#### MAT_VISCO_PLASTIC

###### Material properties
*MAT_VISCO_PLASTIC
"Optional title"
mid, $\rho$, $E$, $\nu$, did, tid
$\sigma_0$, $Q_1$, $C_1$, $Q_2$, $C_2$, cid, $c_{dec}$, $\alpha$
$\beta$, $m$, $T_0$, $T_m$, $n$
##### Parameter definition
VariableDescription
mid Unique material identification number
$\rho$ Density
$E$ Young's modulus
$\nu$ Poisson's ratio
did Damage property command ID
tid Thermal property command ID
$\sigma_0$ Initial yield stress
$Q_1$ Voce hardening coefficient
$C_1$ Voce hardening coefficient
$Q_2$ Voce hardening coefficient
$C_2$ Voce hardening coefficient
cid ID of a FUNCTION or CURVE defining the viscosity of the material
$c_{dec}$ Viscous stress decay coefficient
$\alpha$ Non-linear elastic stifness coefficient
$\beta$ Plastic flow stress triaxiality factor
$m$ Thermal softening parameter
$T_0$ Thermal softening reference temperature
$T_m$ Thermal softening melting temperature
$n$ Non-linear elastic stiffness exponent
default: 2
##### Description

This is a non-linear visco-plastic constitutive model where the total stress $\boldsymbol{\sigma}$ is the sum of three terms:

$\boldsymbol{\sigma} = \boldsymbol{\sigma}^1 + \boldsymbol{\sigma}^2 + \boldsymbol{\sigma}^3$

$\boldsymbol{\sigma}^1$ is a non-linear viscous stress, $\boldsymbol{\sigma}^2$ is a non-linear elastic stress and $\boldsymbol{\sigma}^3$ is a linear elastic stress component. The rheological model is depicted below.

The non-linear viscous stress $\boldsymbol{\sigma}^1$ is defined as:

$\displaystyle{\boldsymbol{\sigma}^1 = f(\dot{\boldsymbol{\bar\varepsilon}}, \boldsymbol{\varepsilon}) \cdot \frac{\dot{\boldsymbol{\bar\varepsilon}}}{\vert\vert \dot{\boldsymbol{\bar\varepsilon}} \vert\vert}}$

where $f$ is a user defined FUNCTION or CURVE with ID cid. FUNCTION allows the viscosity to depend on both the effective geometric strain and the strain rate. If using a CURVE, the viscosity is a function of $\dot{\bar{\boldsymbol{\varepsilon}}}$ only. $\dot{\bar{\boldsymbol{\varepsilon}}}$ is a smeared out strain rate measure:

$\boldsymbol{\dot{\bar\varepsilon}}(t) = \displaystyle{\frac{1}{c_{dec}} \int_0^t} \dot{\boldsymbol{\varepsilon}}(\tau) \mathrm{e}^{-\tau/c_{dec}} \mathrm{d}\tau$

Note that $\dot{\bar{\boldsymbol{\varepsilon}}} = \dot{\boldsymbol{\varepsilon}}$ if $c_{dec}=0$. The non-linear elastic stress $\boldsymbol{\sigma}^2$ is defined to grow quadratically with the total deviatoric strain $\boldsymbol{\varepsilon}_{dev}$:

$\displaystyle{\boldsymbol{\sigma}^2 = 2 \alpha G \left( \displaystyle{\frac{2}{3}} \boldsymbol{\varepsilon}_{dev} : \boldsymbol{\varepsilon}_{dev} \right)^{n/2} \cdot \frac{\boldsymbol{\varepsilon}_{dev}}{\vert\vert \boldsymbol{\varepsilon}_{dev} \vert\vert} }$

$G$ is the linear shear modulus. The linear elastic stress component $\boldsymbol{\sigma}^3$ is defined as:

$\boldsymbol{\sigma}^3 = -p \mathbf{I} + 2 G \boldsymbol{\varepsilon}_{dev}^e$

$p$ is the hydrostatic pressure and $\boldsymbol{\varepsilon}_{dev}^e$ is the deviatoric part of the linear elastic strain tensor $\boldsymbol{\varepsilon}^e$ in the relation:

$\boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}^e + \boldsymbol{\varepsilon}^p$

where $\boldsymbol{\varepsilon}^p$ is a plastic strain tensor. The plasticity model is based on a von Mises effective stress definition and an iso-choric plastic flow law. The plastic flow stress is defined as:

$\sigma_y = \left[ \sigma_0 + \displaystyle{\sum_{i=1}^2} Q_i (1 - \mathrm{e}^{-C_i \varepsilon_{eff}^p}) \right] \cdot \left[ 1 - \frac{T-T_0}{T_m-T_0}^m \right] \cdot \left[1 + \frac{\beta p}{\sigma_0} \right]$

Note that $\beta \neq 0$ leads to a pressure dependent plastic flow stress. $\beta \neq 0$ and an iso-choric plastic flow makes the flow rule non-associated. The hydrostatic pressure $p$ is defined as:

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

where $K$ is the linear 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).

##### Example
###### Gelatin like material

This is a complete model of a rigid sphere impacting a cylinder of a gelatin like visco-elastic material. Plasticity is turned off by setting the initial yield stress to a very large value. The FUNCTION with ID 20 defines the dynamic viscosity as a function of the total effective strain rate "rate" and the total geometrical strain "egeo".

*PARAMETER
R = 2.17e-3
v0 = 90.0
t = 0.03
dens = 1000.0
E = 1.0e6
pr = 0.499
C = 0.1
cdec = 1.0e-5
sigy = 1.0e20
epsf = 0.7
*TIME
0.001
*COMPONENT_PIPE
"gelatin"
1, 1, 15, 20, 20, 0
0, 0, 0, 0, [%t], 0, 0, [0.5*%t]
*COMPONENT_SPHERE
"impactor"
2, 2, 6
0.0, [%t+%R], 0.0, [%R]
*CHANGE_P-ORDER
ALL, 0, 3
*MAT_VISCO_PLASTIC
1, [%dens], [%E], [%pr], 10
[%sigy], 0, 0, 0, 0, 20, [%cdec]
*PROP_DAMAGE_STRAIN
10, 2
0, [%epsf]
*FUNCTION
20
%C*rate*egeo^2
*MAT_RIGID
2, 7800.0
*PART
"gelatin"
1, 1, 0, 0, 50.0
"impactor"
2, 2, 0, 0, 50.0
*INITIAL_VELOCITY
P, 2, 0, [-%v0], 0
*CONTACT
ALL, 0, ALL, 0, 0, -1.0e12
2
*BC_SYMMETRY
Y, 0, 0, 0, 1.0e-6
*END