/MAT/LAW74

Block Format Keyword This law describes the Thermal Hill orthotropic 3D material and is applicable only to solid elements. The yield stress may depend on strain rate, or on both strain rate and temperature.

Format

(1)	(2)	(3)	(5)	(7)	(9)
/MAT/LAW74/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$ $ρ_{i}$
`E`		$υ$ $υ$	$ε_{p}^{\max}$ $ε_{p}^{\max}$	$ε_{t}$ $ε_{t}$	$ε_{m}$ $ε_{m}$
`fct_ID`_E		`E`_inf	`C`_E
	`F`_smooth	`C`_hard	`F`_cut
$σ_{11}^{y}$ $σ_{11}^{y}$		$σ_{22}^{y}$ $σ_{22}^{y}$	$σ_{33}^{y}$ $σ_{33}^{y}$
$σ_{12}^{y}$ $σ_{12}^{y}$		$σ_{23}^{y}$ $σ_{23}^{y}$	$σ_{31}^{y}$ $σ_{31}^{y}$
`Tab_ID`		$σ_{scale}$ $σ_{scale}$	${\dot{ε}}_{scale}$ ${\dot{ε}}_{scale}$
`T`_i		$ρ_{0} C_{p}$ $ρ_{0} C_{p}$

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)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$
`E`	Initial Young's modulus. (Real)	$[Pa]$ $[Pa]$
$υ$ $υ$	Poisson's ratio. (Real)
`fct_ID`_E	Function identifier for the scale factor of Young's modulus, when Young's modulus is function of the plastic strain. = 0(Default) In this case the evolution of Young's depends on `E`_inf and `C`_E. (Integer)
`E`_inf	Saturated Young's modulus for infinitive plastic strain. (Real)	$[Pa]$ $[Pa]$
`C`_E	Parameter for Young's modulus evolution. (Real)
$ε_{p}^{\max}$ $ε_{p}^{\max}$	Failure plastic strain. Default = 10³⁰ (Real)
$ε_{t}$ $ε_{t}$	Tensile failure strain at which stress starts to reduce. Default = 1.0 × 10³⁰ (Real)
$ε_{m}$ $ε_{m}$	Maximum tensile failure strain at which the stress in element is set to zero. Default = 2.0 × 10³⁰ (Real)
`F`_smooth	Smooth strain rate option flag. = 0 (Default) No strain rate smoothing. = 1 Strain rate smoothing active. (Integer)
`C`_hard	Hardening coefficient. = 0 Hardening is full isotropic model. = 1 Hardening uses the kinematic Prager-Ziegler model. = between 0 and 1 Hardening is interpolated between the two models. (Real)
`F`_cut	Cutoff frequency for strain rate filtering. Default = 1.0 × 10³⁰ (Real)	$[Hz]$ $[Hz]$
$σ_{11}^{y}$ $σ_{11}^{y}$	Yield in direction 1. (Real)	$[Pa]$ $[Pa]$
$σ_{22}^{y}$ $σ_{22}^{y}$	Yield in direction 2. (Real)	$[Pa]$ $[Pa]$
$σ_{33}^{y}$ $σ_{33}^{y}$	Yield in direction 3. (Real)	$[Pa]$ $[Pa]$
$σ_{12}^{y}$ $σ_{12}^{y}$	Yield in shear direction 12. (Real)	$[Pa]$ $[Pa]$
$σ_{23}^{y}$ $σ_{23}^{y}$	Yield in shear direction 23. (Real)	$[Pa]$ $[Pa]$
$σ_{31}^{y}$ $σ_{31}^{y}$	Yield in shear direction 31. (Real)	$[Pa]$ $[Pa]$
`Tab_ID`	Table identifier for yield stress definition. 6 (Integer)
$σ_{scale}$ $σ_{scale}$	Yield stress scale factor. Default set to 1.0 (Real)	$[Pa]$ $[Pa]$
${\dot{ε}}_{scale}$ ${\dot{ε}}_{scale}$	Strain rate scale factor. Default set to 1.0 (Real)	$[\frac{1}{s}]$ $[\frac{1}{s}]$
`T`_i	Initial temperature. Default set to 293 K (Real)	$[K]$ $[K]$
$ρ_{0} C_{p}$ $ρ_{0} C_{p}$	Specific heat per volume unit. (Real)	$[\frac{J}{m^{3} \cdot K}]$ $[\frac{J}{m^{3} \cdot K}]$

▸Example (Aluminum)


