/MAT/LAW79 (JOHN_HOLM)
Block Format Keyword This material law describes the behavior of brittle materials, such as ceramics and glass. The implementation is the second JohnsonHolmquist model: JH2.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW79/mat_ID/unit_ID or /MAT/JOHN_HOLM/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  ${\rho}_{0}$  
G  
a  b  m  n  
c  ${\dot{\epsilon}}_{0}$  ${\sigma}_{f\mathrm{max}}^{*}$  F_{cut}  
T  HEL  P_{HEL}  
D_{1}  D_{2}  IDEL  ${\epsilon}_{p}^{max}$  
K_{1}  K_{2}  K_{3}  $\beta $ 
Definition
Field  Contents  SI Unit Example 

mat_ID  Material
identifier. (Integer, maximum 10 digits) 

unit_ID  Unit identifier. (Integer, maximum 10 digits) 

mat_title  Material
title. (Character, maximum 100 characters) 

${\rho}_{i}$  Initial
density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
${\rho}_{0}$  Reference density
used in E.O.S (equation of state). Default = ${\rho}_{0}={\rho}_{i}$ (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
G  Shear
modulus (Real) 
$\left[\text{Pa}\right]$ 
a  Intact normalized
strength constant. 1 (Real) 

b  Fractured
normalized strength constant. 1 (Real) 

m  Fractured strength
pressure exponent. 1 (Real) 

n  Intact strength
pressure exponent. 1 (Real) 

c  Strain rate coefficient.
(Real) 

${\dot{\epsilon}}_{0}$  Reference strain
rate. Usually = 1 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
${\sigma}_{f\mathrm{max}}^{*}$  Maximum normalized
fractured strength. Default = 10^{30} (Real) 

F_{cut}  Cutoff frequency
for strain rate filtering.
(Real) 
$\text{[Hz]}$ 
T  Maximum pressure
tensile strength. Default = 10^{30} (Real) 
$\left[\text{Pa}\right]$ 
HEL  Hugoniot elastic
limit. (Real) 
$\left[\text{Pa}\right]$ 
P_{HEL}  Pressure at
Hugoniot elastic limit. (Real) 
$\left[\text{Pa}\right]$ 
D_{1}  Damage constant.
2 (Real) 

D_{2}  Damage exponent.
2 (Real) 

IDEL  Element deletion flag.
(Integer) 

${\epsilon}_{p}^{max}$  Critical plastic strain for element deletion. Default = 10 ^{20} (Real) 

K_{1}  Bulk
modulus. (Real) 
$\left[\text{Pa}\right]$ 
K_{2}  Pressure
coefficient. 3 (Real) 
$\left[\text{Pa}\right]$ 
K_{3}  Pressure
coefficient. 3 (Real) 
$\left[\text{Pa}\right]$ 
$\beta $  Bulking pressure
coefficient
$0<\beta <1$
. (Real) 
Input Example
B_{4}C [^{2}]  Al_{2}O_{3} [^{1}]  

