/MAT/LAW115 (DESHFLECK)
Block Format Keyword An elastoplastic 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)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW115/mat_ID/unit_ID or /MAT/DESHFLACK/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
E  $\upsilon $  I_{res}  I_{stat} 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

$\alpha $  ${\epsilon}_{vp}^{f}$  ${\sigma}_{1}^{f}$  
${\sigma}_{p}$  $\gamma $  ${\epsilon}_{D}$  ${\alpha}_{2}$  $\beta $ 
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

$\alpha $  ${\epsilon}_{vp}^{f}$  ${\sigma}_{1}^{f}$  ${\rho}_{f0}$  
${C}_{0}^{\sigma}$  ${C}_{1}^{\sigma}$  ${n}^{\sigma}$  
${C}_{0}^{{\alpha}_{2}}$  ${C}_{1}^{{\alpha}_{2}}$  ${n}^{{\alpha}_{2}}$  
${C}_{0}^{\gamma}$  ${C}_{1}^{\gamma}$  ${n}^{\gamma}$  
${C}_{0}^{\beta}$  ${C}_{1}^{\beta}$  ${n}^{\beta}$ 
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) 
$\left[\text{Pa}\right]$ 
$\upsilon $  Poisson's
ratio. (Real) 

I_{res}  Resolution method for plasticity.
(Integer) 

I_{stat}  Activate statistical variation
flag.
(Integer) 

$\alpha $  Yield surface shape
parameter. Default = 0.0 (Real) 

$\gamma $  Linear hardening modulus. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
${\epsilon}_{D}$  Densification strain. Default = 1.0E20 (Real) 

${\alpha}_{2}$  Nonlinear hardening
modulus. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
$\beta $  Nonlinear hardening
parameter. Default = 1.0 (Real) 

${\sigma}_{p}$  Initial flow stress. Default = 1.0E20 (Real) 
$\left[\text{Pa}\right]$ 
${\epsilon}_{vp}^{f}$  Tensile volumic plastic strain at
failure. Default = 0.0 (Real) 

${\sigma}_{1}^{f}$  Maximum principal stress at
failure. Default = 0.0 (Real) 
$\left[\text{Pa}\right]$ 
${\rho}_{f0}$  Density of base material (matrix
material of the foam). Default = 1.0E20 (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
${C}_{0}^{\sigma}$ , ${C}_{1}^{\sigma}$  Statistical law parameters for
initial flow stress. (Real) 
$\left[\text{Pa}\right]$ 
${C}_{0}^{{\alpha}_{2}}$ , ${C}_{1}^{{\alpha}_{2}}$  Statistical law parameters for
nonlinear hardening modulus. (Real) 
$\left[\text{Pa}\right]$ 
${C}_{0}^{\gamma}$ , ${C}_{1}^{\gamma}$  Statistical law parameters for
linear hardening modulus. (Real) 
$\left[\text{Pa}\right]$ 
${C}_{0}^{\beta}$ , ${C}_{1}^{\beta}$  Statistical law parameters for
nonlinear hardening exponent inverse. (Real) 

