What is the maximum value of torque T

Question

1. (a) The state of stress at a given point in a loaded component is given by $\sigma$_x, $\sigma$_y and $\tau$_xy and the principal stresses at the point are s₁ and s₂. If the new axes at the point are changed to x¹ and y¹ such that state of stress w.r.t. the new axes are $\sigma$_x, $\sigma$_y And $\tau$_x'y'.

(i) Show that $\sigma$_x + $\sigma$_y + $\sigma$_y  - $\sigma$₁ + $\sigma$₂.                                                                                             

(ii) Sketch the Mohr's stress circle for pure shear stress state and hydrostatic state of stress.        

(iii) A cast iron bar of 30 mm diameter and 1 m length is subjected to equal & opposite torque T applied at its ends. If tensile strength of cast iron is 80 MPa, what is the maximum value of torque T? show that plane of failure for the applied sends of torque T.