Karlsruhe model

The constitutive model proposed by Fuentes ¹ interrelates the degree of saturation $S^{w}$ with the suction $s$ and couples the void ratio $e$ through an incremental relation.

The input line takes the form:

*Hydraulic = Karlsruhe-Model
alpha^d, n^d, alpha^w, n^w, kappa, n, m^e, S_rd, S_rw

Therein, $α^{d}$ , $n^{d}$ , $α^{w}$ , $n^{w}$ are the van Genuchten parameter for the main drying $⊔^{d}$ and wetting $⊔^{w}$ branch of the soil water retention curve. The parameter $κ$ and $n$ control the scanning curves and $m^{e}$ the void ratio dependence. $S_{r, d}$ and $S_{r, w}$ are the residual degrees of saturation for the main drying curve and the main wetting curve, respectively.

Theory

\begin{aligned} {\dot{S}}^{w} & = \frac{\partial S_{d / i}^{w}}{\partial s} (κ_{w} + (1 - κ_{w}) Y^{d / i}) \dot{s} + \frac{\partial S_{d / i}^{w}}{\partial e} (1 + e) {\dot{ε}}_{v} \\ = (1 - S_{0, d / i}^{w}) \frac{\partial S_{d / i}^{e}}{\partial s} - (1 - S_{0, d / i}^{w}) \frac{\partial S_{d / i}^{e}}{\partial e} (1 + e) {\dot{ε}}_{v} \end{aligned}

where the indices of $d$ and $i$ are related with a drying ( $\dot{s} > 0$ ) and wetting ( $\dot{s} < 0$ ) process respectively, $κ_{w}$ is a material parameter, $e$ is the void ratio, ${\dot{ε}}_{v} = - div (b v)$ is the volumetric strain rate and $Y^{d / i}$ is an interpolation function ( $0 \leq Y^{d / i} \leq 1$ ) defined as:

Y^{d / i} = {\begin{cases} Y^{d} = {[\frac{\log (s / s^{i})}{\log (s^{d} / s^{i})}]}^{n_{w}} & for drying: \dot{s} \geq 0 \\ Y^{i} = {[\frac{\log (s^{d} / s)}{\log (s^{d} / s^{i})}]}^{n_{w}} & for wetting: \dot{s} < 0 \end{cases}

The exponent $n_{w} \approx 1 - 10$ is a material parameter that controls the interpolation. The model considers as boundaries the main drying curve $S_{d}^{w}$ and the main wetting curve $S_{i}^{w}$ described by the van-Genuchten relations extended by the Gallipoli relation which accounts for the void ratio dependence ²:

\begin{aligned} S_{d}^{w} & = S_{0, d}^{w} + (1 - S_{0, d}^{w}) \frac{1}{{[1 + (α^{d} e^{m_{e}} s)^{n^{d}}]}^{1 - 1 / n^{d}}} \\ S_{i}^{w} & = S_{0, i}^{w} + (1 - S_{0, i}^{w}) \frac{1}{{[1 + (α^{i} e^{m_{e}} s)^{n^{i}}]}^{1 - 1 / n^{i}}} \end{aligned}

where $α^{d}, α^{i}$ , $n^{d}$ and $n^{i}$ are parameters of these curves. Differentiation of the relation above with the suction $s$ and void ratio $e$ gives:

\begin{aligned} \frac{\partial S_{d / i}^{w}}{\partial s} & = (s^{d / i} e^{m_{e}} α^{d / i})^{n^{d / i}} (1 + (s^{d / i} e^{m_{e}} α^{d / i})^{n^{d / i}})^{1 / n^{d / i} - 2} (1 - n^{d / i}) / s^{d / i} (1 - S_{0, d / i}^{w}) \\ \frac{\partial S_{d / i}^{w}}{\partial e} & = - m_{e} (s^{d / i} e^{m_{e}} α^{d / i})^{n^{d / i}} (1 + (s^{d / i} e^{m_{e}} α^{d / i})^{n^{d / i}})^{1 / n^{d / i} - 2} (n^{d / i} - 1) / e (1 - S_{0, d / i}^{w}) \end{aligned}

$s^{d}$ and $s^{i}$ are the projected suction on the main branches for a given effective degree of saturation $S_{w}$ . They are computed according to the following relations:

s^{d / i} = {\begin{cases} s^{d} = \frac{{({(1 / (\frac{S^{w} - S_{0, i}^{w}}{1 - S_{0, i}^{w}}))}^{n_{d} / (n_{d} - 1)} - 1)}^{1 / n_{d}}}{α^{d} e^{m_{e}}} & for drying \dot{s} \geq 0 \\ s^{i} = \frac{{({(1 / (\frac{S^{w} - S_{0, i}^{w}}{1 - S_{0, i}^{w}}))}^{n_{i} / (n_{i} - 1)} - 1)}^{1 / n_{i}}}{α^{i} e^{m_{e}}} & for wetting \dot{s} < 0 \end{cases}

The parameters $α^{i}$ , $n^{i}$ and $m_{e}$ are used to describe the main wetting curve while the counterpart parameters $α^{d}$ and $n^{d}$ (with $m_{e}$ ) describe the main drying curve. The hysteretic behavior is controlled through the parameters $n_{w}$ and $κ_{w}$ . A calibration procedure of these parameters can be found in ¹ and ³.

W. Fuentes and Th. Triantafyllidis. Hydro-mechanical hypoplastic models for unsaturated soils under isotropic stress conditions. Computers and Geotechnics, 51:72–82, 2013. URL: https://www.sciencedirect.com/science/article/pii/S0266352X13000281, doi:https://doi.org/10.1016/j.compgeo.2013.02.002. ↩↩
D. Gallipoli, S. J. Wheeler, and M. Karstunen. Modelling the variation of degree of saturation in a deformable unsaturated soil. Géotechnique, 53(1):105–112, 2 2003. doi:10.1680/geot.2003.53.1.105. ↩
W. Fuentes, M. Tafili, and Th. Triantafyllidis. An ISA-plasticity-based model for viscous and non-viscous clays. Acta Geotechnica, pages 1–20, 4 2017. doi:10.1007/s11440-017-0548-y. ↩