/MAT/LAW121 (PLAS_RATE)

Block Format Keyword Elasto-plastic strain-rate dependent material with isotropic von Mises yield criterion. This material law is available for both solids and shells.

Format

(1)	(2)	(3)	(5)	(6)	(7)	(9)
/MAT/LAW121/`mat_ID`/`unit_ID` or /MAT/PLAS_RATE/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`		$ν$	`Ires`	`Ivisc`	`F`_cut	`DTMIN`
`Fct_SIG0`		`Xscale_SIG0`	`Yscale_SIG0`
`Fct_YOUN`		`Xscale_YOUN`	`Yscale_YOUN`
`Fct_TANG`		`Xscale_TANG`	`TANG`
`Fct_FAIL`	`Ifail`	`Xscale_FAIL`	`Yscale_FAIL`

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)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`E`	Young’s modulus. (Real)	$[Pa]$
$ν$	Poisson’s ratio. (Real)
`Ires`	Resolution method for plasticity. = 0 Set to 2. = 1 NICE (Next Increment Correct Error) explicit method. = 2 (Default) Newton iterative semi-implicit method (Cutting plane). (Integer)
`Ivisc`	Strain rate dependency formulation. = 0 (Default) Scaled yield stress. = 1 Visco-plastic formulation. (Integer)
`Ifail`	Failure criterion variable (used only `Fct_FAIL` > 0). = 0 (Default) Equivalent stress. =1 Effective plastic strain. = 2 Maximum principal stress or absolute value of minimum principal stress. = 3 Maximum principal stress. (Integer)
`F`_cut	Cutoff frequency for strain rate filtering (used only `Ivisc` = 0). Default = 10000 Hz (Real)	$[Hz]$
`DTMIN`	Minimum time step size for automatic element deletion. Default = 0.0 (Real)	$[s]$
`Fct_SIG0`	Yield stress versus effective strain rate function identifier. (Integer)
`Xscale_SIG0`	Scale factor for abscissa (strain rate) in `Fct_SIG0`. Default = 1.0 (Real)	$[\frac{1}{s}]$
`Yscale_SIG0`	Scale factor for ordinate (stress) in `Fct_SIG0`. Default = 1.0 (Real)	$[Pa]$
`Fct_YOUN`	Young’s modulus versus effective strain rate function identifier (available only `Ivisc` = 0). Default = 1.0 (Real)
`Xscale_YOUN`	Scale factor for abscissa (strain rate) in `Fct_YOUN`. Default = 1.0 (Real)	$[\frac{1}{s}]$
`Yscale_YOUN`	Scale factor for ordinate (stress) in `Fct_YOUN`. Default = 1.0 (Real)	$[Pa]$
`Fct_TANG`	Plastic tangent modulus versus effective strain rate function identifier. (Integer)
`Xscale_TANG`	Scale factor for abscissa (strain rate) in `Fct_TANG`. Default = 1.0 (Real)	$[\frac{1}{s}]$
`TANG`	If `Fct_TANG` > 0 Scale factor for ordinate (stress) in `Fct_TANG`. Default = 1.0 (Real) If `Fct_TANG` = 0 Plastic tangent `TANG`. Default = 0.0 (Real)	$[Pa]$
`Fct_FAIL`	Failure criterion variable versus effective strain rate function identifier. (Integer)
`Xscale_FAIL`	Scale factor for abscissa (strain rate) in `Fct_FAIL`. Default = 1.0 (Real)	$[\frac{1}{s}]$
`Yscale_FAIL`	Scale factor for ordinate (stress) in `Fct_FAIL`. Default = 1.0 (Real)	$[Pa]$

Example (Steel)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
Unit system
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW121/1/1
Steel
#              RHO_I
             7.85E-9
#                  E                  Nu      Ires     Ivisc                Fcut               DTMIN
            194200.0                 0.3         1         0                1000                   0
# Fct_SIG0                   Xscale_SIG0         Yscale_SIG0
         5                             0                   0
# Fct_YOUN                   Xscale_YOUN         Yscale_YOUN
         6                             0                   0
# Fct_TANG                   Xscale_TANG                TANG
         7                             0                   0
# Fct_FAIL     Ifail         Xscale_FAIL         Yscale_FAIL
         8         1                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/5
Yield stress versus effective strain rate
                 0.0               272.2
              1000.0               572.2
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/6
Young’s modulus versus effective strain rate
                 0.0            192400.0
              1000.0            272400.0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/7
Plastic tangent modulus versus effective strain rate
                 0.0              4500.0
              1000.0              6500.0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/FUNCT/8
Failure criterion variable versus effective strain 
                 0.0                0.15
              1000.0                0.25
#---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 considers isotropic and linear elasticity.
The plastic behavior is described with a classical von Mises plasticity yield function described with:(1)
$f = σ_{V M} - σ_{Y}$

