This document describes the CamClay stress potential based on the Cam-Clay non linear elastic behaviours. The CamClay stress potential is the basis of the CamClayElasticity brick.
The CamClay stress potential provides:
computeElasticPrediction.The hydrostatic pressure \(p\) is defined in this document using the conventions of geomechanics, as:
\[ p=-{{\displaystyle \frac{\displaystyle 1}{\displaystyle 3}}}\,{\mathrm{tr}{\left(\underline{\sigma}\right)}} \]
The bulk modulus is defined as a coefficient of proportionality between the hydrostatic pressure rate \(\dot{p}\) and the trace of the elastic strain rate \(\dot{e}\): \[ \dot{p} = -K\,{\mathrm{tr}{\left(\underline{\dot{\varepsilon}}^{\mathrm{el}}\right)}} \qquad(1)\] with \(e={\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}\)
The bulk modulus \(K\) is assumed dependant of the hydrostatic pressure as follows: \[ K{\left(p\right)} = {{\displaystyle \frac{\displaystyle 1+e_{0}}{\displaystyle \kappa}}}\,p \qquad(2)\] where the hydrostatic pressure is assumed strictly positive and where \(e_{0}\) and \(\kappa\) denotes repectively the initial void ratio and the unloading/reloading slope of the material.
Equation (2) is only valid if the the hydrostatic is strictly positive. To ensure this, a threshold \(p_{\min}\) is introduced.
Below this threshold, the behaviour is assumed linear, as follows: \[ \dot{p} = -K_{\min}\,{\mathrm{tr}{\left(\underline{\dot{\varepsilon}}^{\mathrm{el}}\right)}} \qquad(3)\] where \(K_{\min}\) is defined as: \[ K_{\min} = {{\displaystyle \frac{\displaystyle 1+e_{0}}{\displaystyle \kappa}}}\,p_{\min} \] which ensures that the hydrostatic pressure rate is continuous at \(p=p_{\min}\).
The hypoelastic formulation of the behaviour is cumbersome, in practice. However, Equations (1) and (3) can be integrated explicitly.
Assuming that the material coefficients \(e_{0}\), \(\kappa\) and \(p_{\min}\) are constants, Equation (3) can be integrated as follows: \[ p = -K_{\min}\,{\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}} \qquad(4)\] where we also assumed that the trace of the elastic strain is null for a null hydrostatic pressure
Expression (4) is valid if the pressure is below \(p_{\min}\), i.e. if the trace of the elastic strain satisfies: \[ {{\displaystyle \frac{\displaystyle 1+e_{0}}{\displaystyle \kappa}}}\,{\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}} \geq -1 \qquad(5)\]
If Condition (5) is not met, Equation (1), combined with Equation (2), can be integrated explicitly, which leads to this closed-form expression of the hydrostatic pressure: \[ p = p_{\min}\,\exp{\left(-1 - {{\displaystyle \frac{\displaystyle 1+e_{0}}{\displaystyle \kappa}}}\, {\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}\right)} \qquad(6)\]
Equations (4) and (6) show that a convex free energy \(\phi^{p}{\left({\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}\right)}\) can be defined such that: \[ p = -{\displaystyle \frac{\displaystyle \partial \phi^{p}}{\displaystyle \partial {\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}}} \]
Moreover, in the authors opinion, it makes senses to use Equations (4) and (6) as the starting point for a proper definition of the Cam-Clay non linear elastic behaviour as it allows to introduce dependencies of the material coefficients \(e_{0}\) and \(\kappa\) with respect to external state variables, such as the temperature.
The treatment of the deviatoric part of the Cam-Clay non linear elastic behaviour is given by the following linear relation between the rate of the deviatoric stress \(\underline{\dot{s}}\) and the rate of the deviatoric part of the elastic strain \(\underline{\dot{s}}^{el}\): \[ \underline{\dot{s}} = 2\,\mu\,\underline{\dot{s}}^{el} \qquad(7)\] with \(\underline{s}=\underline{\sigma}+p\,\underline{I}\) and \(\underline{s}^{el}=\underline{\varepsilon}^{\mathrm{el}}-{\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}\,\underline{I}\).
Two choices are made in the litterature:
If the shear modulus is assumed constant, Expression (9) can be integrated as follows: \[ \underline{s} = 2\,\mu\,\underline{s}^{el} \qquad(9)\]
With this assumption, the whole behaviour can be derived from a convex free energy.
Equation (9) could indeed be the starting point of a proper definition of the Cam-Clay non linear elastic behaviour. Moreover, this definition can be used if the shear modulus depends on external state variables surch as the temperature.
However, this choice may lead to unrealistic apparent Poisson ratio.
Using Expression (8) solve the issue of the unrealistic apparent Poisson ratio.
It however leads to major issue regarding thermodynamic requirements as the behaviour can not be derived from a free energy. As a consequence, the derivative \({\displaystyle \frac{\displaystyle \partial \sigma}{\displaystyle \partial \underline{\varepsilon}^{\mathrm{el}}}}\) is not symmetric.
