Flexural strength of sapphire: Weibull statistical analysis of stressed area, surface coating, and polishing procedure effects

The results of fracture testing are usually reported in terms of a measured strength, σ_M=σ_i±Δσ_i, where σ_i is the average of the recorded peak stresses at failure, and Δσ_i represents the standard deviation. This “strength” does not provide an objective measure of the intrisic strength since σ_M depends on the test method and the size of the volume or the surface subjected to tensile stresses. We first clarify issues relating to Weibull’s theory of brittle fracture and then make use of the theory to assess the results of equibiaxial flexure testing that was carried out on a variety of sapphire specimens, at three mechanical test facilities. Specifically, we describe the failure probability distribution in terms of a characteristic strength σ_C—i.e., the effective strength of a uniformly stressed 1 cm² area—which allows us to predict the average stress at failure of a uniformly loaded “window” if the Weibull modulus m is available. A Weibull statistical analysis of biaxial-flexure strength data thus amounts to obtaining the parameters σ_C and m, which is best done by directly fitting estimated cumulative failure probabilities to the appropriate expression derived from Weibull’s theory. We demonstrate that: (a) measurements performed on sapphire test specimens originating from two suppliers confirm the applicability of the area scaling law; for mechanically polished c- and r-plane sapphire, we obtain σ_C≃975 MPa, m=3.40 and σ_C≃550 MPa, m=4.10, respectively. (b) Strongly adhering compressive coatings can augment the characteristic strength by as much as 60%, in accord with predictions based on fracture-mechanics considerations, but degrade the Weibull modulus, which mitigates the benefit of this approach. And (c) Measurements performed at 600 °C on chemomechanically polished c-plane test specimens indicate that proper procedures may enhance the characteristic strength by as much as 150%, with no apparent degradation of the Weibull modulus.