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9.10: When does the sample fail completely?

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    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.

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