/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)	(3)	(5)	(7)
/MAT/LAW79/`mat_ID`/`unit_ID` or /MAT/JOHN_HOLM/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$	$ρ_{0}$
`G`
`a`	`b`	`m`	`n`
`c`	${\dot{ε}}_{0}$	$σ_{f \max}^{*}$
`T`	`HEL`	`P`_HEL
`D`₁	`D`₂
`K`₁	`K`₂	`K`₃	$β$

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)
$ρ_{i}$	Initial density (Real)	$[\frac{kg}{m^{3}}]$
$ρ_{0}$	Reference density used in E.O.S (equation of state). Default = $ρ_{0} = ρ_{i}$ (Real)	$[\frac{kg}{m^{3}}]$
`G`	Shear modulus (Real)	$[Pa]$
`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)
${\dot{ε}}_{0}$	Reference strain rate. Usually = 1 (Real)	$[\frac{1}{s}]$
$σ_{f \max}^{*}$	Maximum normalized fractured strength. Default = 10³⁰ (Real)
`T`	Maximum pressure tensile strength. Default = 10³⁰ (Real)	$[Pa]$
`HEL`	Hugoniot elastic limit. (Real)	$[Pa]$
`P`_HEL	Pressure at Hugoniot elastic limit. (Real)	$[Pa]$
`D`₁	Damage constant. 2 (Real)
`D`₂	Damage exponent. 2 (Real)
`K`₁	Bulk modulus. (Real)	$[Pa]$
`K`₂	Pressure coefficient. 3 (Real)	$[Pa]$
`K`₃	Pressure coefficient. 3 (Real)	$[Pa]$
$β$	Bulking pressure coefficient $0 < β < 1$ . (Real)

Input Example

	`B`₄C [²]	`Al`₂O₃ [¹]
$ρ_{0}$ $[\frac{kg}{m^{3}}]$	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
$σ_{f \max}^{*}$	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
$β$	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----|

Comments

The equation describing the normalized equivalent stress is:(1)
$σ^{*} = (1 - D) σ_{i}^{*} + D σ_{f}^{*}$

with the equivalent stress of the intact material:(2)
$σ_{i}^{*} = a {(P^{*} + T^{*})}^{n} (1 + c \ln \frac{\dot{ε}}{{\dot{ε}}_{0}})$

and the equivalent stress of the failed material:(3)
$σ_{f}^{*} = b {(P^{*})}^{m} (1 + c \ln \frac{\dot{ε}}{{\dot{ε}}_{0}}) < σ_{f \max}^{*}$

Stress are normalized to the stress at the Hugoniot elastic limit:(4)
$σ_{HEL} = \frac{3}{2} (HEL - P_{HEL})$

$σ^{*} = \frac{σ}{σ_{HEL}}$ and pressure are normalized to P_HEL:

$P^{*} = \frac{P}{P_{H E L}}$ and $T^{*} = \frac{T}{P_{H E L}}$
The accumulated damage is:(5)
$D = \frac{\sum Δ ε_{f}^{p}}{ε_{f}^{p}}$

where, the plastic strain to failure is:(6)
$ε_{f}^{p} = D_{1} {(P^{*} + T^{*})}^{D_{2}}$
The Equation of state is: (7)
$P = K_{1} μ + K_{2} μ^{2} + K_{3} μ^{3} + Δ P$

Where, the bulking pressure $Δ P$ is computed as a function of the elastic energy loss $Δ U$ converted into potential hydrostatic energy:(8)
$Δ P_{t + Δ t} = - K_{1} μ + \sqrt{{(K_{1} μ + Δ P_{t})}^{2} + 2 β K_{1} Δ U}$
Time history and animation output is available using these USRi variables:
- USR3: Damage D
- USR4: Bulking Pressure $Δ P$
- 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.