#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/LAW74/1/1
Aluminum
#              RHO_I
               .0027
#                  E                  NU       EPSILON_P_MAX           EPSILON_T           EPSILON_M
               60400                 .33                   0                   0                   0
#   fct_ID                          EINF                  CE
         0                             0                   0
#            FSMOOTH              C_HARD                FCUT
                   1                   0                  10
#           SIGMA11Y            SIGMA22Y            SIGMA33Y
                   1                   1                   1
#           SIGMA12Y            SIGMA23Y            SIGMA31Y
                   1                   1                   1
#    TABLE                   SIGMA_SCALE         EPSPT_SCALE
        10                             0                   0
#                 TI             RHO0_CP
                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  3. FUNCTIONS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/10
table
#DIMENSION
        3
#   fct_ID                             X                   Z
        38                             0                 293                                       
        38                            10                 293                                       
        39                            11                 293                                       
        40                            20                 293                                       
        38                             0                 400                                       
        38                            10                 400                                       
        39                            11                 400                                       
        40                            20                 400                                       
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/38
function_38
#                  X                   Y
                   0                  90                                                           
              2.5E-4                 100                                                           
                .001               104.5                                                           
                .009                 121                                                           
                 .01                 136                                                           
                 .02               143.5                                                           
                 .04                 163                                                           
                 .07               169.5                                                           
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/39
function_39
#                  X                   Y
                   0                 108                                                           
              2.5E-4                 120                                                           
                .001               125.4                                                           
                .009               145.2                                                           
                 .01               163.2                                                           
                 .02               172.2                                                           
                 .04               195.6                                                           
                 .07               203.4                                                           
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/40
function_40
#                  X                   Y
                   0                 126                                                           
              2.5E-4                 140                                                           
                .001               146.3                                                           
                .009               169.4                                                           
                 .01               190.4                                                           
                 .02               200.9                                                           
                 .04               228.2                                                           
                 .07               237.3                                                           
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

This material law must be used with property set /PROP/TYPE6 (SOL_ORTH), /PROP/TYPE14 (SOLID), /PROP/TYPE20 (TSHELL) or /PROP/TYPE21 (TSH_ORTH).
The yield stress is defined by a user function and the yield stress is compared to equivalent stress:(1)
$σ_{eq} = \sqrt{F {(σ_{2} - σ_{3})}^{2} + G {(σ_{3} - σ_{1})}^{2} + H {(σ_{1} - σ_{2})}^{2} + 2 L {σ_{23}}^{2} + 2 M {σ_{31}}^{2} + 2 N {σ_{12}}^{2}}$ $σ_{eq} = \sqrt{F {(σ_{2} - σ_{3})}^{2} + G {(σ_{3} - σ_{1})}^{2} + H {(σ_{1} - σ_{2})}^{2} + 2 L {σ_{23}}^{2} + 2 M {σ_{31}}^{2} + 2 N {σ_{12}}^{2}}$

Where, HILL parameters are:(2)
$F = \frac{1}{2} (\frac{1}{σ_{22}^{2}} + \frac{1}{σ_{33}^{2}} - \frac{1}{σ_{11}^{2}}), G = \frac{1}{2} (\frac{1}{σ_{11}^{2}} + \frac{1}{σ_{33}^{2}} - \frac{1}{σ_{22}^{2}}), H = \frac{1}{2} (\frac{1}{σ_{11}^{2}} + \frac{1}{σ_{22}^{2}} - \frac{1}{σ_{33}^{2}})$ $F = \frac{1}{2} (\frac{1}{σ_{22}^{2}} + \frac{1}{σ_{33}^{2}} - \frac{1}{σ_{11}^{2}}), G = \frac{1}{2} (\frac{1}{σ_{11}^{2}} + \frac{1}{σ_{33}^{2}} - \frac{1}{σ_{22}^{2}}), H = \frac{1}{2} (\frac{1}{σ_{11}^{2}} + \frac{1}{σ_{22}^{2}} - \frac{1}{σ_{33}^{2}})$
(3)
$L = \frac{1}{2 σ_{23}^{2}}, M = \frac{1}{2 σ_{31}^{2}}, N = \frac{1}{2 σ_{12}^{2}}$ $L = \frac{1}{2 σ_{23}^{2}}, M = \frac{1}{2 σ_{31}^{2}}, N = \frac{1}{2 σ_{12}^{2}}$

