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

# 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

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

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