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Stress analysis of the Lithosphere

Dean, D. S.

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Dean, D. S. (1972) Stress analysis of the Lithosphere, Durham theses, Durham University. Available atDurham E-Theses Online: http://etheses.dur.ac.uk/8672/

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STRESS AMA1YSIS OF THE I.ITHOS1-HERE

BY

I) . S. DEAN.

A t h e s i s submit ted f o r the decree of Doctor

of Ph i losophy i n the U n i v e r s i t y of Durham.

rrcy CnHDe.-ije. November 1972.

To my l a t e f a t h e r .

ACKN OV/1 EpJrEJffi N TS

I s h o u l d l i k e to take t h i s opportuni ty of e x p r e s s i n g

my s i n c e r e thanks to I - r o f e s s o r M. H. ! . B o t t f o r h i s

s u p e r v i s i o n d u r i n g the p e r i o d when t h i s r e s e a r c h was

done. He f i r s t i n t r o d u c e d me to the method^and h i s

presence was a lways a source of i n s p i r a t i o n .

1 am a l s o indebted to ffir. A l a n Douglas ( U . K . A . E . A )

and M e s s r s . J-'. B e t t e s s and 1. F a r t o n ( e n g i n e e r i n g

department, U n i v e r s i t y of Durham) f o r t h e i r h e l p f u l

d i s c u s s i o n s of the theory of f i n i t e e lements , D r . R . A . S m i t h

of the mathematics department ( U n i v e r s i t y of Durham) f o r

h e l p w i t h the i n v e r s e L a p l a c e problem, the departments

of l e o l o ^ y and Computing a t the U n i v e r s i t y f o r p r o v i d i n g

the f a c i l i t i e s f o r t h i s r e s e a r c h and f i n a l l y to the

N a t u r a l E n v i r o n m e n t a l R e s e a r c h C o u n c i l f o r p r o v i d i n g

f i n a n c i a l support .

A;;_oTJOiCT_

Computer programs have been w r i t t e n to perform

f i n i t e c lement c a l c u l a t i o n s w i t h 3-noded elements f o r

tnc cases of plane s t r a i n , p lane s t r e s s and ax i syr imetry .

and to produce d a t a f o r these programs. The l a t t e r i s

s p e c i f i c a l ] y o r i e n t a t e d towards g e o p h y s i c a l models and

i n c l u d e s a f a c i l i t y f o r q u a s i - h o r i z o n t a l "boundaries.

These ru-ve been used to inves t iga te - the s t r e s s

f i e l d s ar i f joc iatea w i t h sedimentary b a s i n s and c o n t i n e n t a l

marg ins . Prom these c a l c u l a t i o n s , the p r e f e r e n t i a l

f ormat ion of normal f a u l t s on. and to the c o n t i n e n t a l

a ide of marg ins , w i t h the subsequent f o r m a t i o n of deep

sedimentary b a s i n s , i a e x p l a i n e d .

(Quant i tat ive s t u d i e s on the d i f f u s i o n of s t r e s s e s

through mathemat ica l models of the l i t h o s p h e r e and

as thenosphere have f j i v e n time c o n s t a n t s of 200,OuO y e a r s

f o r p l a t e s of the s i z e of the P a c i f i c . S imula ted f a i l u r e

ha:- been b u i l t i n t o t h i s model and l e a d s to the p r e d i c t i o n

of a 10 y e a r l y f a i l u r e c y c l e w i t h energy r e l e a s e s of

l O ^ e r / j s .

The f i n i t e e lement models have been extended to

co| o v / i th dynamic case s and u s e d to c a l c u l a t e the n a t u r a l

f r e q u e n c i e s t-f f r e e v i b r a t i o n of a sed imentary b a s i n .

]''or a Raylei . ' jh type response the fundamental p e r i o d has

been shown to be d i r e c t l y p r o p o r t i o n a l to the depth of

the b a s i n , 'i'ne n a t u r a l f r e q u e n c i e s c a l c u l a t e d are of the

same oroer a s those measured i n s u r f a c e waves generated ii

by ear thquakes and a Praunhofer a b s o r ] j t i c n e f f e c t i s

su j e s t e d .

CONTENTS.

Fa^e

ClfAlTivK J. INTW0111ICTI0N TO THE FUNBAM^NTAlB OP E1AST1C1TY. 1

( 1 . 3 ) A n a l y s i s of s t r a i n . t (1 .? - ) A n a l y s i s of s t r e s s . t, ( 1 . 3 ) I n t e r r e l a t i o n between s t r e s s and s t r a i n . «j

CHAJ/Tiiii J f F I N I T E ElEftENT THEORY f i -

( 2 . 3 ) Q u a l i t a t i v e a n a l y s i s . 12. ( ? . * : ) Q u a n t i t a t i v e a n a l y s i s . i«l ( 2 . 3 ) A p p l i c a t i o n to e a r t h s c i e n c e s . i«j

CHA1TEH 3 STRESS DISTRI7.-UT3.ONS ASSOCIATED WITH DEEP SEDIMENTARY BASINS.

CHA>TEH 5 STRESS DIFFUSION IN THE UTHOSIHEHE.

3 z

CIIAJ-TER 4 STRESS SYSTEMS AT YOUN> CONTINENTAL MARTENS.

( 5 . 1 ) S p e c i f i c a t i o n of problem 5 c ( 5 - 2 ) F o r m u l a t i o n of problem. S i ( 5 - 3 ) S o l u t i o n . s

( 5 . 4 ) E f f e c t of d i f f e r e n t v e l o c i t y g r a d i e n t i n the a s thenosphere .

( 5 . 5 ) I n c l u s i o n of i n e r t i a l terms. (&&

CI 1AH.'_RR_ C _ STRESS RE3AXATION IN THE LITHOSIIIERE. t ' i

( 6 . 3 ) S i m u l a t i o n of s t r e s s r e l a x a t i o n . ( 6 . 2 ) S p e c i f i c a t i o n of problem. <ot. ( 6 . 3 ) S o l u t i o n . <e

C " A * T M . D I S C U S S I O N . "to

_C!LA1TE_H_ a_ ENEMIY ABSGHBTION BY SEJL)IKENTS T H -

( 8 . 3 ) E x t e n s i o n of f i n i t e element method 7 l +

to dynamic problems. ( 8 . 2 ) The mass -matr ix f o r R a y l e i g h type -ju

response . ( 8 . 3 ) The mass -matr ix f o r Love type response . ~~>1 ( 8 . 4 ) I m p o s i t i o n of c o n s t r a i n t s . 1% ( 8 . 5 ) N a t u r a l f r e q u e n c i e s of sed imentary b a s i n s . t % ( 8 . 6 ) D i s c u s s i o n . ^ a

CHAi-TEtt 9 IffiAlSINT. OF THE IITH0S3 HERE.

A:I ] -END:JX 3 T E N S O R N O T A T I O N AJM-ENDilX 2 E I A S T I C I T Y

AHliNDlX 3 AX1SYKT.;ETR1C AND 6 NODED THEORY. i o 8

Al 3 ENDI.X 4 SO]UT.TONa TO EQUATIONS l.F CHAPTERS 5 b 9. A.1PENDIX 5 SPECIFICATION OF PKO'rRAlKS.

(A) F I N E l (B ) SENPIN (G) IMPROVE BAND WIDTH CD) 3T0fc ( E ) F1NEIG (* ) APINEL Z S I (*) NOMOFV A - ! i (H) 10VE

UIIAPTER 1.

I n t r o o u c t i o n to_the. y^P^JH^^^iL i5i^•_A1^ir ;_' t ;_ ii ;i•J ;X•

S o l u t i o n s of complex s t r e s s analysis .; problem-..:' can

only be attempted w i t h a thorough a p p r e c i a t i o n of the

concepts of s t r e s s and s t r a i n ana t n e i r i n t e r r e l a t i o n ­

s h i p I n p a r t i c u l a r t h i s knowledge i s e s s e n t i a l f o r

an u n d e r s t a n d i n g of the f i n i t e element me thou of s t r e s s

a n a l y s i s , of which much use w i l l be made i n l a t e r

c h a p t e r s . The f o r m u l a t i o n of the mathemat ica l theory

of e l a s t i c i t y has seen the e v o l u t i o n of many n o t a t i o n s .

I t i s the purpose of t h i s c h a p t e r to %ive an i n t r o d u c t i o n

to the theory i n the n o t a t i o n used throughout t h i s

t h e s i s , namely the t e n s o r n o t a t i o n (see Appendix I f o r

the d e f i n i t i o n of a t e n s o r ) .

The aim of the theory of e l a s t i c i t y i s the

g e n e r a t i o n of the v e c t o r f i e l d of r e l a t i v e d i sp lacement

( u ( x , y , z ) = u L t ) of each p o i n t F ( x . y . z ) of a body,

produced by the a c t i o n of j i y e n forcer , wh ich a c t

( a ) on the boundar ies of the body - c a l l e d boundary f o r c e s .

and . (b ) throughout the body e . g . g r a v i t y - c a l l e d

Thus the p o i n t F ( x , y . z , ) of the body i s d i s p l a c e d to

the p o i n t (x+u (. y+u t , z+u s ) by the combined a c t i o n of

When changes take p lace i n tne r e l a t i v e p o s i t i o n s

of the p a r t s of a body, t h a t body i s s a i d to be s t r a i n e d .

body f o r c e s .

boundary and body f o r c e s . Tn genera l u ( , u t , u y are

f u n c t i o n s of x , y , z .

1.1 An_ A£a l^j3 i s_ of S t r a i n .

COn, BOIENOI

t 2 JUNTOS lEono*

Thus the s t r a i n a t a p o i n t w i t h i n a body i. r: a q u a n t i t y

r e l a t e d to tne r e l a t i v e d i sp lacement of the p a r t i c l e

of m a i l e r a t that p o i n t .

I t i s e a s i l y shown t h a t the chan:;e, under a

de format ion u (x ( , x x . x a ) of a s m a l l v e c t o r A j o i n i n g

two n e i g h b o u r i n g po in t s 1 .P 7 of the body i s

(1 1 )

p r o v i d e u the second o r a e r d e r i v a t i v e s of u a t 3r a r e

s m a l l ( a s s u m p t i o n ) . Hence,

= ( 1 2 )

where,

( D l . l )

anu

Note t h a t

( J ) e:- i n symmetric i R . f \ = r— 0 - 3 ) J J

( 1 1 ) w- i s a n t i s y m m e t r i c i e w- = -w- ( 3 . 4 ) j J J

The a n t i s y m m e t r i c component ol tht- trunsfornuj . l iun,

Wjj can be shown to correspond to r i . ; i u bocy motions

i . e . pure t r a n s l a t i o n and r o t a t i o n of the bouy.

Obvious ly these motions 0.0 not produce d i sp lacement of

one p a r t of the body r e l a t i v e to another p a r t . Thus

they r e s u l t i n no s t r a i n and do not i n t e r e s t u s .

The symmetric components of the t r a n s f o r m a t i o n .

. thus r e p r e s e n t pure deformation of the body The

sr.t of s i x d i f f e r e n t e : : (not n ine because e. = e-.-)

i s c a l l e d the s t r a i n t ensor a s s o c i a t e d w i t h the

d i sp lacement f i e l d u and i t s components ,-ire fanet:i.oris

of p o s i t i o n , i . e . e- = e^ ( x , , x 4 , x ) . These components J «J

a l s o depend on the o r i e n t a t i o n of the axes of r e f e r e n c e

The f u n c t i o n u e x p r e s s i n g the d i sp lacement of the

p a r t i c l e a t I i n terms of x, , x x x , the c o - o r d i n a t e s

of F , i s a d i f f e r e n t f u n c t i o n u' ( x ' , x „ ' , x ' ) when e x p r e s s -

i n g the same d i sp lacement , but t h i s time i n terms of

the co- o r d i n a t e s of P i n the primed s y s t e m .

A P i ? . 1.1

A

/ - •> A

a. / \ / /

3> -

The components ol' the s t r a i n t ensor when r< J e r r c o

to the two systems oi' c o - o r d i n a t e s ;?.re

and

from D l . l . and o b v i o u s l y i n g e n e r a l e, - :± <».•: . "J J

Thus the s t r a i n t e n s o r depends on the p o i n t i anu the

o r i e n t a t i o n of the c o - o r d i n a t e axes . The two t e n s o r s

e :| and e(j a r e r e l a t e d 3 and

e - ^ ( i . :>)

v/here

lu = — * J-' l j ~~ ~ * J (ni.3) The v a l i d i t y of ( 1 . 5 ) c h a r a c t e r i s e s the a r r a y as a

t e n s o r

F o r each p o i n t > there e x i s t s one s o t of

c o - o r d i n a t e axes ( i n genernl d i f f e r e n t f o r each 1-)

such t h a t the s t r a i n t e n s o r when r e f e r r e d to these

axes i s a i a j o n a l i . e . e - = o («••*]) • Thei.e axes are c a l l e d

the pjrincjLjrajL_ axes of s t r a i n .

The s i x components of s t r a i n a t a poin o a r e not

i n a e ; e n d e n t end s a t i s f y c o m p a t i b i l i t y equat ions 4

- - V j i - V « = ° ( 1 - 6 )

H- ~ - - 3_

So f a r the s t r a i n t ensor has been developeO

p u r e l y a long mathemat ica l l i n e s . Nov/ we g i v e

p h y s i c a l s i g n i f i c a n c e to the e lements of the ten:;or 5

( a ) . D iagona l terms

The component e„ r e p r e s i n t s the e x t e n s i o n per

u n i t l e n g t h of a s m a l l v e c t o r a t 1_ o r i g i n a l l y p a r a l l e l

to the x, a x i s .

(b ) O f f - d i a g o n a l te rms

The va lue 2e;j r e p r e s e n t s the decrease i n the r i r ; h t

angle between the s m a l l v e c t o r s A B a t 1- which were

i n i t i a l l y ( i . e . b e f o r e deformat ion u ) d i r e c t e d a long

the p o s i t i v e x ; and Xj axes r e s p e c t i v e l y .

E x p r e s s i o n s f o r the components of the s t r a i n

t e n s o r i n other orthogonal c o - o r d i n a t e systems are

g i v e n by

( a ) S p h e r i c a l c o - o r d i n a t e s

C>U- r

d r

( 1 . 7 )

or- r r ^ ©

(b ) Gyj i n d r i . c a l C o- ord inate s .

— -— -r-

r

An_ a n a l y si_s_ of_ S t r e s s_

The s t r e s s a t a p o i n t F w i t h i n a body i n

a s s o c i a t e d w i t h the f o r c e s a c t i n g a t that p o i n t .

C o n s i d e r a s m a l l s u r f a c e . , w i t h i n the beety w i t h

an outward normal v . T h e n the m a t e r i a l on ont r;idr

4, p of i s 1 iu : :h in* ' a :ai.n:;t

14s p±cr i 2 ^ n e EiixtkTifil o n 1 ; n e othi:r a iue

) of ano the m a t e r i a l

on t h i s s i a e i a 'pushing* back w i t h an i qu;„l and

opposi te f o r c e . l e t tne f o r c e thu i the m a t e r i a l on

the s i c e i of i s e x e r t i n g on thi: mato.r:ial on uhe o ther

ai i .e of A% , be P. I f we now take l i m i t s u:: ^S. -*• o

ana w r i t e

|< u i s known ;::s the s t r f s r . v e c t o r a t } f o r the j-liine j£.

fcu can be a T : i i t i n t o components p a r a l l e l to (known as

the normal a t r e u a ) and _k_ar i (known i.u ihe oimur

s t r e s s on the plane |J a t 1 ) . Obv ious ly the s h e a r s t r e s s

a t 1 w i l l ob ta in a maximum v a l u e i o r some o r i e n t a t i o n V

(Note here t h a t P . ana nence R , . i s d i f f e r e n t f o r

d i f f e r e n t l y o r i e n t a t e d s u r f a c e s a t 1 ) . T h i s maximum

s h e a r , and the p lane on which i t a c t s , i s of g r e a t

i n t e r e s t to j e o p h y s i c i s t s .

Note:

( I ) R Y i s independent of the o r i e n t a t i o n of the

c o - o r d i n a t e a x e s .

( I I ) R Y ^ e i y i n g e n e r a l , but there e x i s t s three

m u t u a l l y p e r p e n d i c u l a r d i r e c t i o n s f o r v/nich

//eX V a n ( * these a r e known as the

p r i n c i p a l a x e s . The s t r e s s e s a c t i n g on these

p lanes a r e known a s the p r i n c i p a l s t r e s s e s .

The components of H alon-? the x . y , z , axes ( o r the

x, _ x x x 3 a x e s ) a r e denoted by vx. t , i>x. tj° r T M , i.=

r e s p e c t i v e l y .

i . e . R u = V*- I + » j i * * ( D 1 . 5 )

These components depend on the o r i e n t a t i o n of t;he

c o - o r d i n a t e a x e s .

F o r the complebe s p e c i f i c a t i o n of the s t r e s s a t

any p o i n t of a bouy we should r e q u i r e to know the f o r c e

per u n i t a r e a a c r o s s every p lane drawn through the pu int

( i . e . f o r a l l y ) . The o b j e c t of an a n a l y o i : ; of u trc : . s

i s to determine the na ture of the q u a n t i t i e s by which

the s t r e s s a t a p o i n t can be s p e c i i i e d . I t can be shown

t h a t to J t i l l y o e f i n e the s t r e s s a t a po in t the nine

q u a n t i t i e s

^ - ( - . , ) . , ^ . = * * C T ^ i x C i . y ^ ( O . ii-Cx,,)

and z CTj*.) (reduced to s i x because ; 1 j L

7 ) a r e

L - S - ^ Op|u A.dUA< X p out X~ 5 7 - - -

s u f f i c i e n t . These s t r e s s components i n f a c t c o n s t i t u t e a t ensor a n i i t can he shown* from e q u i l i b r i u m c o n s i d e r a ­t i o n s t h a t i f v has d i r e c t i o n c o s i n e s i*; i n our e o - o r c i n a t e system Oat.w.^ then

Lvi =" " ^ j i ( ] . y )

i . e . f o r ar.y ; iven s u r i a c e ( i . e . '$ivr-n i> ) ^, = T B v

can be obta ined from the s t r e s s t e n s o r i=i,a j = t,s.

When re1 e r r e d to new primed axes the new s t r e s s t e n s o r i s obtained from the o ld by

"Ttj ~ Li U.j Txm (3 .3 0)

where Ly i s d e f i n e d by D1.3- As w i t h the s t r a i n t ensor f o r each p o i n t I a s e t of c o - o r d i n a t e axes e x i s t s c a l l e d

the p r i n c i p a 3 axes of s t r e s s f o r which the s t r e s s tens.:or

when r e f e r r e d to these a x e s , i s d iagona l i . e . Tij = O Ci.+j).

