/MAT/LAW109
Block Format Keyword Elastoplastic material with isotropic von Mises yield criterion with plastic strain rate and temperature depending nonlinear hardening.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW109/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  ν  
C_{p}  $\eta $  Tref  T0  
tab_ID_h  tab_ID_t  Xscale_h  Yscale_h  I_{smooth}  
tab_ID_ $\eta $  Xscale_ $\eta $ 
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) 

${\rho}_{i}$  Initial density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
E  Young’s modulus. (Real) 
$\left[\text{Pa}\right]$ 
ν  Poisson’s ratio. (Real) 

I_{smooth}  Choice of yield function interpolation versus strain rate.
(Integer) 

C_{p}  Specific heat. (Real) 
$\left[\frac{\text{J}}{\text{kg}\cdot \text{K}}\right]$

Tref  Reference temperature. Default = 293K (Real) 
$\left[\text{K}\right]$ 
T0  Initial temperature. Default = Tref (Real) 
$\left[\text{K}\right]$ 
$\eta $  TaylorQuinney coefficient (fraction of plastic work
converted to heat). Value between 0.0 and
1.0. (Real) 

tab_ID_ $\eta $  (Optional) Table identifier defining scale factor for
$\eta $
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_ $\eta $  Abscissa scale factor (strain rate) for
tab_ID_
$\eta $
. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Xscale_h  Abscissa scale factor (strain rate) for
tab_ID_h. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
Yscale_h  Scale factor for ordinate (stress) for
tab_ID_h. Default = 1.0 (Real) 
$\left[\text{Pa}\right]$ 
tab_ID_t  Table identifier for quasistatic yield stress depending on
effective plastic strain and temperature. (Integer Id) 
Examples
#12345678910
/UNIT/2275
unit_Mg_mm_s
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW109/18/2275
Aluminium
# Init. dens.
7.8E9
# 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
#12345678910
/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
#12345678910
/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
#12345678910
/TABLE/1/34
taylorquinney coef = f(strain rate, temp)
#DIMENSION
2
# FCT_ID X Scale_y
35 239 1.0
35 1000 0.9
#12345678910
/FUNCT/35
taylorquinney factor = f(strain.rate)
# X Y
0.000 0
0.002 0
0.04 1
1000000.0 1
#12345678910
/FUNCT/2
ALU Stressstrain
0.00000 310.0
9.3E04 330.8
1.1E03 334.5
2.1E03 339.9
2.6E03 340.9
3.3E03 342.3
6.1E03 344.7
7.8E03 346.0
9.1E03 347.1
1.0E02 348.7
1.2E02 350.7
1.4E02 352.6
1.6E02 354.0
1.8E02 356.5
2.0E02 358.7
3.0E02 369.0
3.5E02 373.5
1.0 410.0
#12345678910
#enddata
#12345678910
Comments
 Yield criterion using isotropic von Mises equivalent stress:
(1) $$\varphi ={\sigma}_{VM}{\sigma}_{y}$$  Yield stress hardening defined by tabulated input as:
(2) $${\sigma}_{y}={\mathrm{f}}_{h}\left({\epsilon}_{p},{\dot{\epsilon}}_{p}\right)\frac{{\mathrm{f}}_{t}\left({\epsilon}_{p},T\right)}{{\mathrm{f}}_{t}\left({\epsilon}_{p},{T}_{ref}\right)}$$Where, ${\mathrm{f}}_{h}\left({\epsilon}_{p},{\dot{\epsilon}}_{p}\right)$
 Function table of yield stresses depending on plastic strain and plastic strain rate.
 ${\mathrm{f}}_{t}\left({\epsilon}_{p},T\right)$
 Table ID of quasistatic yield function depending on plastic strain and temperature.
 ${T}_{ref}$
 Reference temperature. Corresponds to conditions during experimental tests.
 In adiabatic conditions, the temperature is updated using:
(3) $$T={T}_{0}+\frac{\eta \cdot {\mathrm{f}}_{\eta}\left({\epsilon}_{p},{\dot{\epsilon}}_{p},T\right)}{\rho Cp}$$Where, $\eta $ is the constant TaylorQuinney coefficient which may be modified by introducing scalar factor defined by function ${\mathrm{f}}_{\eta}\left({\epsilon}_{p},{\dot{\epsilon}}_{p},T\right)$ .
Otherwise, if /HEAT/MAT is present in the model, the temperature is imposed on all elements and cannot be updated using Equation 3.
Function ${\mathrm{f}}_{\eta}\left({\epsilon}_{p},{\dot{\epsilon}}_{p},T\right)$ may be one dimensional, twodimensional, or threedimensional, but the first abscissa is always strain rate and the second one may be only the temperature.