Geology Reference
In-Depth Information
Therefore,
of the displacement field plays an important role
in the quantitative description of the deformation
process. Let us consider now the curl of
u
:
¨
ij
dx
j
D
©
ijk
k
dx
j
D
.
dr/
i
(7.37)
@
u
3
@x
2
@
u
1
@x
3
@
u
2
@x
3
@
u
3
@x
1
This result shows that ¨ is a rigid rotation
about an axis having the direction of ,so
that no deformation is associated with the anti-
symmetric part of the Jacobian matrix. As a
consequence, any deformation is described by
the strain tensor ©. We note that the compo-
nents of © are non-dimensional quantities that
are calculated from partial derivatives of the dis-
placement field. The diagonal components, ©
kk
,
represent variations of displacement components
along the corresponding directions. For example,
©
11
D
@
u
1
/@
x
1
represents the variation of the
x
-
component of displacement as we move along the
x
axis.
Taking the trace of ©, we obtain the divergence
of the displacement field:
r
u
D
e
1
C
e
2
@
u
2
@x
1
@
u
1
@x
2
C
e
3
(7.41)
It is not difficult to prove that this quantity
is associated only with rigid rotations. This re-
sults promptly from a comparison of expressions
(
7.41
)and(
7.33
). Therefore,
r
u
is non-zero
only when the displacement field includes a com-
ponent of rigid rotation. A deformation that can
be described by a traceless (©
kk
D
0) strain tensor
leaves invariant the volume of a body and is
referred to as a
pure shear
.Asimpleexampleof
pure shear deformation is illustrated in Fig.
7.7
d.
Let us consider now the off-diagonal components
of ©, which arise from variations of displacement
components along transversal axes. For example,
©
12
¤
0whenthe
x
component of
u
varies as
we move in the
y
direction, or when
u
y
changes
as we move in the
x
direction, both possibilities
being admissible at the same time. When in a pure
shear deformation the strain tensor diagonal is
identically zero and the deformation arises from
a single pair of off-diagonal components (e.g., ©
12
and ©
21
), a simultaneous rotation about an axis
orthogonal to the plane of deformation may lead
to the situation illustrated in Fig.
7.7
c, which is
termed
simple shear
.
In simple shear, two orthogonal segments with
a common fixed point, initially parallel to coor-
dinate axes, assume new orientations after defor-
mation (Fig.
7.7
c), with angles
¥
1
and
¥
2
with re-
spect to the coordinate axes. For an infinitesimal
deformation in the
xz
plane, it results:
©
kk
Dr
u
(7.38)
This quantity is called
dilatation
and repre-
sents the volume change per unit volume during
deformation. In fact, assuming that the strain
tensor is diagonal, we have that a volume element
dV
D
dx
1
dx
2
dx
3
is changed as follows:
1
C
dx
1
1
C
dx
2
@
u
1
@x
1
@
u
2
@x
2
dV
0
D
1
C
dx
3
Š
1
C
@
u
3
@x
3
@
u
k
@x
k
dx
1
dx
2
dx
3
1
C
dV
D
.1
C
/dV
@
u
k
@x
k
D
(7.39)
Therefore, the relative variation of volume will
be given by:
@
u
1
@x
3
I
¥
2
Š
tan ¥
2
D
@
u
3
@x
1
(7.42)
¥
1
Š
tan ¥
1
D
dV
0
dV
dV
D
(7.40)
Therefore,
Whenever it results @
u
i
/@
x
i
> 0, we have
exten-
sion
along the axis
x
i
. Conversely, for @
u
i
/@
x
i
< 0
we have contraction. In general, the divergence
@
u
1
@x
3
C
@
u
3
@x
1
D
2
©
13
¥
1
C
¥
2
Š
(7.43)
