HOMEWORK #4
MMAE 530 Advanced Mechanics of Solids
Fall 2014
Prof. M. Vural
(Due 10/16/14)
Problem 1. The components of the stress tensor, σ , in a body are given by
 11  A x12 ,
 22  A x 22 ,
 12  2 A x1 x 2
 13   23   33  0
Calculate the necessary body force for the body to be in equilibrium.
Problem 2. A slab occupies the region
 a  x1  a,
 a  x 2  a,
 h  x3  h
and has the stress distribution
 11  
 12 
p ( x12  x 22 )
a2
2 p x1 x 2
a2
 22 
p ( x12  x 22 )
a2
 13   23   33  0
(a) Examine whether there are any body forces within this slab
(b) Calculate the resultant tractions acting on the faces x1   a , x 2   a
Problem 3. The stress tensor at a point in a solid is given by
T 2 1 
[σ ]   2 0 2
 1 2 0
where T is a constant. Calculate T so that there will be a traction-free plane through the
point and determine the orientation of this plane.
Problem 4. The stress tensor at a point in a solid is given by
3 1 1
[σ ]  1 0 2
1 2 0
in arbitrary units. Calculate the stress vector (traction) on a surface element with normal
(0, 1, 1).
Problem 5. The components of the strain tensor, ε , in a body are given by
 11  A x1 x 2 ,
 22  B x 22 ,
 12  C x 22
 13   23   33  0
Find the displacement field, u i , assuming that u 3  0 and

 0.
x3
Problem 6. In a certain state of stress, the components of the stress tensor, σ , are given by
 11   22   33  0
 12   23   13  
(a) Compute the principal stresses and their principal axes
(b) Compute the three stress invariants and the deviatoric stresses, first in the original axes
frame, then in the principal axes frame.
(c) Compute  oct ,  oct and  max .
Problem 7. The differential element shown is in a state of plane stress:
40 MPa
20 MPa
x2
10 MPa
30º
x1
Use Mohr’s circle to find the principal stresses, maximum shear stress, principal axes and
the stress that act on the plane indicated by the dashed line.

1, i, j = 1