A solid cylinder of circular cross$-$section is compressed axisymmetrically to half its initial height. Calculate the three TRUE STRAINS: AXIAL, DIAMETRAL and CIRCUMFERENCIAL. $(5$ POINTS$).$ Hints: First find the axial true strain. Then use the concept that the summation of the three true strains $($log strains$)$ is zero. Then show that the diametral and the circumferential strains are equal in their magnitudes. Now find both the diametral and circumferential strains. Good luck in solving it$!$

PROVE Mathematically that if the same cylinder is under axial tension, the optimum axial true stain is equal to the strain hardening exponent$/$component$,$ n $,$ of the workpiece material properties when the external axial force, $F,$ reaches a maximum for the useful deformation of the cylinder. $(5$ PTS$).$ Hints: Draw the true stress$-$strain curve of a ductile material; show on it where the force F is maximum and find the corresponding optimum true strain. Use also the concepts Volume Constancy and Ludwik's Law $($stress$-$strain relationship$).$