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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

Therein, \(\alpha^{d}\), \(n^{d}\), \(\alpha^{w}\), \(n^{w}\) are the van Genuchten parameter for the main drying \(\sqcup^d\) and wetting \(\sqcup^w\) branch of the soil water retention curve. The parameter \(\kappa\) and \(n\) control the scanning curves and \(m^e\) the void ratio dependence.

Theory

\[\begin{align*} \dot{S}^w&=\frac{\partial S^w_{d/i}}{\partial s} \left( \kappa_w+(1-\kappa_w)Y^{d/i} \right) \dot{s} + \frac{\partial S^w_{d/i}}{\partial e}(1+e)\dot{\varepsilon}_v\\ & = \left(1-S^w_{0,d/i}\right) \frac{\partial S^e_{d/i}}{\partial s} - \left(1-S^w_{0,d/i}\right) \frac{\partial S^e_{d/i}}{\partial e}(1+e)\dot{\varepsilon}_v \end{align*}\]

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

\[\begin{equation*} Y^{d/i} = \begin{cases} Y^d=\left[\dfrac{\log(s/s^i)}{\log(s^d/s^i)}\right]^{n_w}&\text{for drying:} ~~~ \dot{s} \geq 0 \\ Y^i=\left[\dfrac{\log(s^d/s)}{\log(s^d/s^i)}\right]^{n_w}&\text{for wetting:} ~~~ \dot{s}<0 \end{cases} \end{equation*}\]

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

\[\begin{align*} S^w_{d}&=S^w_{0,d} + \left(1- S^w_{0,d} \right) \dfrac{1}{\left[1+(\alpha^{ d}e^{m_e} s)^{n^{d} }\right]^{1-1/n^{ d}}} \\ S^w_{i}&=S^w_{0,i} + \left(1- S^w_{0,i} \right) \dfrac{1}{\left[1+(\alpha^{ i}e^{m_e} s)^{n^{i}}\right]^{1-1/n^{ i}}} \end{align*}\]

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

\[\begin{equation*}\label{edS1} \begin{split} \frac{\partial S^w_{d/i}}{\partial s}&=(s^{d/i}e^{m_e}\alpha^{ d/i})^{n^{d/i}}(1+(s^{d/i}e^{m_e}\alpha^{ d/i})^{n^{d/i}})^{1/n^{d/i}-2}(1-n^{d/i})/s^{d/i} \left(1-S^w_{0,d/i}\right) \\\ \frac{\partial S^w_{d/i}}{\partial e}&=-m_e(s^{d/i} e^{m_e}\alpha^{d/i})^{n^{d/i}}(1+(s^{d/i} e^{m_e}\alpha^{d/i})^{n^{d/i}})^{1/n^{d/i}-2}(n^{d/i}-1)/e \left(1-S^w_{0,d/i}\right) \end{split} \end{equation*}\]

\(s^d\) 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:

\[\begin{equation*}\label{esdi} s^{d/i} = \begin{cases} s^d=\dfrac{\left({(1/\left(\frac{S^w-S^w_{0,i}}{1-S^w_{0,i}}\right))}^{n_d/(n_d-1)}-1\right)^{1/n_d}}{\alpha^d e^{m_e}} & \text{for drying} ~~~ \dot{s} \geq 0\\ s^i=\dfrac{\left({(1/\left(\frac{S^w-S^w_{0,i}}{1-S^w_{0,i}}\right))}^{n_i/(n_i-1)}-1\right)^{1/n_i}}{\alpha^ie^{m_e}} & \text{for wetting} ~~~ \dot{s}<0 \end{cases} \end{equation*}\]

The parameters \(\alpha^i\), \(n^i\) and \(m_e\) are used to describe the main wetting curve while the counterpart parameters \(\alpha^d\) and \(n^d\) (with \(m_e\)) describe the main drying curve. The hysteretic behavior is controlled through the parameters \(n_w\) and \(\kappa_w\). A calibration procedure of these parameters can be found in 1 and 3.

  1. 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. ↩↩

  2. 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. ↩

  3. 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. ↩