Dear Erik Carton, let us assume the  factors which determine the development of a ring crack caused by tension stress Sy near the edge of axisymmetric Hertzian contact (sphere – elastic half-space). In theory (Frank) the localization of initial crack is postulated on the boundary of contact area, where we calculate maximal value of stress Sy. But really the radius of initial crack varies in a range 1-2 radius of contact area. This depends on function of distribution of microcracks depth. So, the variation of this radius influence on cone crack angle in brittle material. The development of main crack on the first stage of failure process is non-stable. On the second stage (development of long crack) the crack is following to main stress trajectory. Using this line of reasoning, we may conclude that cone crack angle changes in accordance with structure of indented material and coating on the surface, really, plays some role in sense of crack angle formation. Best regards.     

As Serge V. Shil’ko  says, the cone crack just follows the trajectory of maximum tensile stress beneath the contact. This angle doesn't depend on the values of the moduli of the test material (well, a bit with Poisson's ratio, but that only varies from material to material within a very narrow range anyway).  The only way to change the angle is by modifying the shape of the elastic stress field - you could possibly do that via a misfitting coating, introducing stresses; or by a sideways traction on the indenter, making the stress field asymmetric; or by using an indenter of different elastic properties from the testpiece so that frictional tractions are set up within the contact zone (but the friction coefficient is unlikely to be controllable). Why do you want to change the cone crack angle anyway?

Thanks Steve,

However, according to publications after 1995, the cone angle was shown not to follow the path of maximal tensile stress, but rather a direction (close to it) that maximizes the strain-energy release. The latter actually results in another cone angle.

Would a pre-layer between point contact and substrate help to reduce/avoid the ring crack in the substrate? If the ring crack does not occur, the cone crack can not occur, can it?

@ Serge,

"The radius of initial crack depends on function of distribution of microcracks depth. So, the variation of this radius influence on cone crack angle in brittle material."

Could you expand and explain this issue to me?

Does it help to increase or reduce the amount/depth of microcracks near the surface to het a wider ring crack. Or can this be achieved using a coating material?

"However, according to publications after 1995, the cone angle was shown not to follow the path of maximal tensile stress, but rather a direction (close to it) that maximizes the strain-energy release. The latter actually results in another cone angle."

Fair comment - true enough, but doesn't make much difference to the conclusions of my comments.

"Would a pre-layer between point contact and substrate help to reduce/avoid the ring crack in the substrate? If the ring crack does not occur, the cone crack can not occur, can it?"

Yes, of course if you prevent the ring crack forming, then you suppress the cone crack that develops from it. Since the tensile stress that acts on the surface flaws to initiate the ring crack  is very shallow, a surface layer of a less brittle material, or one that is in compression, will suppress cracking.

BTW, I wouldn't call this sort of contact a "point contact". That would be be a very different stress field (Boussineque type), giving radial cracks. Hertzian contact, leading to ring and cone cracks,  is an elastic contact between blunt (ideally spherical) surfaces.

Dear Erik, you have launched a very interesting discussion! Allow me to develop my answer.

"The radius of initial crack depends on function of distribution of microcracks depth. So, the variation of this radius influence on cone crack angle in brittle material."

"Could you expand and explain this issue to me?"

Fair comment: the crack generation in real heterogenious materials is a stohastic process. A number of microcracks create a stohastic picture of their initial distribution near contact boundary and their depth is stohastic too. 

"Does it help to increase or reduce the amount/depth of microcracks near the surface to het a wider ring crack. Or can this be achieved using a coating material?"

Fair comment: The coating allow us to control the stress distribution in brittle material under consideration. This influence on "initiating" tensile stress as well as on "controlling"  main stress which promote further propagation of macrocrack.

Dear Prof. Steve Roberts, your remarks "and (well, a bit with Poisson's ratio, but that only varies from material to material within a very narrow range anyway)" in this context ma be added. There are unuasial materials, so called, auxetics, having a negative or giant (anysotropic case) values of Poisson ratio. Indentability of auxetics is dramatically differs from usial materials with narrow range of PR. The study of cone angle in auxetics may be perspective in spirit of our discussion. Serge.    

Serge,

Do you know of any auxetic materials that are strong or very stiff?

Yes, these are directionally reinforced materials (composites), particularly, epoxy laminates. See, please, papers of Peel L.D., f.e." Exploration of High and Negative Poisson’s Ratio Elastomer-Matrix Laminates / J. Physica Status Solidi (b) 244, No. 3, 988–1003, Jan. 2007."

I hope also, the reviews attached may be useful for you. Good luck, Serge