Seismic Deformation of the Lithosphere - Quantitative Plate Tectonics - page 301

curvature has a maximum › 1 and a minimum › 2
along two orthogonal directions. They are known
as the principal curvatures . The quantity K
› 1 › 2 is called Gaussian curvature and plays a
key role in differential geometry. It can be used
to classify points on S according to the value
andsignof K . At the Earth's surface, which is
biconvex, we have always that K is positive, be-
cause › 1 Š › 2 Š 1/ R D 1/6373 km -1 , R being the
Earth's radius, so that K D 1/ R 2

In a more recent study, Schettino and Tassi
( 2012 ) have argued that the classic “ping-pong
ball” model of Frank ( 1968 ) is not compati-
ble with the kinematics of subduction. First, the
model is in contrast with Euler's theorem, ac-
cording to which the relative motion between any
two plates is a rotation about an axis (see Sect.
2.2 ) . Figure 11.6 a shows the concavo-concave
geometry of a subducting slab in Frank's model.
To obtain this configuration, subduction must
have started at a point with an infinitely small
island arc. Then, the process has continued by
increasing both radius of curvature and width of
the trench zone, in such a way that the lithosphere
has been subducted radially from the initial point.
In this instance, the Wadati-Benioff zone isodepth
lines would have the shape of small circle arcs
having a unique pole and length decreasing with
the depth, as shown in Fig. 11.6 a. Therefore, in
a reference frame fixed to the overriding plate
the relative velocity field of the subducting plate
would be oriented radially with respect to the
trench (Fig. 11.6 c). Clearly, such a kinematics is
not compatible with the plate tectonics paradigm,
not only because it violates Euler's theorem, but
also because it requires the presence of a non-
rigid overriding plate, which extends radially in
so far as subduction proceeds. In any case, the
plate velocities predicted by this model are not
compatible with those effectively observed along
convergent plate boundaries.
The last major problem of Frank's model is
represented by the very strong lateral deforma-
tion (extension or shortening) predicted when
dip angle and radius of curvature assume values
that deviate from an equilibrium combination. It
can be easily proved, using the formulation of
Strobach ( 1973 ), that the theoretical lateral strain
associated with non-equilibrium values of dip and
trench curvature is very high and atta ins a singu -

D 2.462 10 8
km 2 . In Frank's theory, this quantity is con-
served when the lithosphere bends to form the
Wadati-Benioff zone, because we have that for a
slab › 1 Š › 2 Š -1/ R . Therefore, the slab acquires
a biconcave shape, similar to the dent on a ping-
pong ball (Bevis 1986 ). This is in agreement with
an important theorem proved by Gauss in 1827,
which states that K remains unchanged under
continuous deformation of a flexible and inexten-
sible surface. Finally, Frank ( 1968 ) proved that
the simple bending of the lithosphere through an
angle ' can only occur along a small circle having
angular radius of curvature “ D '/2.
Despite its appeal and the capacity to ex-
plain the geometry of subduction zones, Frank's
model has proved to be incorrect. During the
1970s, several authors recognized that Frank's
formula, relating dip angle and radius of curva-
ture, did not seem to be satisfied by the observed
values of dip and trench curvature. The con-
clusion was that subducting slabs were subject
to significant lateral stress as a consequence of
the observed deviations from Frank's equilib-
rium formula (Strobach 1973 ; De Fazio 1974 ;
Laravie 1975 ; Tovish and Schubert 1978 ). An-
other discrepancy was the prediction of Wadati-
Benioff zones having concave downdip curvature
1/ R , whereas most modern subduction zones
show convex radial curvatures, associated with
the downward bending of the lithosphere in the
upper mantle and a general increase of the sub-
duction angle at shallow depths. As pointed out
by Bevis ( 1986 ), modern subducting slabs do
not seem to conserve the Gaussian curvature of
the unsubducted lithosphere, thereby the Earth's
lithosphere cannot be considered as a flexible-
inextensible shell and Frank's theorem is not
applicable.

larity at downdip distance — D ¡= p 1 ¡ 2 =R 2 ,
¡ being the radius of curvature of the trench
(in km).
An alternative model of subduction has been
proposed recently by Schettino and Tassi ( 2012 ).
As illustrated in Fig. 11.6 b, the set of points
placed along a subducting plate margin, which
are entering the trench at an initial time t 0 are