Download Earth`s crust deformations in geosynclines

GEOPHYSICS
EARTH'S CRUST DEFORMATION"S IN GEOSYNCLINES
BY
F. A.
VE~ING
MEINESZ
(Communicated at the meeting of Dec. 17, 1949)
The beIts of strong negative gravity anomalies found in the Indonesian,
Caribbean and Japanese areas have led to the hypothesis 1) that in these
areas the Earth's crust has in recent geological periods buckled downwards thus forming a large bulge of crustal matter at the lower boundary
of the crust. This crustal bulge has pushed away the plastic subcrustal
matter and, being of smaller density, has caused the deficiency of mass
revealed by the gravity anomalies. This explanation of the anomalies
has been more and more wirlely accepted although other views have
also been advanced; it will form the hase of this paper.
For the problems dealt with here we need not go into the question
that the rigid crust probably consists of more than one layer of increasing
density wh en we go downwards, and th at thus the down-huckling gives
rise to more areas in the cru stal cross-section where lighter rocks have
been substituted to den ser ones, each area contributing its share to
the anomalies.
In this paper the writer wants to attack the problem ho\\" this crustal
deformation got into being and this involves the question how far this
deformation had a plastic and how far an elastic character. For this
purpose he shall often make use of the investigations of BYLAARD
about this matter and his conclusions will for a great part coincide with
BYLAARD'S results. He shall especially base his discussions on the
Indonesian Archipelago because so much is known about this area and
among BYLAARD'S papers he may refer particularly to those of 1935
and 1936 which likewise deal with this area.
In the first place we have to choose between the supposition of a
purely elastic beginning of the buckling of the crust and, at least partly,
a plastic start of the deformation. The second assumption, defended by
BYLAARD, has several advantages with regard to the first but it offers
also difficulties; wh en the lateral compression in the crust has increased
to a value th at plastic deformation sets in, the problem arises e.g.
whether the crust will not bulge to both sides and continue to do so in
such a way that isostatic balance will remain valid (fig. lb) instead of
the development of a geosyncline Ieading to a buIging downwards as
fig. la shows. As the gravity anomalies clearly indicate th at in the critical
1)
F. A.
VENING l\'[EINESZ,
1930, 1934, etc.
28
belt isostasy is strongly disturbed and that, therefore, the actual crustal
deformation must be much nearer to la than to lb, it is only possible
to assume a plastic behaviour of the cru st if we can prove that the
deformation in that case will not lead to lb.
~
Fig. la.
Fig. lb.
A second problem we have to consider is whether in case we may
assume deformation la the plastic behaviour of the crust can lead to
a second crustal wave as shown by the left part of fig. 2 which apparently
.:::::::
~
V
~
Fig. 2.
is present in tbe western part of Indonesia viz. in the oilgeosynclines
of Java and Sumatra. 2)
Before starting an investigation of these problems we shall first
examine the difficulties involved in the hypothesis of a purely elastic
beginning of the phenomenon.
In tbe first place we find in that case that the stress needed to bring
about buckling of a rigid crust of a thickness of 30 km must be of the
order of 40.000 to 50.000 kg/cm 23 ) and it seems hardly likely that the
cru st could stand stresses of such an amount. It has, however, already
been pointed out by SMOLUCHOWSKI, 4) who arrived at the same conclusion, that much smaller stresses would be sufficient if we could assume
that the crust consists of severallayers. If we may admit that the friction
between these layers is negligible, we find that the stress required for
buckling would be lln th part of the above figure, n being the number
of layers, jf we assume the layers to have equal thickness. For bringing
down the buckling-stress to a more acceptable value of 3000 kgfcm 2
we would, therefore, have to assume the cru st to consist of some 15
layers and this seems a rather large number. It seems, moreover,
questionable whether the friction between the layers would not prevent
such a development.
There is still another way out of this difficulty. GRIGGS 5) has suggested
2) Called by UMBGROVE idiogeosynclines, p.43 e.s., Umbgrove 1942, or 2nd Ed.
1947, p. 49 e.s.
3) F. A. VENING MElNESZ, 1934, p. 48 or 1948, p. 70.
') M. SMOLUCHOWSKI, 1909.
5) D. A. GRIGGS, 1939.
29
that the downbuiging of the erust might be eaused by the sinking part
of a eonvection system descending below the belt. The drag exerted
by the horizontal parts of this system on the lower boundary of the
crust to both sides of the belt is on eaeh side direeted towards this belt.
This may thus explain the strong eompression of the cru st in the belt.
At the same time the effect of the sinking eurrent on the erust may lead
to the cru stal down-buIging at stresses below the buekling-limit.
This hypothesis is no doubt attraetive aud it would solve our problem.
The writer, however, se es difficulties. In the fi.rst ' place it appears likely
that su eh a eonveetion-eurrent eould not have great dimensions; as
the sinking eurrent ean not have much Iarger cross-section than a few
hundred kilometers and as the distance of the two beits of the Banda
arc is not more than 600 km, the total horizontal dim en sion of the
system could hardly exceed 300-500 km. As from other sources we must
suppose the speed to be at least a few cm per year, it is unlikely th at
sucb a current could continue for the whole period of an active geosyncline
which persists probably over some fifty million years or more; in th at
time it would make more than one complete revolution while the study
of convection in the Earth leads to the conclusion that such currents
never make more than a half turn. This is also GRIGG'S hypothesis as
given in the paper mentionecl.
It would of course be possible to suppose that a CUlTent was only
present during the perioels of actual folding which occur from time to
time during the history of the geosyncline and each of which cover a
much shorter time. But in that case it would be surprising that· the
current would repeat itself at the same place after a first one had occurred.
This first current-system must already have been causing a certain
concentration of crustal matter in the belt above the sinking current
anel the higher percentage of radio-active minerals in th is matter must
thus bring about a gradual rising of temperature in the belt which makes
it unlikely that a sinking current below it would again co me into being.
A second anel still more serious clifficulty with regard to GRIGG'S
hypothesis of local convection-currents causing the down-buIging of the
crust is based on the impression that the beIt in Indonesia as weIl as
that in the Caribbean area each form a great unit extending over many
thousands of kilometers and showing a systematic course and a systematic
oecurring of the anomalies; this makes it unlikely that local eurrentaystems could have brought them about.
Concluding the writer thinks that this mechanism of bringing forth
the down-buIging of the cru st is difficult to admit and that thus the
problem remains how su eh a phenomenon coulel take place if we maintain
the supposition of elastic deformation of the crust at the start of it.
A seeond difficulty is hrought ahout by the hypothesis, in recent times
arrived at by the author, that the substratum itself is no viscous fluid
but that it has a small strength which must be overcome before flow
30
ean set in. 6) It has likewise been advaneed by GRIGGS in his paper of
1939 and sin ce long by JEFFREYS. The writer shall not go here into the
arguments leading to this hypothesis but he may point out that it
obviously makes it diffieult to adhere to an elastieal down-buekling of
the crust. For maintaining this irlea we should need an initial downward
deformation of the cru st whieh would make it possible that the momentum
thus eaused by the resultant of the horizontal stress in the erust would
not only overeome the resistanee against bending of the erust but also
the elastie resistanee offered by the substratum. It is, besides, likely
that this last resistanee will involve a eonsiderably larger value of the
buekling-stress needed in the erust and that thus the number of separate
layers in tbe cru st whieh we had to assume for bringing this stress down
to an admissible value would have to be still greater.
A third problem offered by the hypothesis of a purely elastie beginning
of the crustal deformation in the tectonic areas is to find an explanation
for the eurved and often tortuous course the deformed beIts mostly
show. It is hardly likely that this is eaused by an intrieate pattern of
forces bringing about these deformations. If we ex amine e.g. the main
anomaly belt in lndonesia we eome to the conclusion that it is probably
eaused by a regular field of forees showing an unique or nearly unique
rlirection ') given by the arrows on the map and so it seems out of the
question to attribute the eurved shape of the belt to the forces. It is,
however, possible to account for part of the irregular courses of the
mobile beIts by the hypothesis of pre-existant zones of weakness of
the Earth's crust as advaneed qy the writer in 1943. 8 ) The presenee of
su eh zones must make it likely that the crustal deformation will in part
follow them. The pattern of shearzones derived in the papers referred
to has been based on the assumpt.ion of a polar shift in a very early
stage of the crust's history; the directions derived for lndonesia have
been inrlicated in the map.
Jt is unlikely, however, that the presenee of such zones of weakness
can explain all the features of the eomplicated belt-pattern of the
orogenic areas; in lndonesia e.g. the part parallel to Java and several
other parts have still to be aeeounted for. This is diffieult in case we
assume the pattern to develop as an elastic deformation of the crust.
As BYLAARD has remarked sueh a deformation must be expeeted to
oeeur in beIts at about right angles to the force and although the
eurvature of the Earth might somewhat affect it, it could not deviate
as much as the existing beIts, as e.g. that south of Java, aetually do.
As we shall afterwards examine more in detail a plastic deformation
eould give a mueh better explanation of these direetions. It is well known
that in a plate subjeeted to a single stress-direction the plastic deformation
8)
7)
8)
F. A. VENING MElNESZ, 1948, p. 39.
VENING MElNESZ, Gravity Expeditions at Sea, Vol. II, p. 122, Vol. IV, p. 30 etc.
VENING MElNESZ, 1943 and 1947 (see e.g. 1947, p. 56).
31
after the elastic limit is exceeded will occur under an angle of about
55° with this direction and not at right angles to it; this direction is
e.g. shown by the lines of HARTlI'IANN developing at this moment.
BYLAARD and afterwards VAN ITERso::-r have given an explanation of
th is angle and have also derived the formula for the angle occurring
in the general case of a more complicated stress condition in the plate.
BYLAARD has al ready shown that these directions of the plastic deformation may give a more satisfactory explanation of the pattern of the
tectonic beIts. \Ve shall co me back to this point.
Concluding we may say th at the assumption of an elastic deformation
at the beginning of the phenomenon involves many difficulties which are
avoided by the supposition th at from the start it has a plastic character.
\Ve shall now examine the two problems raised by the plastic hypothesis,
the question whether a downbuckling of the erust can be thus eXplained
and whether it is possible that an adjoining wave develops.
We shall begin by attaeking the first problem. Let us assume that in
a belt over a breadth b the eompression exceeds the elastie limit of the
ernst and that a plastic deformation has set in, leading to a thiekening
of the erust, whieh towards the edges of the belt disappears as represented
by fig. 3. We may probably assume th at the rate ofthickening is extremely
T
~--
Fig. 3.
slow, e.g. of the order of a few mm per year or less, and in that ease we
may praetically neglect the viscous resistance of the sub stratum against
the forming of the bulge at the lower boundary of the erust. 9) As a
eonsequence of this we may expeet the heights ho and hl of the upper
and lower bulges to have a ratio eorresponding to iso statie balanee and
") This may be derived from the speed of Ï!lOstatie readjustment whieh the
writer found to be given by thc formula of p. 662 of his paper of 1937
w = 5 t:, L emfyear
in whieh w
=
vertieal speed of the erust in emjyear
t:, = vertieal deviation of crust's surfaee from the equilibrium position
in km
diameter in 1000 km of equilibrium cleviation
The formula has been derived from the rate of post-glacial uplift of Scandinavia.
L
=
32
if we adopt the usual values for the densities of the crust and the
substratum of 2.67 and 3.27, thîs gives
(I)
This asymmetry is important for our problem because it must involve
a deviation from the regular distribution of stress in a vertical crosssection of the crust. As ho and !tI are no doubt small with regard to the
breadth b of the bulge we shall probably not make a great error in
neglecting the complications in the transition zones at the edges of the
belt and in assuming that the normal stress-component in each crosssection is a linear function of the vertical coordinate z. As we also
know th at the resultant of these components must coincide with the
original axis of the crust, our assumption allows us to derive the
distribution.
If we denote the va lues of the normal stress a at the upper and lower
boundaries by ao and al' their difference by a f:, and their mean value
u:;
alJ
~
4,
- ,I
"4
I
1
Fig. 4.
by a rn which is also the mean value of the stresses a oyer the whoie
eross-section, we can derive
(2a)
and if we use (I) and neglect hl + ho in the denominator
(2b)
" f:, =20.70~.
G,n
T
The differenee a f:, of the norm al stresses at the two erustal boundaries
must lead to a difference in the shortening of the horizontal dimensions
at both boundaries and, therefore, to a down-bending of the crnst. We
see that thus a downward move ment of the ernst may be explained.
We must, however, derive its value to find out whether it can compensate
and even exceed the rising ho of the upper surface so that a geosyncline
is formed .
The downward move ment affects the formules (2a) and (2b). If we
denQte the sinking of the crustal axis in an arbitrary eross-section by
33
zand in the middle of the belt by Ze, the value of al - U o (see fig. 4) is
increased by 2 z, resp. 2 ze' and so instead of (2a) and (2b) we get
(3a)
and using again (1) and neglecting hl
(3b)
+ ho
in the denominator
Ot:; = 12 (1.725h o +z).
T
Om
For the middle cross-section we have to substitute Ze to z.
As the dimensions of the phenomenon in the sense of the belt are
large we may assume the rate of shortening of each partiele of the crust
in the sense of the compression by the stress a (x direction) to be equal
to the rate of thickening in the vertical sen se (z direction). As we suppose
no stress to be working in vertical direction on the crust the usual
stress-strain relation of viscous fluids gives here for the rate of both
deformations
( 4)
where u and ware x and z comp. of displacement and 17 the coefficient of
viscosity.
If we denote the length in x direction after deformation of the whole
deformed area at the surface by bo and at the lower boundary by bI
(undeformed both = b) we get
h
~ (b1-b o)= -i~
b
f a t:; dx= 6~~ f (1.725h o +.z) dx.
o
u
Contenting us with an approximate solution we may - if Z is zero
at both ends of the deformed part - assume th at the mean value of Z
over this area is i Ze. It would, however, be possible that the adjoining
non-plastic parts of the cru st also take part in the ben ding and in that
case we should have to add a figure for the sinking of the ends of the
plastic part. Supposing this to be proportional to Ze we get a mean value
of zover the plastic part of
Introducing this we obtain:
( 5)
We shall now express bI - bo in ze and ho in the time t.
3
34
From fig. 5 we derive:
and so
bT
e=-b
bo
1-
Fig. 5.
and
(6)
Multiplying (5) by hJ8 T we get
(7)
azc
3 Om b
~ = 4 '1T2
2
[
1.7
25
h0
+ ("§ +,u )Ze] •
The value of ho and of the total increase of the crust's thickness which
we shall denote by h (h = 5.45 ho) can be found because we know that the
compressive stress U m and, therefore, the rate of thickening of the crust
since the beginning of the plastic deformation, i.e. since the time t = Ot
has been constant and that this rate is given by formula (4).
'We thus obtain:
(Sa)
35
and integrating:
h=
(8b)
Om
21']
Tt.
Introducing the value of ho = 0.1835 h in (7) we get the following first
order differential equation in ze:
aZe
(9)
(
3)
~- 1+~,u
2
fl
m b
2I']T2 Z e - 0 .475
(flm)2 b2 t _
2~
T
- O.
We can simplify this equatian by choosing a new varia bIe instead of
the time t, yiz. the increase of the crust's thickness h which is proportional
to t. By means of (8) we find
( 10)
This no longer contains explicitly the quantities (1", and 1]. The
solution of the equation is simpIe. Taking into account that ze = 0 for
t = h = 0 we obtain
( 11)
=
Z
c
0.475 [_1_ T3
1 4" 1 iJ f' b2
+
+
(e(1+~I') ~.:
h_
1) - hJ.
For finding the total amount of the sinking of the cru stal axis in the
middle of the plastic belt we have to add the sinking at the edges of
this belt which we assumed to be ,u Ze and so we have to multiply our
formula by 1 ,u. Denoting the result by z' we get
+
( 12)
For ,u = 0, i.e. when the adjoining non-plastic parts of the cru st do
not take part in the bending, the formula becomes
_b:
z'=O.475 [ T3
;:;2- ( eT'
( 13)
h
)
J
- 1 -h .
We obtain the formula for the sinking z~ of the crust's surface in the
middle of the belt by subtracting t.he rising ho = 0.1835 hand so we
get
( 14)
which for ,u = 0 simplifies to
b'
( 15)
,
Zo
=
0.475
T3 ( Ti' h
b2
e
- 1) - 0.658 h.
b2
b2
By multiplying these equations by T3 we see th at T3 z~ is only
functian of -~- h anel ,u, a conclusion which we could already have drawn
36
from the differential equation (10). So we find th at z~ and hare proportiona.I
to the third power of Tand inversely proportional to the square of b.
We may add that by computing numerical values for these quantities
for different values of ft we find th at as a rough approximation both
quantities are inversely proportional to 1 ft, at least inside the limits
o < ft < 1, which will probably not be exceeded by the cases occurring
in practice.
Formulas (14) and (15) give the following values (Tabie I) for three
values of ft , i.e. ft = 0, ft = tand ,u = I. The curve derived from these
+
values for b~~i as a function of ~~ is given by fig. 6; it is practically
the same for the three cases but on different scales. For all three the
vertical dimensions are four times the horizontal ones. The z~ axis is
positive downwards.
TABLE I
p.=o
b2 h
p. = O.i'i
b2 zo'
T3
1'3-
Q.327
0.618
0.654
0.981
- 0.031
0
0.009
0.149
+
+
b2 h
b2 zo'
- j'3
T3
0.212
0.401
0.4240.636
I' = 1.0
- 0.0207
0
0.0056
0.0997
+
+
b2 h
b2 zo'
T3
0.1572
0.2971
0.3144
0.4716
'f3
- 0.0154
0
0.0047
0.0765
+
+
The horizontal axis of fig . 6 gives h which is proportional to t ; it may,
therefore, be considered as a time-axis. The first line of Table I shows
the maximum negative values of z~ , i.e. the maximum rise of the Earth's
,surface in the belt. We see th at after this first rise, the effe et of the
sinking of the crust'g axis begins more and more to dominate the effect
of the rise caused by the plastic thickening of the ernst. At the values
·of h given by the second line of Table I the rise in the centre of the
belt is already neutralized and, as fig. 6 shows, the sinking becomes
.afterwards more and more marked. So we see that our attempt is
successful and that a down-buckling of the cru st as a result of plastic
deformation can not only weIl be explained but appears to be an
unavoidable consequence of it.
We shall now assume values for Tand b in order to derive the values
of z~ for the maximum rise and for other parts of the curve. For T we
shall assume the value of 30 km which follows from the great majority
of the observed gravity values. It is more difficult to make an estimate
of the breadth of the plastic belt b. It is in fact likely that it will be
variabie, even for parts of the same tectonic belt. \Ve shall choose three
values for our computations viz. 50 km, 100 km and 150 km; the last
is probably too large. Table II gives a list of the maximum rise in the
37
centre of the belt for the different cases and of the sinking after a three
times longer time-interval.
TABLE II
maximum rise at time tm
b
50 km
100
"
150
"
1'=0
I' = 0.5
335 m
84 "
37 "
223 m
56 "
25 "
I I' =
1.0
166 m
42
"
18 "
sinking at time 3 tm
1'=0
1610 m
401 "
178 "
I
I' = 0.5
I I' =
1080 m
269 "
120 "
1.0
826 m
207 "
92 "
We shall now make an attempt to find the time-intervals involved
in the phenomenon. We can begin by deriving h by means of the same
Or---------------~~---------
Fig. 6.
assumptions about Tand b as used above. Next we have to apply
formula (8b) for obtaining the corresponding times t. Here, however,
a great deal of uncertainty sets in because of our lack of knowledge
38
about the stress U m and the viscosity 'YJ. The stress U m must slightly
exceed the strength of the crust and so the writer thinks that
(16)
Um
= 3000 kgJcm 2
may be areasonabie value.
Of 'YJ we know practically nothing. It must no doubt considerably
exceed the viscosity of the sub stratum which has been derived from
the post-glacial uplift in Scandinavia; the writer found in this way a
dynamic viscosity 'YJ of 3 X 1022 poises. 10) From the fact, moreover that
never any evidence has been found of flowage of rocks in mountains
in case of strong shearing-stress, an indication of an even higher lower
limit of 'YJ may be derived; we thus find that it is likely to exceed at
least 1023 poises. For our purpose we shall put
(17)
'YJ =
1024 poises.
This value is of course subject to considerable uncertainty and the
same is, therefore, true for the following estimates.
Before applying these assumptions to our main problem we shall
use them for obtaining an idea of the rate of thickening of the crust.
By means of formula (8a) we find
()h = 0.15 cm per year.
ot
( 18)
As far as the order of magnitude is concerned this result appears
acceptable. As we have already mentioned on p. 31, a rate of a few mm
per year allows us practically to neglect the viscous resistance of the
sub stratum and this is necessary, for otherwise this resistance would
play a part in preventing the free development of the down-bending
and down-buIging of the crust.
The above value of (jhJ(jt implies a· rate (jhoJ(jt of the rise at the
surface of 0.03 cm per year.
Returning to our main problem we shall apply formula Sb and use
the values mentioned before of Tand b for changing from the quantity
b2hJT3 to the corresponding time. We shall do so for computing the
time-interval tm in which the maximum rise at the centre of the belt
comes into being. We thus find the values of Table lIl.
TABLE 111
Time tm of maximum rise in years
b
50 km
100
"
150
"
10) . VENING MElNESZ,
#=0
# = 0.5
# = 1.0
2.490.000
620.000
280.000
1.610.000
403.000
179.000
1.200.000
298.000
133.000
1937.
39
These re!mlts for the time elapsed since the beginning of the plastic
deformation of the erust till the maximum rise oecurs, must be doubled
to obtain the time-interval between this maximum rise and the sinking
at the time 3 tm given in Table Il. Most of that sinking is oeeurring in
the seeond half of this last interval, that is to say in an interval equalling
that of the tabie. The order of magnitude of these resuIts seems acceptable.
It is hardly necessary to add that the sinking wiII continue with
increasing speed after t.he time 3 tn! till the whole phenomenon wiII change
when it leads to the entire breaking together of the crust.
Considering the long periods we have found 11) it appears that the
value we assumed for the viscosity 1] of the crust could not be many
times larger; a value of 1025 for in stance would give ten times greater
time-intervals and such intervals seem highly improbable.
Refore leaving our subject we may point out th at in the developing
geosyncline sedimentation must occur and, because of its load and the
isostatic readjustment, t.he rate of sinking must be considerably increased.
In general we may estimate the sediments at four to five times greater
thickness than the depth of the geosyncline if na sediment at ion or
filling up by water had oecurred,
When during the further development the crust breaks together and
is pushed downwards, the buoyancy of the whole structure must become
larger and larger and this must lead to the lifting up of the adjoining
cru stal beIts. In the gravity field th is shows up by beIts of positive
anomalies accompanying on both sides the belt of strong negative
anomalies caused by the downbulge of the crust. This prove8 again the
extremely high viscosity of the crust; if it were less the central belt
would have been ab Ie to rise for readjusting its isostatic equilibrium
and no negative anomalies could have been left. The time elapsed sin ce
the downbuiging of the crust is some ] 5-·20 million years and so during
this whole period no large isostatic readjustment has apparently taken
place.
The second problem to be attacked is whether for a crust in a plastic
state we can explain th at a second syncline develops on the outside of
one of the raised beits adjoining the main syncline; we have already
mentioned that this seems to have occurred in the idio-geosynclines
in Java and Sumatra, i.e. on the inside of the curved arc. As these last
synclines do not appear to lead to an ordinary tectonic belt with great
folding and overthrusting of the surface layers as we have discussed
up to now (although there is evidence of some folding of these layers)
it seems indicated to surmise another phenomenon. In particular it appears
unlikely that the whole cru st has become plastic here and has started
bya plastic thickening.
11) The last line of TableIII will probably not occur as it corresponds to a
larger value of b than is likely.
40
In view of these facts it seems reasonable to suppose that the crust
was near the limit of plastic flow before .the wavè developed ' and that
during this development the part of the croSs-section whéPe the bending
caused a decrease of the compressive stress remained in the elastic state
while the part where the compression increased became in its entirety
subject to plastic flow (fig. 7). We denote the limit of plastic flow by
av and the stresses at surface and lower boundary by ao and al.
