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Introduction
Energy-based analysis of fracture leads to definition of the strain energy release rate to characterise the loading on a crack and of the critical energy release rate as a material toughness property.

The theoretical stress approach to fracture
From ME1 Materials we know that two atoms (or ions) can attract or repel each other, but they remain two entities. Attractive forces exist at long range and short range, but repulsive forces are only significant at short range.
Therefore if we could measure the force between two atoms as the distance between them was varied, we would get a net attractive force at large (by atomic standards) distances and a net repulsive force at small distances, with zero force at some equilibrium distance, ro.

In this context it is also often very useful to consider the potential energy in the bond which is simply given by:

What we want to know is the magnitude of the force and hence the stress required to cause a crystalline solid to fracture across a particular crystallographic plane. To carry out such a calculation we need to consider each atomic bond as a spring element initially at a separation of ro.
Consider the simplest case of 2 layers of atoms, packed in square, on planes perpendicular to the applied force (Fig 2 below). Consider what happens when we try to pull apart two atoms, one in each layer, so that the interatomic spacing increases from ro to ro + ?r:

If the bond is stretched to the max value Fm of F, it will be unstable and will break if the stress continues to increase. Therefore an estimate