Soft Soil Creep Model

The Soft Soil Creep Model (SSCM) [Verneer et al., 1999; PLAXIS 2016] is an elasto-plastic rate dependent model. SSCM describes the mechanical behavior of materials that develop both reversible and irreversible strain, with the last depending on the loading rate.

Theory

The strain increment \(d \boldsymbol{\varepsilon}\) is assumed to be additively decomposed into elastic \(d \boldsymbol{\varepsilon}^e\) and plastic \(d \boldsymbol{\varepsilon}^p\) parts. Dividing both sides by time increment \(d t\), the total strain rate reads:

\(\dot{\boldsymbol{\varepsilon}}=\dot{\boldsymbol{\varepsilon}}^e+\dot{\boldsymbol{\varepsilon}}^p\)

The elastic response, in rate form, reads:

\(\dot{\boldsymbol{\varepsilon}}^e=\dot{\boldsymbol{\varepsilon}}-\dot{\boldsymbol{\varepsilon}}^p = \mathbf{D}^{-1} \dot{\boldsymbol{\sigma}}\)

where \(\dot{\boldsymbol{\sigma}}\) is the rate of the internal stress \(\boldsymbol{\sigma}=\left(\sigma_x,\sigma_y,\sigma_z,\tau_{xy},\tau_{yz},\tau_{xz} \right)^T\) and \(\mathbf{D}\) is the elasticity matrix. Nonlinear isotropic elasticity is assumed and so the elasticity matrix \(\mathbf{D}\) is function of the Poisson coefficient \(\nu\) and the Young modulus \(E\). The Poisson coefficient \(\nu\) is a constant input parameter while E is stress-dependent and updated according to the following law:

\(E = - ( 1 - 2 \nu ) \frac{\sigma_x+\sigma_y+\sigma_z}{k^*}\)

where \(k^*\) is the modified swelling parameter.

The SSCM has not a yield function or a purely elastic domain and the plastic strain is always non-zero. The plastic response is governed by the flow rule:

\(\dot {\boldsymbol \varepsilon}^p = \dot \gamma \frac{\partial \mathrm{p}_c}{\partial \boldsymbol \sigma}\)

where \(\dot \gamma\) is the plastic multiplier rate and the stress \(\mathrm{p}_{c}\) is a representation of the stress state in the plane of the stress invariants (p,q):

\(\mathrm{p}_c = \mathrm{p} + \frac{c}{\tan \phi} + \frac{\mathrm{q}^2}{\mathrm{M}^2 (\mathrm{p} + \frac{c}{\tan \phi})}\)

with:

\(\mathrm{p} = -\frac{\sigma_x+\sigma_y+\sigma_z}{3}\)

\(\mathrm{q} = \sqrt{\sigma_x \left( \sigma_x-\sigma_y \right) + \sigma_y \left( \sigma_y-\sigma_z \right) + \sigma_z \left( \sigma_z-\sigma_x \right) + 3 \left( \tau^2_{xy} + \tau^2_{yz} + \tau^2_{zx} \right) }\)

and where \(c\) is the cohesion, \(\phi\) is the friction angle and \(\mathrm{M}\) is the critical state line.

The plastic multiplier \(\dot \gamma\) is defined as:

\(\dot\gamma = \frac{\mu^{*}}{\tau^{*} \frac{\partial \mathrm{p}_c}{\partial \mathrm{p}}} ( \frac{ \mathrm{p}_c}{\mathrm{p}_{c,r}} )^{\frac{\lambda^*-k^*}{\mu^*}}\)

where \(\tau^*\) is a reference time related to normal-consolidation-state (NCS) in a standard oedometer test, \(\lambda^{*}\) is the modified compression index, \(\mu^*\) is the creep index and \(\mathrm{p}_{c,r}\) is a reference stress that describes the hardening behavior:

\(\mathrm{p}_{c,r}=\mathrm{p}_{c,r,0} exp[ -\frac{ \varepsilon^{p}_{x} + \varepsilon^{p}_{y} + \varepsilon^{p}_{z} }{ \lambda^{*} - k^{*} } ]\)

where \(\mathrm{p}_{c,r,0}\) is a parameter related to the plastic strain developed by the material before loading. The ratio \(\frac{\mathrm{p}_{c,r,0}}{\mathrm{p}_{c,0}}\) is the geotechnical initial Over-Consolidation-Ratio \(OCR_0\). A graphical representation of \(\mathrm{p}_c\) and \(\mathrm{p}_{c,r}\) is shown below :

The stress \(\boldsymbol \sigma\), the reference stress \(\mathrm{p}_{c,r}\) and the plastic multiplier \(\gamma\) describe the current material state in terms of stress and strain (elastic and plastic) and they are defined as the state variables of the model.

Initialization

The initialization can be done using the tab_file or the restart file.

Input Notes

In the input file materials must be set:

• The material type MAT_TYPE = 4.

• The integer parameters MAT_IPAR[NI_MAT_TYPE] with NI_MAT_TYPE = 2:

• The real parameters MAT_RPAR[NR_MAT_TYPE] with NI_MAT_TYPE = 17: