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Continuum Mechanics Aspects
of
Geodynamics
and Rock Fracture Mechanics
8/16/2019 Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics [K. E. Bullen (Auth.),
2/281
NATO ADVANCED STUDY INSTITUTES SERIES
Proceedings
of
the Advanced Study Institute Programme, which aims
at the dissemination
of
advanced knowledge and
the formation of contacts among scientists from different countries
The series is published by an international
board
of publishers in conjunction
with NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New
York
C Mathematical
and D.
Reidel Publishing
Company
Physical Sciences Dordrecht and Boston
D Behavioral and
Sijthoff International Publishing Company
Social Sciences Leiden
E Applied Sciences
Noordhoff
International Publishing
Leiden
Series C - Mathematical
and
Physical Sciences
Volume
12 -
Continuum Mechanics Aspects
of
Geodynamics
and Rock Fracture Mechanics
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Continuum Mechanics Aspects
of
Geodynamics
and Rock Fracture Mechanics
Proceedings of the NATO Advanced Study Institute
held in Reykjavik, Iceland. 11-20 August, 1974
edited by
P. THOFT -CHRISTENSEN
Aalborg Universitetscenter Matematik. Danmarks IngenifJrakademi, Aalborg. Danmark
D. Reidel Publishing Company
Dordrecht-Holland / Boston-U.S.A.
Published in cooperation with NATO Scientific Affairs Division
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ISBN-13: 978-94-010-2270-5 e-ISBN-I3: 978-94-010-2268-2
DOl: 10.1007/978-94-010-2268-2
Published
by
D. Reidel Publishing Company
P.O. Box 17, Dordrecht, Holland
Sold and distributed
in
the U.S.A., Canada, and Mexico
by
D. Reidel Publishing Company, Inc.
306 Dartmouth Street, Boston, Mass. 02116, U.S.A.
All Rights Reserved
Copyright
©
1974
by
D. Reidel Publishing Company, Dordrecht
Softcover reprint of the hardcover 1st edition 1974
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means, without written permission from the publisher
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CONTENTS
Preface
List
of Part ic ipants
Aspects of earthquake energy
K.
E.
Bullen, Universi ty of Sydney
VII
XI
Construct ion of
ear th
models 13
K.
E. Bullen,
Universi ty
of
Sydney
The Fe2 0
theory
of planetary cores
23
K. E.
Bullen, Universi ty of Sydney
Principles
of
f racture
mechanics 29
F. Erdogan,
Lehigh Universi ty
Frac tu re problems in a
nonhomogeneous
medium 45
F. Erdogan,
Lehigh
Univer si ty
Dynamics .
of
f rac ture
propagation
65
F . Erdogan, Lehigh
Universi ty
Nonlocal elast ic i ty
and waves
81
A.
Cemal Er ingen, Pr ince ton Universi ty
On
the
problem of crack t ip
in
nonlocal elast ic i ty 107
A. Cemal Er ingen
and
B.
S.
Kim,
Princeton Universi ty
Stat is t ical problems in
the theory of
elast ic i ty
115
I I
.
Kroner ,
Universi ty
of
Stuttgart
In te rna l -s t resses in
crys ta ls and in
the ear th
135
E.
Krgner , Universi ty
of
Stuttgart
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VI
CONTENTS
The e lements of non-l inear
continuum
mechanics
151
R. S. Rivlin, Lehigh Universi ty
Anisot ropic
elas t ic
and plast ic mater ia l s
177
Tryfan
G.Rogers , Universi ty of Nottingham
/
Symmet r i c
micromorph ic continuum:
Wave propagation, 201
point source solutions and some appl ica t ions to ear th-
quake
processes
Roman Teisseyre ,
Geophysica l Ins t itu te , Poland
Surface
deformat ion in Iceland and crus ta l s t re s s 245
over
a mant le plume
Eyste inn Tryggvason, Universi ty of
Tulsa
Faul t di sp lacement
and ground
t i l t dur ing
smal l
earthquake
s
Eysteinn Tryggvason, Univers i ty of
Tulsa .
Index
255
271
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PREFACE
During
a
NATO
Advanced Study
Inst i tute
in Izmir , T u r
key, July 1973 on
Modern
Developments in Engineering Seis
mology and
Earthquake Engineering i t
emerged
that a debate
on Continuum Mechanics
Aspects
of Geodynamics
and
Rock
Frac tu re
Mechanics
would be
very
welcome.
Therefore ,
i t
was
decided to
seek NATO sponsorship for an Advanced Study In
st i tute
on this subject .
The purpose of the new
Advanced
Study Inst i tute was to
provide
a
l ink
between mechanics of continuum media and geo
dynamic s . By bringing together a group of leading scient is ts
f rom
the above two fields
and
part ic ipants act ively
engaged in
re sea rch and applicat ions in the
same
f ie lds, it
was
believed
tha t
frui t ful
discuss ions
could emerge
to
faci l i tate an
exchange
of knowledge, exper ience and newly-conceived ideas.
The Inst i tute aimed pr imar i ly
at
the solution of such
problems
as connected with
the
study of s t r ess and s t ra in con
dit ions
in the Earth, generic causes of ear thquakes , energy
re lease and
focal mechanism
and se ismic wave propagat ion in
t roducing
modern
methods of continuum
and rock
f racture
mechanics .
Secondly
to
inspire scient is ts working in continuum
mechanics
to open new
avenues
of re sea rch connected with the
above problems, and se ismologis ts
to adapt
modern,
advanced
methods of continuum
and
rock f racture mechanics
to
their
work.
Geophysics is one
of the
most
excit ing
subfields
of
physics . The
main
reason for th is i s perhaps that geophysics
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VIII
PREFACE
is a
r esearch
area,
that general ly
cannot
be
controlled
by
the
observer . Fur ther
this
field i s very fascinating because it re
la te direct ly to
the
relat ionship
between
man and nature.
Fina l
ly
a
character is t ic
aspect
of
this
field
is i ts
problemoriented
nature and that scientists with
very
different
backgrounds in
physics,
mathemat ics , engineering
and
so on here work to
gether
and are forced to look
into
each
others problems.
The
t radi t ional r esearch
in geophysics used perhaps to
be based more on technical and descr ipt ive methods
ra ther
than
on
fundamental
understanding of the natural
phenomena.
But this seems now to have
changed
completely. Geophysics
became a major area
of r esearch
after the f i rs t World
War
due
to the
oil
and
mining
industry, but af ter the
second
World
War the
theory
of
seaf loor spreading has
increased
the
impor t
ance of geophysics so drast ic1y that one can ta lk about
a
r e
volution
in geophysics.
A
completely
new picture
of
the
ear th 's
crus t
with
large
plates floating on the underlying mantle is
developed.
This model
has
open
up
the possibil i ty of
getting a
re l iabel explanat ion of such phenomena
as
continental
drif t ,
sea-f loor spreading, mountain building, seismic
zones
and
volcanics
activity. 'Prediction of the occurence of
ear thquakes
is perhaps
a
possibi l i ty in few
years and
i t
will
some days
perhaps
even be possible
to prevent ear thquakes by injecting
fluids
to
re l ieve
s t ra in
along
rock
fractures .
The
central
idea
in the theory of plate
tectonics is the
excis tens of
a
r igid upper layer , which has
a
considerable
strength
and is roughly 100
ki lometers thick. This
layer
r es t s
or floats on a second layer , which has essential ly
no
strength
and a
thickness of
several
hundred ki lometers . The
second
l ayer i s assumed to offer
practical ly
no
res is tance
to the ho
r izontal
movement of the
upper
layer .
The
upper
layer
is
di
vided into
large
plates which
are
bounded
by
the ocean r idges
and
by
cer ta in
faults.
F r o m
the point
of
view
of
continuum mechanics
the
theory of plate
tectonics
is of great
in te res t and
r a i se
a lot
of interes t ing problems. How
is the
force
sys tem
responsible
for the
movements of the plates ar ranged? Is
the
movement
due to differences in tempera tures under the oceans and the
continents? Is
i t
possible by
considering the ear th as a m e
chanical model to calculate
in
details the motion of the plates ,
the
occurence of ear thquakes e tc . ?
All
aspects
of
modern
continuum
mechanics
a re
needed
to answer such questions. Can
the plates
be considered rigid,
elast ic,
plast ic or
viscoelast ic
or do we need
a more
sophist
icated theory?