1
1
1
1
p=Ta--~)
1
V
I
!
r
+---p
Fig. 7.
The lengthening per unit of length at the lower boundary has an elastic
character and can, therefore, be written
while the speed of the shortening at the surface
IS
given by
and, in case ao remains constant, the total shortening by
t
2 '1.
00
We have two conditions for a(p al and w. In the first place the crosssection must remain plane, and, therefore, if we neglect the thickening
of the cross-section because of the plastic flow in the upper half
( 19)
In the second place the total integral of the stresses over the crosssection added by the bending must be zero. Denoting the increase of
the thickness T of the cru st because of the flow by 8, this condition gives
and as
8= oot.
2'1
we find
(20)
t
(T-w)
41
Eliminating
(21)
G. -
GI
from (19) and (20) we obtain
=
(T~- _)2
W
00 -
0 ••
00
2'7
Et
+
00
E
in which the second term of the right member is brought along by the
plastic thickening of the crust.
Taking the X axis horizontal and the Z axis vertical and denoting the
deviation of the crustal axis from its original position as it is caused by
the ben ding by z we find for the momentum G v Tz of the bending-stresses
of the crustal cross-section (neglecting 8)
( 22)
For the curvature of the crust's axis caused by the bending we obtain
1
è)2z
(23)
è)x 2
=
W
0.-0,
-E·
Combining (22) and (23) we get the differential equation for the curve
of the cru st 's a xis
(24)
which has the same shape as the equation for buckling of the crust in the
elastic state provided w = t T, which in the elastic state is of course
true. The half wave-Iength L of the curve is given by
L2
(25)
=
'll2
~
=
è)2z
;t2
3
w2
~
0v
è)x 2
The distance w is provided by (21); we see that the wave-Iength L
is proportional to it.
Examining (21) we find th at for small values of t we obtain large
values of wand L. These values become smaller for increasing time till
for t infinite we should get the following limits
T
w=---
(26a)
1+ I/~
r
0.
(26b)
which for
T = 30 km
E = 1.000.000 kgfcm 2
G v = 3000 kgfcm 2
w= 1.58 km
give
L= 52 km
We may conclude th at there must be a tendency towards a shortening
of the wavelength with time but we can not assume a gradual shortening
as mentioned here because such a shortening would imply a continual
42
change of Go and so our formula of p. 40for the shortening at the upper
boundary of the crust of Gotf2 1] would no longer be valid; this wonld
invalidate all the deductions.
Formulas (21) and (25) lead to the further important conclusion that
waves of this type can not originate without being started from the
outside by an effect enforcing a cru stal deformation. This must continue
for such a time-interval th at the first term of the right member of (21)
gets down to an acceptable value. In order to obtain an idea about
this question we may point out that for
1] =
E
and t
=
=
102~ poises
l.000.000 kgfcm2
100.000 years
we get
2'1
Et = 0.64.
This wonld lead to reasonable values of wand L. Putting e.g.
Go -
G~ =
120 kgfcm2
we obtain
11: = 4.3 km
and L= 140 km
For smaller values of Go - G v the value of t could of course be smaller.
Resuming we may conclude that this type of wave can only come
into being by an outside disturbance working for a considerable time.
This is of course the case for our problem. The lifting by the main
belt's buoyancy of the adjoining crustal beIts must have a disturbing
effect on the cru st outside these beIts and if this crust would fulfill the
condition of being near the limit of plastic flow it seems probable that
a downward wave of the type here described would develop. As the crust
in this case is not entirely plastic it appears unlikely that such a wave
would lead to a complete giving way and to a down-buIging of the cru st
as occurs in the main tectonic belt and this is in good harmony with
what is known about the idio-geosynclines of Java and Sumatra. As
this type of wave can only come into being in the special condition
mentioned, it is likewise clear that it does not occur everywhere near a
tectonic belt.
We may point out here the possibility that such waves will also
originate when the compression G of the crust is not exactly equal to
G v but slightly below it. Our formulas then become somewhat more
complicated but the principle remains the same. We may safely assume
that such waves will be more difficult to bring about for G differing
more from G v•
We may lastly remark that the tendency towards a shortening of the
wave~length in a later stage is perhaps discernible in the idio-geosyncline
43
in Java, where in the area south of Surabaya it shows clearly a division
in two synclines by a secondary ridge in the middle. A similar breaking
up in smaller synclines appears to occur in the Sumatra idio-geosyncline.
Resuming our investigation we may conclude that a plastic state
of the crust may provide us with a good explanation of a cru stal downbuIging in the main tectonic belt and that it does not make it impossible
to explain the originating of the idio-geosynclines of Java and Sumatra.
In view also of the other advantages there appears, therefore, good
reason to accept th is hypothesis.
As we have al ready l1lentioned it enable!'i us to come to a better
understanding of the complicated pattern of the tectonic beIts. RIJLAARD
has already pointed this out in 1935 12 ) and shown this to be the case 13)
for the whole tectonic area east and Houtheast of Asia including Indonesia.