${\rho}_{0}$ $\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$  2510  3700 
G [GPA]  197  90 
a  0.927  0.93 
b  0.70  0.31 
m  0.85  0.6 
n  0.67  0.6 
c  0.005  0 
${\sigma}_{f\mathrm{max}}^{*}$  0.2   
T [GPA]  0.26  0.2 
HEL [GPA]  19.0  2.8 
P_{HEL} [GPA]  8.71  1.46 
D_{1}  0.001  0.005 
D_{2}  0.5  1 
K_{1} [GPA]  233  131 
K_{2} [GPA]  593  0 
K_{3} [GPA]  2800  0 
$\beta $  1  1 
Example (AL_{2}O_{3})
#RADIOSS STARTER
#12345678910
/UNIT/1
unit for mat
g mm ms
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW79/1/1
Al2O3
# RHO_I RHO_0
.0037 0
# G
90160
# a b m n
.93 0 0 .6
# c EPS0 SIGMA_FMAX
0 .001 1E30
# T HEL PHEL
200 2790 1460
# D1 D2 IDEL EPS_MAX
0 0 1
# K1 K2 K3 BETA
130950 0 0 1
#12345678910
#ENDDATA
/END
#12345678910
Example (B_{4}C)
/UNIT/1
units_for_example_B4C
Mg mm s
/MAT/LAW79/1/1
B4C
# RHO_I RHO_0
2.510E9 0
# G
197000
# a b m n
0.927 0.70 0.85 0.67
# c EPS0 SIGMA_FMAX FCUT
0.005 1.0 200.0 10000.0
# T HEL PHEL
260 19000 8710
# D1 D2 IDEL EPS_MAX
0.001 0.5 2 0.15
# K1 K2 K3 BETA
233000 593000 2800000 1
Comments
 The equation
describing the normalized equivalent stress is:$${\sigma}^{*}=\left(1D\right){\sigma}_{i}^{*}+D{\sigma}_{f}^{*}$$
with the equivalent stress of the intact material:
$${\sigma}_{i}^{*}=a{\left({P}^{*}+{T}^{*}\right)}^{n}\left(1+c\mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)$$and the equivalent stress of the failed material:
$${\sigma}_{f}^{*}=b{\left({P}^{*}\right)}^{m}\left(1+c\mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\right)<{\sigma}_{f\mathrm{max}}^{*}$$The stresses are normalized to the stress at the Hugoniot elastic limit:
$${\sigma}_{\mathit{HEL}}=\frac{3}{2}\left(\mathit{HEL}{P}_{\mathit{HEL}}\right)$$${\sigma}^{*}=\frac{\sigma}{{\sigma}_{\mathit{HEL}}}$ and pressure are normalized to P_{HEL}:
${P}^{*}=\frac{P}{{P}_{HEL}}$ and ${T}^{*}=\frac{T}{{P}_{HEL}}$
 If damage
parameters are not specified (
${D}_{1}={D}_{2}=0$
), plastic strain evolution is not computed
and an instant failure is obtained when the element behavior reaches the
elastic limit. Otherwise, if damage parameters are mentioned, the evolution
of plastic strain is computed and the accumulated damage
is:$$D=\frac{{\displaystyle \sum \mathrm{\text{\Delta}}{\epsilon}_{f}^{p}}}{{\epsilon}_{f}^{p}}$$
where, the plastic strain to failure is:
$${\epsilon}_{f}^{p}={D}_{1}{\left({P}^{*}+{T}^{*}\right)}^{{D}_{2}}$$The maximum pressure tensile strength is decreased during damage as follows:
$${P}^{*}=(1D){T}^{*}$$This equation is not used when IDEL = 1.
 The equation of
state is: $$\left\{\begin{array}{l}P={K}_{1}\mu \\ P={K}_{1}\mu +{K}_{2}{\mu}^{2}+{K}_{3}{\mu}^{3}\end{array}\right.\begin{array}{cc}\text{in}& \text{tension}\\ \text{in}& \text{compression}\end{array}$$
Where,
$$\mu =\frac{\rho}{{\rho}_{0}}1$$When damage starts, a bulking pressure increment $\text{\Delta}P$ is computed as a function of the elastic energy loss $\text{\Delta}U$ converted into potential hydrostatic energy:
$$\mathrm{\text{\Delta}}{P}_{t+\mathrm{\text{\Delta}}t}={K}_{1}\mu +\sqrt{{\left({K}_{1}\mu +\mathrm{\text{\Delta}}{P}_{t}\right)}^{2}+2\beta {K}_{1}\mathrm{\text{\Delta}}U}$$Where, $\text{\Delta}U=U\left(D\right)U\left({D}_{n+1}\right)$ with $U\left(D\right)=\frac{\sigma}{6G}$ .
This increment is then added to the equation of state:
$$\text{\Delta}{P}_{t+\text{\Delta}t}={P}_{t}+\text{\Delta}{P}_{t+\text{\Delta}t}$$  Time history and
animation output is available using these USRi variables:
 USR1: Bulking Pressure $\text{\Delta}P$
 USR2: Old Yield Stress
 Strain rate filtering can be used and activated when a cutoff frequency F_{cut} for filtering is defined.