9.10: When does the sample fail completely?

It is incorrect to say that failure must occur when

$G = R$

There will be some cracking but complete failure (as in tension) also requires that

$\frac{\mathrm{d}^{2} U(c)}{\mathrm{d} c^{2}}<0$

i.e. the energy is at a maximum, or

$\frac{\mathrm{d} G}{\mathrm{d} c}>\frac{\mathrm{d} R}{\mathrm{d} c}$

In other words failure will be catastrophic when the rate of increase of the driving force with crack growth is greater than the change in R with crack growth, which we have taken as a constant.

Alternatively cracking will be stable when

$\frac{\mathrm{d}^{2} U(c)}{\mathrm{d} c^{2}}>0$

i.e. the energy is at a minimum, or

$\frac{\mathrm{d} G}{\mathrm{d} c}<\frac{\mathrm{d} R}{\mathrm{d} c}$

That is, as the crack grows, the resistance to cracking, R, increases faster than the driving force, G.