Are the
plates
homogeneous and i sot ropic?
Is
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PREFACE
i t possible
to
obtain good
solutions
with
regard
to wave
pro
pagation in
the earth?
IX
A new period began in
geophysics
with the theory of
plate tectonics twenty
years ago.
In
continuum
mechanics
a
new
period
began in
1945.
The
new
period is characterized
by
work
on non-l inear phenomena, part icular ly in the case of
large
deformations.
On a sound
basis
the well-known theories
have been supplemented
with
new
theories
able to take
into
considerat ion nearly
all ·situations.
This
new
period in continuum
mechanics can also
be
characterized by the fact
that
continuum
mechanics
to day is
based on
more general pr inciples than i t used to
be. But,
unfortunately
the
physics
behind the
new
theories
often cannot
follow up with the mathemat ical manipulations.
Therefore solving rea l problems
in geophysics
perhaps
may lead
to new improved theories
of
great
pract ical
value.
The problems are there
- the
challenge
is great .
P . Thoft-Christensen.
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SCIENTIFIC DIRECTORS
Thoft -Christensen,
P .
Solnes, J .
LECTURERS
Bullen,
K. E.
Erdogan, F.
Eringen, A. C.
"
roner,
E.
PalInason, G.
Rivlin,
R.
S.
Aalborg
University
Center
Danmarksgade 19
9000 Aalborg
Denmark
University of Iceland
Reykjavik
Iceland
Univer sity of
Sydney
Sydney, N.
S. W. 2006
Austral ia
Lehigh Univer sity
BethleheIn
Pennsylvania 18015
USA
Princeton Universi ty
E-307
Engineering
Quadrangle
Princeton, N.J . 08540
USA
"
nst. fur
Theor .
und
Angew.
Physik "
Univer
si tat Stuttgart
7 Stuttgart I
W. GerInany
LaInont -Doherty Geological
Observatory of ColuInbia
University
Palisades,
N.Y.
10964
USA
Lehigh
University
BethleheIn
Pennsylvania 18015
USA
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XII
Rogers , T. G.
Teisseyre , E.
Tryggvason, E.
PARTICIPANTS
Armand, J . -L.
Atluri ,
S.
Batterman, S. C.
Bj¢rnsson,
S.
Boulanger, P.
Byskov,
E.
LIST OF PARTICIPANTS
Dept. of
Theore t ica l Mecha
n I C S
Universi ty of Nottingham
Universi ty
Par k
Nottingham NG 7 2RD
England
Inst . of Geophysic s
Pol ish
Academy
of Science
Pas teura 3
00-973 Warsaw
Poland
Dept. of
Ear th Sciences
Univer sity
of Tulsa
600
South
College
Tulsa ,
Oklahoma
74104
USA
Dept. of Mechanic s
Ecole
Poly
echnique
12 Avenue Boudon
75016
Par i s
France
Georgia Inst i tute of Tech
nology
225 North Avenue,
N.
W.
Atlanta,
Georgia
30332
USA
Universi ty of Pennsylvania
1 1 1
Towne
Building
Philadelphia
19174
USA
Univer sity of Iceland
Reykjavik
Iceland
Univer
si te
Libre
de
Bruxel les
Depar tement
de Mathematique
Avenue F . -D. Roosevel t , 50
1050
Bruxel les
Belgium
Danmar ks
T
ekni
ske
H¢j
skole
Bygning 118
2800 Lyngby
Denmark
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LIST OF PARTICIPANTS
Caiado, V.
Cetincelik, M.
Drescher , A.
Einarson, T.
Finn, W. D. L.
Gunnlaugsson, G.
A.
Hanagud, S.
Harder , N.A.
Jacobsen, M.
Jensen, Aa. P .
XIII
Geophysical Institute of Lisbon
Universi ty
Rua
da Escola
Poli tecnica
Lisbon
Portugal
Dept.
of Earthquake
Engineer
ing
P. O. Box 400
Kizilay
Ankara
Turkey
Inst. of Fund. Tech. Res.
Pol ish
Academy
of
Sciences
Swietokrzyska
21
00-049 Warsaw
Poland
University of
Iceland
Reykjavik
Iceland
Faculty of
Applied
Science
University of Brit ish
Colum
bia
Vancouver,
B. C.
Canada
University of Iceland
Reykjavik
Iceland
Georgia Insti tute of
Tech
nology
225 North Avenue, N.
W.
Atlanta,
Georgia
30332
USA
Aalborg
University
Center
Danmarksgade 19
9000
Aalborg
Denmark
Aalborg University Center
Danmarksgade 19
9000 Aalborg
Denmark
Danmarks Ingenif/.lrakademi
Bygning
373
2800
Lyngby
Denmark
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XIV
Karaesmen, E.
Karlsson,
T.
Krenk,
S.
Kusznir ,
N.
J.
Neugebauer,
H.
Ramstad , L. J .
Rathkjen, A.
Sabina,
F. J .
Sandbye,
P .
Sawyers, K.N.
LIST OF PARTICIPANTS
Dept. of Civil Engineering
Black Sea Technical Univer
si ty
Trabzon
Turkey
Universi ty of Iceland
Reykjavik
Iceland
Danmarks
Tekniske Hpj skole
Bygning 118
2800
Lyngby
Denmark
Dept.
of
Geological Science
Universi ty of
Durham
Durham
England
Johan ,)\Tol£gang Goethe-Uni
vers i ta t
Fe ldbergs t rasse 47
6 Frankfur t a. M. 1
W. Germany
Inst .
for
Statikk
NTH
Trondheim
Norway
Aalborg Univers i ty Center
Danmarksgade 19
9000 Aalborg
Denmark
Inst i tuto
de
Geofisica
Tor re de Ciencias
Ciudad
Universi tar ia
Mexico
20, D. F .
Danmarks Ingeni9Srakademi
Bygning
373
2800 Lyngby
Denmark
Lehigh Universi ty
Bethlehem
Pennsylvania 18015
USA
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LIST OF PARTICIPANTS
Seide,
P .
Selvadurai , A. P . S.
Sigbj95rnsson, R.
Steketee, J .
A.
Thomsen, L.
Wilson,
R.
C.
Withers ,
R. J .
Woodhouse, J . H.
Universi ty of
Southern
California
Dept.
of
Civil
Engineering
Los Angeles, Cal if .
90007
USA
Dept. of Civil Engineering
Universi ty of
Aston
Gosta Green
Birmingham B4 7ET
England
Universi ty
of
Trondheim
NTH
Trondheim
Norway
Delft
Universi ty of
Tech
nology
Dept. of Aeronaut ical Eng.
Kluyverweg 1
Delft
Netherlands
xv
Dept.
of
Geological
Sciences
State
Universi ty of
N. Y.
Binghamton,
N.
Y. 1
3901
USA
Universi ty of Utah
Salt Lake
City
Utah 84112
USA
Phys ics Department
Universi ty of Alberta
Edmonton
Canada
Dept. of
Applied
Mathema
t ics and Theoret ical Physics
University
of Cambridge
Silver
Street
Cambridge CB3 9EW
England
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ASPECTS
OF
EARTHQUAKE ENERGY
K. E. Bullen
c/o
Department
of Applied Mathematics,
University of Sydney,
Australia
ABSTRACT.
Some aspects
of the energy in
seismic
waves are dis
cussed, with special reference
to
the problem of estimating the
to ta l
energy released in earthquakes. A calculat ion i s presented
connecting
the
energy of a large earthquake with the size of the
region in
which signif icant
deviatoric s t ra in has accumulated
prior
to
the
earthquake.
1. EXPRESSIONS FOR ENERGY IN SIMPLE
ONE-DIMENSIONAL
WAVE TRANS
MISSION
Let
v
be
the velocity of a t ra in of waves advancing along the x
axis
in
a
uniform
deformable medium. The displacement u may be
represented a t time t by the
form
u
f
(x -
v t )
L A
cos{2n(x/A
- t /T ) + E } ,
r r r r
(1 .1)
(the
summation
may
need to be
replaced
by
an
in tegral ) ,
where
Ar
denotes
the
amplitude,
Ar the wave
length,
and Tr the
period
of
a sinoidal
constituent.
