Visco-Elastic
The Maxwell model is the simpler visco-elastic model.
Theory
The strain increment \(d \boldsymbol{\varepsilon}\) is assumed to be additively decomposed into elastic \(d \boldsymbol{\varepsilon}^e\) and viscous \(d \boldsymbol{\varepsilon}^v\) parts. Dividing both sides by time increment \(d t\), the total strain rate becomes:
\(\dot{\boldsymbol{\varepsilon}}=\dot{\boldsymbol{\varepsilon}}^e+\dot{\boldsymbol{\varepsilon}}^v\)
The elastic response, reads:
\(\dot{\boldsymbol{\varepsilon}}^e=\dot{\boldsymbol{\varepsilon}}-\dot{\boldsymbol{\varepsilon}}^v = \mathbf{D}^{-1} \dot{\boldsymbol{\sigma}}\)
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 elastic matrix. The viscous response, reads:
\(\dot{\boldsymbol{\varepsilon}}^v = \mathbf{V} \boldsymbol{\sigma} = \left[ \begin{array}{cccccc} \frac{1}{3 \mu} & -\frac{1}{6 \mu} & -\frac{1}{6 \mu} & 0 & 0 & 0 \\ -\frac{1}{6 \mu} & \frac{1}{3 \mu} & -\frac{1}{6 \mu} & 0 & 0 & 0 \\ -\frac{1}{6 \mu} & -\frac{1}{6 \mu} & \frac{1}{3 \mu} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2 \mu} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2 \mu} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2 \mu} \end{array} \right] \boldsymbol{\sigma}\)
where \(\mu\) is the Maxwell parameter.
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 = 6
. -
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]
withNI_MAT_TYPE = 9
:
INDEX | DESCRIPTION | VALUE | UM |
---|---|---|---|
MAT_RPAR[1] | \(fact_{C_{M}}\) : multipication factor for \(C_{M}\) | > 0.0 |
- |
MAT_RPAR[2] | \(\nu\) : Poisson coefficient | > 0.0 and < 0.5 |
- |
MAT_RPAR[3] | - | 1.0 |
- |
MAT_RPAR[4] | - | 1.0 |
- |
MAT_RPAR[5] | - | 1.0 |
- |
MAT_RPAR[6] | \(c_{b}\) : grain compressibility | >= 0.0 |
\(stress^{-1}\) |
MAT_RPAR[7] | - | 0.0 |
- |
MAT_RPAR[8] | \(\mu\) : Maxwell parameter | > 0.0 |
\((time \cdot stress)^{-1}\) |
MAT_RPAR[9] | \(\theta\) : time integration parameter | 0.5 (Crank-Nicolson) or 1.0 (implicit Euler) |
- |