV theory with moisture migration. This model was given in incremental form, and detailed numerical calculation processes were described. The proposed model was concretized with elliptic-parabolic yield surfaces and validated by tests under different freeze-thaw cycles and different confining pressure. Desirably, the calculated deviatoric stress-axial strain curves and volumetric strain-axial strain curves agreed well with the test results.

Appendix 1

The specific form of the proposed model with elliptic-parabolic yield surfaces can be expressed as:

$$ \frac{1}{K}+\frac{1}{N_1}\frac{\partial {f}_1}{\partial p}\frac{\partial {g}_1}{\partial p}+\frac{1}{N_2}\frac{\partial {f}_2}{\partial p}\frac{\partial {g}_2}{\partial p}=\frac{1}{K}+\frac{1-\frac{q^2}{M_1^2{\left(p+{p}_r\right)}^2}}{\frac{h{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}}+\frac{{\left[\frac{-a{M}_2{q}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2G{\left[{M}_2\left(p+{p}_r\right)-q\right]}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right]}^2}{\frac{a}{G}\sqrt{\frac{q}{M_2\left(p+{p}_r\right)-q}}+\frac{a{q}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{M}_2\left(p+{p}_r\right)}{2G{\left[{M}_2\left(p+{p}_r\right)-q\right]}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}} $$

(63)

$$ \frac{1}{N_1}\frac{\partial {f}_1}{\partial q}\frac{\partial {g}_1}{\partial p}+\frac{1}{N_2}\frac{\partial {f}_2}{\partial q}\frac{\partial {g}_2}{\partial p}=\frac{\frac{2q}{M_1^2\left(p+{p}_r\right)}}{\frac{h{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}}+\frac{-a{M}_2{q}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2G{\left[{M}_2\left(p+{p}_r\right)-q\right]}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} $$

(64)

$$ \frac{C_1}{N_1}+\frac{D_1}{N_2}=\frac{-\left[\frac{\varepsilon_v^{p1}{p}_a}{1-t{\varepsilon}_v^{p1}}{\varsigma}_6+\frac{ht{\varepsilon}_v^{p1}{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}{\varsigma}_7\right]}{\frac{h{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}} $$

(65)

$$ \frac{C_2}{N_1}+\frac{D_2}{N_2}=\frac{-\left[\frac{\varepsilon_v^{p1}{p}_a}{1-t{\varepsilon}_v^{p1}}{\xi}_6+\frac{ht{\varepsilon}_v^{p1}{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}{\xi}_7\right]}{\frac{h{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}} $$

(66)

$$ \frac{1}{N_1}\frac{\partial {f}_1}{\partial p}\frac{\partial {g}_1}{\partial q}+\frac{1}{N_2}\frac{\partial {f}_2}{\partial p}\frac{\partial {g}_2}{\partial q}=\frac{\frac{2q}{M_1^2\left(p+{p}_r\right)}}{\frac{h{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}}+\frac{-a{M}_2{q}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{2G{\left[{M}_2\left(p+{p}_r\right)-q\right]}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} $$

(67)

$$ \frac{1}{3G}+\frac{1}{N_1}\frac{\partial {f}_1}{\partial q}\frac{\partial {g}_1}{\partial q}+\frac{1}{N_2}\frac{\partial {f}_2}{\partial q}\frac{\partial {g}_2}{\partial q}=\frac{1}{3G}+\frac{{\left[\frac{2q}{M_1^2\left(p+{p}_r\right)}\right]}^2}{\frac{h{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}\left[1-\frac{q^2}{M_1^2{\left(p+{p}_r\right)}^2}\right]}+\frac{a}{G}\sqrt{\frac{q}{M_2\left(p+{p}_r\right)-q}}+\frac{a{q}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{M}_2\left(p+{p}_r\right)}{2G{\left[{M}_2\left(p+{p}_r\right)-q\right]}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}} $$

(68)

$$ \frac{C_3}{N_1}+\frac{D_3}{N_2}=\frac{-\frac{2q}{M_1^2\left(p+{p}_r\right)}\left[\frac{\varepsilon_v^{p1}{p}_a}{1-t{\varepsilon}_v^{p1}}{\varsigma}_6+\frac{ht{\varepsilon}_v^{p1}{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}{\varsigma}_7\right]}{\frac{h{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}\left[1-\frac{q^2}{M_1^2{\left(p+{p}_r\right)}^2}\right]} $$

(69)

$$ \frac{C_4}{N_1}+\frac{D_4}{N_2}=\frac{-\frac{2q}{M_1^2\left(p+{p}_r\right)}\left[\frac{\varepsilon_v^{p1}{p}_a}{1-t{\varepsilon}_v^{p1}}{\xi}_6+\frac{ht{\varepsilon}_v^{p1}{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}{\xi}_7\right]}{\frac{h{p}_a}{{\left(1-t{\varepsilon}_v^{p1}\right)}^2}\left[1-\frac{q^2}{M_1^2{\left(p+{p}_r\right)}^2}\right]} $$

(70)

where ς_i are coefficients affected by temperature, ξ_j are coefficients affected by moisture migration.