Geoscience Reference
In-Depth Information
shape undergone by the rectangle is called the
shear component of strain
and is written
e
xy
.In
three dimensions, there are six shear components of strain:
∂
1
2
u
∂
v
e
xy
=
e
yx
=
y
+
(A2.6)
∂
∂
x
∂
u
1
2
∂
z
+
∂
w
(A2.7)
e
xz
=
e
zx
=
∂
x
∂
1
2
w
y
+
∂
v
e
zy
=
e
yz
=
(A2.8)
∂
∂
z
Note that the angle of shear is equal to twice the shear component of strain. As well as
undergoing a change in shape, the whole rectangle is also rotated anticlockwise by an angle
1
2
(
1
−
2
), termed
θ
z
,where
1
2
(
θ
z
=
1
−
2
)
∂
v
1
2
∂
x
−
∂
u
(A2.9)
=
∂
y
θ
z
is an anticlockwise rotation about the
z
axis.
Extending the theory to three dimensions, the deformation (
u
,
v
,
w
)ofany point
(
x
,
y
,
z
) can be expressed as a power series, where, to first order,
∂
u
+
∂
u
+
∂
u
u
=
x
x
y
y
z
z
∂
∂
∂
∂
v
∂
x
x
+
∂
v
∂
y
y
+
∂
v
v
=
∂
z
z
(A2.10)
∂
w
+
∂
w
∂
+
∂
w
∂
w
=
x
x
y
y
z
z
∂
Alternatively, Eqs. (A2.10) can be split into symmetrical and antisymmetrical parts:
u
=
e
xx
x
+
e
xy
y
+
e
xz
z
−
θ
z
y
+
θ
y
z
v
=
e
xy
x
+
e
yy
y
+
e
yz
z
+
θ
z
x
−
θ
x
z
(A2.11)
w
=
e
xz
x
+
e
yz
y
+
e
zz
z
−
θ
y
x
+
θ
x
y
where
1
∂
1
2
w
∂
y
−
∂
v
θ
=
x
∂
z
∂
1
2
u
z
−
∂
w
∂
θ
y
=
(A2.12)
∂
x
∂
v
∂
x
−
∂
u
1
2
θ
z
=
∂
y
In more compact matrix form, Eq. (A2.11)is
&
'
&
'
&
'
&
'
e
xx
e
xy
e
xz
x
0
−
θ
z
θ
y
x
(
u
,
v
,
w
)
=
e
xy
e
yy
e
yz
y
+
θ
z
0
−
θ
x
y
(A2.13)
e
xz
e
yz
e
zz
z
−
θ
y
θ
x
0
z
Strain is a dimensionless quantity. Generally, in seismology, the strain caused by the passage
of a seismic wave is about 10
−
6
in magnitude.
1
The curl of the vector (
u
,
v
,
w
),
∇
∧
u
,isequal to twice the rotation (
θ
x
,
θ
y
,
θ
z
)asdiscussed in
Appendix 1.