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Some theoretical implications of strike-slip faulting Petrak, John A. 1965

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SOME THEORETICAL IMPLICATIONS OP STRIKE-SLIP FAULTING by JOHN A. PETRAK B . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f GEOPHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d : THE UNIVERSITY OF BRITISH COLUMBIA September 1965 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f th e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s . , I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f G e o p h y s i c s The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8 , Canada Date February 2 , 1 9 6 6 i i ABSTRACT \ J When a f a u l t o c c u r s i n the e a r t h ' s c r u s t , the ground i n i t s v i c i n i t y becomes deformed. T h i s t h e s i s uses t h e t h e o r y o f d i s l o c a t i o n s , as d e v e l o p e d by J . A. S t e k e t e e , t o examine t h e n a t u r e o f t h i s d e f o r m a t i o n f o r a v a r i e t y o f s t r i k e - s l i p f a u l t models. The t h e o r y i s d e v e l o p e d f o r c a l c u l a t i n g the d i s p l a c e m e n t f i e l d and s t r e s s changes expected a t any p o i n t around a v e r t i c a l t r a n s c u r r e n t d i s l o c a t i o n s u r f a c e whose net d i s p l a c e m e n t i s c o n s t a n t i n the h o r i z o n t a l d i r e c t i o n , and v a r i e s w i t h d e p t h . The r e s u l t s o b t a i n e d a r e compared g r a p h i c a l l y w i t h g e o d e t i c d a t a , and tho s e d i s c r e p a n c i e s t h a t a r i s e between t h e o r y and o b s e r v a t i o n a r e a t t r i b u t e d t o t h e l i m i t i n g a ssumptions o f the model. The p r i n c i p a l c o n c l u s i o n o f t h i s work i s t h a t the v a r i a b l e -s l i p f a u l t model p r o v i d e s a s i g n i f i c a n t improvement o v e r p r e v i o u s f a u l t models w h i c h assumed a c o n s t a n t d i s p l a c e -ment . i i i ACKNOWLEDGMENTS The author wishes to express- his gratitude to Professor M. A. Chinnery, on whose work t h i s thesis i s based, and whose suggestions and help f u l discussions -have been most valuable. Thanks are also due to Dr. J. A. Jacobs, Head of the Department of Geophysics, U.B.C., who ac-cepted the writer into his group and provided s u f f i -cient National Research Council funds to carry out thi s project. i v TABLE OP CONTENTS Page ABSTRACT i i ACKNOWLEDGMENTS i i i TABLE OF CONTENTS i v LIST OF ILLUSTRATIONS v i i LIST OF TABLES i x INTRODUCTION 1 CHAPTER 1 HISTORICAL REVIEW 3 1.1 The Concept o f F r a c t u r e 3 1.2 A p p l i c a t i o n o f F r a c t u r e t o Geology 4 1.3 B a s i c Assumptions o f D i s l o c a t i o n Theory ' 6 CHAPTER 2 THE DISPLACEMENT FIELD AND STRESS CHANGES DUE TO A RECTANGULAR DIS-LOCATION WITH VARIABLE SLIP 8 2 .1 The E q u a t i o n s o f E l a s t i c i t y 8 2.2 The B a s i c E q u a t i o n s 9 2 .3 G a l e r k i n V e c t o r s 11 2.4 The G a l e r k i n V e c t o r s f o r the D i s p l a c e -ment F i e l d 13 2 .5 I n t e g r a t i o n o v e r the D i s l o c a t i o n S u r f a c e 14 2.6 E x p r e s s i o n s f o r the S t r e s s Change 17 2.7 The Maximum Shear S t r e s s 22 V CHAPTER 3 COMPUTATION OP THE DISPLACEMENTS AND STRESS CHANGES PRODUCED BY VARIOUS FAULT MODELS 25 3.1 The Basic Model ' 25 3 .2 Evaluation of the Integrals Using Numerical Procedures 28 3 .3 Computation of the Displacements and Stresses 30 3.4 P l o t t i n g of the Displacements 31 3 .5 Horizontal Displacements 32 3 .6 - Perpendicular Displacements 38 3.7 V e r t i c a l Displacements 39 3 . 8 P l o t t i n g of the Stress Changes and Maximum Shear Stress 39 3.9 Results of the Horizontal Displacement Analyses 40 3.10 Results of the Stress Analyses 52 CHAPTER 4 A COMPARISON OF THE THEORETICAL DISPLACEMENT FIELD WITH OBSERVATIONS 55 4.1 The Source of Geodetic Data 55 4.2 The San Francisco Earthquake 56 4 .3 The Tango Earthquake 57 4.4 The North Izu Earthquake 57 4 .5 The Imperial Valley Earthquake 58 4.6 The Nevada Earthquake 59 4.7 Application of Theory to Observation 65 4 .8 Interpretation of the F a l l - o f f Curves 67 v i CHAPTER 5 A COMPARISON OP THE THEORETICAL STRESS DISTRIBUTION WITH OBSER-VATION 70 5.1 The Geodetic Data 70 5.2 The Relationship Between the A f t e r -shock Zone and Maximum Shear Stress 72 CHAPTER 6 CONCLUSIONS 78 APPENDIX THE EVALUATION OF THE DISPLACEMENT FIELD AND THE STRESS CHANGES 8 l APPENDIX A THE DISPLACEMENT FIELD 83 APPENDIX B THE STRESS CHANGES 92 APPENDIX C THE MAXIMUM SHEAR STRESS 96 REFERENCES 98 v i i LIST OP ILLUSTRATIONS FIGURE NO. Page 2.1 Dimensions of the Model 18 3.1 Displacement Ui in the Plane u,--0 (Chinnery) J 33 3.2 Displacement u, in the Plane y,-o (Model A) 34 3.3 Displacement U , in the Plane u.=0 (Model B) 35 3.4 Displacement u, in the Plane M,=0 (Model G) 36 3.5 Pall-off of U, with y z along y | S y 3 _ 0 (for various models) 37 3.6 Stress Component T/i*. in the Plane M, = o (Model A) 41 3.7 Stress Component T ,a in the Plane M i = 0 (Model A) 42 3.8 Stress Component in the Plane M,=0 (Model B) 43 3.9 Stress Component "Ci3 in the Plane M i = 0 (Model B) 44 3.10 Stress Component f i z in the Plane Mi = 0 (Model C) 45 3.11 Stress Component T 1 3 in the Plane Mi = 0 (Model C) 46 3.12 F a l l - o f f of T i i with yz along y,=.y3=o (for various models) 47 3.13 Maximum Shear Stress Magnitudes ( y, = 0 , Model A) 48 3.14 - Maximum Shear Stress Magnitudes . ( y, « O , Model B) 49 v i i i 3 . 1 5 Maximum Shear Stress Magnitudes ( y, = o , Model C) 5 0 4.1 F a l l - o f f of U i with distance i n the San Francisco Earthquake 6 0 4.2 F a l l - o f f of U i with distance i n the Tango Earthquake 61 4 . 3 F a l l - o f f of u., with distance i n the North Izu Earthquake 62 4 . 4 F a l l - o f f of u, with distance i n the Imperial Valley Earthquake 6 3 4 . 5 F a l l - o f f of u, with distance i n the Nevada Earthquake 64 5.1 Aftershock Sequence of Rat Island Earthquake 73 5.2 The Hypothetical Fault Trace Asso-ciated with the Rat Island Earthquake 73 5 . 3 Maximum Shear Stress Magnitudes ( y (= 0 > Model D) 74 5 . 4 Maximum Shear Stress Magnitudes ( 0.95L, Model D) 75 i x L IST OP TABLES TABLE NO. Page 4.1 R e s u l t s of t h e Model C a l c u l a t i o n s on Some R e a l F a u l t s 68 A . l T a b l e ( i ) of R e l a t i o n s h i p s 84 A.2 T a b l e ( I I ) of R e l a t i o n s h i p s 85 -1-INTRODUCTION The o b j e c t o f t h i s t h e s i s i s not o n l y t o de t e r m i n e the v a r i o u s components o f the t h e o r e t i c a l d i s p l a c e m e n t f i e l d and s t r e s s changes t h a t a r i s e when a s t r i k e - s l i p f r a c t u r e o c c u r s i n the e a r t h ' s c r u s t , but a l s o t o a p p l y them t o v a r i o u s g e o l o g i c a l s i t u a t i o n s . I t i s our purpose t o d i s c u s s some o f the a s p e c t s o f the t h e o r y o f d i s l o c a t i o n s which may be o f s i g n i f i c a n c e i n g e o p h y s i c a l problems connected w i t h f a u l t p l a n e s t u d i e s o f ear t h q u a k e s and such phenomena as the San Andreas f a u l t i n C a l i f o r n i a . Problems con-n e c t e d w i t h f r a c t u r e zones i n the e a r t h ' s c r u s t and t h e i r c o r r e s p o n d i n g a f t e r s h o c k d i s t r i b u t i o n s , as e x h i -b i t e d by the Rat I s l a n d e a r t h q u a k e ^ i n A l a s k a , a r e o f p a r t i c u l a r I n t e r e s t . However, i t i s e v i d e n t t h a t a l a c k o f p e r t i n e n t g e o l o g i c a l d a t a , among o t h e r t h i n g s , h i n d e r s some o f our a n a l y s e s t o a c o n s i d e r a b l e e x t e n t . P r e v i o u s approaches t o t h i s p roblem u s i n g d i s -l o c a t i o n t h e o r y ( e . g . C h i n n e r y 1962) were made p o s s i b l e o n l y by t h e e f f o r t s o f S t e k e t e e (1957. 1958a, 1958b). The m a t h e m a t i c a l t e c h n i q u e s t h a t were d e r i v e d from t h i s t h e o r y a l l o w us t o c a l c u l a t e the d i f f e r e n c e between the i n i t i a l and f i n a l s t a t e s o f the medium s u r r o u n d i n g the - 2 -f a u l t : to be more precise they permit us to calculate only the a f t e r - e f f e c t s of a s t a t i c fracture s i t u a t i o n . As a departure from the approach of Chinnery, our object w i l l be to develop the formulae f o r the d i s -placement f i e l d and stress changes which are representa-t i v e of a d i s l o c a t i o n surface showing variable v e r t i c a l and constant horizontal s l i p . This allows us to examine a f a u l t model which i s perhaps more r e a l i s t i c . This d i s s e r t a t i o n i s divided into three main sections. The f i r s t (Chapter 2 ) contains the t h e o r e t i -c a l approach to the problem as well as the expressions which are necessary to calculate a n a l y t i c a l solutions. The second part (Chapter 3 ) i s primarily concerned with interpretations of the a n a l y t i c a l r e s u l t s acquired. F i n a l l y , Chapters 4 and 5 deal d i r e c t l y with ap p l i c a t i o n of the th e o r e t i c a l r e s u l t s to various geological obser-vations . -3-CHAPTER 1 HISTORICAL REVIEW We w i l l begin our discussion with a b r i e f outline of the h i s t o r i c a l background of the subject. In p a r t i c u l a r , we s h a l l mention the t h e o r e t i c a l ap-proaches of other authors and show how th e i r work was severely limited by th e i r i n i t i a l assumptions. 1.1 The Concept of Fracture The concept of dislocations (Love, 1944), o r i g i n a l l y referred to as d i s t o r t i o n s , was introduced into e l a s t i c i t y theory by Volterra i n order to describe the deformation that may arise i n a body occupying a multiply-connected region, when the displacements of i t s points are given by many-valued functions of the co-ordi-nates ( l n the absence of body forces). As a re s u l t , very small fractures have been examined by several authors. In g l i s (1913), Sneddon (1946), and G r i f f i t h (1921, 1924) have focussed t h e i r attention on the microscopic causes of fracture. The nature of t h e i r work i s very s i m i l a r to ours, the main difference, being the magnitude of the fractures involved. -4-1.2 A p p l i c a t i o n of F r a c t u r e t o Geology Anderson (1951) u s i n g the t h e o r i e s of N a v i e r , Coulomb, and Mohr c o n s i d e r e d a two d i m e n s i o n a l f a u l t o f f i n i t e l e n g t h and i n f i n i t e d e p t h but to o k no account of the boundary c o n d i t i o n a t the ground s u r f a c e . He adapted the e l l i p t i c a l c r a c k of I n g l i s (1913) as a model f o r h i s s t r i k e - s l i p f r a c t u r e o n l y t o f i n d t h a t few d e d u c t i o n s c o u l d be i n f e r r e d f r o m h i s r e s u l t s . K asahara (1957, 1958, 1959) a f t e r p o s t u l a t i n g a c o n t i n u o u s v a r i a t i o n of d i s p l a c e m e n t w i t h d e p t h , con-s i d e r e d a f a u l t of i n f i n i t e l e n g t h and f i n i t e d e p t h . He t h e n a p p l i e d b a s i c e l a s t i c i t y t h e o r y t o c a l c u l a t e the d i s t u r b a n c e o f the s u r r o u n d i