Page 39 - Introduction to Continuum Mechanics

24 Tensors

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is non-singular) and in this case, [T]" [T] = [T] [T]"" = [I]. Thus, the inverse of a tensor
satisfies the following reciprocal relation:

We can easily show (see Prob. 2B15) that for the tensor inverse the following relations are
satisfied,

and

We note that if the inverse exists then we have the reciprocal relation that

This indicates that when a tensor is invertible there is a one to one mapping of vectors
a and b. On the other hand, if a tensor T does not have an inverse, then, for a given b, there
are in general more than one a which transforms into b. For example, consider the singular
tensor T = cd (the dyadic product of c and d, which does not have an inverse because its
determinant is zero), we have

Now, let h be any vector perpendicular to d (i.e., d • h = 0), then

That is, all vectors a + h transform under T into the same vector b.

2B10 Orthogonal Tensor

An orthogonal tensor is a linear transformation, under which the transformed vectors
preserve their lengths and angles. Let Q denote an orthogonal tensor, then by definition,
| Qa | = | a | and cos(a,b) = cos(Qa,Qb) for any a and b, Thus,

for any a and b.
Using the definitions of the transpose and the product of tensors:

Therefore,

Since a and b are arbitrary, it follows that