Hertzian Fracture Experiments on Abraded Glass Surfaces as Definitive Evidence for an Energy Balance Explanation of Auerbach's Law

Abstract

The question of the physical interpretation of Auerbach's law, that the critical load for production of a Hertzian cone fracture is proportional to the radius of the indenting sphere, has aroused interest in recent years (i) because of its implications concerning the validity of certain brittle fracture criteria and (ii) because of its potential use as a means for measuring fracture surface energies. Two distinct schools of thought, one based on a flaw statistical model and the other on an energy balance concept, have emerged as a result of various attempts to account for this law. This paper describes a theoretical and experimental study aimed at testing the validity of each of these two approaches. Whereas quantitative Hertzian fracture tests have hitherto been extensively made only on surfaces of carefully handled specimens, such as commercial plate glass in the as-received state, in the present case they are made on plate glass surfaces treated by an abrasion process. With as-received surfaces the obscure nature of the distribution of surface flaws, from which the cone cracks initiate, largely precludes a conclusive comparison between the two theoretical approaches, while for abraded surfaces the experimentally justifiable assumption that the flaw distribution is ``uniform'' leads to simple but widely conflicting predictions from the two approaches, thus providing the basis for a definitive experiment. According to a flaw statistical argument, in conjunction with an empirical critical stress criterion for fracture, Auerbach's law is predicted to break down for tests on uniformly damaged surfaces, with the critical load for cone fracture becoming a sensitive function of the depth of the damage layer. On the other hand, a stepwise application of Griffith's energy balance criterion for fracture to the growth of a cone crack through the inhomogeneous Hertzian stress field predicts strict adherence to Auerbach's law within limits of indenter size, with the Auerbach constant of proportionality in this law being independent of flaw statistics. Static and impact tests, made with steel balls ranging from 0.08 to 1.9 cm in radius on plate glass surfaces abraded with slurries of Nos. 240, 320, 400, and 600 SiC abrasive powder, confirm the essential constancy of Auerbach's law, thereby providing strong evidence for the energy balance explanation. Moreover, it is found that the abrasion treatment leads to a drastic reduction in the scatter in results; as a result variations of less than 10% in the fracture surface energy, which is proportional to the Auerbach constant, should be detectable in experiments performed on a given material under different test conditions.