/MAT/LAW115 (DESHFLECK)

Block Format Keyword An elasto-plastic constitutive law using von Mises criterion with pressure dependence. The hardening law is linear – nonlinear with an increasing exponential hardening.

Parameters can either be constant over all element of the part or can be statistically distributed over the elements. This introduces a probabilistic approach in the results of the simulation.

Format

(1)	(3)	(5)	(6)
/MAT/LAW115/`mat_ID`/`unit_ID` or /MAT/DESHFLACK/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`	$υ$	`I`_res	`I`_stat

I_stat= 0: constant parameters over the part

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
$α$		$ε_{v p}^{f}$		$σ_{1}^{f}$
$σ_{p}$		$γ$		$ε_{D}$		$α_{2}$		$β$

I_stat= 1: distributed parameters over the part (used with /PERTURB/PART/SOLID)

(1)	(3)	(5)	(7)
$α$	$ε_{v p}^{f}$	$σ_{1}^{f}$	$ρ_{f 0}$
$C_{0}^{σ}$	$C_{1}^{σ}$	$n^{σ}$
$C_{0}^{α_{2}}$	$C_{1}^{α_{2}}$	$n^{α_{2}}$
$C_{0}^{γ}$	$C_{1}^{γ}$	$n^{γ}$
$C_{0}^{β}$	$C_{1}^{β}$	$n^{β}$

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)	$[Pa]$
$υ$	Poisson's ratio. (Real)
`I`_res	Resolution method for plasticity. = 0 Set to 2. = 1 NICE (Next Increment Correct Error) explicit method. = 2 (Default) Newton iterative implicit method. (Integer)
`I`_stat	Activate statistical variation flag. = 0 Constant parameters over the part. = 1 Distributed parameters over the part (used with /PERTURB/PART/SOLID). (Integer)
$α$	Yield surface shape parameter. Default = 0.0 (Real)
$γ$	Linear hardening modulus. Default = 0.0 (Real)	$[Pa]$
$ε_{D}$	Densification strain. Default = 1.0E20 (Real)
$α_{2}$	Nonlinear hardening modulus. Default = 0.0 (Real)	$[Pa]$
$β$	Nonlinear hardening parameter. Default = 1.0 (Real)
$σ_{p}$	Initial flow stress. Default = 1.0E20 (Real)	$[Pa]$
$ε_{v p}^{f}$	Tensile volumic plastic strain at failure. Default = 0.0 (Real)
$σ_{1}^{f}$	Maximum principal stress at failure. Default = 0.0 (Real)	$[Pa]$
$ρ_{f 0}$	Density of base material (matrix material of the foam). Default = 1.0E20 (Real)	$[\frac{kg}{m^{3}}]$
$C_{0}^{σ}$ , $C_{1}^{σ}$	Statistical law parameters for initial flow stress. (Real)	$[Pa]$
$C_{0}^{α_{2}}$ , $C_{1}^{α_{2}}$	Statistical law parameters for nonlinear hardening modulus. (Real)	$[Pa]$
$C_{0}^{γ}$ , $C_{1}^{γ}$	Statistical law parameters for linear hardening modulus. (Real)	$[Pa]$
$C_{0}^{β}$ , $C_{1}^{β}$	Statistical law parameters for nonlinear hardening exponent inverse. (Real)
$n^{i}$	Statistical law exponent of the parameter. (Real)

Example (Aluminum)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/25
Local unit system
                  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/LAW115/1/25
Aluminum foam constant
#         Init. dens.
             5.1E-10 
#                  E                  Nu      Ires     Istat
              5562.0                 0.3         2         0
#              ALPHA             EPSVP_F              SIGP_F
                2.12                0.11                32.1
#               SIGP               GAMMA                EPSD              ALPHA2                BETA
               14.82                5.37                1.67                66.9                2.99
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Example (Random Noise/Distribution)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/25
Local unit system
                  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/DESHFLECK/1/25
Aluminum foam statistical
#         Init. dens.
             5.1E-10 
#                  E                  Nu      Ires     Istat
              5562.0                 0.3         1         1
#              ALPHA             EPSVP_F              SIGP_F               RHOF0
                2.12                0.11                30.0              2.7E-9
#            SIGP_C0             SIGP_C1              SIGP_N
                   0               590.0                2.21
#          ALPHA2_C0           ALPHA2_C1            ALPHA2_N
                   0               140.0                0.45
#           GAMMA_C0            GAMMA_C1             GAMMA_N
                   0                40.0                 1.4
#        INV_BETA_C0         INV_BETA_C1          INV_BETA_N
                0.22               320.0                4.66
/PERTURB/PART/SOLID/1
set Random Noise with random distribution on Solid density
#         Mean_value           Deviation             Min_cut             Max_cut      Seed   Idistri
                 1.0             0.02471                 0.6                 1.4      1000         2
#grpart_ID           parameter
        46                DENS
/GRPART/PART/46
part
         1
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