The g r e a t e s t s h e a r i n g s t r e s s a t a p o i n t i s equa l

to o n e - n a i f the d i f f e r e n c e between the j r e a t e s t and

l e a s t p r i n c i p a l s t r e s s e s ano a c t s on a plane t h a t

c o n t a i n s the d i r e c t i o n of the i n t e r m e d i a t e p r i n c i p a l

s t r e s s and b i s e c t s the an^le between the d ir t c t i o n s of lo

the l a r g e s t ana s m a l l e s t p r i n c i p a l s t r e s s e s .

E q u i l i b r i u m c o n s i d e r a t i o n s between the s u r f a c e

s t r e s s e s and boay f o r c e s a c t i n g on a s m a l l volume of

m a t e r i a l w i t h i n a body l e a d to the equat ion of

e q u i l i b r i u m i'or s t a t i c deformat ion

o ( 1 . 1 1 )

where P i s the boey f o r c e per u n i t mass and ^ the

m a t e r i a l d e n s i t y .

Sea A^po^cl^K i f ow Y S ' " -7

- — - (, <=1 -l o — n -

lnte_rre_la._t_i_on_ be tween_stre:;_Ei_ and. s t r a i n ^

I n o r a e r to i ' u r t h c r our theory we must make a

second assumption ( o r ax iom) , t h a t there e x i s t s & l i n e a r

r e l a t i o n s h i p between s t r e s s anu s t r a i n (known a s

Hooke's L a w ) . A m a t e r i a l which obeys Hooko's Law i s

known a s e l a s t i c . Thus there e x i s t s a s e t of 3^ c o n s t a n t s

( ^ " < - j K . * » ) Qlote • I t can be shown iro/n energy c o n s i d e r a ­

t i o n s t h a t the c o e f f i c i e n t s of C a r e symmetric and

t h e r e f o r e only 21 c o n s t a n t s a r e needed] s u c h tfiat

or i n m a t r i x format

< ay "Z.X

3 5*

- i . i K . n r t ' ICIV\

- M r >

(1.12li)

( 1 . 1 2 b )

I f we assume t h a t the e l a s t i c body we a r c con­

s i d e r i n g i s i s o t r o p i c ( i . e . i t has the same e l a s t i c

p r o p e r t i e s i n a l l d i r e c t i o n s ) then w« can reuuee the

number of c o n s t a n t s to two ; X and jx c a l l e d l a m e ' s •a.

c o n s t a n t s . The s t r e s s - s t r a i n r e l a t i o n f o r an

i s o t r o p i c m a t e r i a l then reduces to

I : : = r 3 ( 1 . 1 3 )

I2. - Sa«. O-ppe-»«.cA-*06 -2 Xio^v k- I o

lo

where

l u t t i n g i = j i n (1 .13) and summing

© - ( 3 \ + ( 1 . 1 4 )

where G) =. ~T„ -*- i - T 3 i ( D l . 7 )

S u b s t i t u t i o n nov/ g i v e s the i n v e r s e r e l a t i o n

e t l = — 1

ti " - 3 ^ ( 3 V - a ^ 3/* ( 1 - 1 5 )

V/hen working w i t h p r a c t i c a l problems i t i s u s u a l

to use two d i f f e r e n t but r e l a t e d cons tant s c a l l e d

Young's Modulus ( E ) and. l o i s s o n ' s r a t i o These a r e

d e f i n e d a s

and ^ A

1 - ~3cvyI7 ( 1 . 1 7 )

the i n v e r s e r e l a t i o n s be ing

•x -A ~ 0+ )0-2-n^ ( i . i b )

and EL

- j o ^ n ) ( i - i 9 )

The s t r e s s - s t r a i n r e l a t i o n s 1.13 and 1.14 then

assume the iorm

and

e - = ( i ! 2 ) T - I S - & (3.23)

i'rom e'-uaLion 1.13 when i 4 j we have T tj una t n e r e f o r e . v/hcn 6-ij=cO*j) we have Xj=° . Thus a or an i s o t r o p i c m a t e r i a l the p r i n c i p a l n x t i j of utrerju am. .-j t r a i n c oincide.

The p o t e n t i a l s t r a i n energy i or u n i t volume (rf) i n a s t r a i n e a bouy i s

( S o k o l n i k o f f 194t , § 2t-)

An important p r i n c i p l e i n e l a s l i a i l y e u r o c i j - i l y r e l e v a n t t o F i n i t e Element the o r y i r j t l i a t o:l' Saint Venantjjfc which can be s t a t e d as f o l l o w s :

I f some D i s t r i b u t i o n of f o r c e s acbin-j on a p o r t i o n of the surface of a body i s r e j l a c e d by a on l i f e r e n t d i s t r i b u t i o n of f o r c e s a c t i n g on the :jaij>e p o r t i o n of the body then the e i f e c t c of xne two o i s t r i l ; u t i one mi the p a r t s of the body s u f f i c i e n t l y f a r removed .from the r e g i o n of a p r l i c a t i o n of the forw.a are eucjc:ntia33y the same, provided t h a t bhe two d i s t r i b u t i o n s of force-.", are s t a t i c a l l y e q u i v a l e n t .

12.

CJiAl- TER_ _ 2_.

2.1 j^TUjil n t a t i y e Analysis. Pteny problems; ol" :;trccr; a n a l y s i s a r i s i n g i n

nature are completely i n t r a c t a b l e a n a l y t i c a l l y because of the complicated nature of the boundary c o n d i t i o n s i n v o l v e d . F i n i t e element methods provide a numerical means of t a c k l i n g such problems i r r e s r - o c t i v e of the complexity of the boundary c o n d i t i o n s . ( Z i e n k i e w i c z and Cheung 19C7).

I n i t i a l d i s c u s s i o n w i l l be confined t o tne theory of the method a p p l i e d to the s o l u t i o n of • two-dinr-.-nuional problems and l a t e r i t w i l l be widened t o include the n a t u r a l extension t o three-dimensions.

The i n i t i a l procedure i n v o l v e s a d i v i s i o n of the model r e q u i r i n g a n a l y s i s i n t o a s e r i e s of vfcrian ;les n ^ * ^ , (ca l l e d , elements), the corners of which are reierrc-a -' 'r c-r^s 7 . . - i ^ ^ t o as nodal p o i n t s ( o r nodes). See Pi-* 2.1. Each element and node i s sjiven a reference number and values f o r the d e n s i t y , Young's modulus and 3-oisson's r a t i o are assigned t o each element.

The ba«ic unknowns. i n terms of which the problem ±so formul&.tfii'i are the displacements of the nodal points induced by the a c t i o n of ^iven e x t e r n a l l y appJiea f o r c e s . For two-dimensional problems there are two baaic unknowns associated w i t h each nodal p o i n t i e. ?*\ unknowns a l t o g e t h e r , where T. i s the t o t a l , number of nodal p o i n t s .

IS

CM

8 2 CM

H

2 8 T"

i E i

Considering each element i n t u r n i t i s possible t o aefine & unique l i n e a r displacement throughout i n terms of the s i x ba:;ic unknown components oi nocJaJ d i s i l a c e i i i u n t (each element having three noaes). Displacement d e l i n e s s t r a i n thus a l l o w i n g the ii:uacuuatc r:o t e r m i n a t i o n ••>• the s t r a i n throughout each element i n tt.rmo of i t s si.* basi c unknov/n components of nodal displacement. Hooke's law gives the s t r e s s throughout each element i n terms of our basic unknowns.

The boundary strer.;_ses o r i each element are replaced by s t a t i c a l l y e q u i v a l e n t f o r c e s concentrated a t i t s nodes, l e t K denote the concentrated f o r c e introduced a t the nodal point K. r e s u l t i n g from the boundary stresses a c t i n g on element e. Note t h a t the:..:e forces are i n tonus of the unknown nodal displacements.

To solve f o r the 2N unknowns we need'to formula Ue 2M simultaneous equations, l e t us consider the c o n d i t i o n a t :

(a) An i n t e r n a l nocie such as L i n l'i.nj.2 1 Each of the elements ( s i x i n t h i s case) t o which

t h i s no-ue be3 oncjs c o n t r i b u t e s con centra tec" :f. orces The sum of these f o r c e s must be zero, since the boundary for c e s t o which the;/ are e q u i v a l e n t are equal and opposite alomj adjacent faces conver^in. a t i (bewton's t h i r d l a w ) . i . e .

This c o n d i t i o n f i v e s two eciut.tions (2.D i s a v e c t o r equation) invoD v i r i r, the basic unknown:; :i or each i n t e r n a l nocie i .

e O

( b ) A boundary node such as j i n i i ; 2.1 writ-re there i s an e x t e r n a l l y a p p l i e d f o r c e I;.- LirectD y from the d e f i n i t i o n of s t r e s s the boundary strt.-:;we« a c t i n g on the Jiioael anu the e x t e r n a l l y a/, p i e u i o r c e s are e q u i v a l e n t . Thus the sum of the concentrated force: a t the nouai p o i n t j ( c o n t r i b u t e d t o by the boundary streamer; from two elements i n t h i s case) must equal I ; , the e q u i v a l e n t a p p l i e d load.

This c o n d i t i o n gives two equations i n v o l v i n g the. 211 unknown displacements f o r every boundary node j . This completes a set of 2N simultaneous equations l o r the unknown nodal displacements. Solving f o r these unknown completes the a n a l y s i s , the displacements s t r a i n s and stresses having a l r e a d y been defined throughout each element (ana thus throughout the e n t i r e model) i n terms of these basic unknowns. The f o l l o w in.-; p o i n t s about the method are worthy of note: A. C o m p a t i b i l i t y of S t r a i n s

The c o m p a t i b i l i t y equations governing a set of s t r a i n components ( l . C ) ensure t h a t the s t r a i n s represent a continuous displacement. C o n t i n u i t y of displacement e x i s t s throughout a l l elements because i t i s represented by a continuous f u n c t i o n . However, f o r the displacement t o be continuous throughout the body ( i . e . f o r the c o m p a t i b i l i t y equations t o be s a t i s f i e d ) the displacement must be continuous acros

i . e .

- J "P (2.2)

the element boundaries. Consider trie three-noten t r i a n g u l a r element discussed so f t - r . The uis|.lacoii/nt aJ or. j the face j o i n i n ; two nodes, i and j, v a r i e s l i n e a r l y s u b j e c t t o the c o n s t r a i n t a t e i t h e r nocie t h a t i t must equal the noual displacemen t s . Thus the nodal displacements of any two interconnected nodes u n i q u e l y determine the displacement alon-; the i n t e r f a c e between these nones ( i . e . the displace­ment a l on.; the i n t e r f a c e i n independent of the element under c o n s i d e r a t i o n ) . Therefore displacement i s continuous across trian.'nilar element i n t e r f a c e s Such elements are s a i d t o conlorm.

Other conforming elements exist, about which more w i l l be said l a t e r .

S p e c i f i e d Displacements. Sometimes the boundary c o n d i t i o n s of a problem i n v o l v e the s p e c i f i c a t i o n of surface displacements r a t h e r than stresses. Ivor noual p o i n t s where the displacement i s given the number of basic unknown nodal displacements i s correspondingly reduced arid no e q u i l i b r i u m c o n d i t i o n a t these nodes need be formulated.

R i j i d _ Body_ Displacements^ The set of 2N simultaneous equations i o r the unknown nodal displacements has a c h a r a c t e r i s t i c determinant of zero ( i . e . the equations are c i t h e r i n c o n s i s t e n t or have an i n f i n i t e number of possible s o l u t i o n s ) . This i s an expression of the f a c t t h a t the p o s s i b i l i t y of r i g i d body displacements, which do not a l t e r tne s t r e s s s t a t e , has not been eliminaLed i n the formula-

t i o n of the s o l u t i o n . F i x i n g one noual p o i n t A say. ( i . e . s p e c i f y i n g zero displacement) cuts out r i g i d "body t r a n s l a t i o n s and s p e c i f y i n g zero cisT.lacement f o r any other node B i n a d i r e c t i o n per; e n u i c u l a r t o AB stops r i g i d r o t a t i o n . r..']u: tn-.f, of (2K-3) remaining equations has a unique s o l u t i o n f o r the remaining (2N-3) unknown displacements.

Por the type of element Discussed • o f a r 'exact' s o l u t i o n s are obtained on]y f o r u n i f o r m utreo;.; f i e l d s . This i s a d i r e c t consequence of the s p e c i f i c a t i o n of l i n e a r displacement w i t h i n any element, f o r other s t r e s s f i e l d s the r e s u l t s converge t o the exact s o l u t i o n as the.size of the element i s decreased.

For the type of problem t a c k l e d i n t h i s t h e s i s i t was found t h a t elements w i t h sides of the order of 5 Km. s t r u c k an i d e a l balance between convergence of s o l u t i o n and storage needed f o r computation.

E. Maj;rix_ Formulation. vi/hen the i i n i t e element method i s approached q u a n t i t a t i v e l y a m a t r i x f o r m u l a t i o n i s l o c a l . The simultaneous equations reduce t o a m a t r i x equation of the form

( 2.3) 4Nx< jNk. I

(W = T o t a l No. of nodal point.-.')

where J K J i s c a l l e d the s t i l l n e s s m a t r i x ,

contains the unknown nodal displacements and jil' 1} the

e x t e r n a l l y a p p l i e d loads [ K ] turns out t o be a banded, p o s i t i v e d e f i n i t e , symmetric m a t r i x . (¥i j , 2. 7).

: ~ - o : Fi-.2.2. k : !

2N j • ••-

i o ! • i *

* 2hi *

The semi-band v/idth (shadeo) only i s store d . / / i t h computer storage space a t a premium i t i s obviously d e s i r a b l e t o formulate our data i n such a way as t o minimize the semi-band w i d t h . With every nodal p o i n t there i s associated a reference number, c a l l e d the nodal number, r a n ^ i n ^ from 1 t o N. The noaal f o i n t s can be numbered i n N ! d i f f e r e n t ways. On m a t r i x f o r m u l a t i o n i t i s found t h a t the semi-band w i d t h of [ K J increases l i n e a r l y w i t h the greatest d i f f e r * nee, f o r the whole model, between nodal p o i n t numbers belon^in.? t o the same element. J t i s important when numbering the nodes t h e r e f o r e , t h a t we oo so i n a manner t h a t keeps t h i s d i f f e r e n c e minimal. A dro pram f o r doin •; t h i s can be fauna i n appendix 5

The Gholeski method (Jenkins,19^9) i s used t o solve 2.3-

P_t2?_eJ'. yjpes_ of. Element. Rany refinements t o the shape of t?ie element arc

po;j;jible. Sauare elements (4 noaes) and t r i a r i ' j u l a r elements v/ith e x t r a nodes a t the centre o l each sii;e (6 nodes a l t o g e t h e r ) are a l l f e a s i b l e each i n c r e a s i n g the Jreeooir. v/ith which the displacement throughout the element can be chosen ( s u b j e c t t o com?/at:i.bilj.ty c o n d i t i o n s ) . The l a s t element allows the 1 ' i l t i n / ; of a f u l l q u a d r a t i c .

Vor three dimensional analysis.-, the element j.iust have volume, a t e t r a h e d r a l element (4 nodes) bein'.; e q u i v a l e n t t o the t n r t e noded t r i a n j u l a r element a l l o w i n g a l i n e a r displacement t o be f i t t e d ,

j ^ a n t i t a t i ye_ An a 1 ^ s i s ( M a t r i x_ l^ormul a t i o n ) l e t the 2N basic unknowns be represented as a

column m a t r i x

ft] -f s .

M \- _ J

(D2.1)

where ^ i s a 2x1 m a t r i x c o n t a i n i n g the x. y, components of displacement of the nodal p o i n t i . The nodal displacement matrix f o r the element c (nodes i , j . m ) i s d e f i n e d by

(1)2.2)

V/e define the displacement throughout the element i n terms of i t s noual displacements by

i f f = L»Ti$ (2.4)

v/ncie QlJ i n a f u n c t i o n of p o s i t i o n w i t h i n the elf.rncnt e su b j e c t t o the c o n s t r a i n t t h a t {9]*" wnen o vaDualcu a t the nodes i s r e a r c c t i v c l y .

] j e ! i n i : i T the s t r a i n m a t r i x as

( I * . 3 )

./£ o b t a i n the s t r a i n s throughout e i'rtn. as

Hooke's lav/ (1.13) p.vcs the r e l a t i o n be tween

utj-'iss anc. S t r a i n

whe re

H * -

1

33 ~ 3

(D2.4)

We now need an expression f o r the concentraLtd noaal

fo r c e s

5. e q u i v a l e n t s t a t i c a l l y t o the uLniiuuu a c t i n g on the boundaries of e. Consider a v i r t u a l displacement { f * } inmosec on tue element e. The work done by Lhr- boundary s tror.:\pp. i n

Uyj = £ (summed over i )

where.S i s the surface of the element e. ^ the s t r e s s r e s u l t a n t and ds an i n f i n i t e s i m a l p a r t of the surface w i t h outward u n i t normal V . Usinoj (1-9) we have

kJ = i TL. ». f * ds s J J

u J V by the divergence theorem, where V i ; ; the volume of the

element e.

T.j ^ dv v v

Using the e q u i l i b r i u m equation (1.11)

3- + r. - o 1

o ^ c j where p are the body f o r c e components per u n i t volume ani] w r i t i n g

J: — Q_ . -f- eo -^ "J W H E R E B Y

d e f i n i t i o n ( c . f . 1)1.1 and 31.2)

ana

we have

- M r * .

[ } l O t (

V

CjwOij - " ^ j L ^ j i j f (durnmy s u f f i c e s )

(/ X=-X =^ X.- o ) J

Therefore

That i s , the work done per u n i t volume by the- surface stresses i s

(2.7)

(2.8)

u s i n g (2.4) and ( 2 . 5 ) . The work done by the e q u i v a l e n t nodal f o r c e s must be the same t h e r e f o r e

V

This expression i s t r u e f o r a l l v i r t u a l displacements t h e r e f o r e

and from 2.h

(2.9)

where

(2.10)

ana

& a.

— SI

c

• -

(1)2.5)

(2.31)

where i , j m are the nooal p o i n t s comprising the element e.

l e t

where ^ i s a 2 x 1 m a t r i x c o n s i s t i n g of the x and y components of the e x t e r n a l f o r c e a p p l i e d a t the nodal p o i n t i . Applying the e q u i l i b r i u m c o n d i t i o n a t the Q. noae

N» of.

we have from 2.9

c 1 e. b=i «-

where I j ^ i l i s d e f i n e d by D2-5 when both nodal p o i n t s

a and L b e l o n g t o e ane aWsse-ro o t h e r w i s e , i . e .

