The Cauchy stress tensor T at a point x has components Tij relative to an orthonormal basis {eq, ez,P3}, where P2 is one of the 3 principal axis of stress. (a) Show that T33 is the principal stress Tz corresponding to P3 and that T13 = T23 = 0. Consider now that the stress tensor T has the following components 200 400 0 400 800 0 0 0 3 (b) In a 2D Oxy coordinate system, mark the following two points: A = (T11, T12) and B = (T22,-T12). Connect A and B with straight line of length L, and draw a circle M containing both points. Compute the diameter L. (C) Label the point of intersection of line L and the Ox axis as C = (Tavg, 0) and obtain Tavg (d) Label the two points of the circle that intersect the Ox axis as D = (T1,0) and E = (T2,0) and obtain T and T, with respect to T11, T22, and T12 that you picked. (e) Show that T. and T, are actually the other two remaining principal stresses of the stress tensor T. (f) Compute the angle 8 between straight line L and the Ox axis. Show that the other principal axis of stress P. and P2 can be obtained by applying a counter-clockwise rigid body rotation of angle 8/2 to e, and ez respectively. The Cauchy stress tensor T at a point x has components Tij relative to an orthonormal basis {eq, ez,P3}, where P2 is one of the 3 principal axis of stress. (a) Show that T33 is the principal stress Tz corresponding to P3 and that T13 = T23 = 0. Consider now that the stress tensor T has the following components 200 400 0 400 800 0 0 0 3 (b) In a 2D Oxy coordinate system, mark the following two points: A = (T11, T12) and B = (T22,-T12). Connect A and B with straight line of length L, and draw a circle M containing both points. Compute the diameter L. (C) Label the point of intersection of line L and the Ox axis as C = (Tavg, 0) and obtain Tavg (d) Label the two points of the circle that intersect the Ox axis as D = (T1,0) and E = (T2,0) and obtain T and T, with respect to T11, T22, and T12 that you picked. (e) Show that T. and T, are actually the other two remaining principal stresses of the stress tensor T. (f) Compute the angle 8 between straight line L and the Ox axis. Show that the other principal axis of stress P. and P2 can be obtained by applying a counter-clockwise rigid body rotation of angle 8/2 to e, and ez respectively