/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)	(6)	(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`	${\dot{ε}}_{0}$	$σ_{f \max}^{*}$		`F`_cut
`T`	`HEL`	`P`_HEL
`D`₁	`D`₂		`IDEL`	$ε_{p}^{m a x}$
`K`₁	`K`₂	`K`₃		$β$

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)
$ρ_{i}$	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)
`F`_cut	Cutoff frequency for strain rate filtering. = 0 No strain rate filtering. (Real)	$[Hz]$
`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)
`IDEL`	Element deletion flag. = 0 (Default) No element deletion. = 1 Tensile failure when $P^{} + T^{} < 0$ . = 2 Failure when critical plastic strain is reached $ε_{p} > ε_{p}^{\max}$ . = 3 Failure when $D = 1$ . (Integer)
$ε_{p}^{m a x}$	Critical plastic strain for element deletion. Default = 10 ²⁰ (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                IDEL             EPS_MAX
                   0                   0                   1
#                 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----|

Example (B₄C)

/UNIT/1
units_for_example_B4C
                  Mg                  mm                   s                  
/MAT/LAW79/1/1
B4C
#              RHO_I               RHO_0              
            2.510E-9                   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:
$σ^{*} = (1 - D) σ_{i}^{*} + D σ_{f}^{*}$
with the equivalent stress of the intact material:
$σ_{i}^{*} = a {(P^{*} + T^{*})}^{n} (1 + c \ln \frac{\dot{ε}}{{\dot{ε}}_{0}})$
and the equivalent stress of the failed material:
$σ_{f}^{*} = b {(P^{*})}^{m} (1 + c \ln \frac{\dot{ε}}{{\dot{ε}}_{0}}) < σ_{f \max}^{*}$
The stresses are normalized to the stress at the Hugoniot elastic limit:
$σ_{HEL} = \frac{3}{2} (HEL - P_{HEL})$
$σ^{*} = \frac{σ}{σ_{HEL}}$ and pressure are normalized to P_HEL:
$P^{*} = \frac{P}{P_{H E L}}$ and $T^{*} = \frac{T}{P_{H E L}}$
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:
$D = \frac{\sum Δ ε_{f}^{p}}{ε_{f}^{p}}$
where, the plastic strain to failure is:
$ε_{f}^{p} = D_{1} {(P^{*} + T^{*})}^{D_{2}}$
The maximum pressure tensile strength is decreased during damage as follows:
$P^{*} = - (1 - D) T^{*}$
This equation is not used when IDEL = 1.
The equation of state is:
$\{\begin{cases} P = K_{1} μ \\ P = K_{1} μ + K_{2} μ^{2} + K_{3} μ^{3} \end{cases} \begin{matrix} in & tension \\ in & compression \end{matrix}$
Where,
$μ = \frac{ρ}{ρ_{0}} - 1$
When damage starts, a bulking pressure increment $Δ P$ is computed as a function of the elastic energy loss $Δ U$ converted into potential hydrostatic energy:
$Δ P_{t + Δ t} = - K_{1} μ + \sqrt{{(K_{1} μ + Δ P_{t})}^{2} + 2 β K_{1} Δ U}$
Where, $Δ U = U (D) - U (D_{n + 1})$ with $U (D) = \frac{σ}{6 G}$ .
This increment is then added to the equation of state:
$Δ P_{t + Δ t} = P_{t} + Δ P_{t + Δ t}$
Time history and animation output is available using these USRi variables:
- USR1: Bulking Pressure $Δ P$
- USR2: Old Yield Stress
Strain rate filtering can be used and activated when a cutoff frequency F_cut for filtering is defined.

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.