On the theory of Hertzian fracture

Abstract

The fracture of a brittle solid under a spherical indenter is the best studied case of fracture in a strongly inhomogeneous, well defined, stress field. Two principal topics are discussed, the path of a crack in a field of non-uniformly directed stress, and the stability of cracks of various length when the prior stress on the crack path is non-uniform. For the first, it is shown that the crack growth should,to a first approximation, be orthogonal to the most tensile principal stress, and thus correspond, in a torsion-free stress field, to a surface delineated by the trajectories of the other two principal stresses: while, to a second approximation, the crack should deviate from this path by having a larger radius of curvature at every bend, thus exhibiting a pseudonertia even in slow growth. This is in accordance with the known experimental facts about the Hertzian crack, particularly the fact that the crack at the surface forms systematically outside the edge of the circle of contact, at which the maximum tensile stress occurs. On the second question, it is found that there are four crack lengths,c0,c1,c2,c3, corresponding to stationary values of energy.c0andc2represent unstable equilibria, and diminish with increasing load; cx and c3 represent stable equilibria and increase with increasing load. With small indenters,c0soon becomes less than the size of pre-present surface flaws, and an unobserved shallow ring crack of depthc1is produced: the critical condition for observed fracture is then the merging ofc1withc2, allowing unstable growth to the cone crack of depthc3. This explains Auerbach’s law, that the critical load for production of a cone crack is proportional to the radius,r, of the indenter sphere. With larger indenters, of several centimetres radius for a typical case,c1, andc2merge and disappear beforec0exceeds the size of pre-present flaws. The critical load for cone fracture then becomes nearly propor¬tional tor2, as observed. The previous calculations of Roesler (1956a, b) relate to the second stable crack dimension,c3, though his energy scaling principle is also applicable to the critical condition at whichc1andc2merge. The Hertzian fracture test, within the validity range of Auerbach’s law, affords a means of measuring surface energy at the fracture surface independent of knowledge about the pre-present flaws.