{*] - His] * { F } ^

Y/hf.re

(i>2. ti)

a n a

(D2.<j)

The m a t r i x [k]j i s known as the s t i f f n e s s ma.Lrix of the model and must be i n v e r t e d l o r the pol l u t i o n of the fundamental equation 2.12. V/e now look a t the abuve formula l i o n : or the p a r t i c u l a r cafje oi the 3-noded t r i a n . j u l a r element. The_ t r _ i x _ _

3 ' \

>3C

where Ut -anu "J". =

x dis T l a cement of noee i . y displacement of noue i .

From 1)2.2.

I n d e f i n i n g £w] t by 2.4 we can. use 6 undetermined constants since we nave tv/o c o n s t r a i n t s on the displace­ment f u n c t i o n f o r each no0.e. Therefore l e t

(2.13)

define the displacement i n s i d e e. Applying the c o n s t r a i n t we have

(2.14a)

g i v i n g * i * i . ; ntj and

U J otfcy (2.14b)

J

g i v i n g

Solving 2.14 and s u b s t i t u t i n g i n t o (2.13) we o b t a i n

IT =

- ^ { ( ^ ^ ^ a . ^ C V ^ * ^ 1 ! ) ^ (2.25)

2U>

where

J 3i

(1)2.10)

(note: |^| » 2 x area oi' element e)

and.

(D2.31)

aj bj etc. are obtained by c y c l i c i n t e r cnan;e of the s u f f i c e s i - j . m . R e w r i t i n g 2.15 i n m a t r i x n o t a t i o n c f.2.4.

a? = fc} - r«TW

K = * k ^ * ) x = [ I l ]

v/here

and

(2.16)

The_ btrain__K Latrix__ 33 From D2. 3 £ind .1)1.1

r

Q 7 C

c>ir

C.5*-V T J

2-7

and then uaing 2.15

i.e. —

t o bj o o C J "

J "* c«, b„

and comparing w i t h 2; 5

&T - i o C3 b M

o o c j o

- b J

b M

(2.17)

(2.1b)

i For plane s t r a i n we have by d e f i n i t i o n u 3 = o and

6

= o . I t follows from D l . l that e,t = e« 1^ = 0 i-e. the only non-zero components of the s t r a i n termor are e_ , eu , e_ . Therefore, the only non- zero strcus components are (from 1.13) x>: yy, zz. scy. and ui/Ln j 1.20.

Q- tQe t , E

(••"OO- o e

CM) ^

or r e w r i t i n g i n matrix form

(2 19)

arid

('-"OO-a-rO

(2 20)

2%

and comparing w i t h 2.6 we have

i

o i

o o (2.21)

The Element Stiffness' Matrix. CK1

= JW Prom 2.10

where t i s the thickness of element e Both (Bj'and £j)Jft

are independant of x,y (2.1b and 2,18) and therefore

(?.22)

f o r u n i t thickness. A i s defined i n D2.10. Bodv_ Force_ Matrix

Consider the p a r t i c u l a r case of g r a v i t a t i o n a l forces. The body force }? per u n i t volume i s ^ n ^ n e

-ve y d i r e c t i o n (Fig. 2.3). Therefore £sO and f^r -^3 where ^ i s the density of the element e and the acceleration due to g r a v i t y . Using 2.11 ana 2.16 we obtain ...

- a

A J

o N

c> Kl.'

\ dv/

Npv/

where (x,y) i s the centroid of e. Using U 2 . l l .

. - A\A\

Therefore, f o r u n i t element thickness

Wo (2.23)

I n Appendix 3 examples of the matrix formulation of f i n i t e element theory are worked out j o r axisymmctric models and f o r 6-noded t r i a n g u l a r elements.

2.3 _Application of Finite-element methods to the earth sciences.""

Although f i n i t e element analysis has "been used by engineers f o r many years, t h i s powerful method has been generally ignored by earth s c i e n t i s t s despite i t s obvious advantages over a n a l y t i c a l methods w i t h unreal boundaries and s i m p l i f y i n g assumptions of rock homogeneity, isotropy and e l a s t i c i t y . Inhomogeneity and anisotropy can be introduced i n t o the f i n i t e element method without any d i f f i c u l t y whatsoever and non-linear i n e l a s t i c behaviour, such as p l a s t i c i t y , i s dealt w i t h by i t e r a t i v e use of the standard e l a s t i c method (Zienkiewicz and Cheung 1967) .

The method i s not confined to the s o l u t i o n of stress analysis problems and can be applied to any boundary value problem that i s exprer;::ible i n a v a r i a t i o n a l form (Zienkiewicz and Cheung 1967). This includes the f o l l o w i n g which are of i n t e r e s t to the

geophysicist;

(a) the behaviour 03? rocks under s t a t i c and dynamic loads

(b) the propagation and. dispersion of stress waves. (c) heat conduction. (d) f l u i d flow i n porous materials. (e) the d i s t r i b u t i o n of magnetic and g r a v i t a t i o n a l

p o t e n t i a l .

Voight and Saramuelson (1969) have investigated the e f f e c t s of pronounced heterogen&'uy on the stress t r a j e c t o r i e s , the l o c a t i o n of i n i t i a l f r a c t u r e surfaces and the shear stress d i s t r i b u t i o n s produced by the s t a t i c loading of an upper crus t a l model.

Stephansson and Berner (1971) have adapted the method to deal w i t h l i n e a r l y viscous materials and used, i t to investigate the i s o s t a t i c readjustment of a crustal model across the north mid-Atlantic ridge.

Beaumont and Lambert (1972) have shown that t i l t near a surface load i s sensitive to crus t a l structure and conclude* that t i l t measurements can be used to complement g r a v i t y and magnetic data f o r upper crustal modelling.

At-toinprtp—et% Fault propagation studies based on the state of stress immediately before f a i l u r e (llafner 1951) have been attempted. I t i s d i f f i c u l t to assess the relevance of these resu l t s because of the drastic e f f e c t that f a i l u r e has on the i n i t i a l stresses. By simulating f r a c t u r e by small zero strength elencjits i t -i s possible to use the f i n i t e element method i n an i t e r a t i v e proce«du3"e so that at each sta ;e the e f f e c t

of the propagation of the Jault ia allowed f o r by the i n t r o d u c t i o n of more zero strength elements. Dou;las and Service ( i n : ress i n i ) have tie en us in •; f in?, te elements i n t h i s v/ay i n an attempt to obtain a ne onanism f o r r i f t v alley formation.

In the absence of d i r e c t laboratory measurements under conditions closely approximating those encountered at depth i n the earth i t i s impossi b.i • to predict the long term rhcological behaviour of materials. IMrrite element methods should make i t possible to mooel the earth at depth i n an attempt to f i t detailed information of surface response to r e l a t i v e l y well known loading. Rhedomical models obtained i n t h i s way w i l l be the main contribution of f i n i t e element methods to the earth sciences.

CHAl-'™t_.3_-1 S_tre-f3a_ d i s t r i b u t i o n ^ associateO_ with_ deej:

se dimento^as i n s j

Evidence f o r the subsidence of upper cr u s t a l material and the subsequent formation of sedimentary basins i s provided by gravity- magnetic and seismic r e f r a c t i o n surveys. Deep sedimentary basins have been proved on the continental shelves of aseismic margins (Bott and Watts ;1970) some of which extend i n t o continental areas. (Blundellet al,196b). I\;any of these basins are terminated by normal f a u l t s i n d i c a t i n g the one time existence of regional tensions. When analysing the stresses i n the region of a sedimentary basin, the i n t e r a c t i o n of the g r a v i t a t i o n a l forces w i t h l o c a l tensions, and the r e d i s t r i b u t i o n of these tensions by the contrast i n e l a s t i c parameters between the basin and the surrounding basement rocks, are important.

Ignoring the tensions f o r the meantime, the small density contrast between the sediments and basement rocks means that the p r i n c i p a l directions are more or less v e r t i c a l and horizo n t a l . The sediments are less dense than the surrounding crustal rocks and the • v e r t i c a l 1 p r i n c i p a l pressure w i t h i n these increases at a lesser rate v/ith depth than i n the adjacent, crust. This gives conditions under which the rocks adjacent to the sediments are more susceptible to normal f a u l t i n g than rocks at the same depth immediately below the centre of the basin. Because of the ' v e r t i c a l - h o r i z o n t a l ' o r i e n t a t i o n of the p r i n c i p a l axes the imposition of a

3 i

r 8 CM

00

H

CM

CM

uniform horizontal tension on t h i s stress f i e l d w i l l not a l t e r t h i s preferred urea of f a u l t i n g - i t w i l l j u u t increase the l i k e l i h o o d of f a u l t i n g . However, any uniformly applied tension w i l l "be non-uniform i n the region of the basin because of i t s contrasting e l a s t i c parameters. This w i l l r e s u l t i n a possible change i n the region of preferred f a u l t i n g . We use the f i n i t e element method to investigate t h i s modification.

The model used i s shown i n f i g . 3 . 1 - The applied tension i s 1 kbar which can be scaled. The e l a s t i c parameters taken f o r the sediments were calculated by assuming a Toisson s o l i d (X*^/u.) and a density of 2.45 gm/c.c. The Nafe-Drake (1963) curve gave an estimated P-wave v e l o c i t y from which Young's modulus was calculated. This was

E = 0*327 x 10 , a dynes/sq. cia.

For the surrounding material a c r u s t a l density and P-wave v e l o c i t y were used to estimate Young's modulus -

^crust = 0«92o x 10 dynes/sq. cm.

The shape of the basin was based on Bott's (1965) i n t e r p r e t a t i o n of the East I r i s h Sea Eaain. I t s maximum depth i s taken as 6 Km. and i t s width as 6b Km. The c a l c u l a t i o n was treated as a case of plane s t r a i n and g r a v i t a t i o n a l forces ignored. The r e d i s t r i b u t i o n of the stress-differences i n the region of the basin i s represented schematically i n f i g . 3 . 2 . The l e t t e r s represent ranges of stress difference (shown on the

35

8

8 0 0 0 0

0 k. 0 b. 0 kl

(0

IL 0 H 0 X 0

I

M

3b

KEY_ TO FISUHES 3-2, 3.4 4 4_

Symbol S_tres_s_ Range_ _(dynes/s£^ c_.rn._J

A 3 % 10 9

B 9.75 x 10 8 ^ S * 10 9

G 9 . 2 5 x l 0 8 < a A 9.75 x 10 8

2 8.75 x 10 8 * 3 < 9.25 x 10 8

E b.25 x 10 8 ^ 3 •< 8.75 x 10 8

I 1 7.75 x 10 8 a < . 8.25 x 10 8

7.25 x 10 8 * 3 <. 7.75 x 10 8

H 6.75.x 10 8 « 3 < 7.25 x 10 8

I 6.25 x 10 8 3 -c 6.75 x 10 8

J 5.75 x 10 8 £ S -c 6.25 x 10 8

K 5.25 x 10 8 £ 3 < 5.75 x 10 8

I 4.75 x 10 8 £ 3 < 5.25 x 10 8

M 4.25 x 10 8 ^ S < 4.75 x 10 8

N 3.75 x 10 8 -£ S < 4.25 x 10 8

0 3.25 x 10 8 ^ S <: 3.75 x 10 8

P 2 . 7 5 x l 0 8 ^ S < 3.25 x. 10 8

Q 2.25 x l O 8 $ S <. 2.75 x 10 8

R 1.75 x 10 6 ^ 3 * 2.25 x 10 8

S 1.25 x 10 8 S <• 1.75 x 10 8

T 0.75 x 10 8 S S < 1.25 x. 10 8

U 0.25 x 10 8 ^ 3 < 0.75 x 10 8. V S -c 0.25 x 10 8

f i g u r e ) the higher the l e t t e r i n the alphabet the greater the atress-difference i t represents. 1 represents the uniformly applied stress-difference of 500 bar, K a stress-difference greater than t h i s and M one s l i g h t ­l y less.

The stress-differences i n the basement rocks next to the basin deviate from the uniform value seen at large distances from the basin (500 bar) I n general they show an increase the exception being a t the upper surface of the model where a small decrease i s observed. The greatest stress differences (877-5 bar) occur i n

i

the basement rocks below the t h i c k e s t sediments, t h i s , being a 755*. increase i n the regional value of 500 bar.

Thus the r e d i s t r i b u t i o n by the contrasting e l a s t i c parameters has an e f f e c t opposite to that of the low density of the basin. One leads to p r e f e r e n t i a l normal f a u l t i n g conditions a t the sides of the basin while the other gives preferred f a u l t i n g i n the basement rocks immediately below the deepest part of the basin.

The actual p o s i t i o n of i n i t i a l f a u l t i n g w i l l depend on

(a) the size of the regional tension compared w i t h the g r a v i t a t i o n a l forces. and

(b) the e f f e c t of the decrease i n the l i k e l i h o o d of f a u l t i n g w i t h depth because of ( i ) gradual b r i t t l e to d u c t i l e t r a n s i t i o n , ( i i ) higher f r i c t i o n a l resistance across p o t e n t i a l shear planes (Anderson,1942)

3%

HoRIZONTflU T E t j J i c n j Caere. V A ) CONST (MM NED MOWHONTALL*

CM

8

s 2

2

<M

IVM19N03 TONtK

• E CM

According to Anderson the function

P = — S w . ^ © - v * ^ " * - "aT °~ i B J

gives a d i r e c t measure of the likelyhood of f a u l t i n g , where P i s the greatest p r i n c i p a l s t r e s s and R the l e a s t p r i n c i p a l s t r e s s (tensions + ve). Taking />>-' > &3 Anderson suggests l i k e l y from f i e l d observations, we have o - 22£ . Thus

This function does not take into consideration the gradual b r i t t l e to ductile t r a n s i t i o n .

To investigate further t h i s i n t e r a c t i o n a f i n i t e element c a l c u l a t i o n has been performed using the model i n Fig.3.3 for a basin 1 Km. deep and superimposing upon i t the s t r e s s e s for a h o r i z o n t a l l y applied tension of 300 bar. The d i s t r i b u t i o n of F values for the combined f i e l d s i s shown i n Fig.3-4. The r e d i s t r i b u t i o n of the tension i s seen to be dominant and a l l c r u s t a l regions i n contact with the basin, apart from the surface, are the areas most susceptible to f a u l t i n g . This suggests that normal f a u l t i n g w i l l i n i t i a l l y be instigated in the c r u s t a l rocks a t points below the greatest sediment thicknesses. After f a i l u r e s t r e s s concentration w i l l lead to downwards propagation of the f a u l t to depths where d u c t i l i t y i s s u f f i c i e n t for flow to reduce the s t r e s s differences to below the strength of the rocks.

HO

0 s N

Ml

1/

CM

GHAITKR__4_.

JJj_treGL;_ _tjys_tems_ a_t_ Y_ounjj_ p j ^ ^ i ^ J ^ a l ^ MarjinsJ Young continental margins mark the juxtaj-oui kion

of continental and oceanic crust attached together as a part of a single plate of lithosphere. Margins of t h i s type form the boroer of much of the A t l a n t i c ana Indian Oceans. They lack the earthquake a c t i v i t y associated w i t h the release of s t r a i n energy at plate boundaries such as are formed by the active margins around the l a c i f i c Ocean, yet they are t y p i c a l l y associatea w i t h a c h a r a c t e r i s t i c type of tectonic a c t i v i t y involving a substantial amount of subsidence of the shelf ( C o l l e t t e , 19^8) incl u d i n g the iormation of sedimentary basins (Bott 19k 5, Bott and Y/atts 1970). We now use the f i n i k e element method to investigate various stress systems associates w i t h such a margin.

There are at least two ways i n which l o c a l stress systems may be associated w i t h a young margin of aseismic type. F i r s t the body forces associates w i t h the v a r i a t i o n i n thickness of the low density crust and the presence of the ocean on one sine ;ive r i s e to a l a t e r a l l y varying stress system across the marjin. second, i f a regional stress i s applied to the l i t h o u i h e r e , the l a t f . r a l v a r i a t i o n of e l a s t i c parameters acrons the margin w i l l cause the regional stress system to be modified i n the v i c i n i t y of the margin. We us( the f i n i t e element method to investigate q u a n t i t a t i v e l y these types of stress system associated w i t h an ideal margin. Ufcress differences may also be associated w i t h

MO,

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8 WW. n

2 A A 8 I 2 M i l/>

7: /.

8

a d i f f e r e n t i a l temperature - depth d i s t r i b u t i o n , but

these have not been considered.

The s u b d i v i s i o n of the model was done i n such a

way as to preserve the Mohorovicic d i s c o n t i n u i t y (Pig.4.1)

A l l the problems have been t r e a t e d as cases of plane

s t r a i n . I n the model, the c o n t i n e n t a l c r u s t i s 35 Km.

t h i c k and the oceanic c r u s t i s 6.875 Km t h i c k ; the

d e n s i t i e s of the c r u s t a l and mantle m a t e r i a l are taken •

to be 2.85 and 3-25 gm/cm3 r e s p e c t i v e l y . The model

i s thus i n i s o s t a t i c e q u i l i b r i u m a c r o s s the margin.

The e l a s t i c parameters used a r e based on measured P wave

v e l o c i t i e s assuming a t o i s s o n s o l i d i n which lame 7 's

constants ^=/*, and Poisson's r a t i o = 0.25- The v a l u e s

of Young's modulus used were

E . = 0.928 x 1 0 1 2 dyne cm"2

c r u s t E m a n t l e = ° ' 1 7 8 x ^ ^ ^

The t r a n s i t i o n from c o n t i n e n t a l to oceanic c r u s t takes

p l a c e over a h o r i z o n t a l d i s t a n c e of about 60 km. (tforzel,1965)

The d e n s i t y model ( F i g . 4. 2) was used to i n v e s t i g a t e

the e f f e c t of d i f f e r e n t i a l l o a d i n g a c r o s s the margin.

T h i s was obtained by s u b t r a c t i n g the d e n s i t y - depth

d i s t r i b u t i o n beneath the ocean from the whole s t r u c t u r e ,

l e a v i n g a l o a d a t the top of the c o n t i n e n t a l c r u s t and

an equal u p t h r u s t near the base. S t r e s s e s a s s o c i a t e d

w i t h the normal oceanic d e n s i t y - depth d i s t r i b u t i o n can

be superimposed on our model, both ends of which a r e

considered to be c o n s t r a i n e d i n a h o r i z o n t a l d i r e c t i o n .

Because the whole model i s i n i s o s t a t i c e q u i l i b r i u m the

8

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J. 8

8

8

s

H i

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J o

base can be l e f t f r e e of boundary s t r e s s e s . The r e s u l t s

of the f i n i t e element c a l c u l a t i o n are shown i n lvisj.4.3.