${n}^{i}$  Statistical law exponent of the
parameter. (Real) 
Example (Aluminum)
#12345678910
/UNIT/25
Local unit system
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/LAW115/1/25
Aluminum foam constant
# Init. dens.
5.1E10
# 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
#12345678910
#enddata
Example (Random Noise/Distribution)
#12345678910
/UNIT/25
Local unit system
Mg mm s
#12345678910
# 2. MATERIALS:
#12345678910
/MAT/DESHFLECK/1/25
Aluminum foam statistical
# Init. dens.
5.1E10
# E Nu Ires Istat
5562.0 0.3 1 1
# ALPHA EPSVP_F SIGP_F RHOF0
2.12 0.11 30.0 2.7E9
# 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
#12345678910
#enddata
Comments
 The material is supposed to be isotropic linear elastic.
 The law uses the
DeshpandeFleck equivalent stress definition:
(1) $${\sigma}_{eq}=\sqrt{\frac{{\sigma}_{VM}^{2}+{\alpha}^{2}{\sigma}_{m}^{2}}{1+{\left(\frac{\alpha}{3}\right)}^{2}}}$$Where, ${\sigma}_{VM}$
 The von Mises equivalent stress.
 ${\sigma}_{m}$
 The mean stress defined by:
(2) $$\begin{array}{l}s=\sigma {\sigma}_{m}I\begin{array}{ccc}& \text{with}& \end{array}{\sigma}_{m}=\frac{1}{3}tr\left(\sigma \right)\\ {\sigma}_{VM}=\sqrt{\frac{3}{2}s:s}\end{array}$$
The parameter $\alpha $ controls the effect of pressure dependence in the computation of equivalent stress. This parameter must respect the following inequality:(3) $$0\le \alpha \le \sqrt{4.5}$$  The yield
function is denoted
$\text{\Phi}$
, compared to the DeshpandeFleck equivalent
stress to the flow stress,
${\sigma}_{F}$
:
(4) $$\text{\Phi}={\sigma}_{eq}{\sigma}_{F}$$With the flow stress defined by (Figure 1):(5) $${\sigma}_{F}={\sigma}_{p}+\gamma \frac{{\epsilon}_{p}}{{\epsilon}_{D}}+{\alpha}_{2}\mathrm{ln}\left(\frac{1}{1{\left(\frac{{\epsilon}_{p}}{{\epsilon}_{D}}\right)}^{\beta}}\right)$$Where, ${\sigma}_{p}$
 Initial flow stress.
 $\gamma $
 Linear hardening modulus.
 ${\epsilon}_{D}$
 Densification strain.
 ${\alpha}_{2}$
 Nonlinear hardening modulus.
 $\beta $
 Nonlinear hardening exponent.
 Two versions of
the same material are available, depending on the value of the flag
I_{stat}:
 If I_{stat} = 0: parameters ${\sigma}_{p}$ , $\gamma $ , ${\epsilon}_{D}$ , ${\alpha}_{2}$ and $\beta $ are the same for all solid elements of the part.
 If
I_{stat} =
1: parameters
${\sigma}_{p}$
,
$\gamma $
,
${\epsilon}_{D}$
,
${\alpha}_{2}$
and
$\beta $
are not the same for all solid elements of
the part.They are computed from the foam density ${\rho}_{i}$ that is statistically distributed (using /PERTURB/PART/SOLID).
(6) $$\begin{array}{l}{\sigma}_{p}={C}_{0}^{\sigma}+{C}_{1}^{\sigma}{\left(\frac{{\rho}_{i}}{{\rho}_{f0}}\right)}^{{n}^{\sigma}}\begin{array}{cc}& \end{array}{\alpha}_{2}={C}_{0}^{{\alpha}_{2}}+{C}_{1}^{{\alpha}_{2}}{\left(\frac{{\rho}_{i}}{{\rho}_{f0}}\right)}^{{n}^{{\alpha}_{2}}}\\ \gamma ={C}_{0}^{\gamma}+{C}_{1}^{\gamma}{\left(\frac{{\rho}_{i}}{{\rho}_{f0}}\right)}^{{n}^{\gamma}}\begin{array}{cc}& \end{array}\frac{1}{\beta}={C}_{0}^{\beta}+{C}_{1}^{\beta}{\left(\frac{{\rho}_{i}}{{\rho}_{f0}}\right)}^{{n}^{\beta}}\\ {\epsilon}_{D}=\frac{9+{\alpha}^{2}}{3{\alpha}^{2}}\mathrm{ln}\left(\frac{{\rho}_{i}}{{\rho}_{f0}}\right)\end{array}$$Where, ${C}_{0}^{i}$ , ${C}_{1}^{i}$ , and ${n}^{i}$
 Statistical law parameters.
 ${\rho}_{f0}$
 Base material density (for instance, if the material law represents aluminum foam, ${\rho}_{f0}$ will be aluminum density).
 When using I_{stat} = 1, the card /PERTURB/PART/SOLID must be set to create the initial foam density ${\rho}_{i}$ distribution over all elements of the part (Figure 2).
 Most failure criteria are compatible with /MAT/LAW115; however, element deletion can be onset by you, if ${\epsilon}_{vp}^{f}$ and/or ${\sigma}_{1}^{f}$ are different from 0. The element deletion will be onset if the tensile volumetric strain is higher than ${\epsilon}_{vp}^{f}$ and/or, if the first principal stress is higher than ${\sigma}_{1}^{f}$ .