Where, $σ_{11}, σ_{22}, σ_{33}, σ_{12}, σ_{23}$ $σ_{11}, σ_{22}, σ_{33}, σ_{12}, σ_{23}$ and $σ_{31}$ $σ_{31}$ represent the stress components either in the orthotropic frame if an orthotropic property is used, or in the orthogonalized isoparametric frame.
If $ε_{p}$ $ε_{p}$ (plastic strain) reaches $ε_{p}^{m a x}$ $ε_{p}^{m a x}$ , in one integration point, the solid element is deleted.
If largest principal strain $ε_{1} > ε_{t}$ $ε_{1} > ε_{t}$ , stress is reduced using the following relation:(4)
$σ = σ (\frac{ε_{m} - ε_{1}}{ε_{m} - ε_{t}})$ $σ = σ (\frac{ε_{m} - ε_{1}}{ε_{m} - ε_{t}})$
If $ε_{1} > ε_{m}$ $ε_{1} > ε_{m}$ , the stress is reduced to 0 (but the element is not deleted).
The table for yield stress definition can be 2-dimensional or 3-dimensional.
- If the table is 2-dimensional, its parameters are assumed to represent respectively plastic strain and strain rate $(ε^{p}, \dot{ε})$ $(ε^{p}, \dot{ε})$ .
  Then if $ε_{m - 1}^{p} \leq ε^{p} \leq ε_{m}^{p}$ $ε_{m - 1}^{p} \leq ε^{p} \leq ε_{m}^{p}$ and ${\dot{ε}}_{n - 1} \leq \dot{ε} \leq {\dot{ε}}_{n}$ ${\dot{ε}}_{n - 1} \leq \dot{ε} \leq {\dot{ε}}_{n}$ yield is linearly interpolated between the four values of the table corresponding to $(ε_{i}^{p}, {\dot{ε}}_{j}), i = m - 1, m; j = n - 1, n$ $(ε_{i}^{p}, {\dot{ε}}_{j}), i = m - 1, m; j = n - 1, n$ .
- If the table is 3-dimensional, its parameters are assumed to represent respectively plastic strain, strain rate, and temperature $(ε^{p}, \dot{ε}, T)$ $(ε^{p}, \dot{ε}, T)$ .
  Then if $ε_{m - 1}^{p} \leq ε^{p} \leq ε_{m}^{p}$ $ε_{m - 1}^{p} \leq ε^{p} \leq ε_{m}^{p}$ and ${\dot{ε}}_{n - 1} \leq \dot{ε} \leq {\dot{ε}}_{n}$ ${\dot{ε}}_{n - 1} \leq \dot{ε} \leq {\dot{ε}}_{n}$ and $T_{q - 1} \leq T \leq T_{q}$ $T_{q - 1} \leq T \leq T_{q}$ yield is linearly interpolated between the eight values of the table corresponding to $(ε_{i}^{p}, {\dot{ε}}_{j}, T_{k}), i = m - 1, m; j = n - 1, n; k = q - 1, q$ $(ε_{i}^{p}, {\dot{ε}}_{j}, T_{k}), i = m - 1, m; j = n - 1, n; k = q - 1, q$ .
  
  If $(ε^{p}, \dot{ε})$ $(ε^{p}, \dot{ε})$ or $(ε^{p}, \dot{ε}, T)$ $(ε^{p}, \dot{ε}, T)$ falls out of the range of the table, yield stress is obtained by linear extrapolation. Thus, it is necessary to input into the table the static curves corresponding to zero strain rate (entry $\dot{ε} = 0$ $\dot{ε} = 0$ should belong to the table definition).
  
  Values of the table are yield stress values.
If yield stress also depends on temperature, the table is 3-dimensional:
If the /HEAT/MAT option is not associated to the material identifier, adiabatic conditions are assumed and temperature is computed as:(5)
$Τ = T_{i} + \frac{E_{int}}{ρ C_{p} (Volume)}$ $Τ = T_{i} + \frac{E_{int}}{ρ C_{p} (Volume)}$

Where,

E_int

Internal energy computed by $ρ$ $ρ$ ,

Volume

Current density and volume

C_p

Heat capacity per mass unit

Otherwise, the finite element formulation for heat transfer must be asked for (I_form =1 in option /HEAT/MAT); initial temperature and specific heat input in the option /HEAT/MAT will then be used.
The evolution of Young's modulus:
- If fct_ID_E > 0, the curve defines a scale factor for Young's modulus evolution with equivalent plastic strain, which means the Young's modulus is scaled by the function $f ({\bar{ε}}_{p})$ $f ({\bar{ε}}_{p})$ :(6)
  $E (t) = E \cdot f ({\bar{ε}}_{p})$ $E (t) = E \cdot f ({\bar{ε}}_{p})$
  
  The initial value of the scale factor should be equal to 1 and it decreases.
- If fct_ID_E = 0, the Young's modulus is calculated as:(7)
  $E (t) = E - (E - E_{\inf}) [1 - \exp (- C_{E} {\bar{ε}}_{p})]$ $E (t) = E - (E - E_{\inf}) [1 - \exp (- C_{E} {\bar{ε}}_{p})]$
  
  Where, E and E_inf are respectively, the initial and asymptotic value of Young's modulus, ${\bar{ε}}_{p}$ ${\bar{ε}}_{p}$ accumulated equivalent plastic strain.
  
  Note: If fct_ID_E = 0 and C_E = 0, Young's modulus E is kept constant.