Is there a failure criterion/model for brittle spheres?

Hi, everyone

I am a student. I am  dealing with a failure problem of brittle spheres. These have a lot of flaws inside.I did some compression tests. Single spheres are crushed by a pair of parallel plates. I got the crush load distribution (in terms of critical contact force). 

My question is:

is there a failure criterion/model which relates to load(critical contact force), Youngs moduli and Poisson ratios of shpere and plates, radius and friction coefficient?

 Unfortranitaly, I can't get the friction coefficient between plate and sphere which influences the surface stress of spheres significantly as shown in ABAQUS simulation.

 Both theoretical and empirical models are welcome.  

I will appreciate any reply. 

shuo ZHAO