berveiller_zaoui

This behavior defines a model, proposed by Berveiller and Zaoui, as a general formulation of the self-consistent scheme [MS1]. This model is specified for an isotropic elasto-plastic intergranular accommodation for plastically-flowing polycrystals.

(583)\[\sigma_{ij} = \Sigma_{ij} + 2 \alpha \mu (1 - \beta) (E^p_{ij} - \varepsilon^p_{ij})\qquad{\rm where} \qquad \beta = \frac{2(4+5\nu)}{15(1-\nu)}\]

(584)\[\alpha = \frac{1+6\mu h (1+\nu)/(7-5\nu)}{ 1+ 2\mu h (13-5\nu)/15(1-\nu) + 8 \mu^2 h^2(1+\nu)/15(1-\nu)}\,\,\,{\rm and}\,\,\, h =\frac{3 E^p}{2 \Sigma}\]

where \(\Sigma_{ij}\) is the uniform applied stress, \(\sigma_{ij}\) is the uniform stress in one specified grain, \(E^p_{ij}\) is the equivalent macroscopic plastic strain and \(\varepsilon^p_{ij}\) is the uniform plastic strain in the same grain.

Syntax

***behavior berveiller_zaoui \(~\,\) **mu double \(~\,\) **nu double [ **simplified ] \(~\,\) **material  volume_fraction    name | **material_in_file <file> *file file_name \(~\,\) [ *integration method ] \(~\,\) [ *rotation ROTATION ] | [ *rotation_list <ROTATION> ] \(~\,\) [ *volume_fraction_file <file> ] \(~\,\) **material etc … ***return

**mu

is the Lame’s coefficient \(\mu\).

**nu

is the Poisson’s ratio \(\nu\).

**simplified

This option allows the use of an approximate expression of \(\alpha\), which is \(\alpha=1/(1+\mu h)\), instead of the equation given above Eq. (584).

Example

***behavior
  **mu  75000.
  **nu  0.3
  **simplified
  **material  1.000000e+00  grain1
    *file trac_bz.inp 2
    *rotation   -1.496760e+02  1.561819e+01  1.546760e+02
  **material  1.000000e+00  grain2
    *file trac_bz.inp 2
    *rotation   -1.506460e+02  3.386400e+01  1.556460e+02
  **material  1.000000e+00  grain3
    *file trac_bz.inp 2
    *rotation   -1.371380e+02  4.159170e+01  1.421380e+02
***return