/MAT/LAW60 (PLAS_T3)

Block Format Keyword This law models an isotropic elasto-plastic material using user-defined functions for the work-hardening portion of the stress-strain curve (that is, plastic strain versus stress) for different strain rates.

It is similar to LAW36, except yield stress is a nonlinear interpolation from the functions.

Format

(1)	(2)	(3)	(4)	(5)	(7)	(9)
/MAT/LAW60/`mat_ID`/`unit_ID` or /MAT/PLAS_T3/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`		$ν$		$ε_{p}^{m a x}$	$ε_{t}$	$ε_{m}$
`N`_funct	`F`_smooth	`C`_hard		`F`_cut
`fct_ID`_p	`Fscale`		`fct_ID`_E	`E`_inf	`C`_E
`fct_ID`₁	`fct_ID`₂	`fct_ID`₃	`fct_ID`₄	`fct_ID`₅

Read only if 6 ≤ N_funct ≤ 10

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`fct_ID`₆	`fct_ID`₇	`fct_ID`₈	`fct_ID`₉	`fct_ID`₁₀

Always Read

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`Fscale`₁		`Fscale`₂		`Fscale`₃		`Fscale`₄		`Fscale`₅

Read only if 6 < N_funct < 10

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
`Fscale`₆		`Fscale`₇		`Fscale`₈		`Fscale`₉		`Fscale`₁₀

Always Read

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
${\dot{ε}}_{1}$		${\dot{ε}}_{2}$		${\dot{ε}}_{3}$		${\dot{ε}}_{4}$		${\dot{ε}}_{5}$

Read only if 6 ≤ N_funct ≤ 10

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
${\dot{ε}}_{6}$		${\dot{ε}}_{7}$		${\dot{ε}}_{8}$		${\dot{ε}}_{9}$		${\dot{ε}}_{10}$

Definitions

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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`E`	Young's modulus. (Real)	$[Pa]$
$ν$	Poisson's ratio. (Real)
$ε_{p}^{m a x}$	Failure plastic strain. Default = 1.0 × 10³⁰ (Real)
$ε_{t}$	Tensile failure strain at which stress starts to reduce. Default = 1.0 x 10³⁰ (Real)
$ε_{m}$	Maximum tensile failure strain at which the element is deleted. Default = 2.0 x 10³⁰ (Real)
`N`_funct	Number of functions. It should be 4 < `N`_funct < 10. Default ≤ 10 (Integer)
`F`_smooth	Smooth strain rate option flag. = 0 (Default) No strain rate smoothing. = 1 Strain rate smoothing active. (Integer)
`C`_hard	Hardening coefficient. = 0 The hardening is a full isotropic model. = 1 The hardening uses the kinematic Prager-Ziegler model. = value between 0 and 1 The hardening is interpolated between the two models. (Real)
`F`_cut	Cutoff frequency for strain rate filtering. 7 Default = 1.0 × 10³⁰ (Real)	$[Hz]$
`fct_ID`_p	Pressure versus yield factor function. 9 Default = 0 (Integer)
`Fscale`	Scale factor for yield factor in `fct_ID`_p. Default = 1.0 (Real)	$[Pa]$
`fct_ID`_E	Function identifier for the scale factor of Young's modulus, when Young's modulus is function of the plastic strain. 6 Default = 0: in this case the evolution of Young's modulus depends on `E`_inf_. and `C`_E. (Integer)
`E`_inf	Saturated Young's modulus for infinitive plastic strain. (Real)	$[Pa]$
`C`_E	Parameter for Young's modulus evolution. (Real)
`fct_ID`₁	Yield stress function identifier 1 corresponding to strain rate ${\dot{ε}}_{1}$ . (Integer)
`fct_ID`₂	Yield stress function identifier 2 corresponding to strain rate ${\dot{ε}}_{2}$ . (Integer)
`fct_ID`₃	Yield stress function identifier 3 corresponding to strain rate ${\dot{ε}}_{3}$ . (Integer)
`fct_ID`₄	Yield stress function identifier 4 corresponding to strain rate ${\dot{ε}}_{4}$ . (Integer)
`fct_ID`₅	Yield stress function identifier 5 corresponding to strain rate ${\dot{ε}}_{5}$ . (Integer)
`fct_ID`₆	Yield stress function identifier 6 corresponding to strain rate ${\dot{ε}}_{6}$ . (Integer)
`fct_ID`₇	Yield stress function identifier 7 corresponding to strain rate ${\dot{ε}}_{7}$ . (Integer)
`fct_ID`₈	Yield stress function identifier 8 corresponding to strain rate ${\dot{ε}}_{8}$ . (Integer)
`fct_ID`₉	Yield stress function identifier 9 corresponding to strain rate ${\dot{ε}}_{9}$ . (Integer)
`fct_ID`₁₀	Yield stress function identifier 10 corresponding to strain rate ${\dot{ε}}_{10}$ . (Integer)
`Fscale`₁	Scale factor for ordinate (stress) in `fct_ID`₁. Default = 1.0 (Real)	$[Pa]$
`Fscale`₂	Scale factor for ordinate (stress) in `fct_ID`₂. Default = 1.0 (Real)	$[Pa]$
`Fscale`₃	Scale factor for ordinate (stress) in `fct_ID`₃. Default = 1.0 (Real)	$[Pa]$
`Fscale`₄	Scale factor for ordinate (stress) in `fct_ID`₄. Default = 1.0 (Real)	$[Pa]$
`Fscale`₅	Scale factor for ordinate (stress) in `fct_ID`₅. Default = 1.0 (Real)	$[Pa]$
`Fscale`₆	Scale factor for ordinate (stress) in `fct_ID`₆. Default = 1.0 (Real)	$[Pa]$
`Fscale`₇	Scale factor for ordinate (stress) in `fct_ID`₇. Default = 1.0 (Real)	$[Pa]$
`Fscale`₈	Scale factor for ordinate (stress) in `fct_ID`₈. Default = 1.0 (Real)	$[Pa]$
`Fscale`₉	Scale factor for ordinate (stress) in `fct_ID`₉. Default = 1.0 (Real)	$[Pa]$
`Fscale`₁₀	Scale factor for ordinate (stress) in `fct_ID`₁₀. Default = 1.0 (Real)	$[Pa]$
${\dot{ε}}_{1}$	Strain rate 1. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{2}$	Strain rate 2. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{3}$	Strain rate 3. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{4}$	Strain rate 4. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{5}$	Strain rate 5. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{6}$	Strain rate 6. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{7}$	Strain rate 7. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{8}$	Strain rate 8. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{9}$	Strain rate 9. (Real)	$[\frac{1}{s}]$
${\dot{ε}}_{10}$	Strain rate 10. (Real)	$[\frac{1}{s}]$

