/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
ρ i ρ 0            
G              
a b m n  
c ε ˙ 0 σ f max *        
T HEL PHEL        
D1 D2          
K1 K2 K3 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)

[ kg m 3 ]
ρ 0 Reference density used in E.O.S (equation of state).

Default = ρ 0 = ρ i (Real)

[ 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)

 
ε ˙ 0 Reference strain rate.

Usually = 1 (Real)

[ 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 ]
PHEL Pressure at Hugoniot elastic limit.

(Real)

[ Pa ]
D1 Damage constant. 2

(Real)

 
D2 Damage exponent. 2

(Real)

 
K1 Bulk modulus.

(Real)

[ Pa ]
K2 Pressure coefficient. 3

(Real)

[ Pa ]
K3 Pressure coefficient. 3

(Real)

[ Pa ]
β Bulking pressure coefficient 0 < β < 1 .

(Real)

 

Input Example

  B4C [2] Al2O3 [1]
ρ 0 [ 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
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
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

  1. 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 ε ˙ ε ˙ 0 )
    and the equivalent stress of the failed material:(3)
    σ f * = b ( P * ) m ( 1 + c ln ε ˙ ε ˙ 0 ) < σ f max *
    Stress are normalized to the stress at the Hugoniot elastic limit:(4)
    σ HEL = 3 2 ( HEL P HEL )

    σ * = σ σ HEL and pressure are normalized to PHEL:

    P * = P P H E L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaW baaSqabeaacaGGQaaaaOGaeyypa0ZaaSaaaeaacaWGqbaabaGaamiu amaaBaaaleaacaWGibGaamyraiaadYeaaeqaaaaaaaa@3D6D@ and T * = T P HEL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaW baaSqabeaacaGGQaaaaOGaeyypa0ZaaSaaaeaacaWGubaabaGaamiu amaaBaaaleaacaWGibGaamyraiaadYeaaeqaaaaaaaa@3D74@

  2. The accumulated damage is:(5)
    D = Δ ε f p ε f p
    where, the plastic strain to failure is:(6)
    ε f p = D 1 ( P * + T * ) D 2
  3. 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 + ( K 1 μ + Δ P t ) 2 + 2 β K 1 Δ U
  4. 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.