#### MAT_FORMING

###### Material properties
*MAT_FORMING
"Optional title"
mid, $\rho$, $E$, $\nu$, did, tid
cid, $\xi$, $a_0$, $a_1$
$\varepsilon_{1}$, $\varepsilon_{2}$, $\varepsilon_{3}$, $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$
##### Parameter definition
VariableDescription
mid Unique material identification number
$\rho$ Density
$E$ Young's modulus
$\nu$ Poisson's ratio
did Damage model ID
tid Thermal property command ID
cid ID of a CURVE or FUNCTION defining effective plastic flow stress versus plastic strain (equivalent measures)
$\xi$ Kinematical hardening parameter ranging from 0 to 1
default: 0 (pure iso-tropic hardening)
$a_0$ Plastic hardening parameter
$a_1$ Parameter controlling the shape of the yield surface
$\varepsilon_{1}$, $\varepsilon_{2}$, $\varepsilon_{3}$ Initial principal plastic strains
$\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ Initial back stress in principal strain directions
##### Description

A texture-based forming model developed at IMPETUS. The effective stress is defined as:

$\sigma_{eff} = \sqrt{\frac{3}{2} \left[ b_1 \hat{\sigma}_{11}^2 + b_2 \hat{\sigma}_{22}^2 + b_3 \hat{\sigma}_{33}^2 + 2b_0 (\hat{\sigma}_{12}^2 + \hat{\sigma}_{23}^2 + \hat{\sigma}_{31}^2) \right]}$

where:

$\hat{\boldsymbol{\sigma}} = \mathbf{Q} \left[ \boldsymbol{\sigma}_{dev} - \boldsymbol{\sigma}^* \right] \mathbf{Q}^t$

$\boldsymbol{\sigma}_{dev}$ is the deviatoric stress, $\boldsymbol{\sigma}^*$ is the back stress due to kinematical hardening and $\mathbf{Q}$ is a tensor that transforms the stress tensor to principal strain directions. $b_1$, $b_2$ and $b_3$ are parameters that control the difference in flow stress in different principal strain directions. This is motivated by crystallographic texture effects.

$b_i = 1 + a_1 \left( 1 - \frac{\vert \varepsilon_i \vert}{\hat\varepsilon_{geo}} \right) \mathrm{atan} (\hat\varepsilon_{geo})$

where $\hat\varepsilon_{geo}$ is an effective geometric strain measure:

$\displaystyle{ \hat\varepsilon_{geo} = \sqrt{\frac{25}{54} (\varepsilon_1^2 + \varepsilon_2^2 + \varepsilon_3^2)} }$

The definition of $\hat\varepsilon_{geo}$ ensures that the force displacement curve in uni-axial tension does not depend on the parameter $a_1$.

$b_0$ is a parameter that ensures a larger flow stress in the principal strain directions than in other loading directions. This is also motivated by crystallographic texture effects.

$b_0 = 1 + a_0 \hat\varepsilon_{geo}$

The degree of kinematical hardening is controlled by $\xi$, a parameter ranging from 0 to 1. $\xi=0$ results in pure iso-tropic hardening (growing yield surface) and $\xi=1$ in pure kinematical hardening (translating yield surface). The evolution of the back stress is:

$\dot{\boldsymbol{\sigma}}^* = \xi H \dot{\boldsymbol{\varepsilon}}^p$

and the radius of the yield surface grows according to:

$\dot{\sigma}_y = (1-\xi) H \dot{\varepsilon}^p$

$H$ is the tangential hardening and it is defined as:

$\displaystyle{ H = \frac{\mathrm{d}\sigma_y}{\mathrm{d}\varepsilon_{eff}^p}(\sigma_{eff}=\sigma_y) }$

The hydrostatic pressure $p$ is defined as:

$p = -K \varepsilon_v + 3K \alpha_T (T-T_{ref})$

where $K$ is the bulk modulus, $\varepsilon_v$ is the volumetric strain. $\alpha_T$ is the thermal expansion coefficient and $T_{ref}$ is the reference temperature (see PROP_THERMAL).