A micro-mechanically motivated model for the strength of heat-treated glass

In structural design the bending strength of heat-treated glass is in general calculated as the simple sum of the characteristic values of the strength of annealed glass and of the thermally-induced prestress, considered as the 5% quantiles of the corresponding statistical distributions. However, the probability that two stochastic variables attain small values simultaneously is quite low; therefore, it is expected that the 5% quantile of the heat-treated glass strength is higher than the simple sum of the 5% quantiles of the two constituent distributions. Here, we theoretically confirm this result by assuming a two-parameter Weibull distribution for the population of annealed glass strengths and a Gaussian distribution for the thermal stresses. Although recent studies have confirmed that the two-parameter Weibull distribution cannot properly interpret the left-hand-side-tail of the annealed-glass strength population, it is here shown that the statistical competition with the surface prestress lowers the importance of a very precise interpretation of the left-hand-side tail. Remarkably, since glass strength is governed by the opening of surface cracks in mode I, the expected statistical interference is strongly affected by the type of stress state. If the stress state induced by external actions is equibiaxial, all cracks have the same opening stress, but if it is uniaxial, many cracks will remain closed under the sole effect of the prestress, which is in general uniform and equibiaxial. The higher the surface compression is and the closer to the uniaxiality the stress state is, the higher the number of "not-active" cracks will be. We believe that this study will promote and guide the design of ad hoc experimental campaigns for experimental validation.