Geology Reference
In-Depth Information
Fig. 7.3
Positive
components of the stress
tensor
the off-diagonal components of £ generate six
couples with the corresponding components of
traction along the opposite faces. At the equi-
librium, both the total force and the total torque
must be zero, thereby, two torques in direction
e
i
and -
e
i
always cancel out. As a consequence, the
stress tensor is symmetric and we have only six
independent components:
ij
D
ji
(7.2)
The importance of the stress tensor in the
description of the surface force fields within de-
formed bodies arises from its capability to predict
the traction along
any
surface element. This is a
consequence of the following theorem:
Fig. 7.4
The Cauchy tetrahedron
respectively,
n
,-
e
1
,-
e
2
,and-
e
3
. It is easy to
realize that the areas of triangles
dS
i
are given by:
dS
i
D
dS
e
i
D
(
n
e
i
)
dS
D
n
i
dS
. The total force
F
exerted on the tetrahedron is the sum of the
forces exerted on the individual faces. At the
equilibrium, it must be
F
i
D
0. Therefore, taking
into account that
i
-th component of the force
exerted on the surface elements
dS
j
is -£
ij
dS
j
(no
implicit summation) we have:
Cauchy's Theorem
In equilibrium conditions
,
for an arbitrary sur-
face element with normal versor
n
,
the compo-
nents of traction are given by
:
T
i
D
£
ij
n
j
(7.3)
Proof
A surface element of arbitrary shape can
be always divided into a set of triangles. There-
fore, we shall prove the theorem for a triangle
having area
dS
and orientation
n
. Let us choose
a reference frame with the origin
O
very close to
this triangle, as shown in Fig.
7.4
. The triangle
forms a tetrahedron with the three coordinate
axes, with sides
dS
,
dS
1
,
dS
2
,and
dS
3
,
dS
i
being
orthogonal to
e
i
. The unit versors associated with
the four triangles bounding the tetrahedron are,
T
i
dS
£
ij
n
j
dS
D
0
(7.4)
Dividing Eq. (
7.4
)by
dS
gives (
7.3
). This
proves Cauchy's theorem.
Cauchy's theorem shows that the stress ten-
sor completely determines the surface force field
existing within a deformed body. It can be con-
sidered as the linear operator that generates a
