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 = XiXei + 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 PiolaKirchoff stress tensors?
(b) What is the difference between Cauchy, first and second PiolaKirchoff stress tensors?
(c) In which coordinate system is the Cauchy and PiolaKirchoff 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) , 
Responses

CAROLA8 years ago
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