Let W be the mean
energy in the
wave motion, per unit volume
of the medium. Half th is
energy
i s kinet ic
and
half
potent ial
(see
ref .
1, §3.3.6). Thus W i s twice the mean kinetic
energy
per
unit volume.
In a portion of the medium of length b
(say)
paral le l to
the
x-axis, unit cross-sect ional area
and
density p, the kinetic
energy
i s
Tho[t-Christensen (ed.), Continuum Mechanics Aspects o[Geodynamics
and
Rock Fracture Mechanics, 1-12.
All Rights Reserved. Copyright © 1974 by D. Reidel Publishing Company, Dordrecht-Holland.
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2
K.E.BULLEN
f
b 2
o
~ ( a u / a t ) dx.
(1.2)
To obtain W,
we
have
to
divide (1.2)
by
b
and l e t
b ~
00
(in order
to get
the required mean
value) and then
double. On
subst i tut ing
from
(1.1)
and
reducing,
we
obtain
W (1.3)
In par t icu lar , for a purely
s inoidal
wave t ra in , we have,
dropping'subscripts,
W
2 2
-2
2rr
A T p.
(1.4)
The mean
energy
W'
in
a
port ion
of the
medium
of
length
A
and
uni t
cross-sect ional
area
i s
W' =
2 2
-2
2rr
A AT p.
2.
EARLY
METHODS
OF
ESTIMATING SEISMIC WAVE
ENERGY
2.1
Preliminary
remarks
(1.5)
I t
would
be
theoret ical ly
possible, using formulae based on (1.3)
(1.5),
to
estimate
to
closer
precision the
wave
energy,
E
say,
released
in an earthquake i f
suff ic ient ly
well
determined measure
ments
could
be made
on
seismogram
records taken a t
a
suf f ic ient
number
of sui tably dis t r ibuted s tat ions on the Earth ' s surface.
In
pract ice,
many d i f f icu l t ies make the task formidable and, more
over, complicated by greater or less uncertaint ies
a t
several
stages of the process. Following is
an outl ine
of some early
attempts a t approximations.
2.2
Use of
records a t
nearby s tat ions
With some earthquakes, useful estimates of E can be derived from
records
of SH
waves a t
nearby s tat ions. A
s izable f rac t ion
of
the
to ta l bodily wave
energy
can
usually
be expected
to
be in SH
waves,
the treatment
of which i s much
simpler than for
P
and SV since SH
are ref lected and refracted only into SH waves.
For the Jersey
earthquake
of
1926
July 20,
Jeff reys
[2]
noted
tha t
SH bodily
waves
tha t had
t rave l led
through a near-surface
crustal layer were comparatively
large
a t epicentral
distances
up
to 500 kID. He assumed tha t the energy in these
waves
t ravel led
out
from
the
focal
region
(presumed
to
l ie
inside
the
layer)
with
a
cyl indrical
wave
front
inside the layer , and tha t th is energy
approximated
to E. Let
P
and H be
the
density
and
thickness of
the
layer ,
the
angular epicentral
distance of a
recording
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ASPECTS OF EARTHQUAKE ENERGY
s tat ion Q, and l e t rO be the Earth's radius.
Treating
the waves
as
a
sinoidal t ra in of amplitude
A, period T
and to ta l
length L,
and using (1.5),
he
arrived a t
E
3 2
-2
4n
p ( r o s i n ~ ) H L A
T
n p ( r o s i n ~ ) H L V m 2 ,
(2.1)
where vm i s the maximum
velocity
of
the
ground motion.
From records a t each
of several single s ta t ions , Jeffreys
used (2.1) to estimate tha t E 10
12
J
for
the Jersey
earthquake.
Since bodily waves
are not usual ly
closely
sinoidal , a
s l ight ly
more
accurate
formula
would
be
3 J 2 -2
E 4n p(rosin6)HB A T
dt ,
(2.2)
where B is the wave
velocity, and
the
integrat ion
is
over
the
arr iving SH
t ra in .
2.3 Estimation of bodily
wave
energy from
records
a t
dis tant
stat ions
3
Assuming
a
spherical ly symmetrical issue of
bodily
waves from
the
focal
region
F,
and
t reat ing
the
Earth
as uniform,
Gali tz in
derived
a formula which,
as la ter modified, i s
equivalent
to
3 . 2J 2 -2
E 4n p B { 2 r o s 1 n ( ~ / 2 ) } A T
dT,
(2.3)
where A and T re la te to
the
SH waves recorded
a t
a s tat ion Q.
With (2.3),
F
i s assumed to
be
a t
the Earth's
surface, the
wave
energy
therefore issuing
downward
from
F.
With
SH waves, the
calculat ion
i s ass is ted by the theoret ical resu l t tha t
the
ampli
tudes
of
the
waves emerging
a t
Q are
half
those of
the surface
ground
movement. (The
corresponding
resul ts
for
P
and SV
are
more
complicated.
)
When the focal depth
h
i s appreciable (the
wave
energy
now
issuing upward as well as downward from F),
(2.3)
needs to be
replaced by
3 2 . 2 J 2 -2
E Sn pB{h
+ 4r
O
(r
O
-
h)s1n ( ~ / 2 ) }
A T dt .
(2.4)
I f
Q i s
a t
the
epicentre,
(2.4)
reduces to
E
sn3pBh2JA2T-2dt.
(2.5)
A formula equivalent to (2.5) was made the basis
of
a method of
Gutenberg
and Richter
for est imating E.
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2.4 Estimation of
energy
in surface waves
Jeffreys used
simple
Rayleigh wave
theory
to
estimate the
order
of
magnitude
of the
energy in P-SV surface waves. He
arr ived
a t a
formula
of
the
form
(2.2),
with
H
replaced
by
1.1
A,
where
A
is
the wave length and A i s
the horizontal
component of
the surface
ground motion.
(For
deta i l s , see re f . 1, §15.1.3.)
2.5
Application
Depending on
the
charac ter i s t ics
of
an earthquake
(magnitude,
focal
depth,
re la t ive proportions of energy in bodily and surface waves),
formulae
based on those in
§§2.2-2.4
have been much used in ef for t s
to estimate E.
This
applies
in par t icu lar in the pioneering
work
of
Gutenberg
and
Richter
[3]
on
the
Earth 's seismici ty.
3. STEPS TOWARDS IMPROVED PRECISION
The
simplif icat ions
in §2 are of course fa i r ly dras t ic . A s tep
towards
improved precis ion i s
indicated below.
3.1
Taking account
of
continuous
var ia t ion
of
velocity with
depth
Consider (say)
P waves
issuing
symmetrically from a
focal
region
F
and
t ravel l ing to
points
Q
a t
the
Earth ' s
surface
with
velocity
a
which
depends
on
the distance r
from
the centre of the
Earth,
here
assumed spherical ly
symmetrical. Assume
for
the present tha t
the
waves are continuously ref rac ted
between F
and
Q
and encounter
no
internal surfaces of
discontinuity. Let
be
the
angular
epi
central distance of Q, and l e t n = ria. At
any point
of a ray,
l e t e be the angle between the
ray and
the
level
surface through
the
point . For the ray FQ,
l e t
e = e
l
, eO a t F, Q, respectively.
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ASPECTS OF EARTHQUAKE ENERGY
Let I
be the
energy, per
unit
sol id
angle,
in
the
waves in
question
as
they
issue
from F. Then the
energy
dI being t rans
mitted
through
the volume bounded by rays for which e = e l '
e
l
+ del i s given by
dI
2nIIdelicos e
l
•
(3.1)
"The
area
of the
Earth's
surface
a t which
th is energy
emerges i s
2nr02ld6lsin6. The corresponding area dA
of
the
emerging
wave
front i s
dA
2
2.
A I
A
I
rO s ~ n u s ~ n eO
u.
(3.2)
5
Hence, neglecting
al l
per
unit area
of wave
energy dissipat ion
en
route,
front emerging a t
Q
i s
the
energy U (6)
U(6)
dI
dA
I
dell
d6 •
sin eO
(3.3)
Let T be the t ravel time along the ray FQ. Then, by standard
seismic ray
theory,
dT/d6,
(3.4)
whence
(see
ref . 1, §8.l)
Ino
(2
2
U(6) 2. n
l
tan eO
rO n l s ~ n 6
(3.5)
3.2 Limitations
of
the formula
The
formula
(3.5),
though
superior to (2.4)
through
taking
account
of
variat ion
of a with
r , s t i l l ignores several complications tha t
are
signif icant in pract ice.