The writer shall deal here with the latter area for which his views slightly
differ from RIJLAARb's treatment although in many regards he shall
follow the same lines.
Examining the map we see that the direction of the belt of negative
anomalies west of Sumatra fo11ows one of the two directions of the
world 's shear-pattern already l1lentioned on p. 30 and supposed to have
been brought about hy a shift of the poles in an early stage of the crust's
history. The same is approximately true for the belt fo11owing the
eastcoast of the Philippine Islands. In both areas the gravity anomalies,
the seismic movements and the topography are in harmony with the
view that the main component of the relative movel1lent of the crustal
blocks of Indonesia with regard to the outside crust is parallel to these
parts of the belt and that these blocks onIy show a slight overriding
overthe outside cru st combined with a small compression. It is not difficult
to understand th at the tectonic belt in part fo11ows pre-existant zones
of crustal weakness. The character of the relative movement of the
crustal blocks points to a direction of this movel1lent in the sen se of the
arrows, one for the western and one for the eastern half of the archipelago;
the two directions nearly coincide and perhaps the sl1lall di vergen ce
is not more than an error of interpretation.
The part of the anomaly-belt south of Java differs from the crustal
shear-pattern. Here thehypothesis of BIJLAARD, dealt with in our paper
can give a good explanation. As it has already been mentioned the
direction of the belt of plastic deformation for our case of a single stressdirection may be expected to make an angle a of 35° with the crosssection on which this stress works, i.e. an angle of 55° with the stressdirection itself. 14 ) If, besides this main principal stress el' there is a
second smaller principal stress e2 working in a crustal cross-section at
12)
BIJLAARD
13)
BIJLAARD
14)
BIJLAARD
other papers.
1935.
1936, see e.g. map on p. 20.
1931, 1935 and 1936 and many other papers,
lTERSON
1943 and
44
right angles to that of the stress el' the angle a has a different value
for which BIJLAARD derived the formula
(27)
cos 2 a = ~
+
1.'1
1.'2
31.'1 - et
which for ~2 = 0 gives a = 3(j9 (i.e. 55 9 with the stress el itself).
As the map shows this value of a agrees weIl with the belt south of
Java at least with the western part of it. Towards the east the belt
curves slightly and a diminishes towards a value of about 20 9 • This
ma)' perhaps be explained by the direction of flow which we may expect
in the belt in the beginning of the phenomenon when no great downbuIging has yet occurred; according to BIJLAARD the flow in the belt,
if free to do so, would make an angle of a = 35 9 with the stress; th is
direction is indicated in the map by the dotted arrow. This flow-tendency
must probably give rise to a secondary stress in the sen se of the belt
graduaIly increasing towards the east. According to formula (27) a value
of e2 = 0.4 el would bring down the value of a to 20 9 •
The eastern half of the archipelago shows a much more complicated
pattern of the tectonic phenomena. From north to south we have two
beIts of strong negative anomalies of which the southern one is the
prolongation of that south of Java. The northern one ends over east
Celebes; it shows still stronger negative values of the gravity anomaly,
viz - 204 mgal a.gainst about - 130 mgal in the southern belt. We must,
therefore, assume that it means a greater shortening of the cru st ; it
is, ho wever, more irregular in the sense of its axis. lts disappearance in
Celebes must lead us to the supposition that at least in the northern
part of the archipelago a fault-zone must exist dividing the eastern
from the western Indonesian blo ck ; along this fault-zone astrong
relative movement must have taken place.
We shall not further elaborate this problem and only mention a study
by the writer published in Gravity Expeditions at Sea, Vol. IV, p. 31 e.s.,
where he is led to suppose a major fault-zone in one of the two shear
directions already mentioned, running over S.E. Celebes towards the
NNW and ending at the eastern end of the Himalayas. The crustal
shortening of the second Indonesian belt here discussed would thus
represent the continuation of that of the Himalayan zone. The SSE
movement of the eastern Indonesian block would have about the same
direction as this fault-zone; it is given by the arrow of the map.
Returning now to our subject we see on the map that the northern
and southern beIts show similar directions making an angle in the rniddle
and that in both cases these directions from both sides are enclosing
angles of 55 9 with the sense of the force as given by the arrow. So here
again the hypothesis of a plastic beginning of the crustal deformation
may give a satisfactory explanation of the course of the two beIts.
If we can accept this also for the southern belt or, in other words,
45
if we have no accidental coincidence here, we can draw the important
conclusion th at the course of this belt would not primarily be determined
by the shape of the Australian continent as has often been assumed.
This view would be in harmony with the opinion given by many geologists
th at the whole area of the Banda Sea has continental character and
that this sea has fairly recently been formed by the sinking down of
the crust.