n g medium. H i s r e s u l t s were p l a u s i b l e b ut c o u l d not be used t o d e s c r i b e a f a u l t o f f i n i t e l e n g t h , the case most p r e v a l e n t i n the e a r t h ' s c r u s t . B y e r l y and De Noyer (1958) a t t e m p t e d t o d e t e r -mine the de p t h s of s e v e r a l a c t u a l f r a c t u r e s by c a l c u l a t -i n g t h e shear s t r a i n energy i n the e a r t h j u s t b e f o r e f a u l t i n g o c c u r s . They assumed t h a t the d e f o r m a t i o n of s t r a i n a t the e a r t h ' s s u r f a c e remained u n a l t e r e d t o t h a t d e p th a t whi c h t h e f a u l t broke and no deeper. A l t h o u g h t h e i r d e p th e s t i m a t e s seemed v e r y r e a s o n a b l e ( i n c o m p a r i -son w i t h t h o s e d e t e r m i n e d u s i n g d i s l o c a t i o n t h e o r y ) , - 5 -Byerly and De Noyer Indicated that they were not con-vinced by the r e s u l t s . Steketee ( 1 9 5 7 , 1 9 5 8 a ) developed an " e l a s t i c i t y theory of d i s l o c a t i o n s " to Investigate the changes brought about by the occurrence of f a u l t i n g i n nature. He applied a Green's function technique to the funda-mentals of d i s l o c a t i o n theory (Volterra, 1 9 0 7 ) , thus constructing a method for handling surfaces of discon-t i n u i t y i n an e l a s t i c s o l i d . The advantage of t h i s ap-proach i s not only that the condition of zero stress at the ground surface i s inherent i n the theory, but also, that i t allows us to consider normal, reverse, and dip-ping f a u l t s as well as transcurrent s t r i k e - s l i p fractures. Chinnery ( 1 9 5 9 , 1 9 6 2 ) applied Steketee's "elas-t i c i t y of d i s l o c a t i o n s " to a v e r t i c a l rectangular s t r i k e -s l i p f a u l t which may or may not intersect the ground surface. Various aspects of the displacement f i e l d and stress changes were Included In his analyses of t h i s constant s l i p model. Applications of his results to geodetic measurements were considered, especially as a means of determining the depths of various f a u l t s and the magnitudes of the stresses that caused the fractures to occur. Although Chinnery's model was only a mathe-matical approximation to a geological s i t u a t i o n , i t produced several r e a l i s t i c r e s u l t s (Press, 1 9 6 5 ) . Hence, he proceeded to examine a s t r i k e - s l i p f a u l t model that displays a variable horizontal s l i p and a constant ver-t i c a l s l i p . His interpretations of the r e s u l t s have not yet been completed but they should be of consider-able inte r e s t when compared with those outlined i n t h i s thesis. 1.3 Basic Assumptions of Dislo c a t i o n Theory It would be impossible to apply d i s l o c a t i o n theory to obtain a n a l y t i c a l r e s u l t s i f a simple mathe-matical model were not used. It i s usual i n mathemati-c a l analyses to r e t a i n only those features which are believed to be of major importance. In our case t h i s means that we s h a l l concern ourselves with a s e m i - i n f i -nite homogeneous, i s o t r o p i c medium where the laws of the c l a s s i c a l theory of e l a s t i c i t y hold. Such features as the curvature of the earth, i t s g r a v i t a t i o n a l and magnetic f i e l d s , temperature, and non-homogeneity w i l l not be considered. I t i s also important to r e c a l l the assumptions that are i m p l i c i t i n the theory. The most important of these are that the earth's crust may be regarded as an e l a s t i c s o l i d and that i n f i n i t e s i m a l s t r a i n theory i s applicable. -7-Furthermore, we s h a l l r e s t r i c t ourselves to s t a t i c problems which implies that waves and other transient features associated with earthquakes w i l l not be considered. This i s a reasonable assumption since earthquake fractures occur within a matter of a few minutes (Scheidegger, 1 9 5 7 ) , and hence we expect e l a s t i c i t y theory to be applicable to the immediate effects of f a u l t i n g . The process of fracture, of course, i s b a s i c a l l y non-elastic, but i t has been shown (Chinnery, 1964) that f a u l t s appear under applied stresses of as much as two orders of magnitude less than the breaking strength of rock as measured i n the laboratory. Under these circumstances, i t i s p l a u s i -ble that the laws of e l a s t i c i t y should apply even to the material i n the immediate v i c i n i t y of the f a u l t face. CHAPTER 2 THE DISPLACEMENT FIELD AND STRESS CHANGES DUE TO A  RECTANGULAR DISLOCATION WITH VARIABLE SLIP An o u t l i n e o f t h e t h e o r y l e a d i n g up t o the e v a l u a t i o n o f t h e d i s p l a c e m e n t f i e l d , t he s t r e s s changes, and t h e maximum shear s t r e s s t h a t r e s u l t from t h e i n t r o d u c t i o n o f a v a r i a b l e - s l i p d i s l o c a t i o n i n t o a s e m i - i n f i n i t e e l a s t i c medium a r e g i v e n below. A l l e x p r e s s i o n s a r e f i n a l l y reduced t o the case "X=^x and r e f e r e n c e i s made, where a p p l i c a b l e , t o t h e i r p h y s i c a l s i g n i f i c a n c e . The n o t a t i o n used t h r o u g h o u t t h i s t h e s i s f b l i o w s t h a t o f S t e k e t e e ( 1 9 5 7 , 1 9 5 8 a , 1 9 5 8 b ) , S o k o l -n i k o f f ( 1 9 5 6 ) , and C h i n n e r y ( 1 9 6 2 ) . 2 . 1 The E q u a t i o n s o f E l a s t i c i t y I f XLLL - i)2. 43) i s a C a r t e s i a n c o - o r d i n a t e system and u(xc) the d i s p l a c e m e n t f i e l d produced by a f o r c e system w i t h i n a homogeneous i s o t r o p i c e l a s -t i c s o l i d , t h e n t h e components o f the s t r a i n t e n s o r ecj and o f the s t r e s s t e n s o r "Ci.j a r e g i v e n by the r e l a t i o n s - 9 -( 2 . 1 ) ( 2 . 2 ) where A and y. are Lame's Kronecker symbol, and ^ - ^ j -( 2 . 2 ) i s commonly referred to as the generalized form of Hooke's Law (Sokolnikoff 1 9 5 6 ) . On substituting the equations of equilibrium ( i n the absence of body forces - equations 2 . 3 ) J , J ( 2 . 3 Into equation ( 2 . 2 ) we obtain Navier's equations which play the same role i n d i s l o c a t i o n theory that Laplace's equations do In ordinary potential theory. 2 . 2 The Basic Equations one across which there i s a discontinuity i n one or a l l of the components of the displacement vector LLi . Steketee ( 1 9 5 8 a ) uses the Green's function method to deal with the problem of a Volterra d i s l o c a t i o n i n a ( 2 . 4 ) A d i s l o c a t i o n surface ^ i s defined as -10-seml-infinite medium in such a way that the boundary surface of the medium remains stress free. A Volterra dislocation is here defined as a surface across which the displacement components show a discontinuity of the form (2.5) - n - t j = - _ n - t j where Ui and -^-'yj are constants. The relation (2.5b), which is the well-known Weingarten relation (Weingarten, 1907), implies that the discontinuity across ^ is of the form of a rigid body displace-ment. In the following applications we shall assume that the rotation matrix — i s zero. This im-plies that there is no net rotation of the fault faces relative to one another during the process of fracture. By solving this f a i r l y complex boundary value problem which stipulates a condition of zero stress on the surface, Steketee (1958a) obtained the general solution for the displacement f i e l d U K , in a semi-inf i n i t e medium due to the dislocation L\ Lit , in the form U * " * ^ J l i a U i W ' j ( 2-6) -11-Here vy represents the d i r e c t i o n cosines of the normal to the surface element . The Green's functions K LOtj are the displacement f i e l d s i n a semi-infinite medium due to a set of nuclei of s t r a i n defined by i and j (see Love 1944, p. l 8 6 ) j the superscript refers to the d i f f e r e n t f i e l d s . These nuclei of s t r a i n are elementary force systems with moment (for i $j ) and without moment ( f o r L = j ) . A t o t a l of s i x functions UJCJ ( f o r C= 1 ,2 ,3 ; j s 1 ,2 ,3 ; c J = j C ) are needed f o r the solution of the general surface ^ The problem i n which we are interested requires a nucleus that has two double forces i n the plane p a r a l l e l to the boundary. Using Xj - 0 as the surface of the K medium Steketee has evaluated the function tc>(i which s a t i s f i e s these requirements. This allows us to i n t e -grate over the plane surface ^ 2 . - ° with a variable discontinuity i n the component of the displacement vector. Physically, t h i s means that we can determine the displacements produced throughout the medium by a v e r t i c a l s t r i k e - s l i p f a u l t . 2.3 Galerkin Vectors Since several of the expressions with which we must deal may become extremely clumsy, we s h a l l -12-make use of Galerkin vectors (Galerkin 1930; Papcovitch 1932). Although these vectors do not help much i n the actual solution of the problem, they enable us to state the r e s u l t s as compactly as possible. The Galerkin vector Vi i s defined as a vector which gives a displacement f i e l d by d i f f e r e n t -i a t i n g according to the Laplace operator and grad div. Therefore, i f the Galerkin vector f o r a problem i s known, the problem i s solved, as the displacements are given by the expression U< = VK,LL - o< Pc,tk (2.7) where ^ _ 7\ + y. Hence, we have immediately for the strains C LJ and the stresses T i j , (2.8) The substitution of U L into the Navier equations (2.4) now leaves us with the equation V*Vi - - 0 1 f i kK =0 or J J (2.9). -13-whlch shows t h a t each component o f the G a l e r k i n v e c t o r i s b i h a r m o n i c . T h i s i s a d e f i n i t e advantage s i n c e the g e n e r a l s o l u t i o n t o such an e q u a t i o n has a form which i s much s i m p l e r than the c o r r e s p o n d i n g s o l u t i o n o f N a v i e r ' s e q u a t i o n s . 2.4 The G a l e r k i n V e c t o r s f o r the D i s p l a c e m e n t F i e l d S t e k e t e e q uotes the Green's f u n c t i o n u*>\x i n terms o f a G a l e r k i n v e c t o r ( W e stergaard, 1952). I n t h i s case e q u a t i o n (2.6) becomes 8 r r / A j J * (2.10) r-i < where | ,^  i s the G a l e r k i n v e c t o r f o r the d i s p l a c e -ment f i e l d a t an o b s e r v a t i o n p o i n t ( <-ji 5^2.5^3) due t o a n u c l e u s a t the p o i n t (XtjO,X 3) on the d i s l o c a t i o n s u r f a c e . The components o f t h i s G a l e r k i n v e c t o r were g i v e n by S t e k e t e e and g e n e r a l i z e d by C h i n n e r y as f o l l o w s n 3 _ u t ua. sz si (2.11) -14-where: P = * 3 + "Ja t=x,-y, A c o n s i d e r a t i o n o f e q u a t i o n (2.10) i n d i c a t e s t h a t we can now c a l c u l a t e the d i s p l a c e m e n t f i e l d due t o a v e r t i c a l t r a n s c u r r e n t f a u l t model whose d i s l o c a -t i o n i s o v e r a p l a n e s u r f a c e c o n t a i n i n g the ^3 a x i s and p e r p e n d i c u l a r t o e i t h e r the y, o r axes ( f i g u r e 2.1). T h i s i m p l i e s t h a t t h e d i s c o n t i n u i t y U K must be i n the o r y, d i r e c t i o n r e s p e c t i v e l y . 