Mechanics of Honeycombs - Plateau Stress

Relative Density
Elastic Behaviour
Plateau Stress - Elastomers, Elastic Plastic
Crushing and Fast Fracture in Brittle solids.
Now that we have established approximations for the in-plane elastic properties of the honeycombe we next need to determine the yield, fracture or plateau stress for the honeycomb. The actual behaviour of the honeycomb in the second stage of deformation will depend on the type of material from which the honeycomb is made.

Elastomeric Materials


The plateau of the stress strain curve of a honeycomb made from an elastomeric material in compression in the X2 direction is associated with the onset of elastic buckling of the thin walls that are aligned nearly parallel to the applied stress - and is simply the point at which the load acting on the end of the column exceeds the Euler buckling load




The factor n describes the rotational stiffness of the end of the column where the 3 cell walls meet. n can vary from 0.5 in a system which is completely free to rotate to 2 in one in whch the ends are held rigid. For regular hexagons the value is close to 0.69. The load on the end of each column is simply



so the elastic collapse stress in the X2 direction is simply...
for regular hexagons

In the X1 direction the walls continue to bend. In tension, buckling cannot occur and the cell walls just continue to bend andthe cell volume decreases - this leads to a rise in apparent density and hence the stress continues to climb because the number of cell walls per unit area perpendicular to the applied stress increases with increasing strain.

Metals and Plastic Polymers

In an elastic plastic solid the bending moment on the cell walls will eventually reach the point where it becomes fully plastic and the cell wall “gains” a plastic hinge which will slowly fold over with iincreasing strain at an almost constant stress - polymers, or slowly rising stress if there is significant work hardening - metals. On way to determine the stress at which the cell wall plastic hinge develops is to equate the work done by the external stress moving the cell walls inwards with the work done actually bending the cell wall..



The force P, acting on the end of the cell wall in the X1 direction is simply



The distance moved in the X2 direction on each side of the cell is



if f is small (cos(f)=1 and sin(f) is f ). Is this a valid approximation?

The work done in each of the 4 plastic hinges is simly the product of the moment (M) and the angular rotation (f) while the work done bythe external force is the product of force (P) and the total distance moved (2X; X on each side of the cell) so..



The plastic moment for a rectanglar section beam is given by



re-arranging we find..



There is a second way to find the plastic collapse stress and that is by setting the maximum bending moment in the cell wall to the moment required for full plasticity in the cross-section of the beam. The geometry is..

The bending moment is



while the moment for full plasticity is



Combining the two equations leads to exactly the same aanswer as before. For regular hexagons the stress for the onset of plasticity reduces to



Since both approaches (work doen and maximum bending moment produce the same result - we shall just use the latter method for determining the onset of plasticity in the hinges when deforming in the X2 direction.



The bending moment is



while the moment for full plasticity is



Combining the two equations gives..



For regular hexagons the stress for the onset of plasticity reduces to



which is exactly the same as for the X1 direction and so the stress for the onset of plasticity in the walls of the cells is isotropic.

Plastic Collapse in Shear with the honeycomb loaded in the X1-X2 plane then plastic hinges form in the vertical walls. The shear force acting on the walls F , creates a moment M, which must exceed the fully plastic moment of the beam



re-arranging



which for regular hexagons reduces to



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