The p r i n c i p a l s t r e s s e s are n e a r l y h o r i z o n t a l and

v e r t i c a l except beneath the margin. Beneath the

c o n t i n e n t a l r e g i o n the maximum p r i n c i p a l compression

i s v e r t i c a l and the minimum p r i n c i p a l compression i s

h o r i z o n t a l . The p r i n c i p a l s t r e s s i n a d i r e c t i o n

p e r p e n d i c u l a r to the plane of the model i s everywhere

the i n t e r m e d i a t e one. The maximum s t r e s s d i f f e r e n c e

( d e f i n e d as h a l f the d i f f e r e n c e between the g r e a t e s t

and l e a s t p r i n c i p a l s t r e s s and t h e r e f o r e equal to the

g r e a t e s t s h e a r ) occurs a t a depth of 10 Km and reaches

about 370 bar. Als o shown i s the r e l a t i v e v a r i a t i o n of

the g r e a t e s t s h e a r w i t h depth through a t y p i p a l v e r t i c a l

s e c t i o n (marked by arrow) of the c o n t i n e n t a l l i t h o s p h e r e .

Thus the d i f f e r e n t i a l g r a v i t y l o a d i n g a c r o s s a young

c o n t i n e n t a l margin g i v e s r i s e to a s t a t e of s t r e s s more

conducive to normal f a u l t i n g (Anderson,1942) p a r a l l e l to

the margin i n the a d j a c e n t c o n t i n e n t a l c r u s t than i n the

oceanic c r u s t .

V/hen a margin i s s u b j e c t e d to a r e g i o n a l t e n s i o n

or compression, the l a t e r a l change i n e l a s t i c parameters

causes l o c a l v a r i a t i o n from the uniform s t r e s s e s t h a t

would be a s s o c i a t e d w i t h a homogeneous p l a t e . To

i n v e s t i g a t e t h i s e f f e c t a f i n i t e element c a l c u l a t i o n was

performed on a margin s u b j e c t e d to a h o r i z o n t a l t e n s i o n

of ^00 bar. A l l body f o r c e s were neglected. The r e s u l t s

of the c a l c u l a t i o n a r e s c h e m a t i c a l l y i l l u s t r a t e d i n

Pig.4.4. The g r e a t e s t c r u s t a l s t r e s s d i f f e r e n c e s occur

on the c o n t i n e n t a l slope. The s t r e s s d i f f e r e n c e s i n the

z z z z * x « Z 9 s o X *

I s z z z» z * 2 * X 1 ° I 1 z z z * z * 1 • - X x ° •

I 1 z z z* I s 1 * _ z o ° _ M

z * z z z * z * x * _ z o ° -%

z z z 1 z z M 9 x o o ° -»

z z z z* ** z 9 s o o ° -»

z z z ' z x Z 9 s o o ° -*

z z I 1 x x z * z o o ° -»

z z z * X * z • t o o ° ~t

z z z 1 X

X X 9 s o o ° -»

-* z z z* X * z * s o o ° -»

z z z* X * X * s o o ° -»

z z z* X * z * x o o ° z z z* x « z • z o o ° z z z* X * z " o o ° z z_ z* X * z * s o o ° _ —

z z z* «* z x o o ° _ —

z z z* m9 z 9 a o o ° _ -

j z z z z 1 X * z * s o o ° —

z z .z z z 1 z * z * X ° o ° _ — z z z z z 1 z " z 9 _ X «° _ -z Z z z z 1 z x z * X x ° z Z z z z 1 z * 1 * _ X » x

* z z z *z * z * X X / V z z *z * z 1

z * X -1 z J z * z 1 1 •J " z

V- •J -J _J J J z •* J * X

V- _J -J _J -1 -1

-J J z zo

W K _J X 2 * o \ -J y J z X z * o \ J x V

f K J z 1 X * a.

z * x~ K z 2 * o •• z * o x 1 j z 2 * z

z» z * o 1 z K j z 2 * I » z* z * o x X ~ K 2 * z * z* x x o x X ~ K -J 2 * z *• z* x x o x X ~ -J 2 * z o z * X * o x X ~ K -J 2 * z o z« x " o x X ~ K -J

s o z« x x o x X ~ K -J j z X ° z x X * x x X ~ K -t

z o X * x« x x IC j z x o X * X * X 1

-1 K _, z 2 * x« x° x~ 1 M .1 * 2 * x« x ° x x -» J * 2 * X* x° x~ -» • •J 2 * x« o ° x~ K J * 2 * X* o ? —~ K 2 * x° o° _~ -1 J * 2 X x° o ° _— 1 •» -J J x > «° o ° -. * * .1 -J * 2 Z X* o° -,« 1 1 Z

z" x° -,» X z° j -» J -» _ * ^ J

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upper most 5 Kin of the c o n t i n e n t a l c r u s t a r e s l i g h t l y

g r e a t e r than those i n the upper 5 Km of the oceanic

c r u s t .

I n general, both types of s t r e s s system w i l l be

superimposed. V/hen the l i t h o s p h e r e i s s u b j e c t e d to a

l o c a l t e n s i o n , both e f f e c t s enhance each other i n the

a d j a c e n t c o n t i n e n t a l c r u s t producing a s t r e s s regime

h i g h l y favourable to normal f a u l t i n g p a r t i c u l a r l y near

the s l o p e . When the l i t h o s p h e r e i s s u b j e c t e d to r e g i o n a l

compression, then the tv/o e f f e c t s w i l l oppose each other

i n the c o n t i n e n t a l c r u s t , and the sub-oceanic r e g i o n

a d j a c e n t to the margin w i l l be most s u s c e p t i b l e to t h r u s t

f a u l t i n g .

Our computations e x p l a i n the prevalence of normal

f a u l t i n g i n the c o n t i n e n t a l c r u s t on and n e a r the

c o n t i n e n t a l s h e l f , and the corresponding l a c k of evidence

f o r normal f a u l t i n g i n the oceanic c r u s t .

StresjDill\i_s_i_on_ in_ the_ 1 ithosj.herej

The t h e o r i e s oi' c o n t i n e n t a l d r i f t (Wegener ,1912),

based on g e o l o g i c a l and o u t l i n e s i m i l a r i t i e s of con t i n e n t s

thousands of k i l o m e t e r s a p a r t , and ocean f l o o r spreading!

r e s u l t i n g from the study of geomagnetic anomalies of

the s e a f l o o r (Vine and Matthews 1963; Pitman and

Henrtzler 1966). have r e c e n t l y been u n i t e d by the a l l

encompassing theory of p l a t e t e c t o n i c s (McKenzie and

F a r k e r 190 7, Morgan 1968).

The theory of i s o s t a ^ y and s e i s m o l o g i c a l evidence

f o r a low v e l o c i t y . c h a n n e l i n the upper c r u s t both suggest

t h a t the e a r t h has a stro n g outer s h e l l ( l i t h o s p h e r e )

u n d e r l a i n by a weak re g i o n which deforms by flow

(asthenosphere) ( B a r r e l l 1914 a . b ) . I n p l a t e t e c t o n i c s

the e a r t h ' s outer s h e l l i s conceived as a s e r i e s of

r i g i d b l o c k s , r e f e r r e d to as p l a t e s , the s u r f a c e boundaries

of which are defined by the narrow regions of s e i s m i c

a c t i v i t y . These p l a t e s are i n r e l a t i v e motion, s l i d i n g

p a st each other a t transform f a u l t s (Wilson 1965) wi t h

the c r e a t i o n of oceanic c r u s t a t t h e i r r i d g e s and i t s

consumption a t oceanic deeps. By studying the amplitude

and wavelength of bending of the l i t h o s p h e r e i n the

v i c i n i t y of s u p e r c r u s t a l loads, V/alcott (1970) has obtained

estimated t h i c k n e s s e s of the order of 100 Km f o r the

l i t h o s p h e r e .

By f i t t i n g models w i t h an e l a s t i c l a y e r o v e r l y i n g

v i s c o u s l a y e r s McConnell (1965) has formulated a r h e o l o g i c a l

If f

w

M 8 8 3

UJ

4

MM

5o

model of the outer e a r t h to f i t the observed Fennoscandian

r e c o v e r y a f t e r the removal of ice-age l o a d s . He concludes

t h a t the region most s u s c e p t i b l e to deformation l i e s

between ICO to 400 Km depth, the r e s i s t a n c e to deformation

i n c r e a s i n g r a p i d l y below 400 Km. T h i s suggests a base

to the asthenosphcre a t a depth of some 400 Km. The 21

apparent v i s c o s i t y of t h i s l a y e r i s of the o r a e r 3 x 10 I'.

Because of the r e l a t i v e r i g i d i t y of the l i t h o s p h e r e

f o r c e s a p p l i e d a t one boundary are t r a n s m i t t e d to other

boundaries of the p l a t e . Thus, f o r example, f o r c e s a t

the r i d g e of a l i t h o s p h e r i c p l a t e w i l l be f e l t a t i s l a n d

a r e boundaries anu i t i s important to have some idea of

the l i k e l y e f f e c t s of such f o r c e s and of the order of

magnitude of time d e l a y s a s s o c i a t e d w i t h propagation

through the l i t h o s p h e r e . I n t h i s chapter a model has been

s e t up to i n v e s t i g a t e f u r t h e r t h i s problem.

5•1 S p e c i f i c a t i o n of problem.

I f a sudden! f o r c e i s ^ j a p p l i e d and maintained a t the

ridge of a l i t h o s p h e r i c p l a t e then a q u e s t i o n of g r e a t

r e l e v a n c e concerns the e f f e c t s of t h i s f o r c e a t other

p l a t e boundaries.. To obtain i n s i g h t i n t o t h i s problem

the model shown in. f i g u r e (5.1) has been considered.

A more s o p h i s t i c a t e d model i s not warranted because the

e x t r a c o m p l e x i t i e s of mathematical a n a l y s i s introduced

and the u n c e r t a i n t y i n numerical values of T h e o l o g i c a l

parameters would r e s u l t i n l e s s progress towardn an under­

s t a n d i n g of what a c t u a l l y takes p l a c e than t h a t obtained

from c o n s i d e r a t i o n of the crude model. The parameter

v a l u e s shown i n the f i g i i r e have the orders of magnitude

of the corresponding v a l u e s a s s o c i a t e d w i t h a p l a t e the

s i z e of the P a c i f i c . I f a sudden p r e s s u r e , P q I i s a p p l i e d

and maintained a t the 'ridge end', R, of the model-then

the propagation of t h i s p r e s s u r e i s opposed "by:

( 1 ) i n e r t i a l f o r c e s

and (11) v i s c o u s f o r c e s due to the u n d e r l y i n g asthenosphere.

The q u e s t i o n we ask i s 'what i s the pr e s s u r e d i s t r i b u t i o n

along the p l a t e , a time t a f t e r the i n i t i a l a p p l i c a t i o n

of the pressure a t R?'

Formulation of the problem.

Assume th a t the pressure i s a p p l i e d a t the i n s t a n t

i n time t = o. l e t u ( x , t ) be the h o r i z o n t a l displacement,

a t a time t , of the s e c t i o n of p l a t e i n i t i a l l y a t x.

Since a«L. i t i s p o s s i b l e to ignore v e r t i c a l v a r i a t i o n

of t h i s displacement. Consider the p o r t i o n of the p l a t e

i n i t i a l l y between (x - fix) and (x + fix). . A f t e r a time

t i t l i e s between (*.-£*.•*•*.- . S * ) and (sc.*-Sx. * u. •«• fcx. )

and expe r i e n c e s f o r c e s (cxmsi'dering u n i t l e n g t h perpendi-"•" : H •*.•

*" • " 1; • •

c u l a r t o the plane of the model) as shown i n F i g u r e ( 5 . 2 ) .

where T ( x , t ) i s the t e n s i o n a t the point of the p l a t e

o r i g i n a l l y a t x.

MAS* 2f4.£sb

( r ^ t ) - £s*Ja fro*)* £s*]<

F i g . ( 5 . 2 )

A t ' i l y i n j flew ton'., second law wt now have

° ^ f " ah U t / V 1 ^ ) Ax

1'ror.i the oei'ini i,ion of Youn^ s RoduDus

T = E i s c

and i t f o l l o w s t h a t

P-e cause oi the ;rt.at s t r e n g t h (IL) of the j J a t c compareu

v/itn the a p p j i e d p r e s s u r e P , we have I and

(5.2a) reduces to

\ a t " <^c c^x.1- (5.;'b)

where

^ _ ^ (JJ5.D

I f i n i t i a l l y v/r. ignore the i n e r t i a ! f o r c e s co.Tir.arcd

w i t h trie- v i s c o u s ones equation {'j>.2b) reduces in Ihc

d i j fu::i i.n riH Iutj \'-n:

To determine the pre s s u r e d i s t r i b u t i o n p ( x . t ) . i n tfiu

T^.te u.3 t e r a time t we must s o l v e (5.2c) s u b j e c t to the

1 oIlo \ ; j i ! * boundury and i n i t i a l c o n d i t i o n s ,

I X * O cut ^ tr *> o (5.3)

E. '.St - I t > o

and then use the p r e s s u r e form of (5.1) K-*-'^ r ~ E . ( J O

The S o l u t i o n

The P a r t i a l second order d i l f c r e n b i a l equation

(5.2c) can be reduce;.' to an prcinar^r second order

d i f f e r e n t i a l equation by the operation oi a l a j l a c e

t r a n s f o r m a t i o n , and then solvod l o r the l a r l a c c trarisJ urm

of the displacement u ( x , t ) . The p r o W o i s then reduced t

one of performing an i n v e r s e l a r l a c e transformation. The

s o l u t i o n to (5-2c) has been obtained (Appendix A part 1)

i n t h i s way and i s

T h i s displacement d i s t r i b u t i o n gj vi.r; a vrt.-.;:jurc:

t hr ou g h ou t the pi <•!. t e.

5 >

8 8 S J* N

3

8

1

LL

The s t r i t s and (5 4) have been eomputea am*

these computations a r e i l l u s t r a t e d b y the- f i j u r e s (lj.3)«

(!3./|) an6 F i g u r e (5.3) i l l u s t r a t e s the bui.J.d up

of pressure with time a t the f i ^ e d end. k. ol' the p l a t e .

The parameters used, f o r computation are shown on the

graph. The Line taken f o r the pressure to roach, one

h a l f o f t h a t a r.plied f o r a plate 10,000 Km i n length i s

o f the order of 2C O ,000 y e a r s . I t should be noted from

equation (b.'i) t h a t changing the v a l u e s of any of the

parameters dees not a l t e r the shave of the graph. What

i t i n f a c t does do i s change the l a b e l l i n g of the tjmc

a x i s . Because o f the occurrence of the r a t i o ,

i n c r e a s i n g Young's Modulus by a f a c t o r o f w i l l cause

a l l the time l a b e l s i n f i g u r e (5.3) to-be halved unr:

s i m i l a r l y a decrease i n the l e n g t h , 1 of the plate by

a f a c t o r o f 10 w i l l cause the time l a b e l s tu be decreased

by a f a c t o r of 100. F i g u r e s (CJ>.4) and (5.'3) i l l u s t r a t e

the events a t xhe opposite end of the p l a t e H (Y, ~ I ) -

The f i r s t grach snows the dinpl aeement response to the

a p p l i e d f o r c e 3-o a t It, and the second ;rai..h p l o t s the

g r a d i e n t of t h i s graph i . e . the v e l o c i t y response to P e .

The parameterf. used a r e shown on each graph. Note t h a t

the comments rega r d i n g the e f f e c t s of changes i n these

parameters, d i s c u s s e d f o r f i g u r e (5-3) do not :x\ r-l.y to

these l a s t two graphs. The s i t u a t i o n i s j.iorc- compli eated

here, i n v o l v i n g changes i n the dimensioning of both axes..

the warmer of which be in": c v i r - f n t on a c l o s e r c: X;.MI! i n;. t i on

of equation (^.A). The v e l o c i t y curve i n f i jure

shows & ra p i d f a l l o f f with time f o r l-'O 000 .years ;ino

l.iien a s ti-.-ni i <" r1 p e r u x i o'J" abou t < >0, Oi •( • .yr :. !-; -. n i l,li n.

S<*

X n 1 9 8 §8 U* -1 H A

H

u §

57

8

N I *

8

t0

8

v e l o c i t y between 5 cm./yr. and 1 crn./yr.

Tif ie-et op" a_ j i i f f erent_ yplpcity_ JJ-'j^p^^-p^. jp the_ as thenoBjrhore

I n the it ode 1 considered ( f i g u r e ( ' j . l ) ) a J. inc. fir

J i-.l li o:!T of v e l o c i t y w i t h d e i t h wi thin the ;j.rj therjos.s.here i

vie.:: af.isuraed. I f i n s t e a d o f t h i s we apply, t o j e l i i o r

w i t h the c o n t i n u i t y of v e l o c i t y a c r o s s the lithoui-hcre -

aslhenoa;.here boundary a condition or no net flow of

mate r i a l . t h r o u g h any v e r t i c a l cross-.c.ction we can f i t a

q u a d r a t i c v a r i a b l e v e l o c i t y , I f . (Jacoby,ly'70).

Using Jacoby'o n o t a t i o n £ Pi«j.*<-«_ )

wnere

anu

e l ­

and lA-.i \ - 7_Uo Thus, the magnitucie of the

For the cawe b = 2a, we have = M'1 , ci-. --t "z«,-= 75.

tel -v e l o c i t y 'radicnt a t the to), of the airj.henor;i-here i:j

i n c r e a s e Q by a f a c t o r of 7 ( c . f . f o r line.-Ar c u^e).

T h i s i s e q u i v a l e n t to i n c r e a s i n g the v i s c o s i t y 1^ i n the

l i n e a r case ».y a f a c t o r of 7- which, from s e c t i o n 5-3

i s e q u i v a l e n t to i n c r e a s i n g the time a>.is i n f i g u r e s

(5-3) and. (5-4) by a f a c t o r of 7. The e f f e c t on

I I

f >"

ui

Ul

Hi i / i

i/)

6CJ

f i j u re r. Lo renuce the vc-.2of;jty ay . i : : l;.y l:

l ' t . c t o r o f 7 . •

( b . b ) i n c l u : : Lon_ 01 Jnj-2?t;ij-J_ Ternis_

h ' - .u^vion (»>.^b) huu been oolve-u t o invc.:-.l .i ^s-tc the

cJJ'c.ct OJ i j n ^ r i n ; t he i n e r l i . - J t e r i rm . ( A i - : e m i : >. A.

