Thick$-$walled cylinder $($TWC$)$ tests $($as shown in the figure below$)$ is a commonly used experimental technique for determining the mechanical property of rocks under triaxial loading condition in oil industry. For example, in a TWC test on granite samples, two samples are prepared to have a diameter of $50$ mm $,$ height of $100$ mm $,$ with a coaxial cylindrical hole of diameter of $20$ mm $.$ During the test, a uniform confining pressure is provided by hydraulic oil, which is gradually increased to a prescribed value of $10$ MPa for sample $1$ and $20$ MPa for sample $2,$ respectively, and maintained at the prescribed values. Then the axial stress is applied, which is gradually increased until the samples break. The measured failure axial stress is $220$ MPa for sample $1,$ and $280$ MPa for sample $2,$ respectively.

Sample $1$ and $2$ are made of the same granite material with Young's Modulus of $50$GPa and Poisson's ratio of $0\mathrm{.}25.$

find the axial, radial, and tangential stress and strain distribution $($as a function of r $,$ where $r=$ distance to the center$)$ in this thick wall cylinder test at failure pressure for each sample.

assume that the failure could be predicted using Mohr$-$Coulomb criterion, derive the MohrCoulomb properties: cohesion $c,$ and angle of internal friction $.$

Discuss further $($by applying a parametric study$)$ the influence of the confining pressure on the failure stress of the thick wall cylinder.