Page 32 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 17

The result is still valid even in the case when instead of the scalar

function y, a vectorial function of the same r is considered (it is sufficient
to use the previous assertion on each component). At the same time the
conclusion will remain the same if the above condition takes place only
on a set of subdomains (£) with the property that in any neighborhood
of a point from D, there is at least a subdomain from the set (£).

2. General Principles. The Stress Tensor and

Cauchy’s Fundamental Results

2.1 The Forces Acting on a Continuum

Let us consider a material subsystem P of the continuum M, a subsys-
tem imagined at a given moment in a certain configuration D = x(P,t),
which is enclosed in the volume support D of the whole system M. On
this subsystem P of the continuum M, two types of actions are exerted:

(4) contact (surface) actions, of local (molecular) nature, exerted on
the surface S$ of the support D of the subsystem P by the “comple-
mentary” system M\P (as the “pressure or pull” of the boundary, the
“pushing” action through friction on the boundary, etc.)
(ii) distance (external) actions, of an extensive character, exerted on

the bulk portions of the continuum P and arising due to some external
cause (such as gravity, electromagnetic, centrifugal actions, etc.)
But the mechanics principles are formulated, all of them, in the lan-
guage of forces and not of actions. To “translate” the above mentioned

actions into a sharp language of forces we will introduce the so-called
Cauchy’s Principle (Postulate) which states:
“Upon the surface S there exists a distribution of contact forces, of
density T, whose resultant and moment resultant are equipollent to the
whole contact action exerted by M\P.

At the same time there is a distribution of external body or volume
forces of density f, exerted on the whole P or D and whose resultant
and moment resultant are completely equivalent (equipollent) with the
whole distance (external) action exerted on P ”.
The contact forces introduced by this principle are called stresses.

These stresses, of surface density T, at a certain moment ¢, will de-
pend upon the point where they are evaluated and the orientation of
the surface element on which this point is considered, orientation char-
acterized by the outward normal unit vector n on this surface, such that
T = T(r,n,t).
Concerning the external body or volume forces (the gravity forces are

body forces while the electromagnetic forces are volume forces, etc.), of
density f, at a certain time ¢, they depend only on the position vector r