} . ^ " L £ ) • The c u r r t - u ^ Oil d i n j c o J u l i o n o u r e

Ft E

{ J L l ( ! M )

(t>°) anc-

I— n -1 J

where

Vj.'I) C t > o )

1 * — •c

;.nu

(U'J. J )

J<'or c;r.t:J "J rv Z n + una Z r t_ a r c b o t n rcaJ . However l »r

Hin J i c i .• n t J y l a r ;e a we have 0* <

become C O J I UI I . >.. i n t h ' i u ca;ie Z n

am. ^ ir iu Z ,

z n f ind J J I the a b r v e

sn UJ

CM

CM

CM

CM I n I r t

CM 8 O Q «=- CM

CM CM I I I I I I I I I I I I

8 1CM

CM

V)

H CM

co

CM

CM

0. 0» < 1 00 CM <0

equa t ion : :

if-; L rt'j.1 nvan l.» Ly ( s i n c r z*- - i s a lways r e a l ) .

'Jhr. s e r i e s (b.7) I'U'S been computet'; f o r Luc b u i l d

u . '>J' p-vessu're a t x = o. T h i s coin^u l aL i on p n , ;cn to

i:< v t r a l i r o b J r j n a ' . . ' l i ich have b^en d i scu : j ; i i .o i n a i i o n o i x

4 p a r t 3- The s j raph shown i n f i 5. 5-7 i J l u s I r a I ' s U u s

coir . -uta t i o n anc , on coirip&.rif jon w i t h . f.L .:ure ( ! ; . 3 ) ' i t -

i s e v i d e n t t i i a t . d e s p i t e e o n s i i e r a b J e a l t e r a t i o n o l ' the

a a : . l y t i c a l i 'onn o l ' t he s o l u t i o n (corai<L:rt. ( t> .4 ) and

( ! ? . £ ) ano n u t e t h a t ( 5 . C ) does n o t r e cuce t o ( t i . 4 ) on 0

a l l o v / i n - ' • r " * ' 0 ) t he e l l ' e c t o f t h e i n c r t i a l t e rms J o r the:

• j i v e n pt-.ra.'.K-tn*s i s ne j l i . ' s i b l e when the two s o l u t i o n : :

are- comcu ted .

L,

1 Li_ t : " c ; J _ S J - r t - 1 a x a t I o n in_ _Lne_ I . i t h_en;_L he r e_'

IVor tne model d e s c r i b e d i n the p r e v i o u s chai t o r

t he r e i : ; a j r a d u a l i n c r e a s e i n p r e s s u r e a t thr-: end A

u n t i l ! i t r eaches t h e v a l u e o l t h e a p p l i e d p r o c u r e 1 0

( f i g u r e ^.3)- I f t he a p p l i e d , p r e s s u r e i : . : :.;ui "J'i ci.cn ,y

I a r : e t h e n , a f t e r a t i m e T s ay , t he prt . i su rc a t A w i l l

a t t a i n a c r i t i c a l v a i u e l :

c , a t w h i c h j t reL;: : ; r e l a x a t i o n

w i l l t a k e p l a c e . I t i s i n t e r e : . ; t i n * to .nose t h e ipjcr.: on

'How q u i c k l y w i l l t h e screws a t A rebui .1 d a i ' U - r r e l a x a t i o n ' . "

6 . 1 ifiu1 i.._i_on_ of_ atrer.;a_ Re 1 a x a t ion^

I n an a t t e m p t t o s i m u l a t e comple t e r e l a x a t i o n a t A

a f t e r a t i m e T an i n s t a n t a n e o u s d i s p l a c e m e n t ^ has been

imposed on t h e p l a t e . T?iis d i s p l a c e mer.t must s a t i s f y the

f o l l o w i n g c o n d i t i o n s : ( 1 ) i t must have a ce erea: . i r i ' j

e f f e c t a t i n c r e a s i n g d i s t a n c e f r o m A and. (.11) i t mu;;b

g i v e r i s e t o a s t r e s s f i e l d t h a t .- j ives comple t e r e l a x a t i o n

a t A *

l V a t h e m a t i c a l l y these c o n d i t i o n s can be w r i t t e n a s :

( I ) a_ —*>> o anci. ^ * -">- o as — u

( I I ) c U u

The f u n c t i o n "P(?0 ~ - - j r - s a t i s l i t . " thc-st:

c o n i i i t i o n s where a i s a p a r a m e t e r t h a t c o n t r o l s the

r an ' j e o i t h e d i s p l a c e m e n t . To s i m p l i f y t h e bounuary

c o n d i t i o n s i t h e l p s i f we a l s o impost* a r i - j i d . body

d i : : i 1 acemont o f ma- jn i tm-e ' ~ t o t h e p l a t e . Because o f the

E

r i g i d body nature of t h i s extra displacement i t does not change the s t r e s s f i e l d within the plate and consequently does not a l t e r the subsequent b u i l d up of s t r e s s at A. Thus we take

ft s / . — \

<** - ~ ( J - e- ) <*-i>

as our displacement to simulate relaxation. When attempting to solve the equation of motion

of the plate for the given i n i t i a l conditions i t proves advantageous to express (6.1) i n i t s equivalent Fourier s e r i e s form (Appendix 4t, part 3)

u . 2n-\\ an-i) f\-\

where

-2 L P

(6.2)

(1)6.1)

What energy release does t h i s displacement represent per u n i t length (1 cm.) perpendicular to the plane of the model? The potential s t r a i n energy before relaxation (1.22) i s

flu-where u x i s given by (1.40), and afterwards

2 W x d=t /

Thus the energy released i s

- ^ kit cJ&i!«] e ^ d o c

Alt. using (6.1) and (V.40).

2P«,P* ] •O.

n - i

where we have used the facts S<* L and ( t . 3 )

i

/ m

where o * i (_• . Computation l o r the parameter values indicated on figure ( 6 1 ) gives an energy release of

18 0.6 x 10 ergs/cm length perpendicular to the model.

6.2 S p e c i f i c a t i on of_problem We must now solve the equation of motion of tjie

plate (5.2c)

E - V r-r (6.4)

subject to the i n i t i a l and boundary conditions

U = o «vfc 3C- o fc-> o (6.5)

fc > o (6.6)

L L ^ U x + U + o,fc t » o { t 7 )

where from (5.4)

-(»-i)Vei

and u + i s given, by (6.2)

6.3 Solution The solution obtained (appendix 4", part 4) i s

-P. 03t 00

n i l S i .

Kln-iVrxT) J

1 \ e ^ f ^ - ] where

This gives a pressure d i s t r i b u t i o n

( t > o )

(6.9)

4 (6.10)

The pressure build up subsequent to relaxation haw been computed for the end A (x=o) and i s plotted i n figure (6.1) for S=3km. The parameter values used tire shown on the figure. After relaxation the pressure can be seen to rebuild rapidly during the f i r s t year to within 9Cty- of the c r i t i c a l value I which i s l i k e l y to be 3omewhat

c " lowered a f t e r the i n i t i a l f a i l u r e . Afterwards there follows a period of some 10.000 years of steadier increase, during which re-relaxation w i l l occur - probably between one and ten years a f t e r i n i t i a l relaxation.

1

bJ CM bJ 5

CM CM E n

LJ r ~ Iff E £ 9 5 0/ O CM 2 S S 5 ? g f R

* if) o o CM CM

• I ii H to 5 r 1 u ^

in

7> r -

J

H

2

bJ

UJ i3 CM

i

Because of the smallness of t h i s time compared with 1 ( 3 0 0 , 1 X 0 years) the boundary and i n i t i a l conditions a f t e r re-relaxation w i l l be very s i m i l a r to those a f t e r the i n i t i a l f a i l u r e and a cycle of pressure build-up and relaxation with a period of one to ten years w i l l develop. . This i s i l l u s t r a t e d schematically i n fi<jure

IrepsuTe Builjfl up -_ Re 1 axation_Cycle^ The e f f e c t of increasing S i . e . the range of the

instantaneous displacement i s to slow down the i n i t i a l build up of pressure. This w i l l increase the period of the cycle i n Fig. 6.3L. The rebuilding of s t r e s s for S = 10 km. i s plotted on the overlay of fig.6.1. I t should be noted that the time taken for the actual rel a x a t i o n to take place has been t o t a l l y ignored.

(6.2)

nr^^viuura. «Jt A

i

/ l-io > T"-E

Pi_gure__6_. 2

CHAPTER 7. JPJSCUSSI(»r

A new centre for the creation of oceanic lithosphere i s formed by the s p l i t t i n g and forcing apart of a continent by the i n j e c t i o n of b a s a l t i c dykes. As these dykes spread away from the source of basalt they cool to below the Curie point and are permanently magnetised i n the d i r e c t i o n of the earth's magnetic f i e l d . A new ocean basin begins to develop and the two newly formed continental margins, where oceanic and continental crust abut as part of the same l i t h o s p h e r i c plate, begin to separate a t a rate of a few centimeters per year. The continental shelves subside during t h i s period and deep sedimentary basins are formed.

Sleep (1971) suggests that t h i s subsidence i s thermal contraction of the margin as i t moves away from the centre of spreading. Bott (1971) suggests the oceanward creep of lower continental crust as a possible mechanism. The f i n i t e element calculations described i n chapter 4 and i l l u s t r a t e d i n f i g . 4 . 3 show that i n i t i a l l y the v e r t i c a l pressure i n the continental crust increases more rapidly with depth than i n the adjacent oceanic crust. The greatest difference i n pressures occurs a t 10 km and p e r s i s t s to depths of 30 km. This i s the region where ductile behaviour predominates and the layer, within which the hot creep suggested by Bott occurs. The calculations of chapter four lend support to these ideas. The horizontal pressure, although i t increases more rapidly with depth i n the continental crust, does so a t a l e a s e r

rate than the v e r t i c a l pressure. Thus the conditions provided by g r a v i t a t i o n a l forces give preferred normal f a u l t i n g conditions on.the continental side of the margin.

As the new ocean widens with further i n j e c t i o n of material the r e l a t i v e abrupt t r a n s i t i o n from oceanic to continental crust becomes gradational with c r u s t a l thinning beneath the continent and thickening below the ocean. The accompanying subsidence r e s u l t s i n the accumulation of shallow sediments on the continental s h e l f .

The f i n i t e element ca l c u l a t i o n (fig.4.4) shows how a horizontal regional tension i s concentrated on the continental slope. I t has already been argued, that g r a v i t a t i o n a l forces give conditions p r e f e r e n t i a l to normal f a u l t i n g on the continental as opposed to the oceanic side of the margin. Thus any normal faul t i n g produced by regional tensions w i l l be concentrated on the continental slope and i n the adjacent continental crust. This i s exactly i n accordance with observation. Basins already formed as a r e s u l t of subsidence followed by sedimentation w i l l further r e f i n e the s t r e s s f i e l d l o c a l l y . The f i n i t e element c a l c u l a t i o n performed in chapter 3 show*that.this further refinement increases the l i k e l i h o o d of normal f a u l t i n g i n the c r u s t a l rocks immediately below the thickest sediments.

The normal f a u l t s w i l l terminate at depths l e s s than 10 km because rocks below t h i s depth w i l l deform by p l a s t i c flow as opposed to b r i t t l e fracture. The narrowing wedges of crust thus formed can sink into the ductile l a y e r i n i s o s t a t i c response to the sediments

above and deep basins are formed. Normal f a u l t i n g in the oceanic crust adjacent to the margin i s unlikely.

The process of subsidence, normal f a u l t i n g i n response to l o c a l tension and the formation of deep sedimentary basins may go on for 300 - 500 m.y. a f t e r which the ocean can expand no more. Ten thousand kilometers or so of new ocean w i l l have been created i n t h i s time. New oceanic material continues to be added at the spreading centre and pressure begins* to build up throughout the l i t h o s p h e r i c plate r e s u l t i n g i n the ultimate formation of a Beniof zone at the continental margin, where the oceanic plate underthrusts the continental plate. The model considered i n chapter 5 was set up to investigate the c h a r a c t e r i s t i c s of th i s build up i n pressure. A time delay of some hundreds of thousands of years e x i s t s between the ocean ceasing to expand and. the pressure at the continental margin reaching the same order of magnitude of that at which the dykes are being intruded at the spreading centre. The r e d i s t r i b u t i o n of t h i s horizontal pressure.by l a t e r a l changes i n the e l a s t i c parameters across the margin can increase the s t r e s s differences on the slope by as much as 75$ as discussed i n chapter 4. This explains why thrusting i s l i k e l y to be i n i t i a t e d in the region of the continental slope, with the subsequent underthrusting of the continental lithosphere by the oceanic lithosphere. I f the pressure at which the dykes are intruded i s large then thrusting could take place i n the oceanic crust before the pressure has had time to propagate as f a r as

the margin. In t h i s case an i s l a n d arc would develop. Measurements of the energy released and the

p e r i o d i c i t y of earthquakes at regions of underthrusting (Beniof,1955) give s t r e s s releases of around 1 0 ^ ergs a t periods of 1 - 10 years. An explanation of these r e s u l t s can be obtained from the model considered i n chapter 6. Dyke i n j e c t i o n continues a f t e r thrusting and the pressure i n the region of the thrust w i l l s t a r t to rebuild. The discussion i n chapter 6 gives an estimate i n agreement with Beniof's observed time for the pressure to a t t a i n a value s u f f i c i e n t for further underthrusting. The calculations of chapter 6 give

27 energy releases of 0.6 x 10 ergs for each underthrusting for a length of 10,000 km. The earthquakes continue p e r i o d i c a l l y u n t i l such a time as the source of oceanic material i s depleted and i n j e c t i o n ceases.

74-

CHAITEH 8

Energy Absorbti on _by_ _ Sediments.

Free_ vibration takes place when an e l a s t i c system

vibrates under the action of forces inherent i n the system i t s e l f , and i n the absence of external forces. A system under free vibration w i l l vibrate at one or more of i t s natural frequencies. Vibrations that take place under the e x c i t a t i o n of external forces do so at the frequency of the e x c i t i n g force. When t h i s frequency coincides with one of the natural frequencies of the system dangerously high amplitudes may r e s u l t . Thus the c a l c u l a t i o n of natural frequencies i s of great i n t e r e s t .

8.1 General extension of f i n i t e element " theory" tcT dynamic problems"

(Zienkiewicz and Cheung,1967 - note d i f f e r e n t approach)

For s t a t i c e l a s t i c problems the equilibrium equation reduced to

^Bj -i - P = o (1.11) where k

i s the body force/unit mass- For -dynamic problems the i n e r t i a ! terms must be included and the equation of motion

e h f <*fc J J

(8.1)

i s the equivalent equation. The general arguments put. forward i n chapter 2 s t i l l hold good but when replacing the boundary forces acting on each element by s t a t i c a l l y equivalent nodal forces one must be careful to use (8.1) instead of (1.11).

IS

Using the same n o t a t i o n as i n s e c t i o n 2.2, we have

T - e.- civ/ 1 *J

= . J f t>;f*dv Hi* d v * J O T t m t < » w m

V V V

where | l c ] e , are d e f i n t e d i n (2.10) (2.11) and.

(8.2)

C o n s t r u c t i n g ' o v e r a l l ' matrices [kj» £kj and [ f } ^ as i n chapter 2 we have

t* *

For a system under the i n f l u e n c e of no e x t e r n a l forces

(8.4)

When the model v i b r a t e s a t a n a t u r a l frequency a l l p o i n t s v i b r a t e i n phase w i t h a frequency f - Va.-*

l . e. (8.5)

•7U

Equation (8.4) now reduces t o

(8.6)

ax A n».n AO

8.2

(8.6) has non t r i v i a l s o l u t i o n s t f t J o f o r a valuers of w-=u)n . These give the n a t u r a l frequencies of f r e e v i b r a t i o n of the system and the normal modes. \$\ , give the r e l a t i v e nodal amplitudes of v i b r a t i o n a t t h a t frequency.

|j / f J e f o r the 3-noded t r i a n g u l a r element wi th_ Rayl e igh_ type resjj_onse_.

The only non zero displacement components tire i n the plane of the element i . e . u(ac,y) and v ( x , y ) . [ k j e and £n] are the same as i n chapter 2, equations 2.1fc and 2.22.

f

4 J

(8.7)

(see Zienkiewicz #• Cheung, 1967, f> 174) . Thus from the d e f i n i t i o n of | l / f ] e (8.2) we have

I O 'U o '/a. O ! O \/\ O 'A. o i o '/». O '/x O I o 'A o Vx o I O '/a. O / A o

o

c V*. o

(8.8)

<

7*7

8.3 - C M 1 f o r _ ^ o v e type response

The only displacement i of the model W(x,y). Thus v/e define

s perpendicular t o the plane i

CO' J

R J

> =

(D8.1).

(D8.2)

3" (Db.3)

Therefore from 2.6 and 2.4

and

(8.9)

(a.io)

»here N. eto. i s d e f i n e d i n 2.16. Using 8.2. 8.10 and 8. 7 we have

15 ( a . i i )

From Db.l, Db.2 and U2.S

i A C 3^-^) C^i-x^ C*j-x %) (8.31?)

Imposi t i on_ of_ conajLraintG

I f a p a r t i c u l a r noaal p o i n t i s f i x e d then the eauations d e f i n i n g the e q u i l i b r i u m c o n d i t i o n s there are e l i m i n a t e d . The number of normal modeu of v i b r a t i o n i s correspondingly reduced.

Consider ac an example the i node being completely constrained f o r Hayleigh type e x c i t a t i o n . Then both the ( 2 l ) t h and (2V.-I ) t h rows and columns of Qc] and £m] are removed togethe r w i t h the ( l i . ) * n and ) * n elements of { S } 0 . The reduced s e t of equations give (2W-2) n a t u r a l frequencies.

N a t u r a l frequencies of_ scdimentary_ _basin_s Computer programs have been w r i t t e n t o make f i n i t e

element c a l c u l a t i o n s f o r the n a t u r a l frequencies and normal modes of two-dimensional e l a s t i c systems f o r both Rayleigh and love type responses. (Appendix 5 l r o j r a m s NOTJOFV and ICVE r e s p e c t i v e l y ) .

These programs have been used t o c a l c u l a t e the n a t u r a l frequencies of sedimentary basins of various depths- The model used i c shown i n f i g . 8 . 1 . The basin i s considered constrained alon/* i t s surface of contact w i t h the basement rock. Thifs ar;::umpt:ion i t ; vaJid when the contra: J t i n r i g i d i t y bctwi.en tho tsocJ 5 I I K n t:; :uui tin. basemerit i s l a r - j e . I n .general there w i ] I be coupling between the two types of rock and enerjy w i l l be l o s t

CM

CM

10 CM CM

§ 0°

from the bauin. The w i a t h of the bauin was f i x t - 0 ; i t 70 km. and i t s de* t h v a r i e d betv/een 0 and 6 km. Vhe lowest n a t u r a l frequency or the fundamental frequency as i t i t ; o f t e n c a l l e d , i s the most important as t h i s r e q u i r e n leas energy t o e x c i t e i t than the higher n a t u r a l freouencies. The v a r i a t i o n of t h i s frequency w i t h the depth of basin i s p l o t t e d i n lig.fi.? f o r both types of response For the Raylei.gh response trie fundamental frequency increases r a p i d l y w i t h decreasing basin depth t o reach periods of 1 second a t depths l e s s than 1 km. For a deep basin o f some 6 km. the fundamental frequency i s of the order of 0.1 cycles/sec. The shape of the graph s u j ^ e s t s a l i n e a r period-depth r e l a t i o n f o r a f i x e d basin v/idth. This i s v e r i f i e d by the p l o t shown i n f i j j b«3« I n c o n t r a s t the response t o love waves shows only a small increase i n frequency w i t h decreasing basin depth. The fundamental periods ranx'je from C.Hb seconds t o tt.7 seconds.