2.5 I n t e g r a t i o n o v e r the D i s l o c a t i o n S u r f a c e At t h i s s tage i n the a n a l y s i s , i n o r d e r t o s i m p l i f y a l l f u t u r e work we make the a s s u m p t i o n t h a t , which i s not u n u s u a l i n the a p p l i c a t i o n o f e l a s t i c i t y t h e o r y t o g e o p h y s i c s . I t f o l l o w s t h a t <X - 2/3, b = 0, and C = 1/4. The o r i e n t a t i o n and d i m e n s i o n s o f t h e d i s -l o c a t i o n s u r f a c e £ , o v e r which we a r e t o i n t e -g r a t e a r e shown i n f i g u r e 2.1. We assume t h a t the d i s c o n t i n u i t y u K i n d i s p l a c e m e n t a t the f a u l t b - . A -A -15-face,has only ,one component U, which i s a function of the v e r t i c a l p o s i t i o n X3 alone i . e . the f a u l t i s an e n t i r e l y transcurrent, v a r i a b l e - s l i p d i s l o c a t i o n . I f we represent t h i s discontinuity by U,(x 3) then U= u,(o) w i l l be the t o t a l r e l a t i v e displacement of the two sides of the f a u l t on the ground surface. The introduction of these modifications into equations (2.7) and (2.10) requires that we evaluate the displacement f i e l d u K given by L l K - I K,Lt - i t PcjtK 3 L D ( 2 ' 1 2 ) ^ = QIJJL £ L JJ f ~ * u,(x 3)dx, dx 3 The evaluation of equation (2.12b) and i t s subsequent substitution into equation (2.12a) to obtain the d i s -placements i s very long and tedious. As a re s u l t , only an outline of the procedure followed w i l l be given here. Further d e t a i l s are given i n appendix A. F i r s t we introduce the expressions £ K and u)K which are defined respectively by the equations fc< = f Hi dx, J-L (2.13) -16-Such a representation i s possible only because the observation point does not l i e on the d i s l o c a t i o n surface. This allows one to perform d i f f e r e n t i a t i o n s with respect to inside the Integral without d i s t o r t i n g the o r i g i n a l problem. sign For the sake of brevity we s h a l l use the aft e r an equation to represent the substitu-t i o n fU>x 3) = f ( L , x 3 ) - f ( - L , X 3 ) . The components of the term U ) K are then found to be: to. u>a = ! ( F 3 t F») (2.14) where S ^ S ^ t ) S.CS.+t) ^ s l s? sfCs, *> r tf*^ F 3 = S^i-p S, S*(S2.4-p) e-'c ^ sl sr s»* -17-a n d p=x 3 i-y 3 H e n c e t h e c o m p o n e n t s o f t h e d i s p l a c e m e n t v e c t o r a r e f o u n d f r o m t h e e q u a t i o n 8TT u,(x 3) UJK dx 3 (2.15) B e c a u s e o f t h e c o m p l e x i t y o f t h e i n t e g r a n d i n e q u a t i o n s (2.15) i t i s n e c e s s a r y t o e m p l o y n u m e r i c a l m e t h o d s t o o b t a i n s o l u t i o n s t o t h e i n t e g r a l . T h i s w i l l be d i s c u s -s e d i n g r e a t e r d e t a i l i n t h e f o l l o w i n g c h a p t e r . 2.6 E x p r e s s i o n s f o r t h e S t r e s s C h a n g e c h a n g e s t h a t r e s u l t f r o m t h e i n t r o d u c t i o n o f a d i s l o c a -t i o n i n t o a n e l a s t i c m e d i u m a r e i n d e p e n d e n t o f a n y s t r e s s d i s t r i b u t i o n t h a t e x i s t e d b e f o r e h a n d . I n t h i s s e c t i o n we a r e c o n c e r n e d o n l y w i t h t h e s t r e s s c h a n g e s p r o d u c e d i n a s e m i - i n f i n i t e e l a s t i c m e d i u m by a v a r i a b l e S t e k e t e e (1957) h a s s h o w n t h a t t h e s t r e s s -18-F i g u r e 2.1 Dimensions o f the V a r i a b l e - S l i p Model R e c t a n g u l a r D i s l o c a t i o n ( y ^ o O r e p r e s e n t s the ground s u r f a c e ) - 1 9 -s l i p dislocation as shown in figure 2.1. P K We differentiate the Galerkin vector I \i to obtain the displacement components U K , U K - PK,CC - -| PtjCK We may now obtain the stress changes associated with this displacement f i e l d by differentiating accord-ing to Hooke's Law (equation 2.8b). As in the previous section we differentiate with respect to yj inside the integral and use the notation $U.,*0 - f( L,x,) - f (-L,x3) to denote The differentiation of equations (2.14) is algebraically very complex. As a result, we w i l l quote here only the nine derivatives • Further de-t a i l s are given in appendix B. U J COi, ( 2 . 1 6 ) -20-t G 8 <^ 3,3 = f / ^ G y where: S 2 ( S 2 * t ) S , ( S , + t ) G _ 3_+ _ i _ ' " SI s? r _ 3 + _6_ + 3 p ( 8 s ? »9pS» +3 pO _ 60x 3M G 3 = 2 6a t t S 2 3 ( S 2 * t ) 2 2 S.»t S f ( S . + t ) ' sf(s, +.t)' G 5 = ^ D + feq + 3(2 S a +p) _ 3 ( 3 p + + 6 0 x 3 u 3 p -21-^ 6 s E(s a + p) a s,3 rsi(st\Py 2. and P = X 3 i - y 3 ^ = X 3 -t » X, - Lj, We now express Hooke's Law (equation 2 . 2 ) i n terms of the derivatives t / J t , j to obtain the general expression T \ j = -Q^J u,(x3) ( 6tj GOK,k t co,;,] tujj,i) dx3 ( 2 . 1 7 ) - 2 2 -where U, (X3) r e p r e s e n t s the d i s c o n t i n u i t y i n d i s p l a c e -ment as d e f i n e d i n s e c t i o n ( 2 . 5 ) and ~X- JJL . The t e n s o r i s , o f c o u r s e , symmetric ( X^i = XLJ ) and must s a t i s f y t he f o l l o w i n g boundary c o n d i t i o n s a t t h e s u r f a c e o f the medium ( < j 3 - = o ) : f 13 - ^ " z 3 = T 3 3 - O , ( 2 . 1 8 ) We can show t h a t e q u a t i o n s ( 2 . 1 7 ) s a t i s f y t h e s e c o n d i -t i o n s and as a r e s u l t w i l l assume t h a t no a l g e b r a i c e r r o r s e x i s t i n them. At t h i s p o i n t i t i s n e c e s s a r y t o e v a l u a t e the i n t e g r a l s ( 2 . 1 7 ) . A l t h o u g h the e x p r e s s i o n s a re a l g e -b r a i c a l l y l o n g , they' can be e v a l u a t e d w i t h l i t t l e d i f f i -c u l t y , and i n as much a c c u r a c y as p o s s i b l e w i t h t h e a i d of an e l e c t r o n i c computer ( t h e p r e s e n t c a l c u l a t i o n s were performed on the I.B.M. 7 0 4 0 computer). 2 . 7 The Maximum Shear S t r e s s I t i s w e l l known t h a t a s e t of t h r e e homogen--*• eous e q u a t i o n s i n unknown d i r e c t i o n s 0 such as (Xn - X L\) Oi = 0 J J J ( 2 . 1 9 ) has a n o n - t r i v i a l s o l u t i o n i f , and o n l y i f , t h e d e t e r -m i n a n t s o f the c o e f f i c i e n t s o f the Oj i s e q u a l t o -23-z e r o i . e . Tij - X c5lj = O (2.20) T h i s c u b i c e q u a t i o n i n t h e s t r e s s has t h r e e r e a l r o o t s Xx , X-i. , X% , c a l l e d the p r i n c i p a l s t r e s s e s . Knowing t h e s e r o o t s , the c o r r e s p o n d i n g p r i n c i p a l d i r e c -t i o n s o f s t r e s s ^ can be d e t e r m i n e d . I n o t h e r words, the magnitudes and d i r e c t i o n s o f the p r i n c i p a l s t r e s s e s are s i m p l y the e i g e n v a l u e s and e i g e n v e c t o r s , r e s p e c t i v e l y , of the m a t r i x T i j Gi v e n the magnitudes and d i r e c t i o n s o f t h e p r i n -c i p a l s t r e s s e s a t any p o i n t , the c o r r e s p o n d i n g v a l u e s of the maximum s h e a r i n g s t r e s s a re e a s i l y d e t e r m i n e d . The magnitude of t h i s shear s t r e s s i s n u m e r i c a l l y e q u a l t o o n e - 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 e s (assume T 3 > Xz > T i ) and a c t s on t h e p l a n e t h a t b i s e c t s the a n g l e between t h e d i r e c t i o n s o f the l a r g e s t and 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 , i . e . s m a x T 3 - X, z (2.21) S i n c e t h e r e i s no s a t i s f a c t o r y way t o r e p r e -sent a t h r e e d i m e n s i o n a l s t r e s s d i s t r i b u t i o n ( e q u a t i o n -24-2.7) diagrammatically without omitting much of the information contained ±n the f u l l equations, the maxi-mum shear stress is physically the most meaningful quantity for discussing shear fractures. Furthermore, i t i s worth noting that i f an i n i t i a l stress d i s t r i -bution ( Liz in our case) is superimposed on the cal-culated stress change T i j (this is what actually happens in a physical situation), the principal axes of stress and hence the planes of maximum shear stress w i l l assume a completely different form which is of no immediate use to us here. For this reason, in the following discussion we w i l l be concerned only with the magnitude of the maximum shear stress and not with i t s direction. Appendix C contains more detailed i n -formation regarding the maximum shear stress. -25-CHAPTER 3 COMPUTATION OF THE DISPLACEMENTS AND STRESS CHANGES  PRODUCED BY VARIOUS FAULT MODELS I n o r d e r t o p r o v i d e a p h y s i c a l background f o r t he i d e a s o f d i s l o c a t i o n . t h e o r y t h a t have been d e v e l o p e d , we c o n s i d e r here v a r i o u s f a u l t models which a r e a b l e t o account f o r o b s e r v a t i o n s made on a c t u a l e a r t h q u a k e s . I n p a r t i c u l a r , we a r e I n t e r e s t e d i n a f a u l t model which e x h i b i t s a c o n s t a n t s l i p i n t he h o r i z o n t a l d i r e c t i o n and a v a r i a b l e s l i p i n t h e v e r t i c a l p l a n e . D i s c u s s i o n s o f v a r i o u s a s p e c t s o f t h i s p r o -blem may be found l n Kasahara (1958a, 1958b), S t e k e t e e (1958b), B y e r l y and De Noyer (1958) and C h i n n e r y (1959, 1962, 1964). 3.1 The B a s i c Model Anderson (1951) d e f i n e s the s t a n d a r d s t a t e i n t he e a r t h as t h a t o f a h y d r o s t a t i c p r e s s u r e . Thus i t r e p r e s e n t s a c o n d i t i o n i n which t h e l a t e r a l p r e s s u r e from a l l s i d e s i n c r e a s e s s t e a d i l y w i t h depth so as t o be e q u a l everywhere t o the v e r t i c a l p r e s s u r e . The - 2 6 -l a t t e r i s d e t e r m i n e d a p p r o x i m a t e l y by t h e weight o f the o v e r l y i n g r o c k and i s r o u g h l y p r o p o r t i o n a l t o depth p r o v i d e d t h a t t h e r e a r e no g r e a t v a r i a t i o n s i n d e n s i t y . Such v a r i a t i o n s would a r i s e o n l y when the s t r a t a above the p o i n t under c o n s i d e r a t i o n a r e f u n c t i o n i n g as an a r c a s , f o r example, l n the c e n t r e o f an a n t i c l i n e . T h e r e f o r e , i n most cases i t i s r e a -s o n a b l e t o assume t h a t the d i s c o n t i n u i t y U-i w i l l d i e out towards the bottom o f the f a u l t . I f t h e c o e f f i c i e n t o f f r i c t i o n i s c o n s t a n t ( w h i c h i s not known) th e n U,(X 3) s h o u l d d e c r e a s e l i n e a r l y w i t h d e p t h . However, i t has been shown ( C h i n -n e r y , 1962) t h a t a c c u m u l a t i o n s o f s t r e s s around the l o w e r edge o f the f a u l t may cause i t t o extend some-what deeper. Thus i t i s p l a u s i b l e t o assume t h a t t h e r e i s a f u n c t i o n a l v a r i a t i o n o f ( x 3 ) w i t h d e p t h . The a v a i l a b l e energy r e p r e s e n t e d by the con-d i t i o n o f s t r e s s i n t h e e a r t h ' s c r u s t must overcome not o n l y the s t r e n g t h o f the m a t e r i a l but a l s o t h e i n t e r n a l f r i c t i o n o p p o s i n g t h e m o t i o n , i f f r a c t u r e i s t o o c c u r . T h i s f r i c t i o n a l r e s i s t a n c e a l o n g t h e f a u l t p l a n e i s p r o p o r t i o n a l t o t h e normal p r e s s u r e a c r o s s the f a u l t f a c e s ( t h e T22. component o f the s t r e s s change i n t h i s c a s e ) . As a r e s u l t , i t w i l l have a tendency, - 