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

DoITPoMS Teaching & Learning Packages Brittle Fracture Weibull analysis
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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Weibull analysis

The Weibull approach considers a chain of N links. So the survival of the chain under load requires that ALL the links survive.

Pr (survival of CHAIN) = Pr (ALL links survive)

$$\displaylines{ {F_{\rm{C}}} = 1 - {({S_{\rm{L}}})^N} \cr = 1 - {(1 - {F_{\rm{L}}})^N} \cr} $$

What is FL?

$${F_{\rm{L}}} = 1 - \exp - {\left( {{\sigma \over {{\sigma _{\rm{o}}}}}} \right)^m}$$

$${S_{\rm{L}}} = \exp - {\left( {{\sigma \over {{\sigma _{\rm{o}}}}}} \right)^m}$$

$$\displaylines{ {S_{\rm{C}}} = {\left[ {\exp - {{\left( {{\sigma \over {{\sigma _{\rm{o}}}}}} \right)}^m}} \right]^N} \cr = \exp - N{\left( {{\sigma \over {{\sigma _{\rm{o}}}}}} \right)^m} \cr} $$

Taking logarithms

$$\ln {1 \over S} = - N{\left( {{\sigma \over {{\sigma _{\rm{o}}}}}} \right)^m}$$

$$\ln \ln {1 \over S} = m\ln \sigma - (m\ln {\sigma _{\rm{o}}} + \ln N)$$

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