8.6 Discus j i_ori_. A R a y l e i ^ h wave of p e r i o d T seconds t r a v e l l i n g

a l o n ^ the surface of a Toisson s o l i d (X-/*) w i t h a shear wave v e l o c i t y P Km/sec. s u f f e r s a v e r t i c a l (z d i r e c t i o n ) decrease i n (a ) i t s h o r i z o n t a l amplitude p r o p o r t i o n a l

-2.TC • o wz t o exp (8. .-13")

end (b) i t s SV amplitude p r o p o r t i o n a l t o

2-k . o e x

(Bullen,1947)

X

u E

Id

CM

s CP

tn UJ

CM

3 111 S n o g o o O i n o

•j -cX©- K «2x

lii C n t

For T = 10 and p~3-5 the h o r i z o n t a l amplitude i f j decreajed by a f a c t o r ( '(e. ) by an increase i n depth of 6 km. Thus the v e r t i c a l e x t e n t of Rayleigh waves a t t h i s frequency i s of the same order as the depth of the sedimentary basin t h a t w i l l resonate a t t h i s frequency (see f i g . 8.2). The depth a t which the Rayleigh wave amplitude reaches ('/*•) °f i * 3 surface value decreases l i n e a r l y v/ith p e r i o d (equation 8.13). .Pig. 8. 3 shows a l i n e a r f a l l o f f of the fundamental p e r i o d w i t h decreas­i n g depth of basin. Thus there i s a d i r e c t l i n e a r c o r r e l a t i o n between the c h a r a c t e r i s t i c depth of a Rayleigh wave of giv e n frequency and the depth of the sedimentary basin t h a t resonates a t t h a t frequency.

The fundamental frequencies c a l c u l a t e d are of the same order as those observed f o r surface waves generated by earthquakes. A deep sedimentary baoin l y i n g 1 between the p o i n t of observation and the o r i g i n of surface waves w i l l absorb energy a t i t s fundamental frequencies. Thus 'holes' i n the observed energy spectrum would be expected, a t these frequencies i n the same manner t h a t the PraunhOfer l i n e s i n the sun's spectrum coincide v/ith the resonant frequencies of the s o l a r gases-

Griven the r i g h t c o n d i t i o n s ( i . e . h i g h energy content a t the fundamental frequency) the resonant v i b r a t i o n of a sedimentary b a s i n i n the v i c i n i t y of the epi.ccntre of an earthquake w i l l concentrate damage w i t h i n i t s boundaries.

IJ^T&.Z^lBS- P.f_ _the_ Lithosphere'

During a f i n a l year research seminar i t was sug jested t h a t the problem of a p l a t e being dragged along the surface o f a viscous l a y e r by t e n s i o n a l f o r c e s might be worth consiaering. This chapter presents a s o l u t i o n t o t h i s problem but makes no attempt t o discuss i t s relevance.

The model considered i s shown i n f i g . 9 - 1 . The t e n s i o n , T Q, i s a p p l i e d a t time t = o and maintained t h e r e a f t e r . Using the same n o t a t i o n as developed i n chapter 5 and assuming a l i n e a r drop o f f of v e l o c i t y w i t h depth i n the asthenosphere we wish t o solve (see 1)5.1 and 5.2c)

e dZ- = r a t ( 9 1 )

s u b j e c t t o u = o a t t = o o<*-< I (9-2)

= o t > o , (9.3)

t > o (9.4)

Taking the l a p l a c e t r a n s f o r m of the equation and i t s boundary c o n d i t i o n s we have

3 E 3 5 - = (9.5)

subject, t o

VI

ii

UJ

UJ i / i

III UJ

UJ UJ UJ

(/)

UJ

bl UJ

1

(9.7)

The general s o l u t i o n oi' (9.5) i s

(9.6)

(9.7) b = o

Thus

J i 7 * * S U J . J ^ i - (9.8)

Taking the inverse Laplace t r a n s f o r m r Jh-i.00

(9-9)

U-taO

Considering the i n t e g r a l around the contour i n Vi.-;. A4.1 we have poles a t z = o and where S ^ k J ^ L =• o i . e ,

z = - n V E

Note: 1.

2.

As before z = o i s not a branch p o i n t of the i n t e g r a n d . The pole a t the o r i g i n i s not_ s i m p l e ^

Residue a t z = o

xfc

•2.*" where

i i T z + i± I t ) z +

L +

- xt Thus u.c has a second order pole a t z = o w i t h residue (Dennery and K r z y w i c k i , 1969) "J7 (ff£)

(9.10)

_nv_».= Residue a t z=z„ = — -

Fut

Then

• i

and UW — Svi^U. J*|p U

Therefore pole a t z * z w i s simple f o r n = 1,2 3, and has r«.«i

K' G O

I f the i n t e g r a l around ABC of f i g . A 4 - l goes t o zero as r M —* °0 then we have a s o l u t i o n

-T0 C3^v-L_x) ^ "T.t £»»-E < r L - (9-12)

111

CM

8 CM (V CM l

8 Q O Q ~ 2

CM • I •I I I I I I I I I

l£ -J O A F* U Ck^ (T

§ 8

8 in

III u (M

g i v i n g a v e l o c i t y d i s t r i b u t i o n

(9.13)

These s o l u t i o n s have been v e r i f i e d by s u b s t i t u t i o n i n t o ( 9 - 1 ) , ( 9 - 2 ) , ( 9 . 3 ) and (9-4) t h i s being somewhat simple r than p r o v i n g the necessary c o n d i t i o n on the i n t e g r a l around ABC. The v e l o c i t y of the end A (x=o) i s p l o t t e d i n f i g . 9 - 2 f o r the parameter values shown. Some 10.OUO years are seen t o elapse before t h i s end s t a r t s t o move appr e c i a b l y . This i s f o l l o w e d by a pe r i o d of some 50,000 years d u r i n g which the v e l o c i t y a c c e l e r a t e s u n i f o r m l y t o over 2 cm/year. The a c c e l e r a t i o n then slov/s and the v e l o c i t y tends a s y m p t o t i c a l l y t o a value of 3.15 cm/year. The p l a t e now moves w i t h t h i s t e r m i n a l v e l o c i t y w i t h o u t f u r t h e r deformation and the ten s i o n drops o f f u n i f o r m l y along the p l a t e i . e . T ( p t ^ —* " I * f ! l

A l t e r i n g the parameters changes the axes and not the shape of the graph i n f i g . 9 . 2 . By examining the form of equation (9.13) i t i s seen t h a t a r e t u r n f l o w i n the; asthenosphere e q u i v a l e n t t o an increment i n v i s c o s i t y by a f a c t o r 7 would increase the time axes by a f a c t o r of seven and d i v i d e the v e l o c i t y a x i s by the same f a c t o r .

L .

APPENDIX 1.

Cartesian Tensors.

1. Reason When the equations of mathematical physics are w r i t t e n

out i n f u l l c a r t e s i a n form the s t r u c t u r a l s i m p l i c i t y i s o f t e n hidden by the mechanical labour of w r i t i n g every term e x p l i c i t l y . The tensor n o t a t i o n , v/ith the i m p l i e d summation convention, i s a form of shorthand l e a d i n g t o a gr e a t s i m p l i f i c a t i o n of the w r i t i n g . We use c a r t e s i a n co-ordinates so t h a t the d i s t i n c t i o n between c o v a r i a n t and c o n t r a v a r i a n t tensors disappears.

2. Carte s i a n tens_ors__of_ the_ f ^ _(.y?_cj;_9rs_)

I f we have two sets of r e c t a n g u l a r axes (Ox^, 0 x ? > Ox^), (Ox-,, Oxr-. 0 x 7 ) a t the same o r i g i n .

F i g . A l . l

l e t the co-ordinates of a p o i n t P be ( x ^ , x., x ^ ) . (x-^, x2'» x^) i n each system r e s p e c t i v e l y . I f we denote the cosine of the angle between 0x.k and Ox[ by then we can w r i t e

Si

( A l . l a ) IX.

»4 or

L - I

( A l . l b ) J

Let the eo-ordinates of a point F be ( x ^ j ^ , * ^ ) . (x-j.x^.x^) i n each system respectively. I f we denote the cosine of the angle between Ox and Ox by then we can w r i t e

( A l . l a )

o r ^ / U l . l b )

§}^P^^°I}--9PI^y§J9-^2rPI^l. * e raa'ce i t a regular convention that whenever a s u f f i x i n a single term i3 repeated then the term i s to be summed over that s u f f i x from 1 to 3-Using t h i s convention we can now w r i t e the l a s t equation as

~ Lij ^ i . ( A l . l c )

The matrix L = ^ k j ^ i s i n f a c t u n i t a r y i.e. L T= L ' as can be seen by considering the length Of which equals Xj = x- (note implied sum).

J

* L i j x i L,y- ac^ using A l . l c . This i s true f o r a l l points F

and comparing c o e f f i c i e n t s

/ - I • - £ (A1.2a)

or L L.T •= X (Al.2b)

i n matrix notation where I i s the u n i t matrix.

• • f t .

Prom A l . l c

from Al.2a

Therefore the inverse of A l . l c i s i . ' (A1.3)

j J •

In mathematical-physics one often has to deal with sets of three quantities (u^UgjU^) such that i n r e l a t i o n to a d i f f e r e n t set of axes the corresponding quantities are (u-^iUjjfU^) and are rel a t e d "by

i

and = ^ i j w.j i» Such a set of three quantities i s called a tensor of the f i r s t order (or commonly a vector). The i n d i v i d u a l u^,u 2 >u^ are called the components of the f i r s t order tensor.

3. Cartesian tensors of the second order. Suppose we have two f i r s t order tensors u.t and H'K .

'Ve can construct nine quantities 6^- ^i^Vc • l o t us investigate how these quan t i t i e s transform when expressed i n our now axes e.' ( J - - '

A set of nine quan t i t i e s that transform i n t h i s way on r o t a t i o n of the co-ordinate axes i s called a tensor of the second, order (or j u s t tensor). The i n d i v i d u a l are called

components. To obtain the inverse transformation we use Al.2a and Al.4 g i v i n g

IK (A1.5)

1. S t r a i n Tensor? To prove that the components of s t r a i n do i n f a c t

constitute a tensor we must prove that they transform according to (Al.4) when we rotate our frame of reference, I n the unprimed co-ordinate system the s t r a i n components are defined by D l . l

In the primed system we have from Al.3

IK and u.' - < f u i

— rt ^ _

i.e. / . j (A2.1)

Comparing w i t h (Al.4) we see that f ^ i j ) are i n f a c t the components of a second order tensor.

2. Compatibility of Strains

and

Adding

Interchanging j and k we have

Thus

(A2.2a) are the c o m p a t i b i l i t y equations and number 81. Some are i d e n t i c a l l y s a t i s f i e d and others repeated. Only s i x are independent these being:

22 " *1 (

-4- ^T3- _ 62. 6 3 >

c) / ^ 1 - . <^e« <k>. \ (A2.2b) <L 1 6-t. 1

-4-« ^

-f-

-4- a c

These r e s t r i c t i o n s on the s t r a i n tensor ensure that i t gives single-valued continuous displacements.

3 • Ri£id_ Body Displacements. The change i n length $A , of a vector A (length A)

under the deformation (1.1) i s given by

o r ASA = AiSA: ( A 2' 3 )

ignoring small terms of second order. For rii?id_boj displacements SA=o f o r a l l vectors A. Therefore, from ( 1 1 )

MA, - u L jj Afc &• = o Taking A i n the xy-plane i.e. C^»-° ( M ° , ^ we have

S i m i l a r l y =• -U v and u. x < % =• - u i x . Thus a necessary and s u f f i c i e n t condition f o r the transformation (1.1,1.2) to represent a r i g i d body movement i s

" s j - - U j . i ^ j = - ) .

Thus represents the r i g i d body part of the trans­formation. Thus f o r pure deformation (1.1) becomes

(A2.4)

4. Physical Significance of the Strain Components

(a) Diagonal Terms: Prom (A2.4) and (1.2) we have

For pure deformation ^ i j * 0 a n<* therefore ASA - A;Aj Now consider a vector A i n i t i a l l y along the *-,axi3. V/e have

(Not summed)

or TClC

That i s 6.KK represents the change i n length per u n i t length of a vector o r i g i n a l l y p a r a l l e l to the xM axis.

(b) Off-diagonal Terms: Consider 6__ . Take two vectors A = A_.__ and

B = Bjg^ i n i t i a l l y directed along the x x and x,, axes respectively. Upon deformation these become A and B* . (Fig.A2.1),

Fig_.A2_.l_

/ /o B

/

B i s given by

A*. W = A^Sl^ * ^ g f t j

( t o 1st order).

Prom (A2.4)

Therefore

Cos b = s ^ ( I - ^ = -Thus i s equal to the change i n angle between two vectors o r i g i n a l l y p a r a l l e l to the axes, (x^.x,) 5. To show that t T l j l i , j = 1,2.3 f u l l y defines the

stress at a point. To do t h i s we must show how the stress vector H y

(or i t s components) a t a point P can be expressed i n terms of the stress components at F and the d i r e c t i o n cosines U of the surface V . We consider the equilibrium of a small tetrahedron under the action of surface t r a c t i o n s and body forces PL per u n i t mass. (Pig.A2.2)

A' « SA, e,

The angle AOB 1 =

a'W G » S . ^ -

i . e .

91

Force on face ABC 1/tX -+t» Force on face BDC +» Force on face BDA // eJL h»

Force on face ADC //*L Body force //el -K. L Therefore f o r equilib r i u m

L e t t i n g h-»o v/e have

" " ^ = ""^ V j . (A2.5) Thus l" ^ L j | f u l l y defines the stress at P.

6. Equilibrium of Stresses. Consider an a r b i t r a r y volume V f i x e d i n space

(note: not a f i x e d volume of ma t e r i a l ) . (Fig.A2.3) l e t ^C*} be i t s density and F the body force per u n i t mass

Fig.A2.3

The t o t a l force i n i - d i r e c t i o n i s

v j (from A2-5)

loo

(Divergence Theorem)

Therefore f o r equilibrium

v J The volume V was chosen a r b i t r a r i l y , thus

— o (A2.6)

The moment of these forces the x L axis must also vanish

i . e .

J 3* J

or from (A2.5)

J r J K \ J T U L J

ds = o or using Divergence theorem

e. x ; F p dv ••+•

or

d v + J e r h St * V c " ] * v

V v or using A2. 6

o

I CM

The volume V was chosen a r b i t r a r i l y therefore,

or

Hence the stress components are symmetric.

7. 1" } a tensor? Consider two planes with respective normals

both containing the point F (Pig.A2.4)

(A2.7)

Pig.A2.4

R y and Ryare the respective stress vectors at P. The component of R,, i n the d i r e c t i o n V i s

•5«. »' = (?»)• v!

1 0 1 .

i . e .

1

" ^ j ^ from (A2.5)

— l>j iJ- from (A2.7)

Now consider e f f e c t of r o t a t i o n of axes on In primed axes

L ^ j = comp. on surface J l ac. along =c.j

" (3-1). ( ) I • from ( A l . l c )

= ^ Lj. Lj. f r o m ( A 2- 5 )

Thus l Lj T** ( A 2 - 9 )

and constitute the components of a second order tensor. *

I O ? 3

8. P r i n c i p a l Axes.

Consider the case where T ( i- T„ - o axes through an angle <. (Fig.A2.5)

Rotate the

Pig. A2.5. \

\ ->.

We have = Cos. (xjOacj,) therefore i - . s = l B S i = l S | Ljs> = o (

-r ' L,i = O

XI. /

Thus i f the axes are rotated through an angle * given by

have the p r i n c i p a l axes since ""C^j = o -

C< ^ 2 +Bu* ( - T — ) (A2.10)

we

i.e.

f ou>

9- Greatest Shearing Stress. i Suppose we have found the p r i n c i p a l axes and that

the stress tensor referred to these axes are T„ f Tlfc and T3, . We wish to determine the plane V which has the greatest shear stress across i t (Fig.A2.6). The stress vector Rw

Fig.A2.6 ,. , ^ I I

can be s p l i t i n t o a normal component (N) and tagential component (T.). (A2.5) gives (Ry)( = T„ y, f

and («„), = T „ m

Hence

and N ~ R„. p = T„ + T « y* + T 3 L l£

Therefore

We wish to maximise T subject to D,1"-*- L>* -»• V± - 1

i-e.

T 1 = - v: - T , ; o - * )

Putting ^"j} — — ° we have •

V, (T11-T3j[(Tll-T„)-i\j;J"C-C.-T,_) - 2W_-(T„-T 3 l)j = O

and

Solutions

give planes of minimum (zero) shear. Solutions

_ -+ — Hi =

-*- —

_ + 1 - ~ - J Z , Mi :

i -V-

- _L -+ -L

give planes w i t h respective shears

Thus the greatest shearing stress occurs across a plane that bisects the angle between the greatest and least p r i n c i p a l stresses and i n which the intermediate stress l i e s . The magnitude of the shear i s equal to one h a l f the difference between the maximum and minimum p r i n c i p a l stresses.

I

10. The_ J ?J?e_C_ons tanta

The most general l i n e a r r e l a t i o n s h i p between stress and s t r a i n involves 21 constants. For an i s o t r o p i c material i.e. one which has the same e l a s t i c properties i n a l l directions these constants are inter-dependent and can be reduced to 2 independent constants X andy*. known as the lame constants.

For an i s o t r o p i c body we can write the l i n e a r r e l a t i o n as

T T T - Ae^ + Pi (e.„ i- e^) f c«.ll( * c' Ce. e„

I f we reverse the x, axis then ( e 1 % ( "T,„ ( ~T 1 S

change t h e i r sign, while C„ and "T„ stay the same. Therefore C = C = o and B = D = D = o . Thus we have

T„ « Ae„ + A'(eax* e 3 t) etc.

and "T,x - "E>e,11L etc.

I f we now rotate our axes through an angle B about

/o7

the a x i s (i'ig.A2.5) then from section 8 we have

Using (A2.1)

Therefore

T

True f o r a l l values of ©;therefore, comparing c o e f f i c i e n t s i n A and B

A - A 1 - B

I f we write and A = "X we have A- 2p.+\ and the s t r e s s - s t r a i n r e l a t i o n "becomes

T ,

where

A =• e l ( -+- e

I Of.

