Transcribed Text
1.
The eigenvectors and eigenvalues for a given stress state are the following:
01 = 10 MPa; = 1 (e1 + e
O2 =
= 0 n
(3) 1 2 e2)

a.
What is the maximum shear stress?
b. The directions of maximum shear stress bisect the directions ( lines) of maximum and
minimum normal stress. (See figure below.) What are the directions of maximum shear
stress?
c.
Assume the matrix of the stress tensor in the eigenvector basis is ordered with (o1,O2,03)
respectively ordered in the top, middle, and bottom positions on the diagonal.
i.
Write the matrix of the stress tensor in the eigenvector basis.
ii.
Find the traction vector on the plane with unit normal a = 1 with
respect to the eigenvector basis.
iii. What are the values of (N,S), i.e the values of the normal and shear
components of the stress vector?
d. Write the Q matrix that is used to transform vector and higherorder tensor components
from the e, basis to the n (i) basis.
(1 pt extra credit: show Q is an orthogonal matrix.)
X2
(1)
1
n
(e,
+22
2
n
(3) = 1 2 (e, 122 
X1
2
2. Expand the following expressions:
a. bkoik, where Sik are the components of the Kronecker delta operator.
b. Tij,j + Fi = 0 (3 equations  what is the name of this set of equations?)
1
1
1
2
c. SijAij, where [Sij] =
1
2 0
,
[Aij] =
1
1
0 0 1.
2
1
(show work for each term)
1
d. Sijuj, using [Sij] from part 2c, and [vi]
=
2
. (You can use either matrix
1
notation or indicial notation.)
3. In the absence of body forces, does the following displacement field satisfy the equations of
elasticity? What are those equations?
u = Bxyz
v=y
W = sin (ax)
where (a,B,y) are constants. (Work must be shown to receive credit.)
Tij,j + Fi = 0 equilibrium equations straindisplacement equations
22
22
(1+v)V202 2 + ôx² (o, X to, +02) = 0
(1+v)V2Txy+ X + o, + O2 z ) =0
ôxôy
(1+v)V²o, + õy² (onto, + O2) =0 yz ôyôz 22 (OX + of + 02) =0
22
22
(1 + v) V2 T zx +
(o, x +oy + O2 =0
ôzôx
BeltramiMichell compatibility equations (for no body force)
+ = xy
ôxôy
ôyôz Ox Ox ôy Oz
= a de xy )
+ = 2 yz
8²ey
ôzôx by ôy ôz ôx
a den + õe, + de yz )
= 2
õe õe ZX
Oz Oz ôx
SaintVenant compatibility equations
quite ij  o kk S ij
1+v
isotropic Hooke's law
a
Ow
= 0
+ ôx Ov + Ow
=0
Navier's or Lame's equations
Oz
(for no body force)
HV2, W + ôz + ôv + ôw
=0
ôx by Oz
= N =T" n =
Smar=112/0103) where 01 202203
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