2 7 -not only to reduce the magnitude of the s l i p IX, , i n depth ( increases with depth), but also to contribute to the production of heat along the interface. The l a t t e r condition may be associated with the p l a s t i c e f f e cts observed close to the d i s -l o cation surface. Although we have no knowledge of the d i s -continuity over the face of a rea l f a u l t , the geodetic conditions outlined i n the previous three paragraphs gives some j u s t i f i c a t i o n f or the use of a f a u l t model which exhibits a variable s l i p Ui(.Xa) i n the v e r t i -c a l plane. A constant s l i p Ui(X,) = LJ i n the horizontal plane i s also a necessary assumption. For, i f U,(X,) i s allowed to vary from point to point, the computer time required to produce convergent solu-tions to the double numerical integrations (2.10) would be p r o h i b i t i v e . The aforementioned considerations have encou-raged us to choose a v a r i a b l e - s l i p model whose d i s l o c a -t i o n surface i s described by the function U,(X 3) = U e x p where a , b , C are constants which may be varied -28-u n t i l the desired discontinuity i s obtained and i s the maximum value of the s l i p . . This model, however, does not allow the s l i p U| to diminish towards the ends of the f a u l t as we would i n t u i t i v e l y expect; Instead, mathematical s i n g u l a r i t i e s a r i s e i n these regions. Fortunately, the l a t t e r are of l i t t l e con-sequence as long as we do not carry out an a n a l y t i c a l analysis i n t h i s c r i t i c a l area. 3.2 Evaluation of the Integrals Using Numerical Procedures The task of evaluating equations (2.15) and (2.17) which represent the displacement f i e l d and stress changes respectively, would be pr o h i b i t i v e with-out the aid of an electronic computer. Hence, these equations were evaluated on the I.B.M. 7040 at the University of B r i t i s h Columbia. Two quadrature formulae were applied to equa-tions (2.15) and (2.17) i n order to obtain a n a l y t i c a l expressions f o r the displacement f i e l d and the stress changes. The f i r s t of these, Simpson's Rule (see Nielsen, 1957), was ea s i l y adaptable to our equations but did not produce accurate results unless the range of integration was Very f i n e l y divided. This meant that considerable computer time was required to obtain -29-o n l y r e a s o n a b l y a c c u r a t e answers. As a r e s u l t , we used t h e G a u s s i a n q u a d r a t u r e (see N i e l s e n , 1957) t h r o u g h o u t our a n a l y s e s . Such an approach r e q u i r e s t h a t the i n t e g r a t i o n p r o c e d u r e be e n t i r e l y dependent on the z e r o s o f t h e Legendre p o l y n o m i a l s and t h e i r r e s p e c t i v e w e i g h t i n g f u n c t i o n s . The i n t e g r a l s were e v a l u a t e d both a c c u r a t e l y and e f f i c i e n t l y u s i n g t h i s method. , I n most r e g i o n s around the d i s l o c a t i o n s u r -f a c e t h e n u m e r i c a l e v a l u a t i o n o f the i n t e g r a l s was c a r r i e d out w i t h o u t >any t r o u b l e . T h i s i n c l u d e d both t h e area n e a r t h e ground s u r f a c e and t h a t r e g i o n around the bottom o f t h e f a u l t ; i n b o t h cases t h e f a l l - o f f o f t h e i n t e g r a n d s were everywhere c o n t i n u o u s . However, a n a l y t i c a l e x p r e s s i o n s c o u l d not be o b t a i n e d f o r p o i n t s v e r y c l o s e t o t h e f a u l t f a c e . I n such a r e a s the I.B.M. 7040 computer i n d i c a t e d t h a t the c a l c u l a t i o n s i n v o l v e d exceeded th e machine's c a p a c i t y , thus p r o d u c i n g an o v e r -f l o w c o n d i t i o n . T h i s i m p l i e s t h a t the r e g i o n e x t r e m e l y c l o s e t o t h e s u r f a c e o f d i s c o n t i n u i t y , i s r e p r e s e n t e d by e q u a t i o n s c o n t a i n i n g terms o f the form — — , Xi, - y i where Xt and yc a r e v e r y n e a r l y e q u a l . -30-3.3 Computation of the D i s p l a c e m e n t s and S t r e s s e s We have programmed the I.B.M. 7040 computer a t t h e U n i v e r s i t y o f B r i t i s h Columbia t o e v a l u a t e t h e r e l e v a n t a n a l y t i c a l e x p r e s s i o n s . A l l the n e c e s s a r y programmes were c o n s t r u c t e d i n such a manner t h a t : (a) d, D and L are i n p u t p a r a m e t e r s and t h u s can be v a r i e d a t any time, (b) c a l c u l a t i o n s may be ex e c u t e d at any p o i n t i n the LjL c o - o r d i n a t e space by c o n t r o l l i n g d a t a i n p u t , (c) the v a l u e of t h e s l i p U , may be s p e c i f i e d as a r e l a t i v e o r an e x p l i c i t i n p u t parameter, (d) t h e computer o u t p u t c o n t a i n s a l l the i n f o r m a -t i o n p e r t a i n i n g t o a g i v e n f a u l t model. Because e q u a t i o n s (2.15) and (2.17) a r e sym-m e t r i c a l about t h e p l a n e Lj, = O t h e y were computed o n l y f o r p o s i t i v e v a l u e s o f y,, LJJ. and L j 3 . Two b a s i c approaches were used. The f i r s t e n a b l e d us t o c a l -c u l a t e t h e d i s p l a c e m e n t f i e l d and the s t r e s s changes on the ground s u r f a c e f o r v a r i o u s c o - o r d i n a t e s ( • i a. > y 0 = ( O j L J 2 , , o ) ; the second, t o ge n e r a t e a g r i d of any d e s i r e d d e n s i t y i n any p l a n e LJ,= K p r o v i d e d - 3 1 -t h a t |2. K | < L the s e m i l e n g t h o f t h e f a u l t ( K i s a c o n s t a n t ) . Hence, t h e s e a n a l y s e s a l l o w us t o ob-s e r v e t h e f a l l - o f f o f s t r e s s o r d i s p l a c e m e n t on the ground s u r f a c e a l o n g ^'-O a n d t 0 examine c o n t o u r s o f t h e same q u a n t i t i e s i n any v e r t i c a l p l a n e normal t o the f a u l t t r a c e . B o t h o f t h e s e a s p e c t s a r e i l l u s -t r a t e d l a t e r i n t h i s c h a p t e r as w e l l as i n c h a p t e r s 4 and 5 . Four main programmes were s e t up. These were: (a) p a r a l l e l d i s p l a c e m e n t , u, , (b) p e r p e n d i c u l a r d i s p l a c e m e n t , U a , ( c ) v e r t i c a l d i s p l a c e m e n t , U 3 , (d) maximum shear s t r e s s , S ^ a * The l a s t case i s a p p l i c a b l e t o any g e n e r a l s i t u a t i o n . I t s o u t p u t i n c l u d e s a n a l y t i c a l e x p r e s s i o n s f o r VzzjT^^C,3 , X23 , T„ , and t " 3 3 as w e l l as the c o e f f i c i e n t s and r o o t s o f the c u b i c e q u a t i o n o b t a i n e d by d l a g o n a l i z i n g t h e s t r e s s m a t r i x (see appendix C ) . 3 . 4 P l o t t i n g o f the D i s p l a c e m e n t s I n o r d e r t o compare t h e e v a l u a t i o n s o f equa-t i o n s ( 2 . 1 5 ) w i t h g e o d e t i c o b s e r v a t i o n s made i n the v i c i n i t y o f s u r f a c e f a u l t s , the c o n s t a n t d has been s e t e q u a l t o z e r o . A l s o , s i n c e a v a r i a b l e - s l i p model -32 -w i l l be used t o approximate t h e depths of v a r i o u s f a u l t s two new terms w i l l be d e f i n e d . The f i r s t i s the " e f f e c t i v e d e p t h " of a f a u l t w h i c h w i l l be used t o d e s c r i b e t h a t d e p t h a t which the s l i p u., f a l l s t o "^ T o f i t s maximum v a l u e . The second i s the " t o t a l d e p t h " o f a f a u l t which w i l l r e f e r t o t h a t d e p t h a t which the s l i p u ( has been reduced t o about 2.5$ of i t s maximum v a l u e . Where a p p l i c a b l e , t h e s e v a l u e s w i l l be marked "E" and "T", r e s p e c t i v e l y , on a l l of the graphs i n t h i s t h e s i s . Contour maps of the h o r i z o n t a l d i s p l a c e m e n t , were p l o t t e d f o r t h r e e b a s i c models. S i g n i f i c a n t p r o f i l e s were drawn t o show the v a r i a t i o n i n p r o p e r t i e s e x h i b i t e d by t h e s e . I n a l l of them the d i s p l a c e m e n t s are e x p r e s s e d as p e r c e n t a g e s of the maximum s l i p ^ 3.5 H o r i z o n t a l D i s p l a c e m e n t s Contour maps of the h o r i z o n t a l d i s p l a c e m e n t s , u, p a r a l l e l t o the f a u l t are shown i n f i g u r e s 3.1 t o 3.4. The f i r s t of t h e s e (3.1) was p l o t t e d by C h i n n e r y (1961) who used a c o n s t a n t s l i p model as i n d i c a t e d on the p l o t . I n the f o u r c a s e s p r e s e n t e d l i t t l e v a r i a t i o n e x i s t s i n the shape o f the c o n t o u r s ; r a t h e r , the main d i f f e r e n c e i s i n the r a t e of f a l l - o f f of the d i s p l a c e --33-IQ i l l \ _ F i g u r e 3.1 T h e h o r i z o n t a l d i s p l a c e m e n t u ^ a s a f u n c t i o n o f y2 a n d y-^ i n t h e v e r t i c a l p l a n e t h a t b i s e c t s t h e f a u l t a t r i g h t a n g l e s . A l l c o n t o u r s a r e e x p r e s s e d a s p e r c e n t a g e s o f t h e max imum d i s -p l a c e m e n t £ . 2 C h i n n e r y 1961 y = 0 u , • U d = 0 L = 100 E - T = 0.1L -34-F i g u r e 3-2 The h o r i z o n t a l d i s p l a c e m e n t u^ as a f u n c t i o n o f yo and yo i n the v e r t i c a l p l a n e t h a t b i s e c t s the f a u l t a t r i g h t a n g l e s . A l l c o n t o u r s a r e e x p r e s s e d as p e r c e n t a g e s o f t h e maximum d i s -placement U . Model A y i = 0 L = 100 T = 0.12L -35-Ground Surface F i g u r e 3.3 T h e h o r i z o n t a l d i s p l a c e m e n t u^ a s a f u n c t i o n o f y2 a n d y^ i n t h e v e r t i c a l p l a n e t h a t b i s e c t s t h e f a a l t a t r i g h t a n g l e s . A l l c o n t o u r s a r e e x p r e s s e d a s p e r c e n t a g e s o f t h e max imum d i s -p l a c e m e n t U . 2 M o d e l B yn = o 2 •0.5 u-^ = U e x p v l d E 0 - 0.06L L = 100 T = 0.11L -36-Ground Surface Figure 3.4 The horizontal displacement u-i as a function of y 2 and yo i n the v e r t i c a l plane that bisects the f a u l t at right angles. A l l contours are ex-pressed as percentages of the maximum displace-ment U . 2 y i = o d = 0 E = 0.1L L = 100 T = 0.16L -37-Pigure 3.5 Fall-off of the horizontal displacement u-i at the ground surface along a line that bisects the fault at right angles (y^ = 0 ) . I Chinnery 1961 A Model A B Model B C Model C -38-merits w i t h d i s t a n c e from the f a u l t . F i g u r e 3.5 shows the f a l l - o f f o f t h e d i s -placement, u, w i t h yz a l o n g y, = V J 3 = O f o r each o f the above models. P r o f i l e s o f t h i s k i n d may be used t o e s t i m a t e the depths o f a c t u a l f a u l t s ( C h i n n e r y , 1959). We use such an a n a l y s i s i n c h a p t e r 4 t o approximate the depths o f f i v e f r a c t u r e zones. The r e s u l t s o f the survey a r e t a b u l a t e d i n t a b l e 4.1 so t h a t they may r e a d i l y be compared w i t h t h e e s t i m a t e s o f o t h e r a u t h o r s . 