# Derivation of yield stress in the honeycomb

Here we estimate the stress at which the honeycomb starts to deform irreversibly where some of the faces are aligned parallel to the loading axis and the predominant contribution to the strain comes from the bending of the diagonal faces. (see here) The thickness of the cell wall is *t*, its through-thickness width is *w* and the length of each of the hexagonal faces is *l*.

From beam bending theory, the maximum stress, σ_{max}, in a cantilevered beam of length *L* due to a force, *W*, applied normal to the beam is

\[{\sigma _{{\rm{max}}}} = \frac{{6{\rm{ }}W{\rm{ }}L}}{{w{\rm{ }}{t^2}}}\]

Now in our situation *W* = *F* cos θ and *L* = *l*/2, which gives smax as

\[{\sigma _{{\rm{max}}}} = \frac{{3{\rm{ }}F{\rm{ }}l}}{{w{\rm{ }}{t^2}}} \cdot \cos \theta \]

We assume that yielding will start when σ_{max} = σ_{Y}, giving the applied force, *F*, at which yielding starts as

\[F = \frac{{w{\rm{ }}{t^2}}}{{3{\rm{ }}l{\rm{ }}\cos \theta }} \cdot {\sigma _{\rm{Y}}}\]

Now the applied force *F* acts over an area *wl *cos θ, so that the stress, σ, can be expressed as

\[\sigma = \frac{F}{{w{\rm{ }}l\cos \theta }}\]

Substituting for *F* gives the applied stress at which the honeycomb starts to yield in terms of the yield stress of the cell wall material, σ_{Y}

\[\sigma = \frac{1}{3} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot \frac{1}{{{{\cos }^2}\theta }} \cdot {\sigma _{\rm{Y}}}\]

For a regular hexagon, where θ = 30°, this becomes

\[\sigma = \frac{4}{9} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot {\sigma _{\rm{Y}}}\]

At this value of σ, yielding begins at the surfaces of the cell wall, at the cantilevered end of the beam where the stresses will be greatest. For plastic yielding to spread through the thickness of the cell wall requires that the stress is increased by a factor of 1.5, giving a macroscopic flow stress of

\[\sigma = \frac{2}{3} \cdot {\left( {\frac{t}{l}} \right)^2} \cdot {\sigma _{\rm{Y}}}\]