9.7: Another way of expressing the energies
- Page ID
- 30050
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
\[\frac{\mathrm{d} U_{\mathrm{s}}}{\mathrm{d} c}=2 R\]
and
\[\frac{\mathrm{d} U_{\mathrm{M}}}{\mathrm{d} c}=-\frac{2 \pi \sigma^{2}}{E} c=2 G\]
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.