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.
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