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

Unique material identification number
Young's modulus, constant or function of temperature
options: constant, fcn
Poisson's ratio, constant or function of temperature
options: constant, fcn
Damage property command ID
Thermal property command ID
Yield stress, constant or function of temperature
options: constant, fcn
Hardening parameter, constant or function of temperature
options: constant, fcn
Hardening exponent, constant or function of temperature
options: constant, fcn
Viscosity or ID of FUNCTION or CURVE with viscous stress vs strain rate
Viscous decay coefficient
Strength differential parameter
Hosford yield surface exponent (must be an even number)
Threshold strain rate for dynamic softening
Minimum plastic strain at onset of dynamic softening process
Exponent controlling rate of dynamic softening process
Dynamic softening factor
default: no softening
Temperature cap for dynamic softening
Strain rate hardening exponent
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.

*UNIT_SYSTEM SI *PARAMETER 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" *MAT_HSS "Steel" 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] *CURVE "Young's modulus versus temperature" 101 0.0, 210.0e9 700.0, 150.0e9 *CURVE "A versus temperature" 102 0.0, 1200.0e6 200.0, 1200.0e6 700.0, 150.0e6 *CURVE "B versus temperature" 103 0.0, 1000.0e6 200.0, 900.0e6 700.0, 150.0e6 *CURVE "Viscous stress versus strain rate" 104 0.0, 0.0 10.0, 0.0 100.0, 10.0e6 1000.0, 50.0e6 2000.0, 50.0e6 *PROP_DAMAGE_CL 1 [%Wc] *PROP_THERMAL 1, [%alpha], [%Cp], [%lambda], [%k], [%Tref] *END