Page 37 - Introduction to Continuum Mechanics

22 Tensors

i.e.,

We note that another commonly used notation for the dyadic product of a and b is a®b.

2B8 Trace of a Tensor

The trace of any dyad ab is defined to be a scalar given by a • b. That is,

Furthermore, the trace is defined to be a linear operator that satisfies the relation:

Using Eq. (2B7.3b), the trace of T can, therefore, be obtained as

that is,

It is obvious that

Show that for any second-order tensor A and B

Solution. Let C=AB, then C^-A- imB my Thus,

Let D=BA, then Dy=B/ m/4 m/-, and

But Bi ntA mi=B miAf m (change of dummy indices), that is