Page 59 - Introduction to Continuum Mechanics

44 Tensors

degenerate to one independent equation (see Example 2B17.3) so that there are infinitely
many eigenvectors lying on the plane whose normal is %. Therefore, though not unique, there
again exist three mutually perpendicular principal directions.
In the case of a triple root, the above three equations will be automatically satisfied for
a
whatever values of (ai,«2» 3) so that any vector is an eigenvector (see Example 2B17.1).
Thus, for every real symmetric tensor, there always exists at least one triad of principal directions
which are mutually perpendicular.

2B19 Matrix of a Tensor with Respect to Principal Directions
We have shown that for a real symmetric tensor, there always exist three principal directions
which are mutually perpendicular. Let 111,112 and 113 be unit vectors in these directions. Then
using ni,n?,n-> as base vectors, the components of the tensor are

That is

Thus, the matrix is diagonal and the diagonal elements are the eigenvalues of T.
We now show that the principal values of a tensor T include the maximum and minimum
values that the diagonal elements of any matrix of T can have.
First, for any unit vector ej = anj+/3n2+yii3,

i.e.,