The
formula i s
inadequate for waves which have encountered
one or more surfaces of discontinuity
between
F and
Q.
Incident
P waves
may
be
converted a t such
a surface
into
P
and
SV
ref lected
and
P
and
SV refracted
waves.
For anyone of
the
four se ts of
converted waves, an energy ' t ransmission
factor '
has to be applied
to
formulae
of the type (3.5).
Such
factors vary
substant ial ly
with
the
angle of incidence a t
the
discontinuity surface and are
subject
to
uncer ta int ies , which may be considerable,
as to
the
character
and
location of the discontinuity. (For some detai ls
on
transmission
factors,
see ref .
1,
chapters
5 and
8.)
Sufficiently
rapid
changes
of
property
inside
the
Earth
may
also cause
conversion
of
energy. Sometimes, depending on the wave
lengths
involved,
a rapid
change may
be t reated as a discontinuity.
(A
mathematical
discontinuity i s
of
course only a mathematical
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6
K.E. BULLEN
model concept.) Where a rapid
change
cannot be t reated
as
a dis
continuity,
complex
analysis
may be
required.
For an
indication
of the type of
mathematics needed,
see refs . [4], [5], [6] and
[7, §8.4].
Through (3.5), modified i f necessary
by the
inclusion of
transmission
factors
or
the i r equivalent, a
fa i r
estimate
of
the
to ta l
seismic
wave energy can
sometimes
be made
from
data a t a
limited number of s tat ions . In pract ice,
data
from a wide-spread
distr ibut ion of stations
i s l ikely
to be
required because
of
asym
metry
a t
the
focus. The
observational
uncertainties, as well as
uncertaint ies
on the
distributions of
a and S
with
r , contribute
to the
uncertainty
of the estimated energy.
Account
has
also
to
be
taken
of
departures
from
spherical
symmetry in
the
Earth. Departures associated
with
the e l l ip t ic i
t ies
of
surfaces
of
constant
velocity
in
the Earth have
only
minor
effects and could i f needed
be
readily allowed for . But departures
due
to la tera l variat ions of wave velocity, especial ly
in the
crust ,
have
more
serious effects .
There
i s no
ready
way of dealing with
these except by
long t r i a l
and
error ,
and slow accumulation of
evidence on
the three-dimensional
velocity
dis tr ibutions. Limita
t ions of
th is
evidence
add
further to the uncertainties.
Energy losses
also
occur through scat ter ing
(see
e.g .
ref .
8)
and
departures
from
perfect
elas t ic i ty .
B ~ t h
[9]
estimated
that ,
with bodily waves from shallow-focus earthquakes,
the losses
inside
the. crust
(including
losses connected with la tera l variations) may
involve
an
energy
'extinction factor ' as high as
20. The factor
i s
greater
for
S than P waves, and B th regarded the high extinc
t ion
of short-period bodily waves near the focal region as one of
the more
serious
sources of uncertainty
in estimating
earthquake
energy. He also estimated
that
the to ta l extinct ion during t rans
mission inside
the
mantle
is
10-15
per cent of that inside the
crust . For waves t ravel l ing
long
distances D,
attenuation factors
of
the
form
e
kD
are
sometimes
introduced.
(See
again
ref .
9.)
The energy in surface waves i s .not
taken
into
account
in
(3.5).
I f
th is energy
i s
not
independently estimated (see e.g.
§2.4), a further factor
has
to be applied to allow
for
i t . The
factor varies
from earthquake to earthquake
and i s special ly
sensit ive
to
the
focal depth.
The summary is that , although
formulae of
the
type
(3.5)
have
led to some
increase
in precision, it i s
not
yet
possible
to es t i
mate the
energy
of an earthquake
within a factor of at least
2:
usually the uncertainty
factor
i s
appreciably
greater
than
2.
For deta i l
of some further approaches, see Knopoff [10],
Belotelov, Kondorskaya and Savarensky [11], DeNoyer [12] and
Randall [13].
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7
4.
ESTIMATION
OF ENERGY
FROM
STRAIN MEASUREMENTS
A wholly different approach to
the
problem of
earthquake
energy is
through the applicat ion of geodetic data
to
est imating
the
s t ra in
energy
in
the
vicini ty
of
the
focal
region
before
and
af ter
an
earthquake. The s t ra in measurements are made in the vicini ty of
geological faul ts a t the surface, and
assumptions
are made on the
faul t ing
and s t ra in below.
Examples
of
earthquake
energy
calcula
t ions
made
in
th is way are those of Byerly
and
DeNoyer [14]. They
gave
for
the
San
Francisco
earthquake
of
1906
April
18,
E = 0.9
x
10
16
Ji the
Imperial Valley earthquake
of 1940,
0.96 x 1015
J i
and the Nevada earthquake of
1954
December
16,
1-1.5 x
1015
J .
5.
EARTHQUAKE ENERGY
AND
MAGNITUDE
The discussions
in §§2,3
make it evident that
the close
determina
t ion of earthquake
energy
must
have considerable
recourse
to empi
r ica l
methods.
Detai ls of these
methods
are
closely
l inked with
estimations of earthquake magnitudes. The present section
br ief ly
outl ines
some of the principal resu l t s .
The
f i r s t
magnitude
scale
[15]
defined the
magnitude M
in
terms of the maximum amplitude t raced by a standard seismograph
(free period 0.8 S i s ta t i ca l
magnification
2800;
damping
coeff i
cient
0.8)
a t
an
epicentral
distance of
100
km.
Empirical
tables
were set up with a view
to
reducing observations taken
a t
other
distances
and on other types of
seismographs
to
resul ts correspond
ing
to Richter 's standard conditions.
Originally, only shallow
focus
earthquakes
were
considered,
but
the tables
were . later
extended to allow for
s ignif icant
focal
depth. Subsequently,
various modifications
were made to the
magnitude
scale
i t se l f .
On
the
la tes t scale , the largest
earthquakes
have M = 8.9.
In a long ser ies
of
papers, Gutenberg
and Richter sought
to
connect
M
with the
earthquake energy
E
by
the
form
aM (5.1)
bringing vast quant i t ies
of empirical data
to bear.
A recent
revis ion
by
B ~ t h [16] gave
5.24 + 1.44 M,
(5.2)
where
E is
in
joules i th is
gives
E
10
18
J for
M =
8.9, and
E = 1.7 x
105
J
for
a
zero
magnitude earthquake (conventionally
presumed
to
correspond
to
the
smallest
recorded
earthquakes).
Formulae of the type (5.2), along with other observational
evidence,
have
been applied with much success to
estimate
many
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K.E.BULLEN
aspects
of
earthquake
energy release. For example, Gutenberg [17]
estimated that the
to ta l
annual
release
of earthquake energy is
10
18
J , corresponding to
a
rate of work of 10
7
-10
8
kW.
This
is
about 10-
3
times
the
ra te of heat escape from
the Earth's
in ter ior .
( I t
has
sometimes
been
suggested
that
the
Earth
acts
as
a
heat
engine
converting a
small fract ion of
the
escaping heat into s t ra in
energy.)
I t
is interes t ing tha t
the
energy in a major
hurricane
is
of the same order as that
in
an
extreme
earthquake.
Eighty
per
cent
of
the
to ta l
energy
in
al l earthquakes comes
from
those for which E = 10
16
_10
18
J . A table by B ~ t h [16] gives
the following percentages of
earthquake
energy
release in
different
geographic
regions: North America (including Alaska), 10; South
America,
16;
Southwest
Pacif ic and Phil ippines, 26;
Ryukyu-Japan,
16;
Kurile
Islands,
Kamchatka
and
Aleutians,
9;
Central
Asia,
17;
Indian and Atlant ic Oceans, 6.
B ~ t h
and Duda
[18], assuming that the volume V
(m
3
)
of
the
strained region
prior
to
a
large earthquake is about equal to the
volume encompassing the aftershocks, derived empirical ly
3.58
+ 1.47 M.
(5.3)
The formula (5.3)
has
some
in teres t in connection with
the
calcu
la t ions in §7.
6.
