Mechanics of Honeycombs

The overall mechanical behaviour of a honeycomb tested in-plane depends on the type of material from which the honeycomb is made, and whether deformation occurs in tension or compression. However, the stress strain curves have three basic regions, a linear elastic portion, a pleatea and a densification or fracture region.

Relative Density
Elastic Behaviour
Plateau Stress - Elastomers, Elastic Plastic
Crushing and Fast Fracture in Brittle solids.

Compression

The three main stages of deformation, when a stress is applied in-plane to a honeycomb are..
  1. Elastic bending of cell walls
    1. Generalised Elastic behaviour of an Elastomer with temperature,
    2. Generalised Elastic behaviour of an Polymer with temperature
    3. Generalised Elastic behaviour of an Glassy/Brittle Polymer with temperature
  2. Critical strain is attained and the cells start to collapse there are 3 possible collapbse processes that are determined by the material from which the cell walls are constructed.
    1. elastic buckling of the walls,
    2. plastic yielding of walls,
    3. brittle fracture
    During this stage the stress plateaus at the elastic buckling stress, the yield stress, or oscillates about a plateau as a series of brittle fracture events take place in successive cell walls.
  3. Cell walls touch - densification, stress rises rapidly
The Stress-Strain plots are similar for all material types



Fig.1 In-plane stress-strain curves for compression of elastomeric, elastic-plastic and elastic-brittle cell walls

Tension

The in-plane tensile behaviour of the honeycomb is simpler. There are two main stages..
  1. Elastic bending of the cell walls - just in the opposite sense of the compression behaviour
  2. In tension there is no buckling of the thin walls and so the second stage depends on the material
    1. For elastomers, the cell walls continue to rotate into the tensile direction and the stiffness rises because of densification..
    2. plastic yielding of the hinges results in deformation at a constant stress in the material, but because the walls of the cell are approaching there is a gradual increase in density so the stress rises.
    3. hinges or walls will snap in a brittle fashion.
Any increase in density due to the number of cell walls per unit volume increasing (gradual densification) results in increase in elastic moduli, plastic yield stress and brittle fracture stress - just as in compression.


Fig.2 In-plane stress-strain curves for tension of elastomeric, elastic-plastic and elastic-brittle cell walls

Relative density of a generic honeycomb whose vertical sided have length h, whose angled sides have length l and the angle between h and l is 90+q, is calculated as follows;



first determine the area of cell walls and area of cell.



should note that each cell wall is shared by two cells, hence the area of the walls is divided by 2.




when the cells are regular hexagons, then q=30° and h=l, the relative density reduces to


Linear Elasticity

There are 5 elastic constants for in-plane deformation, E1,E2,n12,n21 and G12, however, only 4 are independent since . E1 and E2 are determined as follows, using the geometry set out below where the remote stresses, s1 and s2 act in compression in directions X1 and X2. In the 1 direction the unit of area is in the 2 direction the unit of area on which the stress acts is where b is the depth (out of plane) of the honeycomb.



deformation in the X1 direction...


we can calculate the bending moment M, on the angled beam, from which we can determine the deflection, d,


The component of deflection in the X1 direction is just acting over a length hence the strain in the X1 direction is..



Since modulus is just the ratio of stress to strain, we can re-arrange the above equation to be


where Es is the modulus of the solid of which the beam is made. When the honeycomb consists of regular hexagons in which h=l and q is 30° then the above equation reduces to


Now let’s repeat the exercise for a stress applied in the X2 direction


The bending moment, M and deflection d are...



The component of deflection in the X2 direction is just acting over a length hence the strain in the X2 direction is..



Since modulus is just the ratio of stress to strain, we can re-arrange the above equation to be


where Es is the modulus of the solid of which the beam is made. When the honeycomb consists of regular hexagons in which h=l and q is 30° then the above equation reduces to


which is the same as for E1 and so a regular hexagonal honeycomb is isotropic: they are in practice!

We can now evaluate poisson’s ratio by taking the negative strain in the normal direction and dividing by the strain in the loading direction..



which for a regular hexagon is equal to 1, thus . For non-regular hexagons



Finally, we need to establish the inplane shear modulus G12... In the drawing below we apply a shear to th ehoneycomb and assume that the shear actually takes place by the vertical wall buckling over, while the apeces A,B and C remain fixed relative to each other with only the point D moving laterally.


The bending moment at the bottom of the arm DB is simply the product of force F and distance h/2. This is balanced by the sum of the bending moments in arms AB and BC so . The deflection in the arms AB and BC is then and the angle through which the beam rotates is . The deflection of point D in the horizontal plane is made up of the deflection due to the rotation of the hinge through an angle q, because of the rotation in AB and BC which must be equal to each other as well as that in the arm BD; plus the bending of an end loaded cantilever beam BD due to the force F hence



If q is small the q =tanq so..



The opposite end of the arm BD, point E moves a horizontal distance twice that of the point D and so the shear strain is , while the remote stress is , giving the shear modulus as



when the cells are regular hexagons then


which obeys the standard relation for isotropic solids recalling that v=1
Continue with determination of plateau stresses...