Mechanics of Honeycombs

Mechanics of Honeycombs - Brittle Fracture

Relative Density

Elastic Behaviour

Plateau Stress - Elastomers, Elastic Plastic

Crushing and Fast Fracture in Brittle solids.

Brittle Failure

Brittle failure can occur in both tension and compression if the cell wall material itself is brittle; materials such as glasses, ceramics, brittle plastics like polystyrene, suffer progressive crushing in compression and fast crack groth in tension. We have already demonstrated that in compression the cell walls bend and buckle, leading to the development of tensile stresses on some portions of the outer surfaces of the cell walls - it is these regions of tension that initiate crack propagation through the cell wall and fracture.

The maximum surface stress on the cell wall is simply

if we ignore the axial compression of the cell wall by the component of the applied stress resolved parallel to the cell wall. The moment was calculated in the previous sections and depends on the direction of loading..

hence, if

the fracture strength of the cell wall material, then

which for regular hexagons both reduce to

so the crushing strength is isotropic and scales with the square of the relative density, as do all the modes ofplastic deformation.

Tensile Failure - Fast Fracture

In tension the failure process is different. At some point a cell wall will fail in tension, adjacent walls will also fail as the load transfers to them and create what is effectively a macroscopic crack - an example is shown below. In a fully solid body, the stress field ahead of the crack is defined by

where r is the distance ahead of the crack tip and

is the remote stress perpendicular to the plane of the crack. The opening of the crack creates a bending moment on the cell wall immediately ahead of the crack tip .

The bending moment in the vertical beam ahead of the crack exists because of the gradient in stress ahead of the crack tip which results in a larger force pulling on the first column above the end of the crack, than in the second and third columns, ie. the moments in the two angled beams do not cancel out leaving a resdual moment rotating the vertical beam ahead of the crack tip. This moment creates a tensile stress on the crack side of the vertical cell wall. The bending stress is supplemented by the actual tensile stress in the cell wall itself.

since we shall assume that M_b<<M_a and l>>t then we can ignore the second term and fracture will commence when the stress reaches the failure stess of the cell wall material. On average

, where the the stress is local to the crack tip hence taking r, the distance ahead of the crack tip as half the width of the cell then...

therefore

remembering that this is an approximation. The same approach can be used for crack propagation when stressed in the X₁ direction.

On average, taking the crack to occupy hallf the width of the unit cell, then

If fracture occurs when the stress exceds the fracture strength of the cell wall material, and l>>t then

For a regular haxagonal cell structure, both equations for the fracture strength in tension reduce to the same value, i.e., the fracture strength is isotropic

If we re-write the above equation in terms of fracture toughness, where

then