3.6 P e r p e n d i c u l a r D i s p l a c e m e n t s The p e r p e n d i c u l a r d i s p l a c e m e n t s , U;>. a r e o f l i t t l e i n t e r e s t t o us i n t h i s t h e s i s . However, eva-l u a t i o n s o f t h i s d i s p l a c e m e n t component were c a r r i e d out a t l a r g e d i s t a n c e s from the d i s l o c a t i o n s u r f a c e (about ( y . j L j j ^ y , ) = ( 3 L , 3L , o ) ) so t h a t a c o m p a r i -son c o u l d be made w i t h v a l u e s o f t h e U, component. The a n a l y t i c a l r e s u l t s f o r each o f t h e s e components were n e a r l y i d e n t i c a l . T h i s i s t o be e x p e c t e d a t l a r g e d i s t a n c e s from t h e f a u l t model where each o f t h e f i e l d components u, and u-t s h o u l d "see" the d i s l o c a t i o n s u r f a c e as a s i n g l e n u c l e u s . As a r e s u l t , t h e c a l c u -l a t e d r e s u l t s l e a d us t o b e l i e v e t h a t t h e mathematics -39-t o t h i s p o i n t i n t h e a n a l y s i s i s c o r r e c t . However, such a check o f t h e e q u a t i o n s i s the o n l y u s e f u l p u r -pose o f the LKz e v a l u a t i o n . 3.7 V e r t i c a l D i s p l a c e m e n t s The v e r t i c a l d i s p l a c e m e n t s U 3 , were eva-l u a t e d f o r each o f the f a u l t models e x h i b i t e d i n f i g u r e s 3.2 - 3.4. However, t h e r e s u l t s were so s i m i l a r , ( i n c l u d -i n g t h e change o f s i g n t h a t o c c u r s ) t o tho s e o f C h i n n e r y (1961, 1965) t h a t no f u r t h e r m ention o f the U 3 d i s p l a c e -ment components w i l l be made a t t h i s p o i n t . 3.8 P l o t t i n g o f t h e S t r e s s Changes and Maximum Shear S t r e s s Once a g a i n o n l y s u r f a c e f a u l t s were c o n s i d e r e d and a n a l y s e s were c o n f i n e d t o t h e p l a n e Lj, = O . Contour maps o f the s t r e s s changes tiz and f i 3 were p l o t t e d i n the p l a n e y i = 0 f o r t h r e e b a s i c models. These a r e i l l u s t r a t e d i n f i g u r e s 3.6 t o 3.11. F i g u r e 3.12 demonstrates t h e f a l l - o f f o f t h e s t r e s s change T12. a l o n g LJK Cy>- ^3= O) ^ o r t n e a b ° v e models. C a l c u l a t i o n s c o u l d not be made c l o s e t o the d i s l o c a t i o n s u r f a c e ( c f . C h a p t e r 3.2). Hence, the c u r v e s o f f i g u r e 3.12 were e x t r a -p o l a t e d back ( t h e d o t t e d s e c t i o n s ) t o t h e f a u l t p l a n e - 4 0 -t o e n a b l e us t o o b t a i n a p p r o x imate v a l u e s o f the s t r e s s r e l e a s e a t the o r i g i n , f i * . ( C h i n n e r y , 1964). T h i s t e rm must be combined w i t h the s t r e s s change %z i n o r d e r t h a t the p r i n c i p a l s t r e s s e s and hence the maximum shear s t r e s s can be c a l c u l a t e d . d i i s assumed t o be t h e i n i t i a l s t r e s s component t h a t causes f r a c t u r e t o o c c u r . Contour maps o f t h e maximum shear s t r e s s were t h e n p l o t t e d i n t h e p l a n e ^\-0 f o r each o f the a f o r e -mentioned models ( f i g u r e s 3.13 t o 3.15). T h i s a n a l y s i s i s a p p l i e d t o a s p e c i f i c f r a c t u r e zone i n c h a p t e r 5 i n an attempt t o r e l a t e the. o c c u r r e n c e o f a f t e r s h o c k s t o t r a n s c u r r e n t s t r i k e - s l i p f a u l t i n g . A l s o , t h e v a l u e s ob-t a i n e d fbr t h e s t r e s s r e l e a s e a t the o r i g i n f o r each o f the f i v e f a u l t s d i s c u s s e d i n c h a p t e r 4 have been e n t e r e d i n t a b l e 4.1 f o r t h e sake o f compl e t e n e s s . 3.9 R e s u l t s o f the H o r i z o n t a l D i s p l a c e m e n t A n a l y s e s A c a r e f u l e x a m i n a t i o n o f f i g u r e s 3.1 t o 3.5 i n d i c a t e s t h r e e d e f i n i t e t r e n d s . F i r s t o f a l l the shapes o f t h e c o n t o u r s o f each o f t h e f o u r models ana-l y z e d a r e v e r y s i m i l a r . T h i s i m p l i e s (compare f i g u r e s 3.1 and 3.2) t h a t C h i n n e r y ' s u n i f o r m s l i p model, even though i t has a s i n g u l a r i t y a t t h e bottom edge o f t h e d i s l o c a t i o n s u r f a c e , i s a f a i r a p p r o x i m a t i o n t o t h e -41-Ground Surface 1 — 500 Figure 3 . 6 Contour map of the XK component of the stress change i n the v e r t i c a l plane - 0 . Contour units are 1CP dynes/cm 2. Negative Positive Model A , n d = 0 U = 5m E = 10 km L = 100 km T = 12 km - 3 x 10 cgs -42-G round Surface y2 50. / 100 250 500 N X N N N \ \ \ \ / \ s \ V\ \ X \ V \ \ \ \ F i g u r e 3.7 Contour map o f t h e ^ 3 component o f t h e s t r e s s change i n the v e r t i c a l p l a n e s 0. Contour u n i t s a r e 1(P dynes/cm . N e g a t i v e Model A d = 0 U = 5 m E = 10 km L - .100 km T = 12 km a 3 x 10 1 1 cgs -43-F i g u r e 3.8 C o n t o u r map o f t h e T t z c o m p o n e n t o f t h e s t r e s s c h a n g e i n t h g v e r t i c a l p l a n e y i s 0. C o n t o u r u n i t s a r e 1CP d y n e s / c m 2 . N e g a t i v e P o s i t i v e M o d e l B d == 0 T T - E ™ E = 6 km U = 5 m T - 11 km L = 100 km | i 5 3 x . l 0 c g s -44-Ground Surface F i g u r e 3 . 9 C o n t o u r map o f t h e c o m p o n e n t o f t h e s t r e s s c h a n g e i n t h e v e r t i c a l p l a n e y-, = 0. C o n t o u r u n i t s a r e 105 d y n e s / c m 2 . N e g a t i v e M o d e l B U = 5 m d = 0 E = 6 km T = 11 km L = 100 km 11 yju = 3 x IO** c g s -45-Ground Surface F i g u r e 3.10 C o n t o u r map o f t h e "Cn c o m p o n e n t o f t h e s t r e s s c h a n g e i n t h e v e r t i c a l p l a n e = 0. C o n t o u r u n i t s a r e 105 d y n e s / c m 2 . N e g a t i v e P o s i t i v e M o d e l C U = 5 m L = 100 km d = 0 E = 10 km T = 16 km y- - 3 x 10 1 1 c g s - 4 6 -Ground Surface F i g u r e 3.11 Contour map o f the X\s component o f the s t r e s s change i n t h e v e r t i c a l p l a n e y-^ » 0. Contour u n i t s a r e iq5 dynes/cm 2. N e g a t i v e Model |C U = 5 m L = 100 km d = 0 E = 10 km T = 16 km 11 jx - 3 x 10 cgs -47-Pall-off of the stress component with y 2 at the surface of the medium (y^ = 0), along y-^  = 0. - 48 -Figure 3.13 Contour map of the magnitude of the maximum shear stress i n the v e r t i c a l plane y-^  = 0. Contour units are 10^ dynes/cm 2. 0 1 2 Stress release T i z - 4.95 x 10' dynes/cm 0 10 km 12 km 3 x I O 1 1 cgs Model A U - 5 m L = 100 km d E T - 4 9 -F i g u r e 3 . 1 4 C o n t o u r map o f t h e m a g n i t u d e o f t h e max imum s h e a r s t r e s s o n t h e v e r t i c a l p P l a n e as 0 . C o n t o u r u n i t s a r e 10-> d y n e s / c m . 0 8 2 S t r e s s r e l e a s e tix - 1.5 x 1 0 d y n e s / c m . M o d e l B d = 0 E =• 6 km U = 5 m T = 1 1 km L =• 1 0 0 km JJL - 3 x 1 0 1 1 c g s -50-Ground Sur face F i g u r e 3.15 Contour map o f t h e magnitude o f the maximum shear s t r e s s i n the v e r t i c a l p l a n e = 0. Contour u n i t s a r e 10-5 dynes/cm 2. o 8 2 S t r e s s r e l e a s e X\z - 1.9 x 10 dynes/cm Model C d - 0 E - 10 km U = 5 m T = 16 km 11 L = 100 km y-*- - 3 x 10 cgs -51-more p h y s i c a l l y r e a s o n a b l e v a r i a b l e s l i p model. I n f i g u r e 3.2 we have removed t h e s i n g u l a r i t y by a p p l y i n g a v a r i a b l e - s l i p d i s c o n t i n u i t y i n t h i s u n s t a b l e r e g i o n o n l y t o f i n d r e s u l t s t h a t a r e almost i d e n t i c a l t o tho s e o f C h i n n e r y . F i g u r e 3 .5 , however, shows c l e a r l y t h a t some d i f f e r e n c e s e x i s t i n the r a t e o f f a l l - o f f o f the h o r i z o n t a l d i s p l a c e m e n t s f o r each o f the ca s e s c o n s i -d e r e d . Indeed, t h i s c h a r a c t e r i s t i c 'proves v e r y u s e f u l i n o u r depth a n a l y s e s o f s e v e r a l f a u l t s as d i s c u s s e d i n c h a p t e r 4. The d e p t h d e s c r i b e d by C h i n n e r y 1 s model i s d e f i n i t e l y a f i x e d v a l u e . However, t h i s i s not t h e case f o r t h e v a r i a b l e - s l i p models A, B, and C where t h e r e a r e two v a l u e s o f the de p t h w h i c h a r e o f consequence. These a r e t h e e f f e c t i v e d e p t h (E) and the t o t a l depth (T) o f t h e f a u l t . The e f f e c t i v e depth (compare f i g u r e s 3.1 , 3 .2 , and 3.4) shown i n f i g u r e 3.4 c o r r e s p o n d s i d e n -t i c a l l y w i t h the t o t a l d e p th T o f C h i n n e r y 1 s model and the e f f e c t i v e d e pth E o f model A. A l s o , the c o n t o u r s and the r a t e o f f a l l - o f f o f h o r i z o n t a l d i s p l a c e m e n t f o r each o f t h e s e t h r e e cases a r e v e r y s i m i l a r t o one a n o t h e r . The same o b s e r v a t i o n s , however, do not a p p l y t o model B which was c o n s t r u c t e d so t h a t i t s t o t a l d e p t h would c o r -respond t o t h e t o t a l d e p th o f C h i n n e r y ' s model and the -52-e f f e c t i v e d e p t h o f models A and C. I n model B the h o r i z o n t a l d i s p l a c e m e n t , U, f a l l s o f f more r a p i d l y t h a n i n any o f the o t h e r c a s e s . A l s o , i t s c o n t o u r s a r e c o n s i d e r a b l y s h a l l o w e r t h a n t h o s e e x h i b i t e d i n f i g u r e s 3.1, 3.2 and 3.4. W i t h t h e s e c o n s i d e r a t i o n s i n mind we propose t h a t the f a u l t d e p t h s d e t e r m i n e d by C h i n n e r y (1962) ap-p r o x i m a t e o n l y the e f f e c t i v e d e p t h s and not the t o t a l d epths o f any g i v e n d i s l o c a t i o n s u r f a c e i . e . t h e v a l u e s c a l c u l a t e d by C h i n n e r y are a p p r o x i m a t e l y 50$ t o o s h a l l o w . I n t h i s s e c t i o n we have used o n l y a h y p o t h e t i -c a l f a u l t t o s u p p o r t our argument. C h a p t e r 4, however, c o n t a i n s the depth a n a l y s e s o f f i v e a c t u a l d i s l o c a t i o n s u r f a c e s and a t a b l e (4.1) l i s t i n g d e p t h e s t i m a t e s by Kasahara (1958) and C h i n n e r y (1961, 1962) as w e l l as our new e s t i m a t e s d e t e r m i n e d u s i n g a v a r i a b l e - s l i p model. 3.10 R e s u l t s of the S t r e s s A n a l y s e s Contour maps of the tu. and ^ ' 3 components of t h e s t r e s s change i n the p l a n e O are ex-h i b i t e d i n f i g u r e s 3.6 - 3.11 f o r models A, B, and C. A c l o s e e x a m i n a t i o n of f i g u r e s (3.6, 3.8, 3.10) and f i g u r e s (3.7, 3.9, 3.11) shows t h a t t h e shapes of the c o n t o u r s i n each case are v e r y s i m i l a r . On the o t h e r - 5 3 -hand, f i g u r e 3 . 1 2 c l e a r l y i n d i c a t e s t h a t v a s t d i f f e r -ences i n t h e r a t e o f f a l l - o f f o f the s t r e s s com-r ponent do e x i s t . T h i s means t h a t the, s t r e s s r e l e a s e „ o a t the o r i g i n , l\t , which we assume was t h e s t r e s s c a u s i n g the f r a c t u r e t o o c c u r , v a r i e s c o n s i d e r a b l y from one f a u l t model t o a n o t h e r . F u r t h e r m o r e , e x t r e m e l y h i g h v a l u e s o f Tia° may i m p l y t h a t the c o r r e s p o n d i n g f a u l t models a r e p h y s i c a l l y u n f e a s i b l e . I t i s i n t e r e s t i n g t o n o te t h a t f i g u r e s ( 3 . 6 , 3 . 8 , 3 . 