Simplify the following expression (Recall e is the alternating symbol and is the Aronecker delta): pq3...

 

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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 x₁-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 u₁(x1, x2) and u₂(x1, x2) if u₁1 (0,0) = u2(0,0) = 1 2 [3 4] [1-21 [₁ (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 -M² 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? 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 x₁-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 u₁(x1, x2) and u₂(x1, x2) if u₁1 (0,0) = u2(0,0) = 1 2 [3 4] [1-21 [₁ (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 -M² 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?

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Ill break down and address each part of your question separately 2 To find the stress tensor in a new coordinate system after two rotations we can use the transformation law for secondorder tensors Le... View the full answer