Page 37 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

22 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

stress T, evaluated for the considered moment at a point r, situated
on a surface element of normal n(nj), can be expressed by the relation
T(r,n,t) = T;(r, #)n;, known as Cauchy’s theorem.
The proof is backed by the theorem (principle) of momentum applied
to a tetrahedral continuum element with its vertex at r, the lateral
faces being parallel to the planes of coordinates, its base is parallel to
the plane which is tangent to the surface element where the point r is

located. Considering then that the volume of the tetrahedron tends to
zero and using the mean theorem for each of the coordinates, we get
Cauchy’s theorem. The detailed proof can be found, for instance, in
[33].
Let us now consider, for any moment ¢, the linear mapping [T] of the
Euclidean space £3 into itself, a mapping defined by the collection of

the nine numbers 7;; (r,t), ie, [T]i; = 71; Such a mapping which,
in general, is called a tensor will be, in our case, just the Cauchy stress
tensor, a second order tensor in £3. We will see that by knowing the
tensor [T] which depends, for any instant t, only on r, we have the
complete determination of the stress state at the point r.
Precisely we have

Ti(r,n,t) = T(r, t)n,; = Tig iNj = [Thijn, = [T](r, t)n.

This fundamental relation shows that T depends linearly on n and,
consequently, it will always be continuous with respect to n.
It is also shown that the tensor [T] is an objective tensor, i.e., at a
change of a spatio-temporal frame, change defined by the mapping [Q]
or by the orthogonal proper matrix Q;; = iji;, the following relation
holds:

[T}'(r',¢) = (QU@)[T](r, t)[Q]",# =t +7.

(the proof could be found, for instance, in [33]).
It is also proved that [T] is a symmetric tensor, i.e., [T] = [T]? [33].
This result, besides the fact that it decreases the number of parameters
which define the stress state (from 9 to 6), will also imply the existence,
at every point, of three orthogonal directions, called principal directions,
and vs. them the normal stresses (T - n) take extreme values which are

also the eigenvalues of the tensor (mapping) [T].
The stress tensor symmetry is also known as “the second Cauchy’s
theorem (law)”.