APPENDIX 3•

1. 6-Noded Triangular Elements •

Pig. A3.1

GQ Matrix.

There are two constraints at each of the s i x nodes. Therefore we can use twelve constants to define the displacement throughout e.

or i n matrix form

ft- o o o c» © o o o c o o o

-3 3 2 x 1 2 x 12 12 x 1

Define

I X. LJ O O O O O

o o o o o o 3 * x1 3 (A3.1)

We also have

or

{«r-

1, o o ^ X.«J, 1.

I 4,. =c» «ja o o o o o o o o o o o I 3L.

o o O O o 3* 3CT X, 1. 1 J

Define

Thus

o I o

o o o O O |

O O o o o i

o o o o o

o O O O o .1

(A3.2)

(A3.3)

or

|VI = [VJ H -I (A3-4)

B Matrix.

Ho

Prom T)2.3 and Dl. 1 and using A3 • 3 we have O 1 c> Ox. y o o o o o o e s

O O I © 3C

or — I

(A3.5)

O I O 2ac >-j C O O d ca o O O O O O o ° C. d I ° -ac 2»j

from 2.10

where

| ¥ ] e Matrix

e. or

DCT =. (ET) J I & ^ ^ & ' i d ^ i ^ r ' (Note: In program (Appendix 5) [jsT] =• [ c ] 1 )

(A3-6)

II

to

0

0

0

A o

V

4

0

A

7"

o A o

fj n

rl or

o o

•A

4* ^ M

51 *

8

0

0 4 o

0 4 c

% If 0

0 0 0 I*

i /4 rP

"efl Li

0 0 0 0 0 0 0 0 0 G

0 0

0

0 ct

{ 0 P4

0 ^ 0 if

0 #0

0

rs TI + * 0 / Q 0 c A

A

o A H HP

0 0

f ^ 0 0 0

f

'J

0

rfl f n 0

•3

(0

0

f

0

a:

IT u

0 0 O O O O O O O O 0

M2-

Thus to evaluate |KJ we must evaluate

_ 1^1 1 - 1 ^ 1

( y doc cUj

X 3 = I f where A i s given by D2.10. Using these definitions [ k ] e becomes

• -»

(A3.8)

where [KlTJ e i s given by Fig.A3.3- (see over)

Calculation of I ^ . I g j I ^

• 3

(A3.9)

Fig.A3-4

The equation of the l i n e joining nodes i and m i : 3= Ai*,^ + Bi« provided zx-^^ -3cm

where

A, i.rvi and

l. (VI

(A3-10a)

M l

0

0

0

A i *

0 x f

0 s o

m

IY1

H

IX)

0

0 f

0 4

( 3 - 3 - 0

^ 4 0 <[M 5H

0 0 o ^

A 0

41

M H

/3 5

ft

n

A 3 0 -3" ^

0

0 oM <» IT?

0

0 0 : 0 0 0 0 0 0 . 0 0 0

0 7f |3D

(9

IT?

0 -7T J L 0

0 S

o z

o

Si

0

0

,5 ±

. 2!

0 5

if

0 «, ^ 4 IT?

5 Of

0 0

0 0

IT-) 0 T 0

)H IT")

0 o o -s- • o

5- o '

0 / f

0 0

if H

4

H

0

0

0

/3 > r t IT?

0 0 0

ir , H

Therefore At A i l M JJ

ooi cased

A )

•.ft* in*

ana

a. When x i = xffi these i n t e g r a l s are zero and we can therefore define

^ i i * = = ° ^ ^ ^ i . ( A3.10b)

Therefore, provided i j — * - m goes anticlockwise around the element e i n Fig.A3.4 ( i . e . A> o ) we have

I f i — > j - * m clockwise ( i . e . < o ) then I , i s the negative of thi s answer. Thus i n a l l cases we have

i , = ( ft\) * [i{«u.«-«f h ftjiC»r-«f) - A«j c-f-< )J

' ( A 3 . l l )

ns

S i m i l a r l y ,

m i

(A3.I2) 3 "J

J J

AS 13)

due_ to g r a v i t y Prom 2.11 and A3-4 we have

which reduces to using A j . l r

n ( C c r ' ) « 4

o o o o o o MA

(A3.14)

Axi-symme t r i_c_ Analys i

Pig. A3. 5 t 2

We take the Z-axis as the axis of symmetry i . e . the problem i s independent of .

to t r ix_ j H ] _ e _ Using the 3-noded triangular element, thc- a n a l y s i s

i s the same as i n Chapter 2 with Z replacing ^ and r replacing y.

where

0*1

A

M. J ' J

a- + b:T

T = ( : •)

<Xt =

- r J

(A3.150

J

M a t r i x [ b ] 6

The expressions l o r the components of s t r a i n i n c y l i n d r i c a l co-ordinates are given by 1.8. For axi-symmetrical problems ^ - 4.= 0 and i t follows that the

<3ar non ^.ero components are c r r - gj.

and 2^ - * S?

e A. - - P p - ^

We thus define

(A3.16)

where we have written u_ - u and u = v r z

A3-16, and 2.4 we have

Using

(A3-17)

where

a =

o

r *• r

Comparing with 2.5 we have

o

o

J e t c

(A3.18)

Matrix [ D j e

Prom Hooke's Law 1.13 we have

4>4> •= \ (e r r e,x + ) •-»- 2^

or i n matrix form

(A3.19)

H * -

o

— \ o

X \ o

o o o

where

(erf 5

Z.Z. r r

1

r z (A3 - r>o)

Comparing with 2.G wc have

i>r -x+^ X X o

—. X x+y o X X ^ o o o o

or using 1.18 and 1.19 I

Matrix [KT Prom 2.10

o o

(A3.21b)

fe­

ll we write [B] = [ i j + [B'J where

H&l = • V M (A3.22)

CB-'I = z r>: (A3-?3)

\ O C

o (A3-:-'4)

r ' o r b l r ' o

etc.

.'lO

A, =

o o

a- z + — , r r fc

a o o o

(A3.25)

z -then we have

= i ( r w + • ] * rm )

Using the f a c t that f rdrdz = and fzdrdz = we have

«- i

Therefore |Y|e reduces to

DO* = TCF|^|[B] T1>1|:B] +2 |&'3T[>lEB'J'-draz ( A 3 - 2 6 ) a I f we l e t

r

then XIHBH

d r d Z

I l l

Putting

— 1 = w J 7 d r d z

= ~ d r d-z.

T" -— 3 — d r d-z. r

(A3.27)

and noting that J drt*z =. we have

+ (o.Cr*a r(;)(av f ) + C c ^ v f ) (A3-28)

Calculation of I ^ j l g i l ^

Referring to Pig.A3-6 the equation of the l i n e joining node i to node m i s

MM

where

A: (A3.29a) and

provided r i r-OA w

dlrd-z i *««»

dr

0~~ * L ) (A3-30)

Therefore, provided i-*-m-*j i s clockwise around the element e ( i . e . A > © )

(and the negative of t h i s r e s u l t i f i -* m —*• j anticlock­wise i . e . A < o ) m

Therefore i n general ( £ ° )

(A3.3D

Note 1 I f r. = r m then the i n t e g r a l i n A3.30 i s zero, the equivalent of which i s obtained by putting

Au. - B i M = o r„= r, ^ ^

Note 2 When = o I i s not defined by A3-3-Zero X i n f i n i t y term. But

by

L 1Hopi t a l 1 s rule.

1 M

1 1 ? p. Z«o

and therefore generally

+ ^ ( v O - H ^ C r j - r . ) * ^ ^ ^ ) ( a 3 3 2 )

Note: By L'Hopital's rule the term containing log r. tends to zero as r^ goes to zero.

I f r*r. l«o

(m = 3 d r

r

and 1^ becomes

(A3.33)

Prom 2.11 and A3-15

H • £3

J

o M o M-

o

o

-+• C- T

(A3.34)

where

— s

r

(A3.35)

2-|A| J

Z.r cLr (A3.36)

the rest of the quantities being defined by A3-15-Prom A3 - 8, A3.H, A3-12

(A3-37) ana

* i fBi.flto.0i-r*>+ B ^ t f - r f ) * B.jA-j^-r;)}] _ (A3-38)

FART_1

We wish to solve

^"u- . Sol S S r " ^ i f (A4 • i )

subject to

LA-- o cO: b« o o « x « i — (A4-2)

ext ac=o fc^o (A4.3) (A4-4)

Applying a Laplace transformation to the equation and i t s boundary conditions and denoting the laplace transform of the function u(x, t ) by u(x) =. J"u.C*,t)e_ d-t we have

and .

1 0 f (by parts)

= z u. from A4.2. Therefore the Laplace-transform of A4.1 i s

£ 5"^> ~ 0 - 3 1 • (A4.5)

Boundary condition A4.3 becomes

u = o at x = o (A4.6)

and A4.4 becomes

/ <^L \ =

V^ac-J E x oJb

noting that

' it) - (S) and

J — e. cLt

The general solution of A4-5 i s

where a and b are arbitrary functions of z.

(A4.6) —^> a = o

and (A4.7) gives -ft.

Substituting into A4.8.

-P S^k |0~z _ E LL =

(A4.7)

^ (A4.8)

^ x 3 / l C o s k j S u (A4.9)

Our required solution i s given by taking the inverse Laplace transform of A4-9 (had to be done this way because the inverse of A4-9 could not be found i n standard tables)

t+ ivO r

U.(?C,t) = - I

15-i.co

dirz. C 5 k U (A4.10)

( t > o )

(Spiegel, 196 5.) where 16 i s real and greater than the real parts of a l l the singularities of the integrand in the complex z-plane. At f i r s t sight i t seems that the integranu has a branch point at the origin due to the Jx i n the functions. However, on expansion

CoskJS L. 15 i. cmJ )

( j t 4 * »'C«vf

(\ L

(T2 L-E l i

c Z I— )

which obviously has no branch point i n the complex z-plane. However, there exists a simple pole at the origin with residue

f (AAA O t 1 • J F (A4.:i:i)

12*

Poles also exist where Cosh 1 = 0 i.e. at z = z n = - C -OVe ( A 4 . 1 2 )

What i s the residue of fl§t) - e3*-5***^ at z = z_ ?

l e t

g(z) e. 2. Sw^.k->| e csc-

and b(z) = CoshJ^L-

Both g(z) and fe(z) are regular at z = z R and k ( z n ) = o.

- i u j s aujf

Etc (an-1} Therefore, there i s a simple pole at z = z f c with residue

^ " ^ ; " T I F - ' (A-.)VE^ ^ [ - T l — ^ J — (an-1 E t

(A4.13)

To perform the inverse Laplace transform A4.10 we consider the integral of

-P. *S^LW

around the contour Q-in Pig.A4.1.

Fig.A4.1

z-plane

> Real

<f>(z) i s regular on and within the contour except at the simple poles at the origin and at z = z„ n = 1,2,3,

m. provided we take the radius of the semi-circle ABC to be r , where r m i s chosen so that the contour contains only a f i n i t e number (Mtl) of poles and rM-jz 2^ f o r any n i.e. take r M = . Note that inkeeping with the earlier condition U can be an a r b i t r a r i l y small real positive number. Now, J * i r m *

= J c|>(^chL +- J <K^dx

^ . ^ ftcs'iduAJk o f <f(x)

(Cailchy's Theorem - Dennery & Krzywicki. 1967)

Therefore,

Thus provided we can show ( l a t e r ) that

we have from A4.10 and A4.13..

(A4.14) V/e must now show that i n fact

as ("m-y ao

d-z. 0 i.e. J z 3 , x CoSU Kxjz as

fVi"* 0 0 ( i . e . as m ) whe re

m = 1,2,3,

and

r i s the contour shown i n Fig.A4.2 z-plane

Consider the value of SUA.I\ K,Jx

.Fig.A4.2 —tta-L-m-m n on the

contour T . Apply the transformation u)*"= z Cto» f" f l-,*l) to the w-plane . r"-*- f~ where l ~ i s part of a circle radius r m = - j ^ centre 0 (fig.A4.3)« Now consider the value of | SUjjkjCjUi | on A'B' . (Note: AV= C'B' = o'C =

C O - |=>laow\A.

Fig.A4-3

On A'B'

| S^Ox K,U) | = e. — e_ K,WI I -K,i |e""| - |

132.

and

| Cask Ktu> , -in**

= I Cosk ^ I .

Thus on A'B'

(S^k VC,u)| I C«k )

Since K, i K k and both K,% and are real . In the z plane the li n e A'B'is the parabola (fig.A4.4)

r -Ay .

z-plane

Fig.A4.4

Revising our original contour to f leaves the previous analysis unaltered and on \

S^k

since Rez & o and t > o and

1**1 *

133,

The length of % •<. L»r^ —

Therefore

dx ext Su>.k tc,Jz Z3"- Co&k Ka Jx

d-x

= 1

Thus 1

« xb

J 3 «

1

Cosk K x 4 x dm.

and A4.14 i s i n fact the solution of A4.1 subject to A4.2,. A4.3 and A4.4.

PART 2. We wish to solve

\ *t ^ (14.15)

with boundary and i n i t i a l conditions

u = o at t = o o-ix-sl (A4.16) u = o at x = o t > o (A4.17)

( & \ - - a . &*->Li. = E~ at x = 1 (A4.18)

and ^ = o at t = o o i X ^ L # (A4.19)

Note that the inclusion of the i n e r t i a l term necessitates the inclusion of an extra i n i t i a l condition (A4.19) Applying a Laplace transform to the equation and i t s boundary conditions.

80 - 06 to

" — •

(from previous)

(35

Thus (A4..15) transforms to

i.e.

_ (JL ^ac?- \ H / (A4.20)

with boundary conditions

u = o at x = o t > o (A4.21) and

7~ — ~ a t =c=U t- f o . c^ac J E z ' (A4.22)

The general solution of (D.20) i s

(A4.23) a - a G . s k * • ) + (FT5*) where a and b are arbitrary functions of z.

(A4.21) = > a. = o and (A4.22) gives

Substituting back into (A4.23) gives

Performing the inverse laplace transform

r

- 1

(A4.25)

with the same conditions on t as previous. As before the integrand does not have a branch point at the origin. There i s a pole (simple) at the origin with residue

-_L . ^

Poles also occur where Cosl^^^gM-* L.) = o i.e. where

-+• cr-z. =

Put

- (an-i)VE (A4.26a)

(A4.26b)

and we have poles at

Residue at w' Put g(z) =

~JL t :

and h(z) = Cosh Q L ^ g(z) and fc(z) are both regular at z„ and h(z n) =

(A4.27)

o.

137

II1A

Thus we have a simple pole at z = z A with residue 2KL K'O**)

Hence the residue at z n i s

_ - ' ) Q

As before provided

we have Aftc.

'— n»i *— J

(fc>o) (A4.28)

To prove that this i s a solution rather than attempt to prove I j ^ j < >£t}d"z_ w e shall v e r i f y i t by

substitution back into A4.15 and i t s boundary conditions. A4.28 obviously satisfies A4.17 and A4.18. Prom A4.28

Adding we haven

= o from A4-26a.

Thus A4.28 i s the solution and i t follows immediately

133

that the pressure v a r i a t i o n a t the origin i s

(A4.29)

Computation of (A4.2-9) Suppose that f or n>N P " * - ^ . Consider the

terms i n A4.29 with n>N . Put "p » 4 ^ * * ~ p*-

Then we have from A4.27 terms of the type

Since t > o and r f c s ^ ^ f o r times i n which we are interested these terms are ne g l i g i b l y small.

Consider now the s e r i e s

3 > (-lY* fan- ) \ _

using A4.27- These terms are a l s o negligible. This j u s t leaves

Computation of t h i s s e r i e s as i t stands proves d i f f i c u l t because i t consists of very large numbers (too large i n fa c t to be stored by the computer) of alt e r n a t i n g sign. Thus we must consider the sum of pairs of terms. Let n be odd, then combining the n and (n+1) terms we have

L : ? — -J I? J

For -*1 » ££ttv(?n--»)1' we have

and the combined term reduces to

( a n - O

tn-n) C -)

F J (A4.30)

If V ^P1 — ^ then the exponential terms make the term negligible.

FART 3

Fourier Series f or f ( x ) = l - e " X / ^ a o * x < L

Extend the range by defining

= , ^ v (A4.3D

Let y = £ and put g(y)=f(x).

I f lg> g(y) and le„> ^ ~ then BB 6 I i s a complete orthonormal s e t i n L ^ ( j (Denhery & Krzywicki,1967) Therefore

and T -uMij

a, i . e .

Q M = - \ E ( J " ^

'3 — n 2 1

I o

J

o

— i

Since L » s we can ignore the terms with eT^* giving

Noting o a ~ -a-m a n * a Q = a we have

(A4.32)

l3> =

or

l e t t i n g m = 2n-l

U+ m u s t have gradient 1 a t v

(A4.33)

so IM

where

n -1 (A4-35)

FART 4

Solution of

<*x^ ~ ^ 5 t (A4• 36)

subject to

u = o at x = o t > o (A4.37)

(A4.38)

u = u T + u + at t = o (A4.39)

where

(A4.40) and u + i s given by A4-34. Taking the. Laplace transformation we obtain

(A4.41)

with boundary conditions u = o a t x = o (A4.42)

and

(^B- \ = - j ^ (A4.43)

I f we look for a p a r t i c u l a r solution of A4-41 of the form

(A4.44)

we have on substitution into the equation

1 ?

Comparing c o e f f i c i e n t s we have

e . = (A4.45)

4 - l V ^ > n-1 (A4-46)

x (A4.47)

and •(V>-')VEt

T i l e r o f o re t n o -;cnc-.ra3 s o l u t i o n o f i s

iV»Vv.

M •n O HA. U 4 . ' ! ' j )

where d- and d,, a r e a r b i t r a r y 1 'unc t i oris: o f z .

A4-42 - j i v e s d-j = o

A4 .43 .:;.ives d,> -- o

Thus o u r s o l u t i o n . f o r buO l a i d ;JGO t r a r i : ; f O I T I o f \i{y..l) \:\

s, V Z i--2 L_

( A 4 . !?U)

A l l the terms i n t h i s expression have standard inverse Laplace transforms (see tables - Spiegel 1965). Thus

<=" P V e. ^ n = 1 (A4.5D

APPENDIX 5-

SPECIFICATION OF PROGRAMS.

FINE!

Purpose. FINEI i s a fortran program written to perform two-dimensional f i n i t e element calculations for the cases of plane s t r a i n and plane s t r e s s , using a 3-noded triangular element.

DataInput:

Card 1:

i s by punched card 10

NJOB

NJOB i s an integer contained i n columns 1 - 1 0 (right hand j u s t i f i e d ) specifying the number of f i n i t e element calculations to be done during t h i s run.