NUCLEAR
EXPLOSIONS
AND
EARTHQUAKE ENERGY
Since
nuclear
explosions are in
certain
respects of
the nature
of
controlled
earthquakes, with knowledge
available of
the to ta l
released energy, the source location and
time
of origin, there i s
the theoret ical poss ib i l i ty
of
using them to
derive
information
on
the energy released
in
natural earthquakes.
There
are, however,
several
pract ical
d i f f icu l t ies
in obtaining
useful
resul ts in th is
w ~ .
On
dis tant records
of underground nuclear explosions,
S
and
surface
waves
are
often weak.
For
th is
reason alone, the formula
(5.2)
may
give
log10E
too great
by
unity or
more i f M i s estimated
by
the
usual procedures for natural earthquakes.
More
important
i s the s ize
and var iab i l i ty of
the
seismic
efficiency f (the ra t io
of
the
seismic wave energy caused
by
the
explosion to
the
to ta l
energy
released). Average values of f a r e
as follow: explosions in
the
atmosphere a t alt i tUdes
1-10 km,
0(10-
5
);
a t
the
Earth ' s
surface,
0(10-
4
) ;
300 m
underground,
0(10-
3
);
30 m
underwater,
0(SXlO-
3
) ;
300 m
underwater,
0(10-
2
) .
The
values
vary
widely
with the
source conditions: for an explo
sion
inside
a
large
underground
cavity,
f may be less
than
10-
2
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ASPECTS
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tha t for a well-tamped explosion a t the same
depth.
For further
detai ls ,
see
ref .
1
(chapter 16)
and
ref . 16 (chapter
11).
7.
EARTHQUAKE ENERGY
AND
EXTENT
OF
STRAINED
REGION
9
The extent of the strained
region
prior to a
large
earthquake can
be assessed from
early calculations of
the writer [19,20]
given
below.
Several invest igators, e.g . Tsuboi [21], independently
arrived
la ter a t similar resul ts derived
on
a somewhat
narrower
basis .
7.1 Preliminaries on s t ress-s t ra in re la t ions
and
s t ra in energy
For
present
purposes,
it
i s
suffic ient
to
assume
perfect
elas t ic
i ty , isotropy and
l inear
s t ra in
theory.
Then
the
set of s t ress
s t ra in re la t ions
may
be
writ ten
as
(7.1)
where
the
Pi j
and
ei j are the components of ordinary s t ress
and
s t ra in , e
(= Eekk) is the dilatat ion, 0i j i s the Kronecker del ta ,
k
i s
the incompressibility
and the r ig1dity.
( I t
is preferable
to
use
k
and which have immediate
physical signif icance, ra ther
than pairs
such
as the
Lame
parameters A and
~ . )
The deviator ic s t ress
and
s t ra in components
Pij and
Eij are
defined by
I
P,
,
Pij
-
j'EPkkOij'
1J
(7.2)
E,
. e, . -
}Eekko
i j
•
1J 1J
(7.3)
(All summations
are
from 1
to
3 and are
with
respect to repeated
subscripts . )
By (7.2) and (7.3), the
s t ress-s t ra in
relat ions (7.1)
may
be
re-wri t ten
as
3 k e ~ P . .
1J
2 ~ E ,
, .
1J
(7 .4)
The relat ions (7.4) have
the important
advantage tha t the physically
signif icant parameters k and
appear
in
separate
equations.
The s t ra in energy W per
unit
volume a t a point of a strained
body i s given
[1,
§2.3.5]
by
w
~ e 2 + ~ ( H e , ,2 _ }e
2
) .
1J
By (7.2) and (7.3), th is becomes
w
k2ke
2
+
H E
2
~ i j
(7.5)
(7.6)
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K. E. BULLEN
The two
terms
on the
r ight
side of
(7.6)
give the compressional
and the deviatoric
s t ra in energy
per unit volume, respectively.
7.2 Strength
Let Pi
( i
=
1,2,3)
be the principal stresses a t a point Q of a
stressed body and
l e t primes
indicate values of
s t ress
components
a t the stage when, under increasing stress ,
flow
or fracture
s tar ts to occur a t Q.
Let
p i ~ P2 ~ P3. The strength a t Q i s
commonly defined in terms of the values of certain functions of
the
pi .
Two
different functions have been used:
the
s t ress
difference p i - P3; and
the
Mises function S, where
2 2 2
(p' - p') + (p' - p' ) + (p' - p')
13
21 32·
(7.7)
From
(7.2)
and (7.3), it can be deduced
that
(7 .8)
The
strength
sets an
upper
bound
to
the
possible value of PI - P3'
or
of
1(3EE(P
i j
)2)
on the two defini t ions , respectively.
By simple algebra, it can be shown
tha t
S l i e s
between
1.22
and
1.42 times the stress-difference. Since only orders of magni
tude of S are involved in geophysical applicat ions, it does not
matter
which
definit ion
i s
used.
The
Mises
strength
is
used below.
7.3 Connection
between
energy, strength and r ig id i ty
Just
before a large
earthquake, l e t
V be the volume of
the region
R
(surrounding
the
focus) inside which
there
i s signif icant
deviatoric
s t ra in .
At any point of R,
2
3EEP
. .
l.J
2
as ,
(7 .9)
Corresponding
to (7.6),
the
to ta l
s t ra in energy Es inside R
i s equal
to
~ E d '
where
(>1) i s the ra t io of
Es to
the deviatoric
s t ra in energy Ed'
and
(7.10)
the integrat ion being through V.
Let
E be the energy
released
in
the
form of
seismic waves,
and
write
E
=
~ Y E d . I t i s to
be
expected
tha t
0(1-2)
and
Y ~ 0.5. For
the
purpose of an order of magnitude
calculation
i t
i s
appropriate
to take ~ Y = 0.5. Then
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ASPECTS
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11
E
0 . 5 f f f ~ E E E
.. dT.
~ J
(7.11)
For simplici ty,
and S
wil l be
t reated as constant throughout
V.
Then,
using
(7.4)
and
(7.9),
we have
2 4 ~ E :::
ff
3EEP . . 2dT
~ J
s2fffadT
2
S V
O
' say,
(7.12)
where
Vo
would
be the volume
of
R i f the
material
had been
about
to f racture or
flow
a t every point of
R.
7.4
Implications
of equation
(7.12)
Inside the
range of
depth a t which
earthquakes
originate, the
r igidity ~ is known to l i e (in
effect)
between about
0.4
and
1.5
x 10
1
N/m
2
• For the largest earthquakes, E 10
18
J (§5).
Thus
(7.12) gives
(7.13)
whence S and Vo must both
be
considerable in a large earthquake.
Laboratory
evidence indicates that for rocks
in
the outer
par t
of
the Earth,
S $
0(10
8
N/m
2
) .
Thus
(7.13)
gives
(7.14)
This resu l t seemed surprising
when
f i r s t
derived, though it has
since
been
amply confirmed. I t implies tha t the strained region
would
occupy
a volume
a t
leas t equal to the volume of a
sphere
of
50 km diameter, even i f the material were about to
fracture
or
flow
throughout
th is
volume.
Since
the material
would
actual ly
be well
short of th is
condition throughout most
of R, V
must
be
considerably greater
than
V
O
' perhaps exceeding
the
volume of a
sphere
of
diameter 100 km. Furthermore, it i s improbable that R
would be spherical . Hence one
or
two
of
the dimensions
of
R would
probably be well in
excess of 100
km, thus tending towards the
order of the Earth 's radius.
The
resu l t
(7.14)
may be compared with the
resu l t
obtained
using
the empirical formula (5.3) which, for an earthquake
of
magnitude
8.9, would
give V 5
x 10
16
m
3
• The
resu l t
(7.14)
also
played
an
important
role
in
the reduction
made
by
Gutenberg
and
Richter from
10
20
to 10
18
J as thei r
estimate
of
the
energy
in
an
extreme earthquake.
In
addit ion,
it showed tha t the strength S
cannot
be
much less than
10
8
N/m2 where a
large
earthquake occurs.
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12
K.E.
BULLEN
The finding that
one
or more of the dimensions of V could be
so
large
provided some indirect support for the notion of
possible
causal connections between
globally
wide-spaced
large earthquakes.
Benioff [22]
had suggested
that
earthquakes
for
which
M > 8.0
may
not
be
entirely
independent
events, but are
related to
a
global
s tress
system.
