Geology Reference
In-Depth Information
Fig. 7.7
Geometry of
deformation in two
dimensions.
a
Dilatation,
b
Compression,
c
Simple
shear,
d
Pure shear
This expression says that the off-diagonal
component ©
13
is the average angular variation
in the plane
x
1
x
3
. A similar conclusion can be
drawn for the other off-diagonal components.
Therefore, (
7.43
) furnishes a simple intuitive
interpretation of the off-diagonal components of
the strain tensor.
A consequence of the strain tensor symmetry
is that a system of
principal strain axes
m
i
exists
such that the tensor is diagonal. In this frame, and
assuming no rotations, the variations of displace-
ment
du
i
have the same direction of the variations
of position
dx
i
:
relation between these variables in a geodynamic
context. This can be done adapting the classic
Newton's equations of motion (second law of me-
chanics) to the case of a continuum deformable
body. Let us consider the forces exerted on a
volume element
dV
D
dx
1
dx
2
dx
3
of the region
R
occupied by the body (Fig.
7.8
).
We know that the surface force exerted on a
face of
dV
is given by the traction on that face,
times the area. Therefore, the force
d
F
on a face
at position
r
D
(
x
1
,
x
2
,
x
3
), with normal -
e
j
and
area
dx
r
dx
s
(
r
,
s
¤
j
), has components:
dF
i
e
j
; r
D
T
i
e
j
dx
r
dx
s
D
£
ij
.r/dx
r
dx
s
I
r;s
¤
j
(7.45)
d
u
i
D
©
ij
dx
j
D
œdx
i
(7.44)
The three eigenvectors of (
7.44
) are called
the
principal strains
©
1
, ©
2
,and©
3
. With the
exception of a situation of
hydrostatic strain
,such
that ©
1
D
©
2
D
©
3
, some amount of shear strain is
always present also in the principal strain axes
coordinate system.
Similarly, the force on a face at position
r
C
e
j
dx
j
, with normal
e
j
and area
dx
r
dx
s
(
r
,
s
¤
j
),
has components:
dF
i
e
j
; r
C
e
j
dx
j
D
ij
r
C
e
j
dx
j
dx
r
dx
s
I
r;s
¤
j
(7.46)
7.3
Cauchy Momentum Equation
Clearly, when the stress field is homo-
geneous the net force exerted on
dV
is
zero, because the forces on opposite sides
of the volume element balance each other:
d
F
(
e
j
,
r
)
D
d
F
(
e
j
,
r
C
e
j
dx
j
). Therefore, in
In the previous sections, we have introduced the
concepts of traction, stress, displacement, and
strain for a continuum body in conditions of static
equilibrium. Now we are going to describe the