/MAT/LAW115 (DESHFLECK)

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/UNIT/25
Local unit system
                  Mg                  mm                   s
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#-  2. MATERIALS:
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/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
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#enddata

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/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
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#enddata

1. The material is supposed to be isotropic linear elastic.
2. The law uses the Deshpande-Fleck 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)

   $$0\le \alpha \le \sqrt{4.5}$$
3. The yield function is denoted $\text{Φ}$ , compared to the Deshpande-Fleck equivalent stress to the flow stress, ${\sigma }_F$ : (4)
   $$\text{Φ}={\sigma }_{eq}-{\sigma }_F$$
   With the flow stress defined by (Figure 1): (5)

   $${\sigma }_F={\sigma }_p+\gamma \frac{{\epsilon }_p}{{\epsilon }_D}+{\alpha }_2\ln \left(\frac{1}{1-{\left(\frac{{\epsilon }_p}{{\epsilon }_D}\right)}^{\beta }}\right)$$
   Where,

   ${\sigma }_p$

   Initial flow stress.

   $\gamma$

   Linear hardening modulus.

   ${\epsilon }_D$

   Densification strain.

   ${\alpha }_2$

   Nonlinear hardening modulus.

   $\beta$

   Nonlinear hardening exponent.

   
   
   Figure 1. Flow stress evolution with plastic strain

4. 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)

     $$\begin{array}{l}{\sigma }_p=C_0^{\sigma }+C_1^{\sigma }{\left(\frac{{\rho }_i}{{\rho }_{f0}}\right)}^{n^{\sigma }}\begin{array}{cc}& \end{array}{\alpha }_2=C_0^{{\alpha }_2}+C_1^{{\alpha }_2}{\left(\frac{{\rho }_i}{{\rho }_{f0}}\right)}^{n^{{\alpha }_2}}\\ \gamma =C_0^{\gamma }+C_1^{\gamma }{\left(\frac{{\rho }_i}{{\rho }_{f0}}\right)}^{n^{\gamma }}\begin{array}{cc}& \end{array}\frac{1}{\beta }=C_0^{\beta }+C_1^{\beta }{\left(\frac{{\rho }_i}{{\rho }_{f0}}\right)}^{n^{\beta }}\\ {\epsilon }_D=-\frac{9+{\alpha }^2}{3{\alpha }^2}\ln \left(\frac{{\rho }_i}{{\rho }_{f0}}\right)\end{array}$$
     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).

5. 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).

   
   
   Figure 2. Example: foam density distribution in a foam block for an impact simulation

6. 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 ${\sigma }_1^f$ .