Unified hardening (UH) model for unsaturated expansive clays

Abstract

In this paper, by introducing a new yielding mechanism based on the widely acknowledged double-structure theory, the original UH model for unsaturated soils is extended to capture the behaviour of expansive clays. A novel expansion potential is further established to evaluate the effect of overconsolidation on the volume change of unsaturated expansive clays during wetting. With only one additional parameter, the proposed model can describe the behaviour of both wetting-collapse and wetting-induced swelling for unsaturated clays. Comparisons between model predictions and test results show a good agreement which verifies the capability of the proposed model in charactering the features of unsaturated expansive clays under various stress histories and stress paths.

Appendix: Derivation of three-dimensional elastoplastic stress–strain relationship

As shown in Fig. 4, ${\widehat{\sigma }}_{i}={\sigma }_{i}+{p}_{\mathrm{s}}$ is the principal stress considering the suction and $\widehat{\theta }$ is the corresponding Lode’s angel. The transformation from the general stress space to the transformed stress space can be expressed as

$$\left\{\begin{array}{c}{\widetilde{\sigma }}_{ij}={\widehat{\sigma }}_{ij} , \widehat{q}=0\\ {\widetilde{\sigma }}_{ij}=\widehat{p}{\delta }_{ij}+\frac{{\widehat{q}}_{\mathrm{c}}}{\widehat{q}}({\widehat{\sigma }}_{ij}-\widehat{p}{\delta }_{ij}), \widehat{q}\ne 0\end{array}\right.$$

(45)

where

$$\widehat{p}=\frac{{\widehat{\sigma }}_{1}+{\widehat{\sigma }}_{2}+{\widehat{\sigma }}_{3}}{3}$$

(46)

$$\widehat{q}=\sqrt{\frac{1}{2}\left[{\left({\widehat{\sigma }}_{1}-{\widehat{\sigma }}_{2}\right)}^{2}+{\left({\widehat{\sigma }}_{2}-{\widehat{\sigma }}_{3}\right)}^{2}+{\left({\widehat{\sigma }}_{3}-{\widehat{\sigma }}_{1}\right)}^{2}\right]}$$

(47)

$${\widehat{q}}_{\mathrm{c}}=\frac{2{\widehat{I}}_{1}}{3\sqrt{\frac{{\widehat{I}}_{1}{\widehat{I}}_{2}-{\widehat{I}}_{3}}{{\widehat{I}}_{1}{\widehat{I}}_{2}-9{\widehat{I}}_{3}}-1}}$$

(48)

$${\widehat{I}}_{1}={\widehat{\sigma }}_{ii}$$

(49)

$${\widehat{I}}_{2}=\frac{1}{2}\left[{\left({\widehat{\sigma }}_{ii}\right)}^{2}-{\widehat{\sigma }}_{rs}{\widehat{\sigma }}_{sr}\right]$$

(50)

$${\widehat{I}}_{3}=\frac{1}{3}{\widehat{\sigma }}_{ij}{\widehat{\sigma }}_{jk}{\widehat{\sigma }}_{ki}-\frac{1}{2}{\widehat{\sigma }}_{rs}{\widehat{\sigma }}_{sr}{\widehat{\sigma }}_{mm}+\frac{1}{6}{\left({\widehat{\sigma }}_{nn}\right)}^{3}$$

(51)

Integrating Eq. (45), the UH model for unsaturated expansive clays in 3D stress space can be expressed using tensor as

$$\mathrm{d}{\sigma }_{ij}+\mathrm{d}{\sigma }_{ij}^{{\mathrm{p}}_{\mathrm{s}}}={D}_{ijkl}^{\mathrm{ep}}\left(\mathrm{d}{\varepsilon }_{kl}-\mathrm{d}{\varepsilon }_{kl{\mathrm{p}}_{\mathrm{s}}}^{\mathrm{e}}-\mathrm{d}{\varepsilon }_{kl{\mathrm{p}}_{\mathrm{m}}}^{\mathrm{p}}\right)$$

(52)

where ${\sigma }_{ij}^{{\mathrm{p}}_{\mathrm{s}}}$ is the stress related to ${p}_{\mathrm{s}}$, ${\varepsilon }_{kl{\mathrm{p}}_{\mathrm{s}}}^{\mathrm{e}}$ is the elastic strain induced by ${p}_{\mathrm{s}}$, ${\varepsilon }_{kl{\mathrm{p}}_{\mathrm{m}}}^{\mathrm{p}}$ is the plastic strain induced by the microstructural swelling and ${D}_{ijkl}^{\mathrm{ep}}$ is the elasto-plastic constitutive tensor which can be expressed as:

$$\begin{array}{c}{D}_{ijkl}^{\mathrm{ep}}=L{\delta }_{ij}{\delta }_{kl}+G\left({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}\right)\#\\ -\frac{(L\frac{\partial \widetilde{f}}{\partial {\widetilde{\sigma }}_{mm}}{\delta }_{ij}+2G\frac{\partial \widetilde{f}}{\partial {\widetilde{\sigma }}_{ij}})(L\frac{\partial \widetilde{f}}{\partial {\sigma }_{nn}}{\delta }_{kl}+2G\frac{\partial \widetilde{f}}{\partial {\sigma }_{kl}})}{X}\end{array}$$

(53)

where

$$X=L\frac{\partial \widetilde{f}}{\partial {\sigma }_{ii}}\frac{\partial \widetilde{f}}{\partial {\widetilde{\sigma }}_{kk}}+2G\frac{\partial \widetilde{f}}{\partial {\sigma }_{ij}}\frac{\partial \widetilde{f}}{\partial {\widetilde{\sigma }}_{ij}}-\frac{\partial \widetilde{f}}{\partial {p}_{\mathrm{x}}}\frac{\partial {p}_{\mathrm{x}}}{\partial {p}_{\mathrm{x}}^{*}}\frac{\partial {p}_{\mathrm{x}}^{*}}{\partial H}\frac{1}{\widetilde{\Omega }}$$

(54)

$$L=K-\frac{2}{3}G$$

(55)

$\widetilde{f}$ is the current yield surface in the transformed stress space which can be expressed as

$$\widetilde{f}=\mathrm{ln}\widetilde{p}+\mathrm{ln}\left(1+\frac{{\widetilde{q}}^{2}}{{M}^{2}{\widetilde{p}}^{2}}\right)-\mathrm{ln}({p}_{\mathrm{x}}+{p}_{\mathrm{s}})=0$$

(56)

where

$$\widetilde{p}=\widehat{p}$$

(57)

$$\widetilde{q}={\widehat{q}}_{c}$$

(58)

and

$$\frac{1}{\widetilde{\Omega }}=\frac{{\widetilde{M}}_{\mathrm{Ys}}^{4}-{\widetilde{\eta }}^{4}}{{M}^{4}-{\widetilde{\eta }}^{4}}$$

(59)

$${\widetilde{M}}_{\mathrm{Ys}}=6\left[\sqrt{\frac{\chi }{{\widetilde{R}}_{\mathrm{s}}}\left(1+\frac{\chi }{{\widetilde{R}}_{\mathrm{s}}}\right)}-\frac{\chi }{{\widetilde{R}}_{\mathrm{s}}}\right]$$

(60)

where

$$\widetilde{\eta }=\frac{\widetilde{q}}{\widetilde{p}}$$

(61)

$${\widetilde{R}}_{\mathrm{s}}=\frac{\widetilde{p}}{\widetilde{\overline{p}} }$$

(62)