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9.7: Another way of expressing the energies

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  • What we have been doing so far is to calculate how U varies with c, and then find the equilibrium value of c by differentiating U with respect to c. Looking at the terms they fall into two types:

    Those that come from the loading system, UE and UF: let’s add these together and call the sum UM

    • These give the driving force for cracking

    Those that are associated with the material, US

    • This gives the resistance to cracking

    Equilibrium will occur when

    \[\frac{\mathrm{d} U(c)}{\mathrm{d} c}=0\]

    Breaking our energies into the two different types, gives

    \[-\frac{\mathrm{d} U_{\mathrm{M}}}{\mathrm{d} c}=\frac{\mathrm{d} U_{\mathrm{s}}}{\mathrm{d} c}\]

    From our expression for cracking in tension

    \[\frac{\mathrm{d} U_{\mathrm{s}}}{\mathrm{d} c}=2 R\]


    \[\frac{\mathrm{d} U_{\mathrm{M}}}{\mathrm{d} c}=-\frac{2 \pi \sigma^{2}}{E} c=2 G\]

    where G is the energy per unit area of crack and so is often called the strain energy release rate, or the crack driving force. Now the turning point occurs when

    \[G = R\]

    This is our condition for cracking, that the crack driving force, G, equals the fracture resistance of the material, R.

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