1 0 ) , r e s p e c t i v e l y , show a p o s i t i v e s t r e s s d i s -t r i b u t i o n o f d i m i n i s h i n g magnitude about the bottom o f the f a u l t i n the p l a n e y,= o . The f a c t t h a t t h i s d i s t r i b u t i o n i s a p o s i t i v e one i n d i c a t e s i m m e d i a t e l y t h a t the f a u l t w i l l have a tendency t o p r o p a g a t e down-wards i n an e f f o r t t o r e l i e v e the e x i s t i n g s t r e s s . A b r i e f e x a m i n a t i o n o f each o f t h e models d i s -p l a y e d i n d i c a t e s two t r e n d s : ( l ) a g r a d u a l d e c r e a s e of s l i p w i t h d e p t h i s a s s o c i a t e d w i t h a p o s i t i v e s t r e s s d i s t r i b u t i o n ( Xiz ) o f s m a l l e r magnitude, ( 2 ) f o r deeper f r a c t u r e s , much o f t h e s t r e s s must have been r e l i e v e d i n the p r o c e s s o f downward p r o p a g a t i o n . T h i s would ex-p l a i n t h e r e l a t i o n s h i p between the s l i p , d e p t h , and p o s i t i v e ( T\z ) s t r e s s component f o r the models d i s c u s -sed. C o n t o u r s o f the s t r e s s change T o a r e shown - 5 4 -here f o r the sake of completeness and because they are required i n the c a l c u l a t i o n of the maximum shear stress. A physically more meaningful quantity, the maximum shear stress, has been calculated i n the plane y ( * o for models A, B, and C. Contour maps for each of these models are exhibited i n figures 3.13 to 3.15 . They display shapes very s i m i l a r to those of the X\z stress component and are divided into two basic regions. These are the region of stress r e l i e f which borders the ground surface and the region stressed above the breaking strength of the medium (the breaking strength i s the th e o r e t i c a l approximation f i t ). The latter i s centred about the f a u l t base i n a zone i n which a f t e r -shock a c t i v i t y i s often recorded. Based on the above ideas, we propose that a v e r t i c a l s t r i k e - s l i p f a u l t w i l l tend to propagate down-wards i n order to r e l i e v e the pos i t i v e stress about i t s base. In chapter 5 we attempt to associate the a f t e r -shock zone of an actual earthquake with the aforementioned region of high stress. -55-CHAPTER 4 A COMPARISON OF THE THEORETICAL DISPLACEMENT FIELD WITH OBSERVATIONS We s h a l l now a p p l y the g e n e r a l a n a l y s e s o f t h e p r e v i o u s c h a p t e r t o t h e d i s p l a c e m e n t s o f the ground o b s e r v e d around f i v e s u r f a c e f a u l t s . Of c o u r s e no r e a l f a u l t c o r r e s p o n d s e x a c t l y w i t h any t h e o r e t i c a l model. There w i l l always be t o p o g r a p h i c f e a t u r e s , i n h o m o g e n e i t i e s i n t h e s u r r o u n d -i n g medium, i n a c c u r a t e measurements o f t h e v a r i o u s f a u l t p a r a m e t e r s , and s e v e r a l o t h e r f a c t o r s (see Chap-t e r 1.3) t h a t w i l l p r e v e n t us from s i m u l a t i n g an a c c u -r a t e r e p r o d u c t i o n o f t h e a c t u a l p h y s i c a l c o n d i t i o n s . Hence, i n t h e f o l l o w i n g d i s c u s s i o n we s h a l l t r y t o f i t the g e o d e t i c d a t a w i t h s e v e r a l r e a s o n a b l e t h e o r e t i c a l a p p r o x i m a t i o n s r a t h e r t h a n attempt t o g i v e a s i n g l e i n t e r p r e t a t i o n . 4.1 The Source o f G e o d e t i c Data The h o r i z o n t a l d i s p l a c e m e n t s o f t r i a n g u l a t i o n p o i n t s around an earthquake f a u l t p r o v i d e v a l u a b l e I n f o r m a t i o n about t h e p h y s i c a l c o n d i t i o n s o f the f a u l t --56-i n g . Here, we a p p l y the d i s l o c a t i o n t h e o r y o f a v a r i a b l e -s l i p , s t r i k e - s l i p f a u l t t o some a c t u a l cases i n o r d e r t o d e t e r m i n e t h e i r p r o b a b l e d e p t h s . The f o l l o w i n g ana-l y s e s p e r t a i n t o the San Andreas f a u l t , the Gomura f a u l t , the Tanna f a u l t , t h e I m p e r i a l V a l l e y f a u l t , and t h e P a i r v i e w Peak f a u l t . G e o d e t i c data f o r each of t h e s e have been r e p o r t e d i n c o n s i d e r a b l e d e t a i l by s e v e r a l a u t h o r s . 4.2 The San F r a n c i s c o Earthquake f The San F r a n c i s c o earthquake o f A p r i l 18, 1906, i s a s s o c i a t e d w i t h a l o n g s h a l l o w f r a c t u r e known as the San Andreas f a u l t . T h i s f a u l t i s o f prime i n t e r e s t t o us, not o n l y because i t i s s t r a i g h t and d i s p l a y s v e r y l i t t l e v e r t i c a l movement, but a l s o because a c c u r a t e t r i a n g u l a t i o n s were made to d e t e r m i n e the d i s p l a c e m e n t s o f the ground around i t . Two; s u r v e y s a r e r e l e v a n t t o the a c t u a l f r a c t u r e - t h a t o f 1868 b e f o r e the f r a c t u r e zone e x i s t e d and the r e s u r v e y ( o n l y 47 t r i a n g u l a t i o n p o i n t s ) a f t e r the f r a c t u r e o c c u r r e d . A p l o t o f t h e g e o d e t i c d a t a , g i v e n i n f i g u r e 4.1, shows t h a t t h i s t i me i n t e r v a l o f about t h i r t y - e i g h t years, has been l o n g enough f o r a c o n s i d e r a b l e amount o f s t r a i n t o a c -cumulate i n the r e g i o n . However, o t h e r s m a l l r e g i o n a l f a u l t s , the nearby se a , and i n h o m o g e n e i t i e s i n the -57-medium s h o u l d a l s o be c o n s i d e r e d as f a c t o r s c o n t r i b u t -i n g t o the more o r l e s s random movement o f the ground s u r f a c e . 4.3 The Tango Earthquake The Tango earthquake o c c u r r e d on March 1, 1927 on the n o r t h e r n c o a s t o f c e n t r a l Japan. The two f r a c t u r e s produced a t t h i s time were the Yamada and the Gomura f a u l t s . Our i n t e r e s t l i e s o n l y l n the l a t -t e r f o r which 273 t r i a n g u l a t i o n p o i n t s were r e s u r v e y e d . Some o f t h i s d ata have been p l o t t e d i n f i g u r e 4.2 t o -g e t h e r w i t h an e l e v a t e d base l i n e which has been a t t r i -b uted t o the b l o c k movements t h a t have o c c u r r e d d u r i n g t h e time i n t e r v a l between s u r v e y s (about 30 y e a r s ) . Such a c t i v i t y and a l s o some d i p - s l i p movement (about 0.7 meters) i m p l y t h a t t h i s f a u l t d e p a r t s c o n s i d e r a b l y f r om'the i d e a l i z e d d i s l o c a t i o n s u r f a c e t h a t we have been d i s c u s s i n g . However, a p p r o p r i a t e a d j u s t m e n t s may be made so t h a t our a n a l y s i s u s i n g d i s l o c a t i o n t h e o r y i s a p p l i c a b l e . 4.4 The N o r t h I z u E a r t h q u a k e The N o r t h I z u earthquake o c c u r r e d on Novem-ber 26, 1930 i n the n o r t h e r n p a r t of the I z u P e n i n s u l a o f f the s o u t h e r n c o a s t o f Japan. The Tanna f a u l t , a -58-v e r t i c a l but s l i g h t l y convex ( t o the E a s t ) f r a c t u r e was a s s o c i a t e d w i t h t h i s e a r t h q u a k e . I n the v i c i n i t y o f the f r a c t u r e , 81 t r i a n g u l a t i o n p o i n t s were surveyed and r e s u r v e y e d d u r i n g the seven y e a r i n t e r v a l between the time o f t h e Gre a t Kwanto earthquake o f 1923 and the above event o f 1930. The measured d i f f e r e n c e s a r e assumed t o be due e n t i r e l y t o the Tanna f a u l t a l -though t h e r e i s no way o f d e t e r m i n i n g whether o r not o t h e r s e i s m i c a c t i v i t y i n the s u r r o u n d i n g a r e a was a c o n t r i b u t i n g f a c t o r t o the movements. T h i s d a t a has been p l o t t e d i n f i g u r e 4.3 so t h a t i t may be com-par e d w i t h t h e d i s p l a c e m e n t s o b t a i n e d u s i n g a t h e o r e -t i c a l model. 4.5 The I m p e r i a l V a l l e y E a r thquake A p l o t o f t h e movements o f t h e t r i a n g u l a t i o n s t a t i o n s on e i t h e r s i d e o f the I m p e r i a l V a l l e y f a u l t o f May 18, 1940 were t a k e n from a map p r e p a r e d by C. A. W h i t t e n (1949) o f the Coast and G e o d e t i c S u r v e y . Kasa-h a r a ' s ( 1 9 5 8 a ) i n t e r p r e t a t i o n o f t h i s map shows a de-c r e a s e o f h o r i z o n t a l d i s p l a c e m e n t , U, , w i t h d i s t a n c e , from t h e f a u l t . The g e o d e t i c p o i n t s used by Kasahara have been p l o t t e d i n f i g u r e 4.4. I t i s i m p o r t a n t t o note t h a t the f a l l - o f f i s c o n s i d e r a b l y d i f f e r e n t on the two s i d e s o f t h e f r a c t u r e . T h i s would i n d i c a t e - 5 9 -t h a t t h e f a u l t d i p s s t e e p l y ( t o the E a s t ) , As a r e s u l t , depths o b t a i n e d from t h e d a t a on each s i d e o f t h e f r a c -t u r e must be averaged b e f o r e a m e a n i n g f u l v a l u e can be d e t e r m i n e d . 4 . 6 The Nevada Earthquake On December 1 6 , 1 9 5 4 two e a r t h q u a k e s about f o u r m i n u tes a p a r t shook the neighbourhood o f t h e D i x i e V a l l e y , Nevada. A s s o c i a t e d w i t h the l a r g e s t shock was the F a i r v i e w Peak f a u l t w h i c h d i s p l a y e d b oth v e r t i c a l movement and a d i p ( t o the E a s t ) as w e l l as a l a r g e h o r i z o n t a l s l i p . S c a r p s caused by t h e f a u l t i n g and many minor f r a c t u r e s were a l s o p r e s e n t i n the immediate v i c i n i t y . T h i s a c t i v i t y was s u f f i c i e n t t o obscure the t r a c e o f t h e main f a u l t so b a d l y t h a t i t c o u l d be d e t e r -mined o n l y r o u g h l y from the movements o f t h e t r i a n g u l a -t i o n s t a t i o n s . The l a t t e r were p r e s e n t e d by W h i t t e n ( 1 9 5 7 ) and i n t e r p r e t e d by Kasahara ( 1 9 5 8 a ) . Once a g a i n the f a l l - o f f o f h o r i z o n t a l d i s p l a c e m e n t , u, , (see f i g u r e 4 . 5 ) v a r i e s c o n s i d e r a b l y on each s i d e o f the f r a c t u r e and t h u s r e q u i r e s t h a t we make s u i t a b l e com-p e n s a t i o n when c a l c u l a t i n g t h e d e p t h . The f a l l - o f f d i s c r e p a n c y has been a t t r i b u t e d t o a d i p o f t h e f a u l t p l a n e a t an a n g l e o f 6 2 1 / 2 ° t o the E a s t (Romney 1 9 5 7 ) . -6o-% Average V a l u e C h i n n e r y 1959 -- P e t r a k 1965 F a l l - o f f o f h o r i z o n t a l d i s p l a c e m e n t w i t h d i s t a n c e i n t h e San F r a n c i s c o E a r t h q u a k e . - 6 1 -t Horizontal Displacements (in meters) 4-2.0 Due to Block Movements ][ I 30 20 10 0 10 Distance from Fault — » • F i g u r e 4.2 D i s l o c a t i o n S u r f a c e s 20 30 Km L= 20 Km E « 14 Km T=20Km C h i n n e r y 1 9 5 9 P e t r a k 1 9 6 5 F a l l - o f f o f h o r i z o n t a l d i s p l a c e m e n t w i t h d i s t a n c e i n t h e Tango E a r t h q u a k e . -62-Horizontal D isp lacements ( i n m e t e r s ) - 2 . 0 l . i . i __• I , I i I i L_ 15 10 5 0 5 10 l 5 K m < D i s t a n c e f r o m Fault *-F i g u r e 4.3 D i s l o c a t i o n S u r f a c e s L=I2 E= 12 T* 12 n L=I2 Er20 T=26 L= 15 Km E-20 Km T?26 Km C h i n n e r y 1959 P e t r a k 1965 F a l l - o f f o f h o r i z o n t a l d i s p l a c e m e n t w i t h d i s t a n c e i n N o r t h I z u E a r t h q u a k e . -63-I Hor izonta l D isp lacements i ( in m e t e r s ) 1+2.0 \ \ \ \ \ i T l . 5 \ / - 1 . 