Geology Reference
In-Depth Information
through progressive rotation about the axes
n
0
and
m
0
.For
small values of the bendin
g an
gle ', lateral shortening
occurs along the segment OP after subduction, as the
internal margin of the left and right sectors of the slab
overlap by a quantity
b
, which increases with the distance
— from the hinge line
Fig. 11.7
Lateral deformation associated with the bend-
ing of a non-planar lamina along a segmented hinge line.
(
a
) A planar lamina has been bent along the
x
-axis to
simulate the effect of the Earth's curvature. Then, further
bending is applied along a segmented hinge line. In the
tangent plane, the segments of the hinge line forms an
angle œ with the
y
-axis. (
b
) Slab bending is performed
reaches an equilibrium value. It is easy to prove
that such an equilibrium value of the bending
angle is determined by Frank's formula.
The state of lateral shortening that occurs for
small bending angles is represented in Fig.
11.7
b
by the overlap of the internal margins of the two
slab components by a quantity
b
.If
n
and
m
are the versors describing the projection of the
hinge line onto the tangent plane and
R
(
q
,®)is
the rotation matrix about an axis having versor
q
through an angle ® (positive counter-clockwise),
then the axes of the hinge line are described by
the versors:
components of
R
.
Overlap occurs for
b
< 0, whereas for
b
D
0
we get the equilibrium condition of Frank (
1968
).
Finally, for
b
> 0 we obtain lateral extension
along the internal margin. This result implies that
Frank's formula correctly describes the states of
lateral mechanical equilibrium during the bend-
ing. The angle œ in (
11.23
) must be taken in the
tangent plane (i.e., the plane of
n
and
m
). The re-
lationship of this quantity with the corresponding
angle, œ
0
, in the plane of
n
0
and
m
0
is:
sin
2
œ
0
D
sin
2
œcos
2
¨
2
C
sin
2
¨
(11.24)
n
0
D
R .i;
¨
=2/n
I
m
0
D
R .i;
C
¨
=2/m
(11.22)
2
The application of these expressions to the
case of the Earth's subduction zones can be
made taking the limit as œ
!
0and¨
!
0and
considering that the variation
d
¨ per unit arc
length ` must be equal to the Earth's curvature:
›
e
D
1/
R
D
d
¨/
d
`.Let›
T
D
d
(2œ)/
d
`
D
2
d
œ/
d
`
and ›
D
1/¡
D
2
d
œ
0
/
d
` be
where
i
is the
x
-axis versor and ¨/2 is the angle of
bending of each side of the original lamina about
the
x
axis. In order to determine the overlap
b
,we
rotate the point
P
, having coordinates
p
(—,0,0),
respectively by an angle ' about the axis
n
0
,and
by an angle
' about the axis
m
0
. The amount of
overlap
b
will be given by:
b
D
ǚ
R
n
0
;'
R
m
0
;
'
p
y
D
2—
h
sin œ cos œ cos
¨
respectively
the
trench
curvatures
in
the
tangent
plane
and
n
0
m
0
.From
11.24
)
in
the
plane
of
and
we
easily
obtain
the
relation
between
these
quantities:
2
.1
cos'/
›
R
2
1=2
1
¡
2
R
2
1=2
2
sin '
i
1
1
cosœ sin
¨
›
T
D
D
(11.25)
(11.23)
