## Continuum Mechanics

LMC/DMX/EPFL Prof. Saouma Exam I (Closed notes), March 27, 1998 3 Hours

There are 19 problems worth a total of 63 points. Select any problems you want as long as the total number of corresponding points is equal to or larger than 50.

1. (2 pts) Write in matrix form the following 3rd order tensor Dijk in R2 space. i,j,k range from 1 to 2.

2. (2 pts) Solve for Eijai in indicial notation.

3. (4 pts) if the stress tensor at point P is given by

 10 -2 0 u = -2 4 1 0 1 6

determine the traction (or stress vector) t on the plane passing through P and parallel to the plane ABC where A(6,0,0), B(0,4, 0) and C(0,0, 2).

4. (5 pts) For a plane stress problem charaterized by the following stress tensor

use Mohr's circle to determine the principal stresses, and show on an appropriate figure the orientation of those principal stresses.

5. (4 pts) The stress tensor throughout a continuum is given with respect to Cartesian axes as u

(a) Determine the stress vector (or traction) at the point P(2,1, a/3) of the plane that is tangent to the cylindrical surface x2 + = 4 at P, (b) Are the stresses in equlibrium, explain.

6. (2 pts) A displacement field is given by u = XiX|ei + X2X2e2 + X2X3e3, determine the material deformation gradient F and the material displacement gradient J, and verify that J = F — I.

7. (4 pts) A continuum body undergoes the deformation xi = X1 + AX2, x2 = X2 + AX3, and x3 = X3 + AX1 where A is a constant. Determine: 1) Deformation (or Green) tensor C; and 2) Lagrangian tensor E.

8. (4 pts) Linear and finite strain tensors can be decomposed into the sum or product of two other tensors.

(a) Which strain tensor can be decomposed into a sum, and which other one into a product.

(b) Why is such a decomposition performed?

9. (2 pts) Why do we have a condition imposed on the strain field (compatibility equation)?

10. (6 pts) Stress tensors:

(a) When shall we use the Piola-Kirchoff stress tensors?

(b) What is the difference between Cauchy, first and second Piola-Kirchoff stress tensors?

(c) In which coordinate system is the Cauchy and Piola-Kirchoff stress tensors expressed?

11. (2 pts) What is the difference between the tensorial and engineering strain (Ej, Yij, i = j) ?

12. (3 pts) In the absence of body forces, does the following stress distribution xi, + v(xf — x%) —2vxix2 0

where v is a constant, satisfy equilibrium in the Xi direction?

13. (2 pts) From which principle is the symmetry of the stress tensor derived?

14. (2 pts) How is the First principle obtained from the equation of motion?

15. (4 pts) What are the 1) 15 Equations; and 2) 15 Unknowns in a thermoelastic formulation.

16. (2 pts) What is free energy

17. (2 pts) What is the relationship between strain energy and strain?

18. (5 pts) If a plane of elastic symmetry exists in an anisotropic material,

 ' Tn ' ciiii Ciii2 Cii33 Ciii2 Cii23 Cii3i E11 T22 C2222 C2233 C22i2 C2223 C223i E22 T33 C3333 C33i2 C3323 C333i E33 Ti2 = ci2i2 ci223 ci23i < 2Ei2(Yi2) T23 SYM. C2323 C233i 2E23(Y23) , Tsi , C3i3i ^ 2E3i(Y3i) ,
0 0

### Responses

• CAROLA
When to use the different piola kirchhoff cauchy stress tensors?
8 years ago