/MAT/LAW106 (JCOOK_ALM)

Block Format Keyword This law represents an isotropic elasto-plastic material using the Johnson-Cook material model. This model expresses material stress as a function of strain and temperature.

This law is not compatible with an EOS. The dependence between pressure and volumetric strain is linear. A built-in failure criterion, based on the maximum plastic strain is available. This material law is compatible with solid elements only.

Format

(1)	(3)	(4)	(5)	(6)	(7)	(9)
/MAT/LAW106/`mat_ID`/`unit_ID` or /MAT/JCOOK_ALM/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$ $ρ_{i}$	$ρ_{0}$ $ρ_{0}$
`E`	$ν$ $ν$		`fct_ID`₁	`fct_ID`₂	`fct_ID`₃
`a`	`b`		`n`		$ε_{p}^{m a x}$ $ε_{p}^{m a x}$	$σ_{\max}$ $σ_{\max}$
`P`_min		`N`_max	`Tol`
			`m`		`T`_melt	`T`_max
$ρ_{0} C_{p}$ $ρ_{0} C_{p}$					`T`_r

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)
$ρ_{i}$ $ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$
$ρ_{0}$ $ρ_{0}$	Reference density used in EOS (equation of state). Default = $ρ_{0} = ρ_{i}$ $ρ_{0} = ρ_{i}$ (Real)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$
`E`	If `fct_ID`₁ = 0: Young's modulus. If `fct_ID`₁ ≠ 0: Ordinate scale factor of `fct_ID`₁ and `fct_ID`₂. (Real)	$[Pa]$ $[Pa]$
$ν$ $ν$	If `fct_ID`₃ = 0: Poisson's ratio. If `fct_ID`₃ ≠ 0: Ordinate scale factor of `fct_ID`₃. (Real)
`fct_ID`₁	Function identifier defining Young’s modulus versus temperature when heating. (Integer)
`fct_ID`₂	Function identifier defining Young’s modulus versus temperature when cooling. (Integer)
`fct_ID`₃	Function identifier defining Poisson’s ratio versus temperature. (Integer)
`a`	Yield stress. (Real)	$[Pa]$ $[Pa]$
`b`	Plastic hardening parameter. (Real)	$[Pa]$ $[Pa]$
`n`	Plastic hardening exponent. Default = 1 (Real)
$ε_{p}^{m a x}$ $ε_{p}^{m a x}$	Failure plastic strain. Default = 10³⁰ (Real)
$σ_{\max}$ $σ_{\max}$	Maximum stress. Default = 10³⁰ (Real)	$[Pa]$ $[Pa]$
`P`_min	Pressure cutoff (< 0). Default = -10³⁰ (Real)	$[Pa]$ $[Pa]$
`N`_max	Maximum number of iterations to compute plastic strains. Default = 1 (Integer)
`Tol`	Tolerance. Default = 10^-7 (Real)
`m`	Temperature exponent. Default = 1.0 (Real)
`T`_melt	Melting temperature. = 0 No temperature effect. Default = 10³⁰ (Real)	$[K]$ $[K]$
`T`_max	For T > `T`_max: `m` = 1 is used. Default = 10³⁰ (Real)	$[K]$ $[K]$
$ρ_{0} C_{p}$ $ρ_{0} C_{p}$	Specific heat per unit volume. (Real)	$[\frac{J}{m^{3} \cdot K}]$ $[\frac{J}{m^{3} \cdot K}]$
`T`_r	Reference temperature. Default = 300K (Real)	$[K]$ $[K]$

▸Example (Metal)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW106/1/1
Metal
#              RHO_I
                8E-9                   0
#                  E                  Nu   fct_ID1   fct_ID2   fct_ID3
              200000                 0.3         4         5         6
#                  a                   b                   n             EPS_max            SIG_max0
                 400                 500                  .5                   0                   0
#               Pmin                NMAX                 TOL     
                   0                   0                   0
#                                                          m              T_melt               T_max
                                                           3                2500                3000
#              RhoCP                                                        Tref
                 3.5                                                         298
/HEAT/MAT/1
#                 T0             RHO0_CP                  AS                  BS     IFORM
                 298                 3.5                  20                   0         1
#                 T1                  AL                  BL               EFRAC
                2500                  20                   0                  .9
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/4
Young modulus factor versus temperature during heating
#                  X                   Y
                   0                   1
                 300                   1
                1500                  .1
                2000                  .1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/5
Young modulus factor versus temperature during cooling
#                  X                   Y
                   0                   1
                 300                   1
                1500                  .1
                2000                  .1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/6
Poisson's Ratio factor versus temperature
#                  X                   Y
                   0                   1
                 300                   1
                1500                 1.5
                2000                 1.5
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

In this model, the material behavior is elastic-plastic and the yield stress is calculated as:(1)
$σ = (a + b ε_{p}^{n}) (1 - {(T^{*})}^{m})$ $σ = (a + b ε_{p}^{n}) (1 - {(T^{*})}^{m})$

Where,(2)
$T^{*} = \frac{T - T_{r}}{T_{m e l t} - T_{r}}$ $T^{*} = \frac{T - T_{r}}{T_{m e l t} - T_{r}}$

Where,

$ε_{p}$ $ε_{p}$

Equivalent plastic strain

$T$ $T$

Temperature

$T_{r}$ $T_{r}$

Reference temperature

$T_{m e l t}$ $T_{m e l t}$

Melting temperature

The material behaves as a linear-elastic material when the equivalent stress is lower than the yield stress.

When /HEAT/MAT (with I_form =1) references this material model, the values of T_r and T_melt defined in this card will be overwritten by the corresponding T₀ and T_melt defined in /HEAT/MAT.

When the temperature is not initialized using /HEAT/MAT or /INITEMP, the reference temperature (T_r) is also the initial temperature.
The plastic yield stress should always be greater than zero. To model pure elastic behavior, the plastic yield stres,s a can be set to 10³⁰.
When $ε_{p}$ $ε_{p}$ reaches the value of $ε_{p}^{m a x}$ $ε_{p}^{m a x}$ (for tension, compression or shear), in one integration point, then the deviatoric stress of the corresponding integration point is permanently set to 0; however, the solid element is not deleted.
The plastic hardening exponent must be $n \leq 1$ $n \leq 1$ .
The hydrostatic pressure is linearly proportional to volumetric strain:(3)
$P = K μ$ $P = K μ$

Where,

$K = \frac{E}{3 (1 - 2 v)}$ $K = \frac{E}{3 (1 - 2 v)}$

Bulk modulus

$μ = \frac{ρ}{ρ_{0}} - 1$ $μ = \frac{ρ}{ρ_{0}} - 1$

Volumetric strain
This material can be used with the material options /HEAT/MAT and /VISC.