0 I _ > • \ \ • \ \ \ \ 0.5 3 0 2 0 10 0 10 < — D i s t a n c e f r o m Faul t F i g u r e 4.4 D i s l o c a t i o n S u r f a c e s 2 0 3 0 K m L=40 Km E= 10 Km T=16 Km P e t r a k 1965 F a l l - o f f o f h o r i z o n t a l d i s p l a c e m e n t w i t h d i s t a n c e i n the I m p e r i a l V a l l e y E a r t h q u a k e . -64-+2.0 1.5 / / 1 I / / / / / X / I. Horizontal D isp lacements ( i n m e t e r s ) \ \ \ \ 10 \ \ N 4-0.5 x 3 0 2 0 10 0 10 -s D i s t a n c e f r o m Fault F i g u r e 4.5 D i s l o c a t i o n S u r f a c e s 2 0 Km 3 0 L = 3 0 Km E = 2 0 Km T = 2 6 Km P e t r a k 1965 F a l l - o f f o f h o r i z o n t a l d i s p l a c e m e n t with, d i s t a n c e l n the Nevada E a r t h q u a k e . - 6 5 -4.7 Application of Theory to Observation When ca l c u l a t i n g the displacements using d i s l o c a t i o n theory, each change i n the f a u l t dimen-sions produces some noticeable change i n the a n a l y t i -c a l r e s u l t s . Such an approach i s the only way we have of determining any trends or c h a r a c t e r i s t i c s ex-hibited by a given f a u l t model. In such analyses the mathematician rarely concerns himself with the accur-acy of the geodetic data which he uses as a basis for his model. Rather, he tends to become completely i n -volved i n an examination of the t h e o r e t i c a l r e s u l t s . This s i t u a t i o n was not allowed to develop i n our analyses of the various f a u l t models. Care was taken to remember that such parameters as the f a u l t semi-length L and i t s s l i p U can be measured only approximately i n the f i e l d . For instance, i n resurveying the ground surface a base l i n e must be chosen on the assumption that i t has not been affected by the disturbance. This of course i s never j u s t i f i e d since the resurvey must include the area very close to the fracture. As a r e s u l t , errors i n both the magnitude and the d i r e c t i o n of the observed displace-ments must be expected. - 6 6 -The measurements o f t h e l e n g t h 2L o f some o f t h e f i v e s u r f a c e f a u l t s examined were o n l y rough e s t i m a t e s . F o r example, i t has been suggested t h a t the San Andreas f a u l t a c t u a l l y extends north-westwards i n t o t h e P a c i f i c , and t h a t the I m p e r i a l V a l l e y f a u l t may ex t e n d w e l l i n t o M e x i c o . The l a t t e r i d e a i s sup-p o r t e d by t h e f a c t t h a t t h e maximum f a u l t d i s p l a c e -ment o c c u r s near t h e Mexican b o r d e r . Such d i s c r e p a n c i e s become e v i d e n t f rom a t h e o r e t i c a l v i e w p o i n t on an e x a m i n a t i o n o f the u, f a l l - o f f c u r v e s produced by a v a r i a b l e - s l i p model r e p -r e s e n t a t i o n o f the Tanna f a u l t ( L ~ D, K a s a h a r a ) . I n t h i s i n s t a n c e , they showed v e r y l i t t l e s i m i l a r i t y t o the g e o d e t i c r e s u l t s u n t i l we i n c r e a s e d the l e n g t h o f the model by about 75$ ( k e e p i n g t h e depth c o n s t a n t ) , i n d i c a t i n g t h a t a g r o s s e r r o r i n t h e measurement o f the l e n g t h , 2L, o f t h e f a u l t i s i n h e r e n t i n t h e d a t a . Such a s i t u a t i o n i s i n d i c a t i v e o f some o f t h e problems t h a t a r i s e i n a comparison o f t h e o r y w i t h o b s e r v a t i o n . To make t h e s e m a t t e r s even more c o n f u s i n g , we have no means o f o b t a i n i n g a c t u a l d e pth measurements o f any d i s l o c a t i o n s u r f a c e l o c a t e d i n the e a r t h ' s c r u s t . - 6 7 -4 . 8 I n t e r p r e t a t i o n o f th e F a l l - o f f Curves I n f i g u r e s 4 .1 - 4.5 we have superimposed on t h e g e o d e t i c d a t a s e v e r a l t h e o r e t i c a l f a l l - o f f c u r v e s . Among t h e s e ( f i g u r e s 4 .1 - 4.3) a r e the ap-p r o x i m a t i o n s o f C h i n n e r y (1959) c a l c u l a t e d from h i s c o n s t a n t - s l i p model. No d i s c u s s i o n o f a c t u a l d e p t h measurements w i l l be g i v e n i n t h i s s e c t i o n . They have been l i s t e d , however, i n t a b l e 4 .1 a l o n g w i t h p r e v i o u s e s t i m a t e s p r e p a r e d by C h i n n e r y (1962) and KaSahara ( 1958) . From a g e n e r a l a n a l y s i s o f t h e t r e n d s e x h i b i t e d i n f i g u r e s 4 .1 - 4 . 5 , s e v e r a l changes i n d e p t h e s t i m a t e s a r e e v i d e n t . Those v a l u e s a p p r o x i m a t e d by C h i n n e r y (1962) a r e g e n e r a l l y 40 - 60$ l e s s t h a n our e f f e c t i v e f a u l t d e p t h s and about 100$ l e s s t h a n o ur t o t a l d e p t h s . These d i s c r e p a n c i e s a r e t o be e x p e c t e d , however, s i n c e t h e tke . c o n s t a n t s l i p model does i n v o l v e t h e movement of„most ground. Kasahara (1958), on the o t h e r hand, has produced e s t i m a t e s ( e x c e p t i n t h e case o f th e Tanna f a u l t ) e q u i -v a l e n t t o our e f f e c t i v e d e p t h s . Such s i m i l a r i t i e s a r e not s u r p r i s i n g because Kasahara's model, as our own, e x h i b i t s d i s p l a c e m e n t s w h i c h f a l l o f f w i t h d e p t h . Those depths suggested f o r the San Andreas, I m p e r i a l V a l l e y , and F a i r v i e w Peak f a u l t s by B y e r l y and De Noyer (1958) ( t h e i r c a l c u l a t i o n s were based on a c o n s i d e r a t i o n o f -68-l o I i 0 •P t 0 0 >o a s? 3 s. J n * a> O X VS r -O X o •< CO In 1 O cn to o X cn 00 O X 1 o X « o X rsl O 1 o0 CO I >S rvl v9 A O rsl o A O O $ O CO 00 O o CO in O v9 O oo In rxi f^ -* • nj oi • Vn Vo • %s o-V *> £ i. • o V £ o - j "3 O —_j • s h "55 V »? £ <* ^ a) O V CD o V # o * \ i I o 00 o V o V O 00 o p-o cn ^ <2 :° u> o o al 9J 0 t4 — 1 .—1 _ - * "o ,3 r o p~ o ro o co 1 o v s 00 V0 Q •x CO o V o X ex? i n In I In O o i o A vS o J « » * o x t xr 8 O OO O CO O i \ 8 lo O x9 In In * rvj in In • in d Q) OJ V7 , 00 o * CD o In O # <P O X • O X o lo 00 CD O O O no O N 00 ° CO q In 4 / i / d to -d -M ^ 1/7 o S 0 r-3 h JD o « n t T - r s i * u _ i -+* ^ > i If) - 6 9 -s t r a i n e n e r g i e s and earthquake i n t e n s i t i e s ) agree v e r y w e l l w i t h our t o t a l d e p t h e s t i m a t e s i n a l l t h r e e c a s e s . However, we have no way o f knowing whether o r not t h i s agreement i s a c o i n c i d e n c e or r e a l i t y , e s p e c i a l l y s i n c e B y e r l y and De Noyer s t a t e "The v a l u e o f 10 Km was d i s t a s t e f u l t o u s , but the r a p i d d i e o f f o f i n t e n s i t y from t h e f a u l t seems t o demand a s h a l l o w break" ( t h e r e f e r e n c e here i s t o the San Andreas f a u l t ) . The s i m i l a r i t y o f t h e s e r e s u l t s t o our own r e m a i n s , n e v e r t h e l e s s , and s h o u l d be i n v e s t i g a t e d f u r t h e r . -70-CHAPTER 5 A COMPARISON OF THE THEORETICAL STRESS  DISTRIBUTION WITH OBSERVATION I n t h i s s e c t i o n we a p p l y t h e g e n e r a l ana-l y s e s o f Cha p t e r 3 t o a s p e c i f i c g e o d e t i c s i t u a t i o n . The c o n t o u r s o f maximum shear s t r e s s d i s p l a y e d t h e r e l e a d us t o e n q u i r e whether o r not t h e r e i s any connec-t i o n between them and a f t e r s h o c k s . The Rat I s l a n d E a r t h q u a k e ^ (-off t h e A l a s k a n c o a s t ) i s used t o i n v e s t i -g a t e t h i s p o s s i b i l i t y . A l t h o u g h no d e f i n i t e r e s u l t s a r e o b t a i n e d from t h i s a n a l y s i s , f u r t h e r work may r e v e a l some m a t t e r s o f i n t e r e s t . S i n c e t h e sea c o v e r s most o f t h e a f f e c t e d r e g i o n , t h e e x a c t p o s i t i o n o f the f a u l t t r a c e cannot be l o c a t e d . F u r t h e r m o r e , n e i t h e r the f a u l t t y p e n o r i t s r e l a t i v e d i m e n s i o n s have been measured. Thus the f o l l o w i n g comparison o f t h e o r y and o b s e r v a t i o n must be t r e a t e d as a h y p o t h e s i s r a t h e r t h a n as an e x a c t i n t e r -p r e t a t i o n o f t h e a v a i l a b l e g e o d e t i c d a t a . 5.1 The G e o d e t i c Data The U.S. Coast and G e o d e t i c Survey l o c a t e d 870 a f t e r s h o c k s f o l l o w i n g t h e Rat I s l a n d earthquake -71-o f F e b r u a r y 4, 1965. The e p i c e n t r e s ( f i g u r e 5.1) are l o c a t e d between t h e a x i s o f the A l e u t i a n t r e n c h and the i s l a n d c h a i n i n the r e g i o n of the A l e u t i a n I s l a n d a r c and f o r m an a p p r o x i m a t e l y r e c t a n g u l a r zone about 650 km l o n g and 200 km wide. I n t h i s r e g i o n b o t h tem-p o r a r y and permanent seismograph s t a t i o n s were used t o d e t e r m i n e t h e h y p o c e n t r e s of t h e main shock (about 40 km i n depth) and twenty a f t e r s h o c k s . The l a t t e r have been p l o t t e d i n p l a n a l o n g w i t h a h y p o t h e t i c a l f a u l t t r a c e on f i g u r e 5.2 and i n e l e v a t i o n on f i g u r e s 5.3 and 5.4. The p o s i t i o n of the f a u l t t r a c e drawn on f i g u r e 5.2 was not chosen i n an a r b i t r a r y manner. R a t h e r , a f t e r s h o c k r e g i o n s d i s p l a y e d by R i c h t e r e t a l . 1958, Tocher 1956, and P l a f k e r 1965 were c o n s u l t e d . Each of t h e s e s i t u a t i o n s showed an a r r a y of a f t e r s h o c k s , the p o s i t i o n of the main shock, and the r e l e v a n t f a u l t t r a c e . I t was n o t i c e d t h a t t h e a f t e r s h o c k zone was d i v i d e d i n a r a t i o of about 1:2 w i t h the main e p i c e n t r e l y i n g n ot on the f a u l t t r a c e but r a t h e r n e a r i t a t one end o f t h e zone. These o b s e r v a t i o n s were t h e n a p p l i e d t o t h e Rat I s l a n d e a r t h quake so t h a t i t s f a u l t t r a c e c o u l d be drawn i n a p h y s i c a l l y s i g n i f i c a n t l o c a t i o n . The l e n g t h and depth o f t h i s f r a c t u r e were assumed t o c o r r e s p o n d r e s p e c t i v e l y , w i t h t h e e x t e n t of t h e a f t e r -- 7 2 -shock zone and th e depth l o c a t i o n o f th e main shock. A h o r i z o n t a l s l i p , U was t h e n a r b i t r a r i l y chosen i n o r d e r t o c a r r y out model c a l c u l a t i o n s r e p r e s e n t a -t i v e o f the g e o d e t i c s i t u a t i o n . I n f i g u r e 5 « 3 the h y p o c e n t r e s o f th e twenty a f t e r s h o c k s noted i n f i g u r e 5 - 2 have been p l o t t e d on top o f a c o n t o u r map o f the maximum shear s t r e s s i n the v e r t i c a l p l a n e t - j i -O . C a l c u l a t i o n s o f the maximum shear s t r e s s were a l s o performed i n t h e p l a n e s y,= 0.75L and tj, = 0.95L. S i g n i f i c a n t changes i n the c o n t o u r shapes and p o s i t i o n s were n o t i c e d o n l y i n t h e l a t t e r p l a n e which i s i l l u s t r a t e d i n f i g u r e 5 . 4 a l o n g w i t h the twenty h y p o c e n t r e s . F a u l t models o f o t h e r d e p t h s , s l i p s , and l e n g t h s were a l s o examined but t h e r e s u l t s showed no r e l e v a n t changes. 5 . 