Hypo-Plastic
The hypo-plastic (also referred to hypo-elastic) model is a constitutive law characterized by a stress dependent vertical compressibility \(C_M(\sigma_z)\), in order to take into account the increase of stiffness of the material due to the increase of vertical stress. Furthermore the model can describe - in a simple but efficient fashion - the variation of stiffness due to loading-unloading cycles, experimented for example during reservoir exploitation.
Theory
The hypo constitutive model can be considered as an enhancement of the elastic model with the vertical compressibility \(C_M\) depending on the vertical stress \(\sigma_z\), hence
\(d\boldsymbol{\sigma} = C^{-1}_M(\sigma_z) \mathbf{D} d\boldsymbol{\varepsilon}\)
Therefore, the model is fully defined once the relation \(C^{-1}_M = C^{-1}_M(\sigma_z)\) is provided. Depending on the maximum experienced vertical stress \(\sigma_{z,max}\), two different conditions are assumed:
- Loading condition (I cycle): \(\sigma_z > \sigma_{z,max}\)
\(C^{I}_M = a \sigma_z^b\)
where \(a\) and \(b\) are constant parameters that can be either assigned or obtained by interpolation.
- Unloading-reloading condition (II cycle): \(\sigma_z < \sigma_{z,max}\)
\(C^{II}_M = \frac{M}{\left[(1+e_f)-M\log_{10}\left(\frac{\sigma_z}{\sigma_{z,max}}\right)\right] \ln(10)\sigma_z}\)
where \(M = \frac{C^I_{M,f} (1+e_f)\sigma_{z,max}\ln(10)}{C_r}\) with \(C^I_{M,f}\) and \(e_f\) respectively the value of \(C_{M}\) and void ratio at \(\sigma_{z,max}\), and \(C_r = C^{I}_{M,f} /C^{II}_{M,f}\).
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 = 2. -
The integer parameters
MAT_IPAR[NI_MAT_TYPE]withNI_MAT_TYPE = 1:
| INDEX | DESCRIPTION | VALUE | UM |
|---|---|---|---|
| MAT_IPAR[1] | index of the table_file in tab_list input file | >= 1 and < N_TAB |
- |
- The real parameters
MAT_RPAR[NR_MAT_TYPE]withNR_MAT_TYPE = 11:
| INDEX | DESCRIPTION | VALUE | UM |
|---|---|---|---|
| MAT_RPAR[1] | \(fact_{C_{M}}\) : multipication factor for \(C_{M}\) | > 0.0 |
- |
| MAT_RPAR[2] | \(\nu_{Z}\) : vertical Poisson coefficient | > 0.0 and < 0.5 |
- |
| MAT_RPAR[3] | \(\beta = E_{H}/E_{Z}\) | > 0.0 |
- |
| MAT_RPAR[4] | \(\gamma = \nu_{H}/\nu_{Z}\) | > 0.0 |
- |
| MAT_RPAR[5] | \(\theta = G_{H}/G_{Z}\) | > 0.0 |
- |
| MAT_RPAR[6] | \(c_{b}\) : grain compressibility | >= 0.0 |
\(stress^{-1}\) |
| MAT_RPAR[7] | \(C_{r}\) : ratio \(C^{I}_{M,f} /C^{II}_{M,f}\) | >= 1.0 |
- |
| MAT_RPAR[8] | \(limcr\) : tolerance at inversion point | >= 0.0 |
\(stress\) |
| MAT_RPAR[9] | \(\sigma_{z}^{lim}\) : \(\sigma_{z}\) limit below which linear elasticity is addressed | >= 0.0 |
\(stress\) |
| MAT_RPAR[10] | \(\delta \sigma_{z}^{tol}\) : \(\delta \sigma_{z}\) limit below which linear elasticity is addressed | >= 0.0 |
\(stress\) |
| MAT_RPAR[11] | - | 0.0 |
- |