Card 2.

Card 3-where

T i t l e card for f i r s t set of data I I f II l o l o

NPOIN NEIEM lo » • So &o

NPOIN i s number of nodes. NEIEM i s number of elements.

A =

B =

C =

D =

Card 4:

0 plane s t r a i n 1 plane s t r e s s

0 is o s t a c y forces not to be included. 1 i s o s t a c y forces to be included.

0 No v i s u a l display of s t r e s s differences. 1 Visual display of s t r e s s differences required.

0 1

10 l l

NR.EF

No punched output punched output for graphplot.

(NPOIN of them) Nodal point data * • 3» 3* >*> •»>. lo • *** *\ IB 60 To B™01 *4

F«o-o fci&o Eio-o

i 5 o

NREF - reference number of nodal', point (range 1 to NP01N) •

X - X co-ordinate of nodal point i n Km. Y - Y co-ordinate of npdal point i n Km.

M _ 0 - displacement i n x di r e c t i o n not specified. 1 - displacement i n x dir e c t i o n i s specified.

DX - ( i f N = 1) s p e c i f i e d displacement i n x direction i n metres.

I. 0 - displacement i n y dir e c t i o n not specified. * ~ 1 - displacement i n y dir e c t i o n i s specified.

DY - ( i f M = 1) s p e c i f i e d displacement i n direction i n metres.

R (1) - x component of exter n a l l y applied force acting at node ( i n dynes)

R (2) - y-component of ext e r n a l l y applied force acting a t node ( i n dynes;

Card_5^ (Only included i f B = 1 on card 3)

YSUB] ROSUBJ" I

where < YSUB - Y co-ordinate of base of model ( i n Km.)

ROSUB - density of substratum (gm/cc. )

Card 6: Element data (NE1EM cards) i io I n 3 » »•> So bo la

NA[ " ~"NBT Ncf i f " v \ R ' 6 | IEREF X i o 1 i O 310 I i o E '0 -0 F i o . o fiO'O

IEREP - Reference number of element.

NA,NB,NC - Reference numbers of the three, nodes

comprising the element (order i r r e l e v a n t )

E - Young's modulus (dynes/sq. cm. )

V - Poisson's r a t i o RO - density (gm/cc.)

15/

Card 7: (only included i f c = 1 on card 3)

' ? 2S a* ac aim a i a& a^ [bABC 7.T .". .™~UV ""["" NR [ NC ""{

i l a i a b ABC NR NC

Storage:

blank > UV - l i t e r a l l y Number of rows i n model Number of columns i n model.

NPOIN - No. of nodal points. NE1EM - No. of elements. RNODAR (NF0IN.4) - Nodal point data

1 2 3 4

x co-ord y co-ord x displacement i f spec i f i e d , y displacement i f sp e c i f i e d .

ISDISF (NPOIN,2) - displacement markers

ISDISF (NFOIN.l) =

ISDISP (NPOIN,2) =

IREPEL (NELEM,3) -

ELACON ( NELEM,2) -

1 2

RO (NELEM) RPORCE (2NPOIN)

0 - No displacement sp e c i f i e d i n x dir e c t i o n

1 - displacement s p e c i f i e d i n x direction.

0 - No displacement sp e c i f i e d i n a direction

1 - displacement s p e c i f i e d i n JJ direction.

Nodal points comprising the elements.

E l a s t i c constants

Young's Modulus. Poiason's r a t i o .

density of elements applied nod:il forces.

After CHOLS i t contains the dio placemen to of l-hi.- rioiiaJ. points. MARK (1) - Isostacy marker (1 'r yes; 0 - No)

IS5L

MARK (2) - Visual display marker (1 - yes; 0 - no) MARK (3) - lunched output marker (1 - yes; 0 - no) IMK - plane s t r a i n or plane s t r e s s marker. NBW - computed semi-band width of s t i f f n e s s matrix B (3,6) - B - matrix c.f. 2.5-D (3,3) - D - matrix c.f. 2.6. ST( ) - S t i f f n e s s matrix (semi-band width only stored) YSUB - co-ordinate of base of model. ROSUB - density of underlying material. NBASEF - number of nodes on the base of model. IBASEP (NBASEP) - the reference numbers of these nodes.

I<5£

Choleski Routine

Used for solving the stiffness'equation [ k ] { X ] = { M

f o r the unknown column matrix fx} , where det k * o and jV] i s symmetric. For such a £k] we can f i n d an upper trian g u l a r matrix U such that

Thus

and w r i t i n g

we have

Del = u T. o

- t i t

(A5.1)

(A5.2)

(A5.3)

Having found the triangular matrix U the process consists of solving A5.3 for \y] and then solving A5.2 for ^x}_

U i s computed using

'J

i SU­

IT ote that 0 can have? purely complex components but i f one element of a row: i s complex then the whole row i s complex, ^ y } i s evaluated from A5.3 using

U. n U-0

L b

( A 5 . 5 )

Note that i f row i of U i s purely complex then so i s y ; .

\x\ i s evaluated from A5-2 using

=S = 2* ( A 5 . 6 )

Advantages of the method for f i n i t e _ element calculations.

1. In a l l f i n i t e element calculations [ k } i s positive de d e f i n i t e as w e l l as being symmetric. This means that U and [y] are r e a l matrices. The routine presented here i s general and includes the complex case as t h i s can be done with l i t t l e increase i n storage requirements.

2. The matrix [kQ i s banded for f i n i t e element calcu l a t i o n s . I t i s e a s i l y seen that both U and [k] have the same band width.

•IBM

155

To save storage only the semi-band width (shaded i n (Pig.A 5 . 1 ) of Qc] i s stored. With storage space at a premium the great advantage of the Choieski routine i s that when calcu l a t i n g U from k i t can be stored by overwriting k (see A 5 . 4 ) . 1*1 can be overwritten by ^y} (see A 5 - 5 ) which i n turn i s overwritten by { x ^ (see A 5 . 6 ) . Thus no extra storage space i s required for the matrices C u3, and

Notes on routine:

Solving [A] [X] = J c } Hole of MAR ( 1 )

(a) = 0 then row 1 of U and y ; are purely r e a l . (b) = 1 then row 1 of u and y ; are purely imaginary.

The r e a l or imaginary part i s stored for any

component of ii together with MAR. Thus i f u ^ i s r e a l

then we store as follows ( A 5 - 4 , A 5 - 5 , A 5 . 6 » )

2 -

IBM Hi

"J _ J mm.{ J IHsi J J

U KK ( A 5 . 7 )

K-l

u

K.I

U KK.

I

and i f i s imaginary we store

K-i

rW.M

•C-1

M s I

u u u LA. M

J J 1*1=1

J U. KK. ( A 5 . 8 )

K-l K-I

1 M « l NIK

3 KK

1 M « K 1 1

IL9?®.i. T n e s t i f f n e s s element i s stored i n ST (IN ( I , J ) ).

I *"7

Function _IN

The s t i f f n e s s matrix i s symmetric and banded with a semi-hand width NBW. To ensure economy oi' storage space the semi-hand s i d t h only i s stored i n the l i n e a r array ST, i . e . the shaded area of k only i s stored ( P i g . A 5 . 2 ) . This part i s stored row by row i n ST.

• \ ' o ' I r

• *

i P i g . A 5 . 2 O -* !

*

i • • I -

L_. . . 1

Thus the ( l j ) t n element of Qc] i s stored i n

S T ( { (<- - O * NBW + Cj - I •+ I ) \ )

or ST (IN ( I , J ) )

where I N ( I , J ) = ( I - 1 ) x NBW + ( J - I + l ) (fts.s )

I I

Subroutine Is o s : j

F i g . A 5 . 3 t

-4 N

I f i s o s t a t i c forces are introduced on the base of the model then these are proportional to the displacement of the base. I f the density of the underlying material i s f and the displacement of the i base point (node b L ) in the y - di r e c t i o n i s £ , then the i s o s t a t i c force exerted on t h i s node i s

r s . i»s,-*o | M*5 (A5.10a)

unless i t i s the f i r s t or l a s t base point, i n which case i t i s

and he

(A5.10b)

(A5.10c)

respectively.

i

Writing the i s o s t a t i e force on the nodal point i as — A S M where A. = o i f i i s not a base point, we have by adding these extra loads on to the applied force matrix.

where

A ,

( A 5 - 1 2 )

Thus replacing Qc] by \k] + JV| introduces the required i s o s t a t i c forces.

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c 1

ft' '.1 CI

— -

i -1—• I —

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> 1.

B. GrENFIN

^^.pJi .PPJ. 3ENFIN generates the mesh data f o r the

two-dimensional f i n i t e element program FINE1. FINEL

r e q u i r e s a c r o s s - s e c t i o n of the model to be d i v i d e d i n t o

a network of t r i a n g l e s ( c a l l e d elements). The corners

of the t r i a n g l e s are numbered so t h a t the g r e a t e s t

d i f f e r e n c e f o r a l l elements between the nodal numbers of

any one element i s kept as s m a l l as p o s s i b l e .

Fig.A$ . 4 -

_ . x . O-.

f,

It ...

'A'

6

A

8"

standard s t a t e

f.

GENFIN assumes t h a t the c r o s s - s e c t i o n i s r e c t a n g u l a r

w i t h a s m a l l p e r t u r b a t i o n on the top s u r f a c e (OABC)

a l l o w a b l e . T h i s s u r f a c e and any i n t e r n a l boundary such

a s (O'A'B'C) are defined a t 5 km. i n t e r v a l s i n the i n p u t

data. The co-ordinates of C" give the o v e r a l l s i z e of

the s e c t i o n . For optimum band-width we r e q u i r e 0 O 0 0 " .

The d e n s i t y and e l a s t i c constants of each l a y e r must a l s o

be g i v e n as data.

A f a c i l i t y e x i s t s f o r the v a r i a t i o n of the s i z e of

the mesh generated as one proceeds along the model i n a

h o r i z o n t a l d i r e c t i o n . (Note t h a t i t i s not p o s s i b l e to

vary the s i z e of the mesh i n a v e r t i c a l d i r e c t i o n w i t h the

present program). To do t h i s one must s p l i t the c r o s s -

s e c t i o n i n t o a s e r i e s of "blocks (00"A"A, AA"B"B, BB"C"0

i n fig. 4 ^ 4 ) by d e f i n i n g the x co-ordinate of the l i n e s

AA", BB" and then s p e c i f y the h o r i z o n t a l and v e r t i c a l

dimensions of the elements i n each block (SPH, SPV,

r e s p e c t i v e l y ) . Note here t h a t because of the d i s c r e t e

way the boundaries a r e s p e c i f i e d (every 5 km) the l i n e s

AA", BB" must c o i n c i d e w i t h one of these 5 km l i n e s and

a l s o the h o r i z o n t a l dimension of any element (SPH) must

be an i n t e g r a l d e v i s o r or m u l t i p l e of 5 km. A c o n s t r a i n t

a l s o e x i s t s on the p o s s i b l e t r a n s i t i o n v a l u e s of the

v e r t i c a l dimension of the element s i z e . I t can 'change'

i n one of three ways as a new block i s entered:

1. Double 2. Halve 3. Stay the same ( i . e . SPH changes a l o n e )

Thus a d r a s t i c change i n element s i z e must, be achieved

by u s i n g a s e r i e s of s m a l l b l o c k s , the v e r t i c a l dimension

of the element being halved a t each block, i f the s i z e i s

being decreased, and doubled i f being i n c r e a s e d . Note t h a t

no s i m i l a r c o n s t r a i n t e x i s t s f o r changes i n SPV-.

Yet another f a c i l i t y a l l o w s the s u b t r a c t i o n of a

standard s t a t e ( f i g . A 5 « 4 ) - The standard s t a t e i s defined

by h o r i z o n t a l l a y e r s of given d e n s i t y . The d e n s i t y g i v e n

to the element i s then equal to the d e n s i t y of the a c t u a l

l a y e r i n which i t l i e s minus the d e n s i t y of the standard

s t a t e l a y e r i n which i t l i e s . T h i s i s u s e f u l i f the e f f e c t

of the d e n s i t y c o n t r a s t of the roo t i n FigA5.4 i s r e q u i r e d .

The d e n s i t y given to the root i s f " f

1*1

Data Input: i s by card.

10

Card 1: NJOB n o

NJOB - No. of jobs to be processed.

Card 2: to

Mxf 3.Q

NY! N B !

u-o Nl'i

1 to 3 i o 7.M:

where

(MX,NY)

NB

NL

Card 3:

Co-ordinates of bottom r i g h t hand corner C" of r e c t a n g l e i n Km. (Note: NY u s u a l l y -ve and MX i s m u l t i p l e of 5)

No.of boundaries ( i n c l u d i n g top s u r f a c e although not base)

No.of l a y e r s i n standard s t a t e (Note: N12&1, t h e r e f o r e , i f no standard s t a t e wanted we have NL=1 and give l a y e r zero d e n s i t y ) .

Boundary data ( MX/5 + 1 c a r d s )

2 ID

| W Go-oRfc O F • H Co-oftfe O F a*" 1 " " I

I- I Ci • o

Cc-anfe O F ~H Co-cRb" OF a' J o

*o«MBft£l ATx*Stc«« ;B^Kj^ftftw AT x = s i u , ' 1 J , e i c

1

Card 4:

ROBll i~ •• a- o

2.o

H0B2 F i o . u

.3 o

H0B3! etc.

where,

ROBI i s the d e n s i t y of the m a t e r i a l immediately below

boundary I (gm./c. c. )

Card_ 5j

' _ E l E2 j E3] e t c .

where,

E I i s the Young's Modulus of the m a t e r i a l immediately

below boundary I (dynes/sq.cm.)

Gard_6j_

_; ! o J.O 3u ! ; VI; V2' V3'' e t c .

p I Q . B Pus. o F"'o-o

where V I i s the value of Poisson's r a t i o f o r the m a t e r i a l

immediately below boundary I ( d i m e n s i o n l e s s ) .

£a_rd_7_L

L a y e r data f o r standard s t a t e ( N l c a r d s )

|_ D ( l ) j R0SS(1).

where D ( I ) i s the y co-ordinate ( u s u a l l y - ve) of the

base of l a y e r I and ROSS(I) i s the d e n s i t y of l a y e r I .

U n i t s km. and gm/cc.

Card. Bt_

Spacing data. • ' *!0 ' a

X S ( I ) j S P V ( I ) ' SPH(I);

where X S ( I ) i s the x co-ordinate (km) of the end of

block I and the elements w i t h i n t h i s block are to have

v e r t i c a l dimensions S P V ( I ) and h o r i z o n t a l dimensions S P H ( I ) .

The data input i s terminated as soon as a card w i t h

X S ( I ) = EX i s read.

Sample data and run.

We wish to produce the model shown i n Fig . A 5 . 5 of a mountain and root

Fig . A 5 5.

4 \

To i I I Y

—~- 130

— A-

l i o

_ Motto "~C

w i t h

p a r t f = 2.85

f77} p a r t f> = -0.4

= 0.0 C D P a r t P =

E l a s t i c parameters

E c r u s t = 0 , 9 2 8 x l o 1 2 dynes/sq.cm.

I c r u s t = 0 , 2 5

E m a n t l e = 0 , 1 7 8 x l o 1 3 dynes/sq. cm.

^mantle = 0 , 2 5

Element s i z e : A and C • I O -

B

To a c h i e v e t h i s we choose boundaries and standard

s t a t e as i n f i g . A 5 . 6 .

Real_ parameters. C=^-1S E r - O.q»xio

E r o-1*7^ v i o 17.

F i f l . A $ . 6.

Standard s t a t e

f = o • o

The output data from GENFIN i s p l o t t e d i n Pig . A 5 . 7 -

i

i l l lTi " i W Hfftfhi?")

2 in a O

8 ri

LU < a.

uj <

7 U 7 III a

Eg

_ _ . ] _

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i 3:

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s

a O

S n

i

5=

I

03 5r CO

+1- trr In I S

LYIEJIUF^EIE^fcJfcjfcJLiUJLjlC^

F r T / 9 , 7

no

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1

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t 01

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I I • I I

t -4

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i; i — t » i — S

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o o

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! T H E fceouji&eb H O R I X S M T M . - .

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t 3- ZZJ ...

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! j .

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( A T I-EAST)

T N Qe> u K I ( i f t f u ) - N _ \

N s N - * I N C Q U C U ) » N C a l - O - ) * I

fPUMCM uA|«L-« C A t t f r ' !

M « N 4 I

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^ „ • ! P U N C H S o u N f e A A i CACfcj

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I F o « M T H E ! E L E M E N T t l ( \ T f i !

N E i - E M =• I (.NO. P P S i - K M E N T ^ N T o P •=. | (NofeM. "Vol K I T M o . o i r Motors v. ? B I N T f\T T o P o r

PRESENT COk-UMN.^

. ®1 I ( N O . OF HfUslt* C O U . ^

OF iwsc>N<; fVC«T) SMIkU.

LNEX-r = L .+ I . H & - 2. j T i - * \

J T + " ~

- ( M C o t , C u . > : i N i C c u ( u N E V - r ) - f }

_ _ . 1 * . ." '

i * . N M i l M . "PCI'MT • NO. AT T o p o f N E X T C O I - U M H

j"SE-r" T B S " - rMAV i ; i & N E X T 8>cuNb*<^!

So Tv i f t r 0ouMbMl'"f

is } M : N T o P + N t 6 L ( t ) j NbhttL. f N T . N o j ftT T o P OP NEXT . | C o i _ .

1 & >& K1EVT £iouNLl\i01 • i.:." " "

A •

ISjofcAL F»oiNT No. AT T o p O P KJEXT C B L ,

! &1ET X f i ^ THAT

. . . 1 T l ' s N T o P - t - X - l j J a = N T o t » * x j T 3 - M + T - I ;

3" I t M ^ X - I

7 3 •= M T O I » ^ - X - | ^ $ _ j ' i~C?i... :....^.'r.O.I £ L , L ) J ( j l i i * ^ frfcJ1-*1^

_ fcfiosiri I '

X L - T L f l ! ' lb-—

i :

; J i s. NTOPH-T • ' U * - * \ - * X - l ;

5 3 « x

; X I . = SI.-*-I .

< — i — , .

. S & A L _

• PwMCH El-CMZMT

M»PKoPR«ATE

' . . . J

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Purpose

To produce data compatible w i t h FINEI6 u s i n g data

f o r PINEI.

l a t a Input

The mesh data f o r F I N E I .

Data Output

The mesh data f o r PINEL6.

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Purpose

To perform a f i n i t e element c a l c u l a t i o n w i t h a

6-noded element f o r the cases of plane s t r e s s and

s t r a i n .

Data Input

As obtained from 3T06.

For data storage and theory see Appendix 3- For flow c h a r t s

see PINEI.

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