Question $4(25$ points$)$

A brittle material with a single spherical void of $10$ microns in diameter is placed in uniaxial compression. The sample can be modeled as an infinite plate.

a$.(10$ points$)$ Explain how the sample will fail in terms of crack initiation and propagation when no confining stresses are present.

Suppose that the material failed at a compressive load of $20$ MPa $,$ where a tensile confining stress of $2$ MPa was present.

b$.(10$ points$)$ Comment on how the sample would have failed and any similarities$/$differences with the behavior explained in part $($a$).$

c$.(5$ points$)$ Calculate the critical crack length at which point the sample failed spontaneously.