# /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 Johnson-Holmquist model: JH-2.

## 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 ${\stackrel{˙}{\epsilon }}_{0}$ ${\sigma }_{f\mathrm{max}}^{*}$
T HEL PHEL
D1 D2
K1 K2 K3 $\beta$

## Definitions

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)

$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$
${\rho }_{0}$ Reference density used in E.O.S (equation of state).

Default = ${\rho }_{0}={\rho }_{i}$ (Real)

$\left[\frac{\text{kg}}{{\text{m}}^{3}}\right]$
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.
= 0 (Default)
No strain rate effect.

(Real)

${\stackrel{˙}{\epsilon }}_{0}$ Reference strain rate.

Usually = 1 (Real)

$\left[\frac{\text{1}}{\text{s}}\right]$
${\sigma }_{f\mathrm{max}}^{*}$ Maximum normalized fractured strength.

Default = 1030 (Real)

T Maximum pressure tensile strength.

Default = 1030 (Real)

$\left[\text{Pa}\right]$
HEL Hugoniot elastic limit.

(Real)

$\left[\text{Pa}\right]$
PHEL Pressure at Hugoniot elastic limit.

(Real)

$\left[\text{Pa}\right]$
D1 Damage constant. 2

(Real)

D2 Damage exponent. 2

(Real)

K1 Bulk modulus.

(Real)

$\left[\text{Pa}\right]$
K2 Pressure coefficient. 3

(Real)

$\left[\text{Pa}\right]$
K3 Pressure coefficient. 3

(Real)

$\left[\text{Pa}\right]$
$\beta$ Bulking pressure coefficient $0<\beta <1$ .

(Real)

## Input Example

B4C [2] Al2O3 [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
PHEL [GPA] 8.71 1.46
D1 0.001 0.005
D2 0.5 1
K1 [GPA] 233 131
K2 [GPA] -593 0
K3 [GPA] 2800 0
$\beta$ 1 1

## Example (AL2O3)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/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               1E-30
#                  T                 HEL                PHEL
200                2790                1460
#                 D1                  D2
0                   0
#                 K1                  K2                  K3                BETA
130950                   0                   0                   1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

1. The equation describing the normalized equivalent stress is:(1)
${\sigma }^{*}=\left(1-D\right){\sigma }_{i}^{*}+D{\sigma }_{f}^{*}$
with the equivalent stress of the intact material:(2)
${\sigma }_{i}^{*}=a{\left({P}^{*}+{T}^{*}\right)}^{n}\left(1+c\mathrm{ln}\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)$
and the equivalent stress of the failed material:(3)
${\sigma }_{f}^{*}=b{\left({P}^{*}\right)}^{m}\left(1+c\mathrm{ln}\frac{\stackrel{˙}{\epsilon }}{{\stackrel{˙}{\epsilon }}_{0}}\right)<{\sigma }_{f\mathrm{max}}^{*}$
Stress are normalized to the stress at the Hugoniot elastic limit:(4)
${\sigma }_{\mathit{HEL}}=\frac{3}{2}\left(\mathit{HEL}-{P}_{\mathit{HEL}}\right)$

${\sigma }^{*}=\frac{\sigma }{{\sigma }_{\mathit{HEL}}}$ and pressure are normalized to PHEL:

${P}^{*}=\frac{P}{{P}_{HEL}}$ and ${T}^{*}=\frac{T}{{P}_{HEL}}$

2. The accumulated damage is:(5)
$D=\frac{\sum \mathrm{\text{Δ}}{\epsilon }_{f}^{p}}{{\epsilon }_{f}^{p}}$
where, the plastic strain to failure is:(6)
${\epsilon }_{f}^{p}={D}_{1}{\left({P}^{*}+{T}^{*}\right)}^{{D}_{2}}$
3. The Equation of state is: (7)
$P={K}_{1}\mu +{K}_{2}{\mu }^{2}+{K}_{3}{\mu }^{3}+\mathrm{\text{Δ}}P$
Where, the bulking pressure $\text{Δ}P$ is computed as a function of the elastic energy loss $\text{Δ}U$ converted into potential hydrostatic energy:(8)
$\mathrm{\text{Δ}}{P}_{t+\mathrm{\text{Δ}}t}=-{K}_{1}\mu +\sqrt{{\left({K}_{1}\mu +\mathrm{\text{Δ}}{P}_{t}\right)}^{2}+2\beta {K}_{1}\mathrm{\text{Δ}}U}$
4. Time history and animation output is available using these USRi variables:
• USR3: Damage D
• USR4: Bulking Pressure $\text{Δ}P$
• USR5: Yield Stress
1 An improved computational constitutive model for brittle materials, G.R. Johnson, T.J. Holmquist, American Institute of Physics, 1994.
2 Response of boron carbide subjected to large strains, high strain rates, and high pressures G.R. Johnson, T.J. Holmquist, Journal of Applied Physics, Volume 85, #12, June 1999.