#### Command list

• Input handling
• Solution control and techniques
• Output
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• Material properties
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• SPH

### MAT_HSS

###### Material properties
Attention: This command is in the beta stage and the format may change over time.
*MAT_HSS
"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

VariableDescription
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
$\dot{\varepsilon}_s$ 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

#### Description

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 $\mathbf{\sigma}^e$ and a viscous stress $\mathbf{\sigma}^v$.

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

where

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

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

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

$p$ is the material pressure, $\mathbf{\varepsilon}_{dev}^e$ is the deviatoric part of the elastic strain tensor and $\dot{\mathbf{\varepsilon}}_{dev}$ 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.

$\displaystyle{ \bar\varepsilon_s = \frac{\varepsilon_s}{\dot\varepsilon_s} \int_0^t \dot{\varepsilon}_{eff}^p (\tau) \cdot \mathrm{e}^\frac{\dot{\varepsilon}_s \cdot (\tau-t)}{\varepsilon_s}\mathrm{d} \tau}$

#### Example

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