Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Teaching & Learning Packages Brittle Fracture Another way of expressing the energies
Brittle Fracture AimsBefore you startIntroductionWhen do atomic bonds break?Why do cracks weaken a material?Inglis and the crack tip stress ideaCan we calculate the energy changes?What about tension?Another way of expressing the energiesAnother way of calculating the energiesWhy bother if they are the same?Coping with a scatter in strengthSimulation of Weibull modulus experimentWhen does the sample fail completely?Sub-critical crack growth and R-curvesSummaryQuestionsGoing further
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Another way of expressing the energies

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

$${{{\rm{d}}U{\rm{(}}c{\rm{)}}} \over {{\rm{d}}c}} = 0$$

Breaking our energies into the two different types, gives

$$ - {{{\rm{d}}{U_{\rm{M}}}} \over {{\rm{d}}c}} = {{{\rm{d}}{U_{\rm{S}}}} \over {{\rm{d}}c}}$$

From our expression for cracking in tension

$${{{\rm{d}}{U_{\rm{S}}}} \over {{\rm{d}}c}} = 2R$$

and

$${{{\rm{d}}{U_{\rm{M}}}} \over {{\rm{d}}c}} = - {{2\pi {\sigma ^2}} \over E}{\rm{ }}c = 2G$$

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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