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/LAW60/1/1
Aluminium_example
#              RHO_I
               .0027 
#                  E                  Nu           Eps_p_max               Eps_t               Eps_m
               60400                 .33                   0                   0                   0
#  N_funct  F_smooth              C_hard               F_cut                
         4         0                   0                   0                    
#  fct_IDp              Fscale   Fct_IDE                EInf                  CE
         0                   0         0                   0                   0
# Funtions
         1         2         3         4
# Scale factors          Fscale_5
                   1                 1.2                 1.4                 1.6
# Strain rates          Eps_dot_5
                   0                  20                  30                  40
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  3. FUNCTIONS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/1
function_36
#                  X                   Y
                   0                  90                                                            
              2.5E-4                 100                                                           
                .001                 104                                                           
                .009                 121                                                            
                .017                 136                                                            
                .021                 143                                                           
                .036                 156                                                           
                .045                 162                                                         
                .055                 165                                                           
                .072                 170                                                           
                .075                 170                                                             
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/2
function_36
#                  X                   Y
                   0                  90                                                            
              2.5E-4                 100                                                           
                .001                 104                                                           
                .009                 121                                                            
                .017                 136                                                            
                .021                 143                                                           
                .036                 156                                                           
                .045                 162                                                         
                .055                 165                                                           
                .072                 170                                                           
                .075                 170                                                             
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/3
function_36
#                  X                   Y
                   0                  90                                                            
              2.5E-4                 100                                                           
                .001                 104                                                           
                .009                 121                                                            
                .017                 136                                                            
                .021                 143                                                           
                .036                 156                                                           
                .045                 162                                                         
                .055                 165                                                           
                .072                 170                                                           
                .075                 170                                                             
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/4
function_36
#                  X                   Y
                   0                  90                                                            
              2.5E-4                 100                                                           
                .001                 104                                                           
                .009                 121                                                            
                .017                 136                                                            
                .021                 143                                                           
                .036                 156                                                           
                .045                 162                                                         
                .055                 165                                                           
                .072                 170                                                           
                .075                 170                                                             
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The first point of yield stress functions (plastic strain versus stress) should have a plastic strain value of zero. If the last point of the first (static) function equals 0 in stress, default value of $ε_{p}^{m a x}$ is set to the corresponding value of $ε_{p}$ .
If $ε_{p}$ (plastic strain) reaches $ε_{p}^{m a x}$ , in one integration point, the element is deleted.
If (largest principal strain) $ε_{1} > ε_{t}$ , stress is reduced using:(1)
$σ = σ (\frac{ε_{m} - ε_{1}}{ε_{m} - ε_{t}})$
If $ε_{1} > ε_{m}$ , the element is deleted.
The kinematic hardening model is not available in global formulation (hardening is fully isotropic).
For kinematic hardening and strain rate dependency, yield stress depends on the strain rate.
Strain rate filtering input (F_cut) is only available for shell and solid elements.
Strain rate filtering is used to smooth strain rates.
fct_ID_p is used to distinguish the behavior in tension and compression for certain materials (that is, pressure dependent yield). This is available for solid elements only. The effective yield stress is then obtained by multiplying the nominal yield stress by the yield factor corresponding to the actual pressure.

Figure 1.
If ${\dot{ε}}_{n} \leq \dot{ε} \leq {\dot{ε}}_{n + 1}$ , yield stress is a cubic interpolation between functions f_n-1, f_n, f_n+1 and f_n+2.
If $\dot{ε} \leq {\dot{ε}}_{1}$ , yield stress is interpolated between functions $f_{1}$ , $f_{2}$ and $f_{3}$ .
If ${\dot{ε}}_{Nfunc - 1} \leq \dot{ε} \leq {\dot{ε}}_{Nfunc}$ , yield is extrapolated between functions f_Nfunc-3, f_Nfunc-2, f_Nfunc-1 and f_Nfunc.
If $\dot{ε} > {\dot{ε}}_{Nfunc}$ , yield is extrapolated between functions f_Nfunc-2, f_Nfunc-1 and f_Nfunc.

Figure 2.
Functions describing strain dependence must be defined for different strain rates values.
Strain rate values must be given in strictly ascending order.
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 $f ({\bar{ε}}_{p})$ :
  - $E (t) = E \cdot f ({\bar{ε}}_{p})$
    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:(2)
    $E (t) = E - (E - E_{\inf}) [1 - exp (- C_{E} {\bar{ε}}_{p})]$
Where,

E and E_inf are respectively the initial and asymptotic value of Young's modulus, and ${\bar{ε}}_{p}$ is the accumulated equivalent plastic strain.
Note:
If fct_ID_E = 0 and C_E = 0, Young's modulus E is kept constant.