Page 36 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 21

D
5 | evede = | ex + pa; + pujdiv v) dv = [ exide.

D D D

Under these circumstances, the above equations become

[ cade = [taa+ | ptav

D S D
and

[=x padv= [rx Tada+ fr x pao,

D Ss D

both equalities being valid for any subsystem P C M and implicitly for
any domain DC D.
A direct application of the momentum variation principle is Cau-
chy’s lemma which establishes that, at any moment and at any point r
from a surface element of orientation given by n, the stress vector T,

supposed continuous in r, satisfies the action and reaction principle, Le.,
[33] T (r,n,t) = —T (r, —n,?).

2.4 The Cauchy Stress Tensor

As the stress vector T, evaluated at a point r, does not depend only
on r and # but also on the orientation of the surface element where the
point is considered (i.e., on n), this vector cannot define the stress state
at the respective point. In fact, at the same point r, but considered
on differently oriented surface elements, the vectors T could also be

different. This inconvenience could be overcome by the introduction,
instead of an unique vector T, of a triplet of stress vectors T; whose
components with respect to the coordinates axes will form a so-called
tensor of order 2. This stress tensor, introduced by Cauchy, is the first
tensor quantity reported by science history.
The triplet of stress vectors thus introduced will be associated, at

every moment, to the same point r but considered on three distinct
surface elements having, respectively, the outward normal parallel with
the unit vectors i; of the reference system, namely T; = T(r,i,;,¢) (@ =
1,2,3). Let us denote by 4%; (¢ = 1,2,3) the components on the axes
Oz; of the vector T;, Le., Tj = 743 ij.
We will show, in what follows, that the stress state at a point r, at
every moment ¢, will be characterized by the triplet of these vectors

T; or, synonymously, by the set of the nine scalars 7; (¢ = 1,2,3;
gj = 1,2,3) which depend only on r. Precisely, we will show that the