REFERENCES
1. K. E.
Bullen,
Introduction
to
the Theory of Seismology,
University Press, Cambridge,
3rd
ed, 1965.
2.
H.
Jeffreys, Mon. Not. Astr. Soc.*l, 483, 1927.
3.
B. Gutenberg and C.
F.
Richter, seismicity of the
Earth
and
associated
phenomena,
University
Press,
Princeton,
2nd
ed,
1954.
.
4.
J .
G. J . Scholte, Kon. Ned.
Meteorol.
Inst .
65,
1, 1957.
5.
L. Cagniard, Reflexion e t Refraction
des
Ondes
Seismiques
Progressives,
Gauthier-Villars,
Paris ,
1939.
(English t rans.
by E. A. Flinn
and
C. H. Dix,
McGraw-Hill,
New
York,
1962.)
6. B. L. v. d. Waerden, Reflection and
Refraction of
Seismic
Waves, Shell
Development
Company, 54
pp. , 1957.
7. Mi.:Brth, Mathematical
Aspects
of Seismology, Elsevier ,
Amsterdam, 1968.
8. R. A. W. Haddon
and
J . R. Cleary, Phys.
Earth
Planet.
Interiors
8,
211, 1974.
9.
M.
B ~ t h ,
in
Contributions
in Geophysics,
Pergamon, London,
pp. 1-16,
1958.
10. L. Knopoff, Geophys. J . , Roy.
Astr.
Soc.,
1,
44, 1958.
11. V. L. Belotelov, N. V. Kondorskaya and E. T. Savarensky,
Ann. di
Geofis. 14, 57,
1961.
12.
J .
DeNoyer, Bull:=Seismol. Soc. Amer. 48, 353, 1958, and
49, 1, 1959. --
13. M. J . Randall, Bull.
Seismol.
Soc.
Amer. 63, 1133,
1963.
14. P.
Byerly
and
J .
DeNoyer,
in
C o n t r i b u t i o n ~ i n
Geophysics,
Pergamon,
London,
pp. 17-35, 1958.
15. C.
F.
Richter, Bull.
Seismol.
Soc. Amer. 25, 1, 1935.
16. M.
Bath, Introduction to
Seismology,
Birkhiuser
Verlag, Basel,
1973.
17. B.
G ~ t e n b e r g , Quart. J .
Geol. Soc. Lond.
112, 1,
1956.
18. M.
Bath and S. J . Duda,
Ann.
di Geofis.
17,353,
1964.
19.
K.
E. Bullen,
Trans. Amer.
G e o ~ h : i s .
Un.
34,
107,
1953.
20.
K. E. Bullen, Bull. Seismol. Soc. Amer.
45,
43,
1955.
21. C.
Tsuboi, J .
P h ~ s .
Earth i ,
63,
1956.
=
22. H.
Benioff,
Bull.
Geol. Soc. Amer.
65,
385,
1954.
* Geophys.
Suppl.
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CONSTRUCTION
OF EARTH
MODELS
K. E.
Bullen
c/o
Department of Applied Mathematics,
University of Sydney, Australia
ABSTRACT. An
outline
is given of methods used to construct model
distributions of the density, pressure, ' incompressibil i ty,
r igid
i ty ,
gravitational
intensi ty
and P and S seismic
veloci t ies
in
the
Earth 's
inter ior . Spherical
symmetry
is
assumed.
Reference
is
made
to
the
problem of
formulating a
standard Earth
model. A table
giving values
of
various
propert ies
of
the
Earth 's
in ter ior a t
selected depths is included.
1. INTRODUCTION
The Earth models
to be discussed
in
th is
paper give model
dis t r i
butions of the
density
p,
pressure p,
incompressibil i ty k, r igidi ty
gravitational intensity g,
and
P
and
S seismic veloci t ies a
and
S
in
the
Earth 's
inter ior .
An ultimate aspiration is to
derive rel iable
values
of
these
propert ies a t
points of the in ter ior
whose
posit ions
are specif ied
in
terms
of three space variables. Consideration wil l , however,
here
be limited to
spherically
symmetrical models,
the
propert ies
being thus expressed in
terms
of
the
distance r from the centre 0,
or depth z below the surface. Data
are available
[1] from which
models taking
account
of the el1ipt ic i t ies of surfaces of constant
density within
the Earth
can be
readily derived; but
fine deta i l
taking account of
other
deviations from spherical symmetry is
not
adequately
available
as
yet. Thus the
models give,
in some
sense,
la tera l ly averaged values of properties. Incidentally, non-symme
t r ica l
models
would
(apart
from el l ip t ic i ty) involve further compli
cations;
e.g. the s t ress
would
not be adequately represented
by
the
single
parameter
p - in
solid
regions
there
would
be
non-zero
deviatoric
stresses.
Thoft-Christensen (ed.), Continuum Mechanics Aspects o f Geodynamics and Rock Fracture Mechanics, 13-21.
AllRights Reserved. Copyright © 1974 by D. Reidel Publishing Company, Dordrecht-Holland.
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14
K. E. BULLEN
The
sets of observational
evidence brought to
bear
in
const
ructing
Earth
models
include:
(i)
Data on the Earth 's mean
radius
R, mass M
and
mean moment
of
iner t ia I .
The
uncertaint ies
of
these
data are
now
suffic iently
small,
compared with other uncertainties, to
be neglected.
(i i) Data derived from records of seismic
bodily
and
surface
waves,
and
free Earth osci l lat ions. The data (i)
and
(i i)
occupy
a
dominant, though not exclusive, place
in
the
model
constructions.
( i i i )
Evidence from a
wide
range of other sources, including
data on Earth
t ides,
thermal data, invest igat ions on the variation
of k with
p,
f in i te -s t ra in and sol id-state
theory,
laboratory expe
riments
on
rocks, including
shock-wave
experiments
a t
pressures
up
to 4 x lOll N/m
2
, and
evidence
from geodesy,
planetary physics,
geology
and geochemistry.
This thi rd
body
of
evidence, though
mostly less precisely determined than the seismic data,
ass i s t s
in
assessing the plaus ibi l i t ies of
models which
f i t the
seismic
data
within
the uncertainties, and usefully
supplements the
seismic
data
where
the
uncertainties
are
unusually
large.
In the historical evolution of Earth
models,
density has been
the
key
property.
The dis tr ibutions of other propert ies
are fai r ly
readily derivable when the density dis tr ibution has
been determined.
Attention
will
therefore
f i r s t
be devoted
to
the density
determina
t ion.
2. THEORY ON DENSITY VARIATION
The density p
in the
Earth
i s
a function of p, the temperature T
and
parameters qi
representing
chemical composition
and phase.
Thus
dp 2.£. dp + 2.£. d T + I: .1E- dqi
dz ap dz aT dz
aqi
dz '
(2.1)
= P + T + Q,
say. I t
transpires that the term P can be evaluated
more
accurately
than T and
Q. Also,
T/P and Q/P
are fai r ly small
for most
z.
Hence the usual procedure has been to
s ta r t
by assuming
dp/dz =
P and
then
proceed by
successive approximation.
2.1 The Williamson-Adams
equation
Let G
be
the gravitation
constant
and m
the
mass
within the
sphere
of radius r
and
centre O. Since ap/ap
=
p/k, dp/dz
=
gp
=
GmP/r
2
,
a
2
=
(k +
4p/3)/p
and
e
2
=
pIp, the
equation
dp/dz
=
P becomes
dp/dz
2
Gmp/r cp,
(2.2)
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CONSTRUCTION OF EARTH MODELS
15
where
kip.
(2.3)
An
equation equivalent
to
(2.2)
was
used
in theoret ical
work
las t
century. I t
i s now
called
the
Williamson-Adams
equation; these
authors
[2]
substi tuted
values of derived from seismic data
into
(2.2)
to
estimate dp/dz numerically
inside the
Earth.
The
application of (2.2) to
an
internal region of
the Earth
requires,
in addition to knowledge of a and 8, knowledge from non
seismic sources of
values
of p
and
m
a t
some
level
of the region.
For
example,
in t reat ing the
immediate
subcrust,
a
variety of
evi
dence
has led
to the
assumption of
about 3.3 g/cm
3
for the
value
p'
of
p
a t
the top.
