How to do a Weibull statistical analysis of flexural strength data: Application to AION, diamond, zinc selenide, and zinc sulfide

For the purpose of assessing the strength of engineering ceramics, it is common practice to interpret the measured stresses at fracture in the light of a semi-empirical expression derived from Weibull's theory of brittle fracture, i.e., ln [- ln(1 - P)] = -m ln(sigma(N)) + m ln(sigma), where P is the cumulative failure probability, a is the applied tensile stress, m is the Weibull modulus, and sigma(N) is the nominal strength. The strength sigma(N), however, does not represent a "true" mea, sure because it depends not only on the test method but also on the size of the volume or the surface subjected to tensile stresses. In this paper we intend to first clarify issues relating to the application of Weibull's theory of fracture and then make use of the theory to assess the results of equibiaxial flexure testing that was carried out on polycrystalline infrared-transmitting materials. These materials are brittle ceramics, which most frequently fail as a consequence of tensile stresses acting on surface flaws. Since equibiaxial flexure testing is the preferred method of measuring the strength of optical ceramics, we propose to formulate the failure-probability equation in terms of a characteritic strength, sigma(C), for biaxial loadings, i.e., P = 1 - exp {- pi (r(o) /cm)(2) [Gamma (1 + 1/m)](m) (sigma/sigma(C))(m)}, where r(o) is the radius of the loading ring (in centimeter) and Gamma(z) designates the gamma function. A Weibull statistical analysis of equibiaxial. strength data thus amounts to obtaining the parameters m and sigma(C), which is best done by directly fitting estimated P-i vs i data to the failure-probability equation; this procedure avoids distorting the distribution through logarithmic linearization and can be implemented by performing a non-linear bivariate regression. Concentric-ring fracture testing performed on five sets of Raytran(TM) materials validates the procedure in the sense that the two-parameter model appears to describe the experimental failure-probability distributions remarkably well. Specifically, we demonstrate that the wide divergence in published CVD-diamond strength data reflects the poor Weibull modulus of this material and must be attributed to the size effect rather than the "quality" of the deposits. Finally, the problem of obtaining correct failure stresses from the measured failure loads is examined in the Appendix.