The further course of the southern tectonic belt to the north of the
part here considered, i.e. from the Key Is towards the eastpoint of
Ceram is given by the NNW direction of the crustal shear pattern and
adjoining this part we get the Ceram-Buru stretch which may perhaps
again be considered to make an angle of 55° with the arrow. Near
Buru, where the negative anomalies become weak, it beg ins to deviate
from th is direction.
The northern belt of negative anomalies practically stops towards
the north where the part considered ahove comes to an end.
Resuming we may say that the intricate pattern of the tectonic beIts
east of the supposed major shear-zone through S.-E. Celebes and the
Mangkalihat PeninsuIa can for the greatest part be weIl explained by
a combination of the crust's shear-zone pattern and the direction with
regard to the force in which plastic deformation will be likely to occur.
There is still one part of the southern belt not yet discussed, viz.
that between W. Sumba and E. Timor. In the same way as it is the
case for the part further to the east which we have already examined,
the inner Banda arc is more regular here than the outer arc where we
find the belt of negative anomalies anel the great crustal deformation.
More than one supposition can be made to explain the curve of the latter
belt in the Sumba-Timor area but the writer thinks that the data are
not yet sufficient for coming to a satisfactory conclusion.
Resuming our investigations we may say th at BIJLAARD'S hypothesis
about the crust's deformation in the tectonic beIts having from the
beginning a plastic character can explain many questions otherwise
difficult to account for.
Summary.
In comparing the hypothesis that the great deformations of the
Earth's cru st in the tectonic beIts begin by having an elastic or a plastic
character the writer comes to the same conclusion as formerly BIJLAARD
already arrived at th at the latter is more likely. By assuming a
deformation of the plastic kind throughout the whole phenomenon a
satisfactory explanation can be given of the downbuIging of the crust
which the beIts of negative gravity anomalies lead us to suppose in these
beIts. At the same time we can thus explain the intricate pattern of
the beIts in Indonesia anel it appears likely that this is also possible in
other orogenic areas.
46
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
BIJLAARD, P. P., Weerstand van een verzwakte scheeve doorsnede van een
getrokken plaat, berekend volgens HUBER·HENCKY. Ingenieur, 37 (1931).
- - - , Beschouwingen over de knikzekerheid en de plastische vervormingen
van de aardkorst in verband met de geologie van den O. Indischen
Archipel. Summary in german. Ingenieur in Ned. Indie, 11 (1935).
- - - , Théorie des déformations plastiques et locales par rapport aux
anomalies négatives de la gravitation, aux fosses océaniennes,
aux géosyncIinaux, au volcanisme, à l'orogénie et à la 'g éologie
de l' océan pacifique occidental. Rapport congrès d'Edimbourg de
I'Union Géodésique et Géophysique Intle, (1936).
GRIGGS, D. A., Theory of Mountain buildi~g. Americ. Journ. o. Sc., 237 (1939).
ITERsoN, F. K. TH. VAN, Bijdrage tot de plasticiteitstheorie. Short summaries in english, frenoh, german. Versl. Nederlandsche Akad.
v. Wetensch., 52, (1943).
SMOLUCHOWSKI, M., Anzeiger d. Akad. d. 'Viss. Krakau, Math. Naturw.
Kl. 2 (1909).
UMBGROVE, J. H. F., The Pulse of the Earth, Ie Ed. 1942, 2e Ed. (1947).
VENING MEINESZ, F. A., Maritime Gravity Survey in the Netherl. East
Indies; tentative interpretation of the results. Proc. Kon. Akad.
v. Wetenseh., Amsterdam, 33, 6 (1930).
VENING MElNESZ, F. A., UMBGROVE, J. H. F., KUENEN, PH. H.; Gravity
Expeditions at Sea, 1923-1932, Vol. 11, Netherl. Geodetic. Comm.,
Delft (1934).
VENING MEINESZ, F. A., The determination of the Earth's plasticity from
the post-glacial uplift in Scandinavia. Proc. Kon. Akad. v.
Wetensch. Amsterdam, 40 (1937).
Spanningen in de aardkorst tengevolge v. poolverschuivingen, short,
summaries in english, french, german. Versl. Ned. Akad. v.
Wetensch. Amsterdam, 52 (1943).
- - , Shear Pattern of the Earth's crust. Transaction" Americ. Geophys.
Union, 1 (1947).
- - - , Gravity Expeditions at Sea, 1923 - 1938, Vol IV. Netherlands
Geodetic Comm., Delft (1948).
F. A. VENING MEJNESZ: Earfh's crust dejorrnafions
o · .. .
B
RE.GIONAL ISOSTATIC ANOMALlE.S .
[:<.::::J
+100 -
+150 milligal.
[2J
+ 50 -
+100
0-
+ 50
50 -
0
-100-
- 50
. . -150-
-100
rnIlII - 200 -
- 150
_
-200
0
ITIITill _
-250-
..
.
,
"
"
~[ARROWS GIVE THE DIRtCTlOIJS OF THf
SUPPOSfO COMPRfSSIVf FORCES
DIRECTIONS CRUsr's
SHEAR
PATTERN
o
R
N
E