Values
of
m
as
a
function
of
r
are
derived
star t ing from the surface, where m M,
and
using dm
=
4nr2p dr
along with (2.2); the
condition
m =
0 a t
r =
0
has also to
be
sat isf ied.
2.2 Temperature correction
Birch [3] derived
T
(2.4)
where
y
is
the
coeff icient
of
thermal
expansion
a t
constant
pres
sure,
and ~ is
the ' super-adiabatic ' temperature gradient.
(For
a short derivation of (2.4),
see
Bullen [4].)
For numerical
detai ls
on the application of (2.4),
see
Bullen
[5].
2.3
Generalization of
the Williamson-Adams equation
Information on variat ions of chemical composition
and
phase
in the
Earth i s not
suffic iently
well
determined
to enable the las t
term
Q of (2.l) to be evaluated
direct ly.
But the following generaliza
t ion
(Bullen
[6])
of
the
Williamson-Adams
equation takes
account
of
Q.
I t
is to be understood below
that
dp/dp stands for
(dp/dz)/(dp/dz);
and similarly with dk/dp.
From (2.3), we have
(dk/dp)dp/dz
~ d P / d z +
p d ~ / d z ,
whence, on
putting
dp/dz = gp
and
dividing by
dp/dz
where
n
n g p / ~
2
nGmp/r ~ ,
-1
dk/dp - g d ~ / d z .
(2.5)
(2.6)
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16
K. E. BULLEN
The coeff ic ient n also sa t i s f ies
dP/dp
= np/k. The
W.A.
equation
(2.2)
i s the part icular
case
of (2.5) for
which
n =
1;
in th is
case (Bullen
[7])
dk/dp
-1
1
+
g d
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CONSTRUCTION OF EARTH MODELS
17
evidence has also placed fa i r ly close bounds
to
the extent
to
which
k
i s
l ikely to
deviate
from smooth
variat ion with p.
The
k-p
hypothesis places additional
rest r ict ions
on
the
allowable
variat ions of
k,
p
and
V
in
various
parts
of
the Earth.
I t enta i ls sol id i ty in the
inner core
(Bullen
[10]) corresponding
to
the
sizable
jump in a
found
by Lehmann [11] from the
outer to
the
inner
core; th is follows from the relat ion a
2
p = k +
4v/3
and
the fact
tha t
to a f i r s t approximation p
must
increase with z
throughout the
Earth.
The hypothesis also throws l ight on a l ikely
abnormal
density variat ion inside the
lowest
200 km
of
the
mantle.
In conjunction with Birch's estimate [12],
from
shock-wave
experiments a t
pressures
exceeding lOll N/m2,
that
the
Earth 's
central
density
does
not
much
exceed
13
g/cm
3 ,
the
k-p
hypothesis
enabled Earth model
distr ibut ions
to be rel iably
continued
from
z
=
5000
km
to the centre.
4. APPLICATION
OF
SEISMIC SURFACE
WAVE
AND FREE EARTH OSCILLATION
DATA
The seismic
bodily-wave
data
yield
evidence on
p,
k and V only
in
the combinations kip and vip. The
deta i l
in §§2,3 enables evidence
from outside seismology to
be
brought to bear in deriving
values
of
p
separately
from k
and
v.
Seismic
surface
wave
and
free
Earth
oscil la t ion data, however, provide independent evidence
on
p in
combinations other
than
kip and vip.
This
evidence has
enabled
some refinements to be added to Earth models
constructed
using
the
principles of §§2,3.
Examples
of
recent
Earth models
incorporating
evidence
from
free
Earth
oscil la t ions are the models HBl (Haddon
and
Bullen
[13])
and B497
(Dziewonski
and Gilbert
[14]).
The model.
HBI
meets
most
requirements for the
Earth 's
mantle, but has a simple f luid
core.
The model B497
used
l a te r
observations
of
certain
free Earth
osci
l la t ion
overtones to estimate
r ig id i ty
in the inner core.
5.
USE OF SEISMOLOGICAL INVERSION PROCEDURES
Until
recently, the
main approach to the construction
of Earth
models
has
been through successive approximation. For example,
the
model HBl was
arrived
a t through a
well-defined
sequence,
s tar t ing
from the original Model A and
incorporating successively
evidence
on the variat ion of k with p, evidence from shock-wave data to
improve
the
lower-core
density
dis tr ibution,
revised data
on
I ,
and
data
on
free
Earth
oscil la t ions .
Successive approximation brought
the rel iabi l i t ies
of models
to
the point
where
procedures towards
further
inprovements
are
often
reducible to
l inear
theory.
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18
K.E.BULLEN
Some recent procedures aim to apply the to ta l
seismic
data
(bodily wave,
surface
wave and
free
Earth
osci l lat ion data)
- i . e .
to ' invert ' the data
-
to arrive, independently of
past Earth
models, a t ranges of
values
within
which
p ,
k and must l ie
for
each
z.
In one of the newer approaches (Keilis-Borok
and
others [15],
Press
[16]),
huge numbers
of
models
are randomly
generated inside
a computer by a Monte Carlo technique and
subjected
to
tes ts
which
include f i t t ing
the different sections of
seismic
data within pres
cribed
l imits, f i t t ing
data
on I , etc . , and
meeting various other
s t ipulat ions , e.g. on the core-radius. Of the
millions
of models
that
may
be
generated, the computer
prints out
only a very small
number which pass
a l l the
tes ts . Although freedom from
bias
is
sometimes
claimed
for
this
procedure
because
of
the
absence
of
dependence
on ear l ie r Earth models, the procedure as so far applied
has i t s
own biases.
For
example,
it has favoured
models more
complicated than the
data warrant. In practice,
it
has, moreover,
sometimes
fai led
to
find
classes of models which
are
otherwise
known to
f i t the data
assumed. But
the
procedure is being
developed
and
has considerable potent ial i ty .
Other,
more general,
inversion
procedures use sophisticated
mathematics
with the aim of covering comprehensively a l l
models
which are compatible with
wide
sets
of data. The procedures some
times introduce ' c redibi l i ty ' cr i ter ia with
a
view
to
arriving
a t
an optimum model. Although considerable progress has
been
made
with
the
auxiliary
theory,
the stage has
not
yet
been
reached
where
much information has
been
added
to that derived through successive
approximation. But some useful
information
has already
been
pro
vided
on the
relat ive
uncertainties
of
values
of p, e tc . , a t
different
depths. For
detai ls
on
various
analytical
aspects of
general inversion
procedures, see
refs .
17-21.
6.
CRITERIA
FOR
A
STANDARD
EARTH
MODEL
The developments to
date have
made
it
desirable to formulate a
standard Earth model f o ~ general reference purposes both inside
and
outside
geophysics. A committee for
this purpose
was
se t
up
by
the International
Union
of
Geodesy and
Geophysics
in 1971.
A cardinal requirement
of
a
standard
model
i s simplici ty.
This
requirement is so paramount that the specification of a
standard
model
may well involve fewer parameters than the
minimum
number s t r ic t ly demanded by the data. The
problem
of determining
a
standard
model is therefore
dis t inct
from
that of determining an
optimum model on a given
se t of data.
Examples of points on which
decisions
have to be made are:
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CONSTRUCTION OF EARTH MODELS
19
(i)
What
i s the appropriate
representat ion in a range of depth
where
there is evidence of a rapid
or
sudden
change
of property?
Should
the model show a
mathematical
discontinuity
or
a rapid conti
nuous change? I t is usual
to t rea t
the Mohorovicic discontinuity
and
the mantle-core
and
inner-core
boundaries
as mathematical
dis
continuit ies.
But
the decision i s more di f f i cu l t to make in other
parts
of the
Earth's
in ter ior . 'When
in
doubt,
smooth' has
often
been
stated as
a
guiding
principle , but
the di f f icu l t question
i s
real ly 'how smooth?'.
(i i) How should relat ive degrees
of
smoothing
be determined
among different properties? For example, should pr ior i ty be
given
to simplicity
in
the
variat ion
of
p
or of
6 over a range
of depth
in
which
there is
interplay
( ' t rade-off ' )
between
p
and
6 when a
model
i s
perturbed?
Should
n
(§2.3)
be
kept constant
inside
a
part icular region
of a
model,
or
should
n
be
l e t fluctuate as a
consequence of
smoothing
procedures
applied
to a and 6?