While Expression (7) seems to be the most used in the litterature, the authors would also also highlight that there is no major reason to prefer using Equation (9) and Equation (7).
The user of the CamClay stress potential can select one of this two Equations using the incremental_deviatoric_part option. By default, Equation (7) is used.
In pratice, Equation (9) is treated incrementally as follows: \[ \Delta\,\underline{s}=2\,\mu{\left({\left.p\right|_{t+\Delta\,t}}\right)}\,\Delta\,\underline{\varepsilon}^{\mathrm{el}} \] and the stress tensor at the end of the time step is computed as: \[ {\left.\underline{\sigma}\right|_{t+\Delta\,t}}={\left.\underline{s}\right|_{t}}+2\,\mu{\left({\left.p\right|_{t+\Delta\,t}}\right)}\,\Delta\,\underline{s}^{el}-p\,\underline{I} \]
Note that even the deviatoric part is treated incrementally, the hydrostatic part is treated using (4) and (6).
The derivative \({\displaystyle \frac{\displaystyle \partial {\left.\underline{\sigma}\right|_{t+\Delta\,t}}}{\displaystyle \partial \Delta\,{\left.\underline{\varepsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}}}\) is thus given by: \[ {\displaystyle \frac{\displaystyle \partial {\left.\underline{\sigma}\right|_{t+\Delta\,t}}}{\displaystyle \partial {\left.\underline{\varepsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}}}= 2\,\mu{\left({\left.p\right|_{t+\Delta\,t}}\right)}\,\underline{\underline{\mathbf{K}}}+ 2\,{\displaystyle \frac{\displaystyle \partial \mu}{\displaystyle \partial {\left.{\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}\right|_{t+\Delta\,t}}}}\, {\left(\Delta\,\underline{s}^{el}\,\otimes\,\underline{I}\right)} -{\displaystyle \frac{\displaystyle \partial p}{\displaystyle \partial {\left.{\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}\right|_{t+\Delta\,t}}}}\,\underline{I}\otimes\underline{I} \] which is non symmetric du to the term propotionnal to \(\Delta\,\underline{s}^{el}\,\otimes\,\underline{I}\).
The derivative \({\displaystyle \frac{\displaystyle \partial {\left.\underline{\sigma}\right|_{t+\Delta\,t}}}{\displaystyle \partial {\left.\underline{\varepsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}}}\) is thus given by: \[ {\displaystyle \frac{\displaystyle \partial {\left.\underline{\sigma}\right|_{t+\Delta\,t}}}{\displaystyle \partial {\left.\underline{\varepsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}}}= 2\,\mu\,\underline{\underline{\mathbf{K}}}+ 2\,{\displaystyle \frac{\displaystyle \partial \mu}{\displaystyle \partial {\left.{\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}\right|_{t+\Delta\,t}}}}\, {\left({\left.\underline{s}^{el}\right|_{t+\Delta\,t}}\,\otimes\,\underline{I}\right)} -{\displaystyle \frac{\displaystyle \partial p}{\displaystyle \partial {\left.{\mathrm{tr}{\left(\underline{\varepsilon}^{\mathrm{el}}\right)}}\right|_{t+\Delta\,t}}}}\,\underline{I}\otimes\underline{I} \]
The following options are available:
unloading_reloading_slope: unloading/reloading slope \(\kappa\).initial_void_ratio: initial void ratio \(e0\).poisson_ratio: Poisson ratio of the material \(\nu\).shear_modulus: shear modulus of the material \(\mu\).pressure_threshold: threshold below which the behaviour is linear elastic. The value of this threshold must be. strictly positive.deduce_shear_modulus_from_poisson_ratio: boolean stating if the shear modulus shall be deduced from the Poisson ratio (Case 2). If this option is true, the Poisson ratio is automatically declared as a material property. Otherwise, the shear modulus is automatically declared as a material property.incremental_deviatoric_part: boolean stating if the deviatoric part of the stress/strain relationship shall be treated incrementally. This option is only valid if the shear modulus is deduced from the Poisson ratio. In this case, an incremental treatment given by Equation (7) is assumed by default. If false, Equation (9) is used.`The options poisson_ratio, shear_modulus and deduce_shear_modulus_from_poisson_ratio are mutually exclusive.
The incremental_deviatoric_part is not compatible with the shear_modulus option.
CamClayElasticity brick@Brick CamClayElasticity{
unloading_reloading_slope : 0.01,
initial_void_ratio : 0.5,
poisson_ratio : 0.3,
pressure_threshold : 1
};StandardElastoViscoPlasticity brick@Brick StandardElastoViscoPlasticity{
stress_potential : "CamClay" {
unloading_reloading_slope : 0.01,
initial_void_ratio : 0.5,
poisson_ratio : 0.3,
pressure_threshold : 1
}
};