beta_multimat

This method considers piecewise constant eigenstrain in each material phase and proposes a localization rule for each material sub-volume [M21]. The total strain tensor in each phase is computed according to:

(585)\[\Delta \ten{\epsilon}_i = \Delta \ten{\epsilon}^p_i + \Delta \ten{E} - \Delta \ten{E}^p + \left(\tenf L_i\right) \left(\Delta \ten{B}-\Delta \ten{\beta}_i\right)\]

where \(\beta_i\) is the eigenstrain in the sub-volume \(i\) and \(\Delta \ten B\) is deduced from the following equation:

(586)\[\Delta \ten{B} = \left\langle \tenf L_i \right\rangle^{-1} \left\langle \left(\tenf L\,c_i\right):\Delta \ten\beta_i \right\rangle\]

and \(<u>\) denotes the volume average of u.

Syntax

***behavior beta_multimat \(~\,\) **material  volume_fraction name | **material_in_file <file> \(~\,~\,\) *file file_name \(~\,\) [ *integration method ] \(~\,\) [ *rotation ROTATION ] | [ *rotation_list <ROTATION> ] \(~\,\) [ *volume_fraction_file <file> ] \(~\,\) **material etc … \(~\,\) **beta name <BETA_TFA> \(~\,\) **localization name \(~\,~\,~\,\) <LOCALIZATION2> ***return

**localization

specify localization rule for calculating the strain localisation tensor \(\tenf L\).

**beta

define the evolution law for the \(\beta\) internal variable in each sub-volume.

The localization rule and the evolution law must be defined for each volume. If the different sub-volumes have the same localization rule or the same evolution rule, it is possible to use “all” instead of specifying the names of sub-volumes after **beta and **localization. In this case, **beta and **localization must be defined before behaviors.

Example

***behavior beta_multimat
 **beta all delta D 300. delta 0.3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this example, both phases austen and ferrit have the same evolution law for $\beta$
variable (**beta). Thus, the option "all" is used and the **beta is defined before
behaviors (**material).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 **material .7 austen
  *file bugat1.austenite
  *rotation x3  -0.166667 0.649830 0.741582
            x1  -0.074915 0.741582 -0.666667
 **material .3 ferrit
  *file bugat1.ferrite
 **localization austen kroner poisson 0.3 ratio 1.
 **localization ferrit kroner poisson 0.3 ratio 1.
***return

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
beta_test1.mat
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

***behavior beta_multimat
 **localization all voigt mu 10100.
 **material .5 gr1
  *file beta_test1.mat 2
  *rotation -149.060000 18.010000 164.060000
 **beta      gr1 delta D 300.0 delta 0.3
**material .5 gr2
  *file beta_test1.mat 2
  *rotation -.060000 18.010000 164.060000
 **beta         gr2 delta D 300.0 delta 0.3
***return