The material is supposed to be isotropic linear elastic.
The law uses the Deshpande-Fleck equivalent stress definition: (1)
$σ_{e q} = \sqrt{\frac{σ_{V M}^{2} + α^{2} σ_{m}^{2}}{1 + {(\frac{α}{3})}^{2}}}$

Where,

$σ_{V M}$

The von Mises equivalent stress.

$σ_{m}$

The mean stress defined by: (2)
$\begin{array}{l} s = σ - σ_{m} I \begin{matrix} with \end{matrix} σ_{m} = \frac{1}{3} t r (σ) \\ σ_{V M} = \sqrt{\frac{3}{2} s : s} \end{array}$

The parameter $α$ controls the effect of pressure dependence in the computation of equivalent stress. This parameter must respect the following inequality: (3)
$0 \leq α \leq \sqrt{4.5}$
The yield function is denoted $Φ$ , compared to the Deshpande-Fleck equivalent stress to the flow stress, $σ_{F}$ : (4)
$Φ = σ_{e q} - σ_{F}$

With the flow stress defined by (Figure 1): (5)
$σ_{F} = σ_{p} + γ \frac{ε_{p}}{ε_{D}} + α_{2} \ln (\frac{1}{1 - {(\frac{ε_{p}}{ε_{D}})}^{β}})$

Where,

$σ_{p}$

Initial flow stress.

$γ$

Linear hardening modulus.

$ε_{D}$

Densification strain.

$α_{2}$

Nonlinear hardening modulus.

$β$

Nonlinear hardening exponent.

Figure 1. Flow stress evolution with plastic strain
Two versions of the same material are available, depending on the value of the flag I_stat:
- If I_stat = 0: parameters $σ_{p}$ , $γ$ , $ε_{D}$ , $α_{2}$ and $β$ are the same for all solid elements of the part.
- If I_stat = 1: parameters $σ_{p}$ , $γ$ , $ε_{D}$ , $α_{2}$ and $β$ are not the same for all solid elements of the part.
  They are computed from the foam density $ρ_{i}$ that is statistically distributed (using /PERTURB/PART/SOLID). (6)
  $\begin{array}{l} σ_{p} = C_{0}^{σ} + C_{1}^{σ} {(\frac{ρ_{i}}{ρ_{f 0}})}^{n^{σ}} \begin{matrix} \end{matrix} α_{2} = C_{0}^{α_{2}} + C_{1}^{α_{2}} {(\frac{ρ_{i}}{ρ_{f 0}})}^{n^{α_{2}}} \\ γ = C_{0}^{γ} + C_{1}^{γ} {(\frac{ρ_{i}}{ρ_{f 0}})}^{n^{γ}} \begin{matrix} \end{matrix} \frac{1}{β} = C_{0}^{β} + C_{1}^{β} {(\frac{ρ_{i}}{ρ_{f 0}})}^{n^{β}} \\ ε_{D} = - \frac{9 + α^{2}}{3 α^{2}} \ln (\frac{ρ_{i}}{ρ_{f 0}}) \end{array}$
  
  Where,
  
  $C_{0}^{i}$ , $C_{1}^{i}$ , and $n^{i}$
  
  Statistical law parameters.
  
  $ρ_{f 0}$
  
  Base material density (for instance, if the material law represents aluminum foam, $ρ_{f 0}$ will be aluminum density).
When using I_stat = 1, the card /PERTURB/PART/SOLID must be set to create the initial foam density $ρ_{i}$ distribution over all elements of the part (Figure 2).

Figure 2. Example: foam density distribution in a foam block for an impact simulation
Most failure criteria are compatible with /MAT/LAW115; however, element deletion can be onset by you, if $ε_{v p}^{f}$ and/or $σ_{1}^{f}$ are different from 0. The element deletion will be onset if the tensile volumetric strain is higher than $ε_{v p}^{f}$ and/or, if the first principal stress is higher than $σ_{1}^{f}$ .