CONTINUUM MECHANICS for ENGINEERS

Solution
Here Eq 2.5-13 may be used. Thus in matrix form
[ ′ ] =
⎡1/
2 0 −1/
2⎤⎡2
6 4⎤⎡1/
2 0 1/ 2⎤
⎢
⎥ ⎢ ⎥ ⎢
⎥
Tij ⎢ 0 1 0 ⎥ ⎢
0 8 0
⎥ ⎢ 0 1 0 ⎥
⎢
⎥
⎣
1/ 2 0 1/ 2
⎦ ⎣
⎢4
2 0⎦
⎥ ⎢
⎣
−1/
2 0 1/ 2⎥
⎦
⎡−3
4/ 2 1⎤
⎢
⎥
= ⎢ 0 8 0⎥
⎢
⎥
⎣
1 8/ 2 5
⎦
2.6 Principal Values and Principal Directions of Symmetric
Second-Order Tensors
First, let us note that in view of the form of the inner product of a secondorder
tensor T with the arbitrary vector u (which we write here in both the
indicial and symbolic notation),
Tu ij j = vi
or T · u = v (2.6-1)
any second-order tensor may be thought of as a linear transformation which
transforms the antecedent vector u into the image vector v in a Euclidean
three-space. In particular, for every symmetric tensor T having real components
T ij, and defined at some point in physical space, there is associated
with each direction at that point (identified by the unit vector n i), an image
vector v i given by
Tn ij j = vi
or T⋅ nˆ = v
(2.6-2)
If the vector v i determined by Eq 2.6-2 happens to be a scalar multiple of n i,
that is, if
Tn ij j = λ ni
or T⋅ nˆ = λnˆ
(2.6-3)
the direction defined by n i is called a principal direction, or eigenvector, of T,
and the scalar λ is called a principal value, or eigenvalue of T. Using the
substitution property of the Kronecker delta, Eq 2.6-3 may be rewritten as