Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics [K. E. Bullen (Auth.),

8/16/2019 Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics [K. E. Bullen (Auth.),

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Continuum Mechanics Aspects

of

Geodynamics

and Rock Fracture Mechanics


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 -


of

Geodynamics



3/281


of

Geodynamics


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


4/281

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


5/281

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


6/281

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


7/281

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


8/281

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


9/281

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.


10/281

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


11/281

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


12/281


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


13/281

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.


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


14/281


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


15/281

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.


16/281

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


17/281

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.


18/281

4

K.E.BULLEN

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.


19/281

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


20/281

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].


21/281

ASPECTS

OF

EARTHQUAKE ENERGY

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


22/281

8

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


23/281

ASPECTS

OF

EARTHQUAKE ENERGY

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)


24/281

10

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


25/281

ASPECTS

OF

EARTHQUAKE ENERGY

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.


26/281

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.


27/281

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.


28/281

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)


29/281

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)


30/281

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


31/281


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.


32/281

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:


33/281


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].


34/281

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.

F. Birch ,

J .

K.

E.

Bul len ,

K. E. Bul len ,

1956.

Geo,eh;r:s.

Res.

Trans.

Amer.

Mon. Not. R.

~ ,

227,

1952.

GeoEh;r:s.

Un. 34, 107, 1953.

Astr .

Soc. ,

Geo,ehx

s

•

Su,EE1. ,

6.

7.

K.

E.

Bul len ,

GeoEhx

s

.

J .

,

R.

Astr .

Soc. , 7,

584, 1963.

K.

E.

Bul len , Mon.

Not. R. Astr .

Soc.

,

GeOEhx

s

•

Suppl .

,

1949.

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

Press , Cambridge,

3rd

ed,

1965.

10. K. E. Bullen, Nature , Lond., 157,

405, 1946.

7:,

214,

~ ,

355,

11. I .

Lehmann,

Publ . Bur.

C e n t r . ~ i s m o l . In t e rna t . A,

14, 3,

1936.

12. F. Birch ,

GeoEhys.

J . , R. Astr . Soc . , 4, 295, 1961. ==

13.

R. A.

W. Haddon

and K. E. Bul len , P h y s ~ Earth Planet .

In t e r io r s

.?' 35, 1969.

14. A. M. Dzievwnski and F. Gilber t , Geo,ehxs. J., R. Astr . Soc . ,

~ ,

401,

1973.

15. ~

J . Asbel ,

V.

I . Keil is -Borok

and

T. B. Yanovskaya, Geophys.

J . , R.

Astr . Soc. , ~ ,

25, 1966.

16. F. Press , J . G e o , e h x ~ Res.

~ ,

5223,

1968.

17.

L.

B.

Sl i ch te r , Proc. R. S o ~

Lond.

A, 224,

43,

1954.

18. G. Backus and F. Gilber t , Geophys. J . , ~ A s t r . Soc. ,

16,

169,

1968.


35/281


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.


36/281

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.


37/281

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

Documents

Continuum Mechanics Aspects of Geodynamics and Rock Fracture Mechanics [K. E. Bullen (Auth.),