Behaviour of Partially Saturated Soil Under Isotropic and Triaxial Condition Using Modified BBM

Abstract

How suction influence the shear strength of partially saturated soil is one of the important challenges in many geotechnical problems. As compared to saturated soil, the triaxial compression stress path for partially saturated soil is more complex. In the present study a single element finite element model has been developed and implemented in FORTRAN code. An explicit integration algorithm is adapted to Barcelona basic model with some modification considering locus of apparent tensile strength in the p′–s plane as non-linear, anisotropy to the yield surface and dependence of degree of saturation on volumetric strain. The performance of the code is examined and verified with suction controlled triaxial experimental results. Application of isotropic compression, wetting and triaxial compression stress path on model demonstrated significant influence of stress paths on strength and deformation behaviour of partially saturated soil.

Appendices

Appendix 1: Flowchart

Flow chart for the calculation of the stress and plastic strain increments and the changes in hardening parameters.

Appendix 2: Incremental Stress Strain Relation

The total strains takes the form along with flow rule:

$$\begin{aligned} {\text{d}}{\varvec{\upvarepsilon}} & = {\text{d}}{\varvec{\upvarepsilon}}^{\text{e}} + {\text{d}}{\varvec{\upvarepsilon}}^{\text{p}} + {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{e}} + {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{p}} \\ & = {\text{d}}{\varvec{\upvarepsilon}}^{\text{e}} + \delta \chi \frac{\partial g}{\partial \sigma } + {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{e}} + {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{p}} \\ \end{aligned}$$

(15)

where ${\text{d}}{\varvec{\upvarepsilon}}^{\text{e}}$ is the elastic strain due to change in net stress, ${\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{e}}$ and ${\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{p}}$ are the elastic and plastic strain due to change in suction and $\delta_{\chi }$ is the plastic multiplier (flow rule).

$${\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{e}} = \frac{{\kappa_{s} }}{{\nu (s + p_{\text{atm}} )}}{\text{d}}s$$

(16)

$${\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{e}} = \left\{ {\begin{array}{*{20}l} {\frac{{\lambda_{s} - \kappa_{s} }}{{\nu (s + p_{\text{atm}} )}}} \hfill & {{\text{if}}\,{\text{suction}}\,{\text{increase}}\,{\text{yield}}\,{\text{surface}}\,{\text{is}}\,{\text{active}}} \hfill \\ 0 \hfill & {{\text{if}}\,{\text{suction}}\,{\text{increase}}\,{\text{yield}}\,{\text{surface}}\,{\text{is}}\,{\text{not}}\,{\text{active}}} \hfill \\ \end{array} } \right.$$

(17)

The elastic strain and net stress change take the form:

$${\text{d}}{\varvec{\upsigma}} = {\mathbf{D}}^{\text{e}} {\text{d}}{\varvec{\upvarepsilon}}^{\text{e}}$$

(18)

where D ^e is the elastic constitutive matrix.

Combining Eqs. 15 and 16:

$${\text{d}}{\varvec{\upsigma}} = {\mathbf{D}}^{\text{e}} \left( {{\text{d}}{\varvec{\upvarepsilon}} - \delta \chi \frac{\partial g}{\partial \sigma } - {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{e}} - {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{p}} } \right)$$

(19)

The consistency condition for partially saturated soils can be written as:

$$\begin{aligned} {\text{d}}f & = \left( {\frac{\partial f}{{\partial {\varvec{\upsigma}}}}} \right)^{\text{T}} {\text{d}}{\varvec{\upsigma}} + \frac{\partial f}{\partial s}{\text{d}}s + \frac{\partial f}{{\partial \varepsilon_{\text{v}}^{\text{p}} }}{\text{d}}\varepsilon_{\text{v}}^{\text{p}} = 0 \\ & = \left( {\frac{\partial f}{{\partial {\varvec{\upsigma}}}}} \right)^{\text{T}} {\text{d}}{\varvec{\upsigma}} + \frac{\partial f}{{\partial p_{\text{mc}} }}\frac{{\partial p_{\text{mc}} }}{\partial s}{\text{d}}s + \frac{\partial f}{{\partial p_{s} }}\frac{{\partial p_{s} }}{\partial s}{\text{d}}s + \frac{\partial f}{{\partial p_{\text{mc}} }}\frac{{\partial p_{\text{mc}} }}{{\partial \varepsilon_{\text{v}}^{\text{p}} }}{\text{d}}\varepsilon_{\text{v}}^{\text{p}} = 0 \\ \end{aligned}$$

(20)

The plastic multiplier can be solved by combining Eqs. 17 and 18

$$\delta \chi = \frac{{\left( {\frac{\partial f}{{\partial {\varvec{\upsigma}}}}} \right)^{\text{T}} {\mathbf{D}}^{\text{e}} \left( {{\text{d}}{\varvec{\upvarepsilon}} - {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{e}} - {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{p}} } \right) + \left( {\frac{\partial f}{{\partial p_{\text{mc}} }}\frac{{\partial p_{\text{mc}} }}{\partial s} + \frac{\partial f}{{\partial p_{s} }}\frac{{\partial p_{s} }}{\partial s}} \right){\text{d}}s}}{{\left( {\frac{\partial f}{{\partial {\varvec{\upsigma}}}}} \right)^{\text{T}} {\mathbf{D}}^{\text{e}} \frac{\partial g}{{\partial {\varvec{\upsigma}}}} + H}}$$

(21)

where

$$H = - \frac{\partial f}{{\partial p_{\text{mc}} }}\frac{{\partial p_{\text{mc}} }}{{\partial \varepsilon_{\text{v}}^{\text{p}} }}\frac{\partial g}{{\partial p^{{\prime }} }}$$

(22)

Plastic volumetric and shear strain due to stress change may be calculate as:

$$d\varepsilon_{v}^{p} = \delta \chi \frac{\partial g}{{\partial p^{{\prime }} }}\quad \& \quad d\varepsilon_{s}^{p} = \delta \chi \frac{\partial g}{\partial q}$$

(23)

Substituting Eq. 21 into Eq. 19 gives:

$${\text{d}}{\varvec{\upsigma}} = {\mathbf{D}}^{\text{ep}} \left( {{\text{d}}{\varvec{\upvarepsilon}} - {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{e}} - {\text{d}}{\varvec{\upvarepsilon}}_{s}^{\text{p}} } \right) - {\mathbf{W}}{\text{d}}s$$

(24)

where

$${\mathbf{D}}^{\text{ep}} = {\mathbf{D}}^{\text{e}} - \frac{{{\mathbf{D}}^{\text{e}} \frac{\partial g}{{\partial {\varvec{\upsigma}}}}\left( {\frac{\partial f}{{\partial {\varvec{\upsigma}}}}} \right)^{\text{T}} {\mathbf{D}}^{\text{e}} }}{{\left( {\frac{\partial f}{{\partial {\varvec{\upsigma}}}}} \right)^{\text{T}} {\mathbf{D}}^{\text{e}} \frac{\partial g}{{\partial {\varvec{\upsigma}}}} + H}}$$

(25)

is the elasto-plastic constitutive matrix for the case of fully saturated soil, and

$${\mathbf{W}} = \frac{{{\mathbf{D}}^{\text{e}} \frac{\partial g}{{\partial {\varvec{\upsigma}}}}\left( {\frac{\partial f}{\partial s}} \right)^{\text{T}} }}{{\left( {\frac{\partial f}{{\partial {\varvec{\upsigma}}}}} \right)^{\text{T}} {\mathbf{D}}^{\text{e}} \frac{\partial g}{{\partial {\varvec{\upsigma}}}} + H}}$$

(26)