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Karlsruhe model

The Karlsruhe model, developed by Fuentes and Triantafyllidis (2013)1, is an advanced hydro-mechanical constitutive model for unsaturated soils. Its primary features are the ability to simulate hysteretic water retention behavior (drying and wetting scanning curves) and to couple the retention properties with the soil's mechanical state through the void ratio (\(e\)).

Model Formulation

The model is formulated in an incremental, or rate-based, form. The theory is built top-down, starting from the main bounding curves that define the limits of the hysteretic behavior.

Bounding Curves: Void-Ratio-Dependent SWRC

The hysteretic behavior is bounded by a main drying curve (\(S^w_d\)) and a main wetting curve (\(S^w_i\)). These curves are described by the van Genuchten model, extended to include a dependency on the void ratio \(e\) as proposed by Gallipoli et al. (2003)2:

\[ S^w_{d} = S_{rd} + \left(1 - S_{rd} \right) \left[1+(\alpha^{d} e^{m_e} p^c)^{n^{d} }\right]^{-\left(1-1/n^{d}\right)} \]
\[ S^w_{i} = S_{rw} + \left(1 - S_{rw} \right) \left[1+(\alpha^{w} e^{m_e} p^c)^{n^{w}}\right]^{-\left(1-1/n^{w}\right)} \]

Hysteretic Scanning Curves

For any state inside the main hysteretic loop, the change in saturation is defined by an incremental relation that interpolates between the bounding curves. The rate of change of the degree of saturation, \(\dot{S}^w\), is given by:

\[ \dot{S}^w = \frac{\partial S^w_{d/i}}{\partial p^c} \left( \kappa_w+(1-\kappa_w)Y^{d/i} \right) \dot{p}^c + \frac{\partial S^w_{d/i}}{\partial e}(1+e)\dot{\varepsilon}_{vol} \]

This equation consists of two parts: the change due to the rate of capillary pressure (\(\dot{p}^c\)) and the change due to the rate of volumetric strain (\(\dot{\varepsilon}_{vol}\)). The indices \(d\) and \(i\) refer to drying (\(\dot{p}^c \geq 0\)) and wetting (\(\dot{p}^c < 0\)) processes, respectively.

The shape of the scanning curves between the main branches is controlled by the interpolation function \(Y^{d/i}\):

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

The interpolation function depends on the history parameters \((p^c)^d\) and \((p^c)^i\). These represent the capillary pressure values on the main drying and wetting curves, respectively, for the current degree of saturation \(S^w\). They are found by inverting the bounding curve equations:

\[ (p^c)^{d/i} = \begin{cases} (p^c)^d=\dfrac{\left(\left(\frac{S^w-S_{rw}}{1-S_{rw}}\right)^{-\frac{n_d}{n_d-1}}-1\right)^{\frac{1}{n_d}}}{\alpha^d e^{m_e}} & \text{for drying} \\ (p^c)^i=\dfrac{\left(\left(\frac{S^w-S_{rd}}{1-S_{rd}}\right)^{-\frac{n_i}{n_i-1}}-1\right)^{\frac{1}{n_i}}}{\alpha^i e^{m_e}} & \text{for wetting} \end{cases} \]

Model Parameters

The model requires a comprehensive set of parameters to define the hydro-mechanical behavior. A full calibration procedure can be found in 1 and 3.

  • Bounding Curve Parameters:

    • alpha^d, n^d, S_rd: Van Genuchten parameters for the main drying curve.
    • alpha^w, n^w, S_rw: Van Genuchten parameters for the main wetting curve.

  • Hydro-Mechanical Coupling Parameter:

    • m^e: Exponent controlling the influence of void ratio on the retention curves.

  • Hysteresis Parameters:

    • kappa (\(\kappa_w\) in equations): Controls the slope of the initial reversal from a bounding curve.
    • n (\(n_w\) in equations): Controls the shape of the scanning curves during interpolation.

Jacobian Contributions

The partial derivatives of the bounding curves with respect to capillary pressure \(p^c\) and void ratio \(e\) are required for the incremental formulation. These are given by:

\[ \frac{\partial S^w_{d/i}}{\partial p^c} = - (1-S_{rd/i}) (1-1/n^{d/i}) \left( \alpha^{d/i} e^{m_e} \right) n^{d/i} \left( \alpha^{d/i} e^{m_e} p^c \right)^{n^{d/i}-1} \left[ 1 + (\alpha^{d/i} e^{m_e} p^c)^{n^{d/i}} \right]^{1/n^{d/i}-2} \]
\[ \frac{\partial S^w_{d/i}}{\partial e} = - (1-S_{rd/i}) (1-1/n^{d/i}) \frac{m_e}{e} n^{d/i} \left( \alpha^{d/i} e^{m_e} p^c \right)^{n^{d/i}} \left[ 1 + (\alpha^{d/i} e^{m_e} p^c)^{n^{d/i}} \right]^{1/n^{d/i}-2} \]

  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. ↩