# 16.4: Yielding and Plateau Behaviour

The aluminium honeycomb will start to plastically deform if the stress in the faces anywhere exceeds the flow stress, σY, of the aluminium cell wall. We have already shown that the predominant contribution to the elastic strain is the bending of the diagonal faces. (see here). Furthermore we could estimate this if each face were considered to be made up of two beams, each of length l/2, cantilevered at the end connected to the vertical cell wall and acted upon by a force of magnitude F cos θ, where F is the force applied at the ends of the sample and θ is the angle between the diagonal face and the horizontal. It is clear then that the stress will be a maximum where the moment is greatest, that is at the vertices of the hexagonal cells.

It can be shown that the applied stress, σ, when the maximum stress in each face reaches the flow stress, σY, of the material making up the cell walls is given by (derivation)

$\sigma=\frac{4}{9} \cdot\left(\frac{t}{l}\right)^{2} \sigma_{Y}$

Using the measured values of t ( = 0.09 mm), l ( = 6.30 mm) and σY ( = 100 MPa), predicts the yield strength of the honeycomb, σ, to be 9 kPa, somewhat lower than the measured value of 15 kPa. The stress we have estimated is the stress at which plastic flow will start in the outer surfaces at the cantilevered point. To enable plastic flow to spread through the thickness of the cell face requires that the stress is increased further by a factor of 1.5, giving a macroscopic flow stress of 13.5 kPa, much closer to the measured value.

Once the material has started to yield the cell walls begin to collapse. This occurs at an approximately constant stress until the cell walls impinge on one another when the stress begins to rise more rapidly with increasing strain.