Graphics Reference
In-Depth Information
Note that the stress tensor only depends on position, not on the surface
normal. It can also be proven from conservation of angular momentum
that the stress tensor must be symmetric:
σ
=
σ
T
.
Since the unit normal has no units, the stress tensor is also measured as
force per area, just like traction. However, it's a little harder to interpret;
it's easier instead to think in terms of traction on a specific plane of contact.
As a concrete example using continuum materials that are a little easier
to experience, put your hand down on a flat desk. The flesh of your hand
and the wood of the desk are essentially each a continuum, and thus there
is conceptually a stress tensor in each. The net force you apply on the desk
with your hand is the integral of the traction over the area of contact. The
normal in this case is the vertical vector (0
,
1
,
0), so the traction at any
point is
⎤
⎡
⎤
⎡
⎤
⎡
σ
12
σ
22
σ
32
σ
11
σ
12
σ
13
0
1
0
⎦
⎣
⎦
=
⎣
⎦
.
t
=
σn
=
⎣
σ
21
σ
22
σ
23
σ
31
σ
32
σ
33
Note that the normal force comes from the vertical component
σ
22
of
traction—how hard you are pushing down on the desk. The other com-
ponents of the traction,
σ
12
and
σ
32
, are tangential—how hard you are
pushing the desk forwards, backwards, or to the side.
Those tangential forces are due to friction; without it there could only be
a normal force. Viscosity is in many ways similar to friction, in particular
that a fluid without viscosity only exerts forces in the normal direction.
That is, the traction
t
=
σn
in an inviscid fluid is always in the normal
direction: it must be parallel to
n
. Since this is true for any normal vector,
it can be proven that the stress tensor of an inviscid fluid must be a scalar
times the identity. That scalar is, in fact, the negative of pressure. Thus,
for the inviscid case we have considered up until now, the stress tensor is
just
σ
=
−pδ,
(8.1)
where we use
δ
to mean the identity tensor. When we model viscosity, we
will end up with a more complicated stress tensor.
8.2 Applying Stress
The net force due to stress on a volume Ω of fluid is the surface integral of
traction:
F
=
∂
Ω
σn.
