Material properties
"Optional title"
mid, $\rho$, $E$, $\nu$, did, tid
$A$, $B$, $n$, $c$, $c_{dec}$, $s$, $m$
$\dot{\varepsilon}_s$, ${\varepsilon}_s$, $c_s$, ${\eta}_s$, $T^{max}_s$
$c_r$, $\dot{\varepsilon}_r$
Parameter definition
mid Unique material identification number
$\rho$ Density
$E$ Young's modulus, constant or function of temperature
options: constant, fcn
$\nu$ Poisson's ratio, constant or function of temperature
options: constant, fcn
did Damage property command ID
tid Thermal property command ID
$A$ Yield stress, constant or function of temperature
options: constant, fcn
$B$ Hardening parameter, constant or function of temperature
options: constant, fcn
$n$ Hardening exponent, constant or function of temperature
options: constant, fcn
$c$ Viscosity or ID of FUNCTION or CURVE with viscous stress vs strain rate
$c_{dec}$ Viscous decay coefficient
$s$ Strength differential parameter
$m$ Hosford yield surface exponent (must be an even number)
$\dot{\varepsilon}_s$ Threshold strain rate for dynamic softening
${\varepsilon}_s$ Minimum plastic strain at onset of dynamic softening process
$c_s$ Exponent controlling rate of dynamic softening process
${\eta}_s$ Dynamic softening factor
default: no softening
$T^{max}_s$ Temperature cap for dynamic softening
$c_r$ Strain rate hardening exponent
$\dot{\varepsilon}_r$ Reference rate for strain rate hardening
default: 1

This is a material model for high strength steels and hard metals in ballistic applications. The total stress is the sum of an elastic stress $\boldsymbol{\sigma}^e$ and a viscous stress $\boldsymbol{\sigma}^v$.

$\displaystyle{ \boldsymbol{\sigma} = \boldsymbol{\sigma}^e + \boldsymbol{\sigma}^v }$


$\displaystyle{ \boldsymbol{\sigma}^e = -p {\bf{I}} + G \boldsymbol{\varepsilon}_{dev}^e }$
$\displaystyle{ \boldsymbol{\sigma}^v = \frac{1}{c_{dec}} \int_0^t g(\vert \dot{\boldsymbol{\varepsilon}}_{dev} \vert) \frac{\dot{\boldsymbol{\varepsilon}}_{dev}} {\vert \dot{\boldsymbol{\varepsilon}}_{dev} \vert} \mathrm{e}^{\frac{\tau-t}{c_{dec}}} \mathrm{d}\tau }$

$g$ is either a FUNCTION of viscous stress versus strain rate or, if defined through a constant $c$:

$g(\vert \dot{\boldsymbol{\varepsilon}}_{dev} \vert) = c \cdot \vert \dot{\boldsymbol{\varepsilon}}_{dev} \vert $

$p$ is the material pressure, $\boldsymbol{\varepsilon}_{dev}^e$ is the deviatoric part of the elastic strain tensor and $\dot{\boldsymbol{\varepsilon}}_{dev}$ is the total deviatoric strain rate. $G$ is the shear modulus.

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

$K$ is the bulk modulus, $\alpha$ is the thermal heat expansion coefficient and $T_{ref}$ is the reference temperature (defined in PROP_THERMAL). Note that the elastic properties can either be defined as constants or as functions of temperature. Yielding occurs when the effective elastic stress $\sigma_{eff}^e$ reaches the yield stress $\sigma_y$.

$\displaystyle{ \sigma_{eff}^e = \left[ \frac{1}{2}\left( \sigma_1^e - \sigma_2^e \right)^m + \frac{1}{2}\left( \sigma_2^e - \sigma_3^e \right)^m + \frac{1}{2}\left( \sigma_3^e - \sigma_1^e \right)^m \right]^{1/m} }$
$\displaystyle{ \sigma_y = \left[ \left( A(T) + B(T)\left(\varepsilon_{eff}^p\right)^{n(T)} \right) \cdot \left( 1 + s \right) + 3s \cdot p \right] \cdot \left[ 1 + \frac{\dot{\varepsilon}_{eff}^p}{\dot{\varepsilon}_r} \right]^{c_r} \cdot h(\bar\varepsilon_s) }$

In the yield stress expression $h(\bar\varepsilon_s)$ is a material softening function. Its purpose is to capture the effects of dynamic recrystallization or the formation of adiabatic shear bands.

$\displaystyle{ h(\bar\varepsilon_s) = \left\{ \begin{array}{ccc} 1 & : & \bar\varepsilon_s \leq \varepsilon_s \\ 1 - \eta_s\cdot(1-\mathrm{e}^{-c_s\cdot(\bar\varepsilon_s-\varepsilon_s)}) & : & \bar\varepsilon_s \gt \varepsilon_s \end{array} \right.}$

$\bar\varepsilon_s$ is a strain measure that grows faster at high effective plastic strain rates. It will never reach $\varepsilon_s$ (limit for onset of the softening process) if the plastic strain rate is below $\dot\varepsilon_s$. The curves below show how $\bar\varepsilon_s$ and the softening factor $h$ develop at different strain rates for a given set of input parameters.

$\displaystyle{ \bar\varepsilon_s = \int_0^t \dot{\varepsilon}_{eff}^p (\tau) \cdot \mathrm{e}^\frac{\dot{\varepsilon}_s \cdot (\tau-t)}{\varepsilon_s}\mathrm{d} \tau}$
Dynamic softening at different plastic strain rates.
Dynamic softening at different plastic strain rates.
A material that accounts for thermal softening

A simple example showing how certain parameters can be defined as functions of temperature. Note that this example does not represent a real alloy.

dens = 7800.0, "density"
nu = 0.3, "Poisson's ratio"
cdec = 1.0e-7, "viscous decay coefficient"
s = 0.03, "strength differential parameter"
m = 8, "Hosford yield surface exponent"
edot_s = 180, "threshold strain rate for softening"
eps_s = 0.22, "threshold strain for softening"
c_s = 5.0, "material softening exponent"
eta_s = 0.8, "material softening parameter"
Wc = 1.0e9, "ductile failure parameter"
alpha = 1.2e-5, "thermal expansion parameter"
Cp = 450.0, "heat capacity"
lambda = 35.0, "thermal conductivity"
k = 0.9, "Taylor-Quinney coefficient"
Tref = 25, "reference temperature"
1, [%dens], fcn(101), 0.3, 1, 1
fcn(102), fcn(103), 0.3, fcn(104), [%cdec], [%s], [%m]
[%edot_s], [%eps_s], [%c_s], [%eta_s]
"Young's modulus versus temperature"
0.0, 210.0e9
700.0, 150.0e9
"A versus temperature"
0.0, 1200.0e6
200.0, 1200.0e6
700.0, 150.0e6
"B versus temperature"
0.0, 1000.0e6
200.0, 900.0e6
700.0, 150.0e6
"Viscous stress versus strain rate"
0.0, 0.0
10.0, 0.0
100.0, 10.0e6
1000.0, 50.0e6
2000.0, 50.0e6
1, [%alpha], [%Cp], [%lambda], [%k], [%Tref]