A new statistical distribution function for fracture toughness

Abstract
A statistical view of fracture at cracks is presented that is also appropriate for failures in singularity-dominated, self-similar fields other than those at crack tips. Consideration of the behaviour of the distributions of stress and strain near crack tips results in the development of a new two-parameter distribution function for the probability of failure. The two fundamental premises on which the function is based are, firstly, that the failure of any part of the material near to the crack tip leads to total failure along the whole crack front or at least represents total failure; and secondly, that the variability of strength in material is due to micro-structural inhomogeneity. The new function is tested by means of several large sets of toughness data from other workers, and is found to give with only two parameters better fits than can the three-parameter distribution function of Weibull. The Weibull function is capable of giving reasonable fits in its extremely flexible three-parameter form, but that very flexibility means that these fits may be no more than descriptions without theoretical foundation. It is found also that the new function is applicable equally to ceramics and steels. The very good fits afforded by the new function are further support for previous findings in two basic areas in the science of fracture. Firstly, previous work concerned with the distributions of stress and strain at crack tips and with crack-opening displacement upon which the new function is based, is supported, as is the idea that the crack-opening displacement is fundamental in determining the possibility and prob­ability of failure. Secondly, the present work is in agreement with widely accepted ideas concerned with stress-controlled mechanisms of failure in materials.