( i i i )
Should
the poss ib i l i ty
of
a t ransi t ion
layer
between
the
outer and inner core be
ignored
in a standard model?
(iv)
Should 6 be taken constant in the inner core? I t i s to be
noted tha t i f 6 is constant,
~
cannot
be
constant; for 6
2
= ~ / p
and p is not constant.
For
some
fur ther
detai ls
on
the
problem
of
a
standard
Earth
model,
see ref . 22.
7. SOME NUMERICAL RESULTS
Table 1 gives values,
derived
from a select ion
of recent
Earth
models, of
p,
p,
g,
k,
a and 6
a t
various depths z (km)
below
the Earth ' s surface. The
units are
g/cm
3
for
p;
lOll
N/m for
p,
k a n d ~ ; m/s2
for
g; and km/s
for
a and 6.
For further numerical
deta i l ,
and
a
comprehensive
account of
the whole subject
of
Earth models,
see Bullen [23].
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20
K. E. BULLEN
Table 1.
Model values
o f proper t i es o f
the
Ear th ' s i n t e r i o r
a t
se lec ted
depths z .
z
p
p g k
f1
a
S
0
2.84
0 9.82 0 .65 0 .36
6.3 3.6
30
3.32
0.01 9.84 1 .07 0.72 7.80 4.65
200
3.39
0.06
9.90
1. 39
0.69
8.26
4.50
400
3.70
0.14 9.96
1 .
89
0.84 8.92 4.72
650 4.17 0.23 9 .99
2.68 1.43
10.48
5.80
1000 4.54 0.39
9.96
3.49
1.84 11. 44 6.36
2000
5.09
0.87
10.02
5.07 2 .44 12.79 6 .92
2886 5.69 1 .
35
10.8
6.54
3.04 13.64 7 .30
2890
9.95 1 .
35
10.8
6.54 0 .00
8.12
0.00
4000
11.
39
2.48
7.9
10.34 0 .00 9 .53
0.00
5120 12.70 3 .34 4.4
13.50 0 .00 10.33
0.00
5160 12.7
3.34 4.3
13.6
1 .7
11.25 3.7
6371
13.0 3.67 0
15.0 1 .3
11.25
3.2
REFERENCES
1.
K.
E.
Bullen and R. A.
W.
Haddon, Phys. Earth Planet . In t e r io r s
7, 199, 1973.
2.
E.
D. Williamson and
L.
H.
Adams,
J . Wash. Acad.
Sci . 13,
413,
1923.
3.
4.
5.
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K.
E.
Bul len ,
K. E. Bul len ,
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Geo,eh;r:s.
Res.
Trans.
Amer.
Mon. Not. R.
~ ,
227,
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GeoEh;r:s.
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Astr .
Soc. ,
Geo,ehx
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•
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6.
7.
K.
E.
Bul len ,
GeoEhx
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J .
,
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Astr .
Soc. , 7,
584, 1963.
K.
E.
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•
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8. K.
E.
Bullen, Geo,ehxs. J . , R. Astr . Soc. , l l , 459, 1967.
9.
K.
E.
Bullen,
In t roduc t ion
to
the
Theorx
or-Seismology,
Univers i ty
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3rd
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1965.
10. K. E. Bullen, Nature , Lond., 157,
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7:,
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12. F. Birch ,
GeoEhys.
J . , R. Astr . Soc . , 4, 295, 1961. ==
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16,
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CONSTRUCTION OF EARTH MODELS
21
19.
H.
Takeuchi and
K.
Sudo, J . Geophys.
Res.
73, 3801, 1968.
20. V. I . Keilis-Borok (Ed), Computational
Metnods
in Seismology
(English
trans. by E. A.
Flinn,
Consultants Bureau, New York),
1972.
21. L. Knopoff and D. D.
Jackson,
The
analysis of undetermined
and
overdetermined systems,
in
course of
publication.
22.
R.
D. Adams, K. E. Bullen, J .
R.
Cleary, A. M. Dziewonski,
E. R.
Engdahl, R.
A.
W.
Haddon, A. L.
Hales, R.
Lapwood and
others: A
se t of
papers on
Standard Earth
Model, in
course
of publication, Phys. Earth Planet. Inter iors .
23.
K. E.
Bullen,
The
Earth 's Density,
Chapman
&
Hall, London,
to
be
published,
April
1975.
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THE FeZO THEORY
OF
PLANETARY
CORES
K.E.Bul len
c /o
Depar tment of
Applied Mathematics ,
Universi ty of Sydney, Austral ia
ABSTRACT.
A·
r ~ s u m e
is
given of
the evidence for
the
theory
that
the
outer
cores (when they exist)
of
t e r res t r ia l
type
planets consis t of
the i ron oxide FeZO, which is known
to be unstable at
ordinary
pressu res
but
stable at pressu res
equal to
those in
the Ear th 's core . The
theory,
while
avoid
ing the
main
object ions
to the ear l ier phase- t ransi t ion theory
of
planetary cores ,
permits
the Earth, Venus and Mars to
have a
common overal l composi t ion. Essent ia l to the theory
is the assumption
that
the pressure at the
Ear th 's
mant le
core boundary
is
a cr i t ica l
pressu re common
to all
planets
which have
outer
cores . Brief comments
are made
on M e r
cury and the Moon.
1. THEORIES
ON
THE COMPOSITIONS OF
THE
CORES OF
THE
TERRESTRIAL
PLANETS
By
1906,
i t was well establ ished that the
Ear th
has a
dense
central
core ,
and in 1936 that this
core consists
of
a
(so
called)
outer
core
and an inner core . An ear ly view,
pr inc
ipal ly
based
on meteor i t e
evidence, was
that the central
core
is composed of
i ron
and nickel .
Later
invest igations conf i rm
ed that
this composit ion applies to
the
inner core with high
probability,
but
indicated that the outer core has
a
density
too low
(at
the pressures involved) to consis t
of
pure i ron and
nickel .
I t
was
thereupon suggested that the
outer
core
con
sis ts of i ron
alloyed with
less
dense
elements (e.
g.
silicon,
carbon,
sulphur). On the
hypothesis that
the core consis ts
predominantly o f iron,
and
thus has a dist inct chemical com-
Thoft-Christensen(ed.), Continuum Mechanics Aspects o f Geodynamics and
Rock
Fracture Mechanics. 23-28.
All Rights Reserved. Copyright © 1974 by D. Reidel Publishing Company. Dordrecht-Holland.
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24
K.
E. BULLEN
posi t ion
f rom
the
mantle.
it
was
shown [1.2] that the
t e r r e s
t r ia l - type planets Ear th . Yenus and Mars cannot have the
same overa l l
composi t ion:
for
example.
if Yenus and Mars
a re assumed
to possess s imilar ly composed
mantles and
cores to those
in
the Earth.
thei r
mant le-core
m a s s rat ios
would
be
3.6
and
5.4. as against 2. 1 for
the
Earth;
these
rat ios would be entai led using observational data on the mas s
es M
and
radii R
of Yenus
and Mars . In the
following.
the
subscr ip ts
Y and M will
indicate
proper t ies
of Yenus
and
Mars . respect ively.
In 1948-9. it was shown by Ramsey [3] and
Bullen
[4]
independently that
if.
in contrast to the predominant ly- i ron
core
theory. the change at the Ear th ' s mant le-core boundary
N
were
a
pres sure
phenomenon
(the
outer
core
thus
cons i s t
ing of a high-density metal l ic
phase
of
the lower
-mant le
mater ia l ) . the observat ional
values
(at the t ime) of
My. MM'
R y
and RM. as
well
as
the
moment of
iner t ia coefficient
YM'
would
be
compat ible with Ear th . Yenus
and Mars
having
the
same overa l l
composi t ion.
An essent ial point is that if the
phase t ransi t ion occurs a t the same
cri t ical
p res su re Pc
in
all three planets. the mant le-core
m a s s
rat io would increase
with decreas ing planetary size.
The phase - t ransi t ion
theory
appeared at the t ime to fit
all the relevant
observational data
remarkably well. but
the
theory
la ter
met severa l difficul t ies. chiefly: (i) having r ega rd
to
the packing
of oxygen atoms in the Ear th ' s
lower
-mant le