/MAT/LAW109

Format

(1)	(2)	(3)	(5)	(7)	(10)
/MAT/LAW109/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$ $ρ_{i}$
`E`		`ν`
`C`_p		$η$ $η$	`Tref`	`T0`
`tab_ID_h`	`tab_ID_t`	`Xscale_h`	`Yscale_h`		`I`_smooth
`tab_ID_` $η$ $η$	`Xscale_` $η$ $η$

Definition

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	(Optional) Unit Identifier. (Integer, maximum 10 digits)
`mat_title`	Material title. (Character, maximum 100 characters)
$ρ_{i}$ $ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$
`E`	Young’s modulus. (Real)	$[Pa]$ $[Pa]$
`ν`	Poisson’s ratio. (Real)
`I`_smooth	Choice of yield function interpolation versus strain rate. = 1 (Default) Linear interpolation. = 2 Logarithmic interpolation (base 10). = 3 Logarithmic interpolation (base n). (Integer)
`C`_p	Specific heat. (Real)	$[\frac{J}{kg \cdot K}]$ $[\frac{J}{kg \cdot K}]$
`Tref`	Reference temperature. Default = 293K (Real)	$[K]$ $[K]$
`T0`	Initial temperature. Default = `Tref` (Real)	$[K]$ $[K]$
$η$ $η$	Taylor-Quinney coefficient (fraction of plastic work converted to heat). Value between 0.0 and 1.0. (Real)
`tab_ID_` $η$ $η$	(Optional) Table identifier defining scale factor for $η$ $η$ depending on strain rate, temperature, and plastic strain. Value between 0.0 and 1.0. (Integer Id)
`tab_ID_h`	Table identifier for yield stress depending on effective plastic strain and strain rate. (Integer)
`Xscale_` $η$ $η$	Abscissa scale factor (strain rate) for `tab_ID_` $η$ $η$ . Default = 1.0 (Real)	$[\frac{1}{s}]$ $[\frac{1}{s}]$
`Xscale_h`	Abscissa scale factor (strain rate) for `tab_ID_h`. Default = 1.0 (Real)	$[\frac{1}{s}]$ $[\frac{1}{s}]$
`Yscale_h`	Scale factor for ordinate (stress) for `tab_ID_h`. Default = 1.0 (Real)	$[Pa]$ $[Pa]$
`tab_ID_t`	Table identifier for quasi-static yield stress depending on effective plastic strain and temperature. (Integer Id)

▸Example (Aluminum)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/2275
unit_Mg_mm_s
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW109/18/2275
Aluminium
#        Init. dens.         
              7.8E-9
#                  E                  Nu
             70000.0                  .3         
#                 CP                 Eta                Tref                Tini
              0.45E9                0.95               293.0               293.0        
#  Tab_Yld  Tab_Temp              Xscale              Yscale                                 Ismooth
        25        26                 1.0                 1.0                                       1
#  tab_eta          xcsale_eta
        34                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/25
Yld Functions : plastic strain + strain rate dependency
#DIMENSION
         2
#   FCT_ID                             X                                                     Scale_y
         2                           0.0                                                        1.0
         2                      100000.0                                                        1.35
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/26
 Yld Functions (quasistatic): plastic strain + temperature dependency
#DIMENSION
         2
#   FCT_ID                             X                                                     Scale_y
         2                         293.0                                                        1.00
         2                        1000.0                                                        0.70
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/34
taylor-quinney coef = f(strain rate, temp)
#DIMENSION
         2
#   FCT_ID                             X                                                     Scale_y
        35                           239                                                         1.0
        35                          1000                                                         0.9
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/35
taylor-quinney factor = f(strain.rate)
#                  X                   Y
               0.000                   0                                                            
               0.002                   0                                                            
                0.04                   1                                                            
           1000000.0                   1                                                            
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/2
ALU Stress-strain
          0.00000            310.0
          9.3E-04            330.8
          1.1E-03            334.5
          2.1E-03            339.9
          2.6E-03            340.9
          3.3E-03            342.3
          6.1E-03            344.7
          7.8E-03            346.0
          9.1E-03            347.1
          1.0E-02            348.7
          1.2E-02            350.7 
          1.4E-02            352.6 
          1.6E-02            354.0 
          1.8E-02            356.5 
          2.0E-02            358.7 
          3.0E-02            369.0 
          3.5E-02            373.5 
          1.0                410.0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

Yield criterion using isotropic von Mises equivalent stress:(1)
$ϕ = σ_{V M} - σ_{y}$ $ϕ = σ_{V M} - σ_{y}$
Yield stress hardening defined by tabulated input as:(2)
$σ_{y} = f_{h} (ε_{p}, {\dot{ε}}_{p}) \frac{f_{t} (ε_{p}, T)}{f_{t} (ε_{p}, T_{r e f})}$ $σ_{y} = f_{h} (ε_{p}, {\dot{ε}}_{p}) \frac{f_{t} (ε_{p}, T)}{f_{t} (ε_{p}, T_{r e f})}$

Where,

$f_{h} (ε_{p}, {\dot{ε}}_{p})$ $f_{h} (ε_{p}, {\dot{ε}}_{p})$

Function table of yield stresses depending on plastic strain and plastic strain rate.

$f_{t} (ε_{p}, T)$ $f_{t} (ε_{p}, T)$

Table ID of quasi-static yield function depending on plastic strain and temperature.

$T_{r e f}$ $T_{r e f}$

Reference temperature. Corresponds to conditions during experimental tests.
In adiabatic conditions, the temperature is updated using:(3)
$T = T_{0} + \frac{η \cdot f_{η} (ε_{p}, {\dot{ε}}_{p}, T)}{ρ C p}$ $T = T_{0} + \frac{η \cdot f_{η} (ε_{p}, {\dot{ε}}_{p}, T)}{ρ C p}$

Where, $η$ $η$ is the constant Taylor-Quinney coefficient which may be modified by introducing scalar factor defined by function $f_{η} (ε_{p}, {\dot{ε}}_{p}, T)$ $f_{η} (ε_{p}, {\dot{ε}}_{p}, T)$ .

Otherwise, if /HEAT/MAT is present in the model, the temperature is imposed on all elements and cannot be updated using Equation 3.

Function $f_{η} (ε_{p}, {\dot{ε}}_{p}, T)$ $f_{η} (ε_{p}, {\dot{ε}}_{p}, T)$ may be one dimensional, two-dimensional, or three-dimensional, but the first abscissa is always strain rate and the second one may be only the temperature.