# /MAT/LAW109

Block Format Keyword Elasto-plastic 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
Cp $\eta$ Tref T0
tab_ID_h tab_ID_t Xscale_h Yscale_h     Ismooth
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)

Ismooth Choice of yield function interpolation versus strain rate.
= 1 (Default)
Linear interpolation.
= 2
Logarithmic interpolation (base 10).
= 3
Logarithmic interpolation (base n).

(Integer)

Cp 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)

$\left[\text{K}\right]$
$\eta$ Taylor-Quinney 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 quasi-static yield stress depending on effective plastic strain and temperature.

(Integer Id)

## Examples

#---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

1. Yield criterion using isotropic von Mises equivalent stress:(1)
$\varphi ={\sigma }_{VM}-{\sigma }_{y}$
2. Yield stress hardening defined by tabulated input as:(2)
${\sigma }_{y}={\mathrm{f}}_{h}\left({\epsilon }_{p},{\stackrel{˙}{\epsilon }}_{p}\right)\frac{{\mathrm{f}}_{t}\left({\epsilon }_{p},T\right)}{{\mathrm{f}}_{t}\left({\epsilon }_{p},{T}_{ref}\right)}$
Where,
${\mathrm{f}}_{h}\left({\epsilon }_{p},{\stackrel{˙}{\epsilon }}_{p}\right)$
Function table of yield stresses depending on plastic strain and plastic strain rate.
${\mathrm{f}}_{t}\left({\epsilon }_{p},T\right)$
Table ID of quasi-static yield function depending on plastic strain and temperature.
${T}_{ref}$
Reference temperature. Corresponds to conditions during experimental tests.
3. In adiabatic conditions, the temperature is updated using:(3)
$T={T}_{0}+\frac{\eta \cdot {\mathrm{f}}_{\eta }\left({\epsilon }_{p},{\stackrel{˙}{\epsilon }}_{p},T\right)}{\rho Cp}$

Where, $\eta$ is the constant Taylor-Quinney coefficient which may be modified by introducing scalar factor defined by function ${\mathrm{f}}_{\eta }\left({\epsilon }_{p},{\stackrel{˙}{\epsilon }}_{p},T\right)$ .

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

Function ${\mathrm{f}}_{\eta }\left({\epsilon }_{p},{\stackrel{˙}{\epsilon }}_{p},T\right)$ 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.