Limit analyses for surface crack instantaneous propagation angles in elastic hertzian field

This paper presents the limit analyses of instantaneous crack propagation angle in the classical linear-elastic Hertzian field. The formation and propagation of Hertzian cone crack have been studied extensively in the open literature. Many experiments have been carried out on soda-lime glass under static loading. The observed final Hertzian cone angle is usually around 22°, and it tends to increase with increasing indentation velocity, The goal of this paper is to use the crack tip stress field derived in linear elastic fracture mechanics (LEFM) to bound the instantaneous surface ring crack extension angles in the Hertzian field. The upper and lower bounds for the instantaneous crack extension angle are expressed in terms of the ratios of the mode I and mode II stress intensity factors at the crack tip field. It is shown that under a pure mode I dominant stress field, instantaneous crack growth would follow the original surface crack direction; and under a perfect mode II stress field, the instantaneous crack extension angle has a limiting angle of 19.5°. The actual instantaneous crack extension angle will fall between these two limit states depending on the ratio of K_I/K_II at the crack tip.