Simplify the following expression (Recall e is the alternating symbol and is the Aronecker delta): pq3 8p1 bq 2. (10 pts.) Assume the following Cartesian stress tensor is isotropic. and the rotation vector is What is the stress tensor in a new coordinate system found through a rotation of 57 about the x-axis and then a rotation 13 about the 3-axis? 3. (25 pts.) Consider the following strain field in a loaded body: Jij -3 7 eij lij = = 14 -3 7 -2 11 11 4 = (a) Does this strain field produce continuous, single-valued displacements? Why or Why not? (b) Determine the displacement components u(x1, x2) and u(x1, x2) if u1 (0,0) = u2(0,0) = 1 2 [3 4] [1-21 [ (0,0) = 0 4. (40 pts.) Consider the following strain and stress fields in Cartesian coordinates: 3 -0.05] -0.05 7 0 0 22x1 ; Jij Pa Determine (a) the principal strains (b) the principal directions associated with the principal strains (c) the traction vector on a surface with a unit normal vector equal to the eigenvector associated with the smallest principal strain. (d) the von Mises stress for that same surface. 5. (10 pts.) Consider a cantilever beam with circular cross-section subjected to a concentrated torsional moment. An elementary Mechanics of Materials Theory analysis gives the stress field within the beam as Jij MzMy 0 0 My 0 0 0 -M where M is the applied moment and I is the polar moment of inertia. Does this stress field satisfy equilibrium if body forces are neglected? What does this say about the assumptions made for the Mechanics of Material Theory?