9.5: Can we calculate the energy changes?

In the beginning, we calculated the force, and hence the stress, for failure by considering the energy changes as the atoms moved apart.

Why not consider cracks like this?

The first person to do this was A.A. Griffith in 1920. He considered a body in tension, but tension is rather complicated, so let’s consider another stress-state – wedging.

Wedging is what you do when you split a piece of wood, or try to peel paper off the wall by getting your fingernail underneath it. It’s shown in the animation where a block of thickness h, is being driven in under a layer of thickness d.

So why does the energy, U, of the body change with crack length?

Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.

From the animation you can see there are 3 contributions:

Pushing the wedge in causes the peeling layer to bend more, increasing the strain energy, U_E. And only the layer between the crack tip and the point where the layer touches the wedge is bending. Using the TLP on beam bending (a more detailed derivation can be found here) gives U_E as

\[U_{\mathrm{E}}=\frac{E d^{3} h^{2}}{8 c^{3}}\]

What the about work done by the applied force, U_F?

We can see that the action of the wedge is the same as a force applied at the point where the wedge touches the peeling ligament. So as the crack grows the force moves sideways, i.e. perpendicular to its line of action, so no work is done. So

\[U_{F}=0\]

Finally, as the crack length changes, the energy of the surfaces, U_S, changes, giving

\[U_{\mathrm{s}}=c R\]

where R is known as the fracture energy, the energy required to create new surfaces.

The total energy, U, of the body is just these 3 terms added up.

So what does this have to do with cracking?

Again explore the animation above and start with the default values.

What will be the crack length, c*, where the body has the lowest energy? We can find c* by differentiating the expression for u with respect to c and setting dU / dc = 0.

What will happen if the crack is shorter than this?

The energy of the body would increase if c < c*, so the crack would grow until c = c*, and then stop.

This is called stable cracking – it happens where there is a stable equilibrium in the energy change with crack length.

But what happens if the initial crack length is longer than the equilibrium value, or if we pull the wedge out?

What is predicted?

What happens if the wedge is pulled out?

If we pull the wedge out, the crack length should decrease; in other words, the crack should heal. To test this, Obreimoff cleaved crystals of muscovite mica by wedging. On pushing the wedge in he found that the crack grew at a constant distance ahead of the crack tip, as predicted, and estimated a value of R for mica close to that measured elsewhere.

Then he pulled the wedge out. His first tests in air showed no crack healing.

Then he tried testing in vacuum – healing really did occur.

Obreimoff thought that this occurred because in air chemical groups attached themselves to the fresh mica surfaces, just as fluff sticks on a toffee in your pocket, so that the surfaces no longer stick.

This showed cracking really was thermodynamic in nature – totally different from the critical stress ideas.

This reversibility, crack healing, seems a strange idea at first. But we know surfaces stick together – that is the cause of friction – and there have been many observations of this on metal sheets, on probes in atomic force microscopes. And it is what we would expect if we think of a surface in terms of unfulfilled chemical bonds, desperate for other electrons.