Page 191 - Introduction to Continuum Mechanics

176 Stress

faster than the terms on the left where the stress vectors are multiplied by areas, the product
of two infinitesimal lengths. Thus, in the limit, the acceleration term drops out exactly from
Eq. (i) (We note also that any body force e.g. weight that is acting will be of the same order
of magnitude as that of the acceleration term and will also drop out). Thus,

Let the unit normal vector of the inclined plane ABC be

The areas A/tj_,A^2» and A/^, being the projections of AA n are related to A/4 n by

Using Eq. (4,2.3), Eq. (ii) becomes

But from the law of action and reaction,

Thus, Eq. (4.2.4) becomes

Now, using Eq. (4.2.2) and Eq. (4.2.5), Eq. (4.2.1) becomes

That is, the transformation T defined by

is a linear transformation [see Eq. (2B1.2)]. It is called the stress tensor, or Cauchy stress
tensor.

4.3 Components of Stress Tensor

According to Eq. (4.2.7) of the previous section, the stress vectors tg. on the three coordinate
planes (the e rplanes ) are related to the stress tensor T by

By the definition of the components of a tensor, Eq. (2B2.1b), we have

Thus,