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] with NI_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] with NI_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) -