With:(2)
$σ_{V M} = \sqrt{\frac{3}{2} s : s}$

Where, $s$ is the deviatoric stress tensor.
The yield stress is strain rate dependent and described as:(3)
$σ_{Y} = σ_{0} (\dot{ε}) + \frac{E (\dot{ε}) E_{t} (\dot{ε})}{E (\dot{ε}) - E_{t} (\dot{ε})} ε_{p}$

Where,

$σ_{0}$

Strain rate dependent initial yield stress defined by function Fct_SIG0.

$E (\dot{ε})$

Young’s modulus that may optionally depend on the strain rate (if Fct_YOUN is defined).

$E_{t} (\dot{ε})$

Plastic tangent modulus that may optionally depend on the strain rate (if Fct_TANG is defined).

$ε_{p}$

Cumulative plastic strain.

Note: The inequality $E (\dot{ε}) > E_{t} (\dot{ε})$ must always be fulfilled. If not, the solver will automatically limit the value of the tangent modulus $E_{t} (\dot{ε}) = 0.99 \times E (\dot{ε})$ .
Two strain rate dependency formulations are available depending on the value of the flag Ivisc.
- Ivisc = 0: scaled yield stress formulation. In this case, the strain rate corresponds to the total effective strain-rate computed as:(4)
  $\dot{ε} = \sqrt{\frac{2}{3} \dot{ε}' : \dot{ε}'}$
  
  Where, $\dot{ε}'$ is the deviatoric total strain rate tensor.
  
  For this formulation, all strain rate dependent variables are updated at the beginning of the timestep and are constant during the return mapping algorithm. All strain rate dependent variables are allowed using this formulation. The strain rate is, in this case, filtered by default as: (5)
  ${\dot{ε}}_{_{f}}^{n} = α {\dot{ε}}^{n} + (1 - α) {\dot{ε}}_{f}^{n - 1}$
  
  Where, $α = 2 π F_{c u t} Δ t$ .
  
  You can then adapt the filtering by specifying a value for the cutoff frequency F_cut. This formulation is lighter and faster than the viscoplastic one described below.
- Ivisc = 1: viscoplastic formulation. In this case, the strain rate corresponds to the plastic strain rate. (6)
  $\dot{ε} = {\dot{ε}}_{p}$
  
  In this case, the strain rate is initially null, and is computed and updated during the return mapping procedure. All strain rate dependent variable variations are taken into account leading to a heavier cost of computation. However, no strain rate filtering is needed, and you can speed up the simulation using the NICE explicit return mapping (Ires = 1). This formulation, the Young’s modulus is not allowed to vary with strain rate.
You can choose the algorithm of return mapping to find a compromise between precision and cost. Two return mapping methods are available.
- Ires = 1: the NICE explicit method (Next Increment Correct Error). This procedure only requires 1 iteration and uses a self-correcting ability to still preserve a good precision. This method can dramatically speed up the simulation.
- Ires = 2: the Cutting-Plane (Newton iteration) method. This procedure requires several iterations (usually between 3 and 5) to solve the material behavior nonlinear equations. It is more expensive than the NICE explicit method but offers very good precision. This method is set by default.
You can choose to add failure in the material behavior. To do so, a function Fct_FAIL must be defined to describe the evolution of the failure criterion variable with the strain rate. The nature of the failure criterion variable is chosen using the flag Ifail:
- Ifail = 0: the criterion is fulfilled when the von Mises stress reaches the critical value defined in the function
- Ifail = 1: the criterion is fulfilled when the cumulative plastic strain reaches the value defined in the function
- Ifail = 2: the criterion is fulfilled when the maximum principal stress or the absolute value of the minimum principal stress reaches the critical value defined in the function
- Ifail = 3: the criterion is fulfilled when the maximum principal stress reaches the critical value defined in the function
Automatic element deletion can also be set if the element timestep decreases to reach a critical value defined by DTMIN.