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# Continuum properties of aluminium honeycomb

Honeycomb cores are often used in composite structures as an alternative core material to e.g. polymeric foams or end-grain balsa.

In FEA we want to be able to treat honeycomb as a continuous material instead of having to model individual cells. Otherwise even simple FEA models involving honeycomb would become unmanageably large.

Luckily, formulas exist with which we can calculate the nine engineering constants for a honeycomb.

In this example we will cover hexagonal honeycomb with cell size 1/4” made from EN AW-3003 aluminium foil of 0.0030” thickness. The material properties of this foil are:

• `E` = 69500 MPa
• `ν` = 0.33
• `ρ` = 2730 kg/m³

The length of the edge of such a cell is named `L`. The thickness of a single wall is `t`. The cell size `S` is then 2·L·cosθ. The density of the honeycomb is given as 5.2 lb/ft³, or 83 kg/m³.

We can now calculate the properties of the honeycomb as follows:

```In [1]: %precision %.3g
Out[1]: '%.3g'

In [2]: import math

In [4]: sθ = math.sin(θ)
Out[4]: 0.5

In [5]: cθ = math.cos(θ)
Out[5]: 0.866

In [6]: L = 25.4/(4*2*cθ)
Out[6]: 3.67

In [7]: t = 30/10000 * 25.4
Out[7]: 0.0762

In [8]: r = 0;

In [9]: E = 69.5e9;  # Pa

In [10]: ν = 0.33;

In [11]: G = E/(2*(1+ν))
Out[11]: 2.61e+10

In [12]: Gxz = (r*cθ*sθ+r*θ+L*cθ**2+L)/(L*sθ*(1+cθ))*(t/L)*G  # Pa
Out[12]: 1.02e+09

In [13]: Gyz = -(r*cθ*sθ-r*θ+L*cθ**2-L)/(L*sθ*(1+cθ))*(t/L)*G  # Pa
Out[13]: 1.46e+08

In [14]: Gxy = (1+sθ)/(3*cθ)*(t/L)**3 * G  # Pa
Out[14]: 1.35e+05

In [15]: t2θ = math.tan(θ)**2
Out[15]: 0.333

In [16]: Ex = (cθ/((1-ν**2)*sθ*sθ*(1+sθ)*(1+(2/(1-ν)+1/t2θ)*(t/L)**2)))*(t/L)**3*E
Out[16]: 1.61e+06

In [17]: Ey = Ex;

In [18]: Ez = (t/L)*E  # Pa
Out[18]: 1.44e+09

In [19]: ν_xy = ((1+sθ)*sθ)/(cθ**2)*(1+(1.4+1.5*ν)*(t/L)**2)/(1+(5.4+1.5*ν)*(t/L)**2)
Out[19]: 0.998
```

Is is assumed that `ν_xz` and `ν_yz` are both zero.

The formulas used for the continuum properties are from “Finite Element Analysis of Honeycomb Sandwich Panel for Continuum Properties Evaluation and Core Heigh Influence on the Dynamic Behavior” by Rahman et. al.

We can now formulate material properties for CalculiX/Abaqus:

```*MATERIAL, NAME=Mhc_alu_5.2
*ELASTIC,TYPE=ENGINEERING CONSTANTS
1.613e6,1.613e6,1.445e9,0.998,0.0,0.0,1.354e5,1.019e9
1.455e8
*DENSITY
83
```

In the same way, we can calculate the properties of other densities:

```*MATERIAL, NAME=Mhc_alu_3.6
*ELASTIC,TYPE=ENGINEERING CONSTANTS
4.786e5,4.786e5,9.630e8,0.999,0.0,0.0,4.013e4,6.791e8
9.701e7
*DENSITY
58

*MATERIAL, NAME=Mhc_alu_2.5
*ELASTIC,TYPE=ENGINEERING CONSTANTS
2.020e5,2.020e5,7.223e8,1.000,0.0,0.0,1.693e4,5.093e8
7.276e7
*DENSITY
40

*MATERIAL, NAME=Mhc_alu_1.8
*ELASTIC,TYPE=ENGINEERING CONSTANTS
5.988e4,5.988e4,4.815e8,1.000,0.0,0.0,5.017e3,3.395e8
4.850e7
*DENSITY
29

*MATERIAL, NAME=Mhc_alu_1.2
*ELASTIC,TYPE=ENGINEERING CONSTANTS
2.527e4,2.527e4,3.611e8,1.000,0.0,0.0,2.116e3,2.546e8
3.638e7
*DENSITY
19
```