2 The R e l a t i o n s h i p Between t h e A f t e r s h o c k Zone and Maximum Shear S t r e s s The c o n t o u r s o f maximum shear s t r e s s whose magnitudes a r e g r e a t e r t h a n the s t r e s s r e l e a s e L \% exceed the e s t i m a t e d b r e a k i n g s t r e n g t h o f the s u r r o u n d -i n g medium ( 3 . 8 3 x 10 c.g.s. i n t h i s c a s e ) . I t w i l l be assumed t h a t t h e downwards p r o p a g a t i o n o f the f r a c -t u r e mentioned i n S e c t i o n 3.10 w i l l t end t o r e l e a s e - 7 3 -^ A T T U L ^ FEBRUARY 4, 1965 EARTHQUAKE A N D AFTERSHOCK SEQUENCE RAT ISLANDS, ALEUTIAN ISLANDS THROUGH MARCH 20 (45 DAYS) •—v- " ^ • ^ • - : ' j ^ ^ r * * ™* 7 • • * ~—; • • • -mi • • <* U.S. COAST AW M0OCTIC SUSVtY F l Q 5 I M a p °* e a r t n c i l i a k e o n 4 F e b r u a r y 1965, earthquake and aftershock sequence of the Rat Islands, Aleutian Islands. 3 -74-Ground Surface F i g u r e 5.3 Contour map o f t h e magnitude o f the maximum shear s t r e s s i n t h e v e r t i c a l p l a n e y-^  = 0. Contour u n i t s a r e 105 dynes/cm 2. o T P S t r e s s r e l e a s e X,z - 3.83 x 10' dynes/cm . Model D U - 10 m d - 0 E - 30 k m T = 46 k m L = 325 km - 3 x 10 cgs -75-Ground Surface F i g u r e 5 .4 Contour map o f the magnitude o f the maximum sh e a r s t r e s s i n t h g v e r t i c a l p l a n e y, = O.95L. Contour u n i t s a r e 10-3 dynes/cm 2. o 7 2 S t r e s s r e l e a s e T|i = 3 .83 x 10' dynes/cm Model D d E U = 10 m T L = 325 km L = 0 = 30 Km = 46 K m 11 = 3 x 10 cgs -76-t h i s s t r e s s b u i l d up i n t h e form o f a f t e r s h o c k a c t i -v i t y . As a r e s u l t , a l l t h e a f t e r s h o c k s would be ex-p e c t e d t o l i e w i t h i n t h e h i g h l y s t r e s s e d r e g i o n about the f a u l t base. I t i s q u i t e e v i d e n t t h a t t h i s p r o p o s a l does not a p p l y t o the' g e o d e t i c s i t u a t i o n shown i n f i g u r e 5.3 ( y> = O ) where a p p r o x i m a t e l y o n e - t h i r d o f the h y p o c e n t r e s a r e l o c a t e d w i t h i n t h e h i g h l y s t r e s s e d r e g i o n . However, t h i s i s not t h e case i n the p l a n e 0 . 9 5 L where about 8 0 $ o f the a f t e r s h o c k s l i e i n t h e d e s i r e d p o s i t i o n ( f i g u r e 5 . 4 ) . There a r e s e v e r a l p r o b a b l e e x p l a n a t i o n s why t h e a f t e r s h o c k h y p o c e n t r e s do not a l l l i e i n t h e h i g h l y s t r e s s e d r e g i o n . Indeed, the f a u l t may not be o f the s t r i k e - s l i p v a r i e t y . T h i s i s i n d i c a t e d by the f a c t t h a t fewer a f t e r s h o c k s appear on one s i d e of t h e main e p i c e n t r e t h a n on t h e o t h e r i . e . the r e g i o n t o the n o r t h - e a s t ( f i g u r e 5 . 2 ) appears t o be i n c o m p r e s s i o n w h i l e t h a t t o t h e south-west appears t o be i n t e n s i o n . A s i t u a t i o n s i m i l a r t o t h i s a r o s e from t h e A l a s k a n e a r t h quake o f March 27, 1 9 6 4 ( P l a f k e r , 1 9 6 5 ) where v e r t i c a l d i s p l a c e m e n t s o f up t o 10m were measured. I f t h e same e x p l a n a t i o n does not a p p l y t o t h e Rat I s l a n d e a r t h q u a k e , t h e n th e f a u l t must extend w e l l beyond the a f t e r s h o c k zone and e x h i b i t b o t h a -77-v a r i a b l e h o r i z o n t a l as w e l l as a v a r i a b l e v e r t i c a l s l i p . A h o r i z o n t a l s l i p o f t e a r d r o p shape would be a r e a s o n -a b l e d i s c o n t i n u i t y ( C h i n n e r y , p r i v a t e communication) s i n c e i t would r e p r e s e n t a l a r g e d i s p l a c e m e n t i n the r e g i o n near the main e p i c e n t r e which would d i m i n i s h w i t h d i s t a n c e from the i n i t i a l shock. I t i s a l s o e v i -dent by comparing f i g u r e s 5»3 and 5.4 t h a t d i s l o c a t i o n t h e o r y s u p p o r t s t h i s p r o p o s a l - most o f the a f t e r s h o c k s l i e i n t h e r e g i o n o f h i g h s t r e s s o n l y n e a r the end o f t h e f a u l t ( f i g u r e 5.4) and not i t s c e n t r a l p l a n e ( f i g u r e 5.3). T h i s would a l s o i n d i c a t e t h a t t h e a f t e r s h o c k sequence c o u l d be a s s o c i a t e d w i t h the s t r e s s r e l e a s e d when the f r a c t u r e extends i t s e l f l e n g t h w i s e . Any d e f i n i t e c o n c l u s i o n s i n t h i s r e s p e c t , however, would r e q u i r e a c o n s i d e r a b l e amount o f work i n c l u d i n g the r e c o n s t r u c t i o n o f the s p e c i f i c e q u a t i o n s f o r t h e d i s -placement f i e l d and s t r e s s changes. -78-CHAPTER 6 CONCLUSIONS Many a s p e c t s o f g e o p h y s i c s d e a l w i t h i n a c -c e s s i b l e a r e a s such as the depths o f the i n t e r i o r o f t h e e a r t h , t h e o r e t i c a l l y a c c e s s i b l e r e g i o n s o f which a r e o n l y p a r t i a l l y c o v e r e d by g e o d e t i c o b s e r v a t i o n s . No m a t h e m a t i c a l model e v e r r e p r e s e n t s r e a l i t y , y e t , g i v e n a c e r t a i n s e t o f a s s u m p t i o n s , i t i s u s u a l l y pos-s i b l e t o e s t a b l i s h a v e r y c l o s e agreement between t h e o r y and o b s e r v a t i o n by v a r y i n g the p a r a m e t e r s . There a r e u s u a l l y s e v e r a l models t h a t can be j u s t i f i e d t h e o r e -t i c a l l y but o f t h e s e o n l y one approximates r e a l i t y . The problem o f e l i m i n a t i n g t h e m u l t i p l i c i t y o f p o s s i b i l i t i e s and f i n d i n g t h e one model t h a t conforms t o a l l a v a i l a b l e g e o d e t i c d a t a i s i d e a l l y s u i t e d t o the modern methods o f computer a n a l y s i s . The computer can accommodate the o b s e r v a t i o n s and s i m i l a t e t h e p r o c e s s e s g i v e n by the p e r t i n e n t e q u a t i o n s . I t can p e r f o r m t h e s e o p e r a t i o n s v e r y r a p i d l y , and I n c o r r e c t s o l u t i o n s t h a t f a i l t o con-form t o r e a l i t y can be improved I n a r e a s o n a b l e time by amended e q u a t i o n s , by new d a t a o r by b o t h . These arguments, t o g e t h e r w i t h the r e s u l t s o f t h e p r e v i o u s two c h a p t e r s , i n d i c a t e t h a t our attempt t o r e p r e s e n t -- 7 9 -t h e n a t u r a l p r o c e s s e s o f f a u l t i n g , u s i n g d i s l o c a t i o n t h e o r y as a g e n e r a l m a t h e m a t i c a l model, i s r e a s o n a b l e . The use o f S t e k e t e e ' s " E l a s t i c i t y Theory o f D i s l o c a t i o n s " r e s t r i c t s our a n a l y s e s t o m a t e r i a l s which obey Hooke's Law f a i r l y c l o s e l y . T h i s I n d i c a t e s t h a t t h e n o n - e l a s t i c o r p l a s t i c e f f e c t s a s s o c i a t e d w i t h the p r o c e s s o f f r a c t u r e a r e c o n f i n e d t o a c o m p a r a t i v e l y s m a l l zone i n t h e immediate v i c i n i t y o f the f a u l t , and t h a t t h i s zone may be r e p r e s e n t e d by a d i s l o c a t i o n s u r f a c e . The medium, o f c o u r s e , i s assumed t o c o n s i s t o f a homogeneous i s o t r o p i c m a t e r i a l t h a t i s semi-i n f i n i t e w i t h r e s p e c t t o the f r a c t u r e zone. The t h e o r y , however, does have l i m i t a t i o n s o t h e r t h a n t h o s e mentioned above and t h o s e d i s c u s s e d i n C h a p t e r 1. The h o r i z o n t a l s l i p U, was assumed t o v a r y o n l y i n the v e r t i c a l and not i n the h o r i z o n t a l p l a n e . A l s o , the s t r e s s r e l e a s e a t the o r i g i n , 1-12. i s s t r i c t l y a t h e o r e t i c a l a p p r o x i m a t i o n . No p h y s i c a l measurements o f t h i s f r a c t u r e - c a u s i n g s t r e s s have e v e r been made. Hence, i t i s i m p o s s i b l e t o account f o r t h e r e a s o n s o f t h e i n i t i a l f r a c t u r e and the magnitude o f t h e s t r e s s t h a t cause i t . -80-N e v e r t h e l e s s , as l o n g as we c o n s i d e r t h e s e l i m i t a t i o n s i n our i n t e r p r e t a t i o n o f the n u m e r i c a l r e s u l t s , we a r e j u s t i f i e d i n e x p e c t i n g t h e t h e o r y t o p r o v i d e much I n f o r m a t i o n about the d i s p l a c e m e n t f i e l d and s t r e s s changes produced by f r a c t u r i n g . Such know-le d g e a l l o w s us t o e s t i m a t e t h e depth o f a g i v e n f r a c -t u r e and t o examine t h e c o n t o u r s o f t h e maximum shear s t r e s s i n t h e d i s t u r b e d r e g i o n i . e . i t a l l o w s us t o p r e d i c t a p l a u s i b l e a f t e r s h o c k sequence a s s o c i a t e d w i t h a new f r a c t u r e zone. P o s s i b l y , t h i s approach can be de v e l o p e d f u r t h e r so t h a t t h e f o r e s h o c k sequence and the p r e d i c t i o n o f ear t h q u a k e s may be f a c i l i t a t e d . F i n a l l y , t h e r e i s an e x t e n s i o n o f the t h e o r y w hich may be o f s p e c i a l i n t e r e s t v i z . t h e c o n s i d e r a t i o n ' o f a d i s l o c a t i o n s u r f a c e t h a t e x h i b i t s b o t h v a r i a b l e v e r t i c a l and h o r i z o n t a l s l i p . W i th t h e t h e o r y a l l o w i n g f o r b oth o f t h e s e f e a t u r e s , the a n a l y t i c a l r e s u l t s might be a b l e t o suggest a p o s s i b l e c o n n e c t i o n between s t r i k e -s l i p f a u l t i n g and mountain b u i l d i n g ( C h i n n e r y , 1965). I f such a r e l a t i o n s h i p c o u l d be s u b s t a n t i a t e d by a c t u a l g e o l o g i c a l f e a t u r e s , the im p o r t a n c e o f the use o f d i s -l o c a t i o n t h e o r y i n a s s o c i a t i o n w i t h t e c t o n i c changes i n t h e e a r t h ' s c r u s t would be f u l l y r e a l i z e d . -81-APPENDIX THE EVALUATION OF THE DISPLACEMENT FIELD  AND THE STRESS CHANGES We have to determine the displacement f i e l d U L K and the stress changes tcj at points P ( y , ^cj3) due to a d i s t r i b u t i o n of nuclei at points Q(x,,0,X 3) on the d i s l o c a t i o n surface shown i n figure 2.1. The r e s u l t i n g algebra i s very involved and tedious. To simplify the expressions, the following terms w i l l be used, p = *s +• ya V x » - ^ t « X, - y , Table A . l contains an array of expressions that undergo s p e c i f i c transformations when d i f f e r e n -t i a t e d with respect to y c { L ~ 1^2,3). In p a r t i c u -l a r we f i n d the following three relationships, ^2K = t AK "by, ^ > y a -82-i l * . -Z w h i c h a r e v a l i d f o r b o t h s t a r r e d and n o n - s t a r r e d v a l u e s o f Z, k, B, and C. A second t a b l e (A.2) has a l s o been c o n s t r u c t e d t o i l l u s t r a t e a s i m i l a r s e t o f t r a n s f o r m a -t i o n s . I n t h i s case two b a s i c r e l a t i o n s h i p s e x i s t f o r b o t h s t a r r e d and n o n - s t a r r e d v a l u e s o f I , J, K} and L, v i z . : The f o l l o w i n g two r e l a t i o n s h i p s a r e a l s o a p p l i c a b l e t o t a b l e (A.2), I -83-A. THE DISPLACEMENT FIELD From equations (2.12), u K is given by (1) r K - 8TT I , i u , (x s ) dx, d x 3 (2) The Galerkin vector i " ^ for a nucleus at the point ( X, , o , x3 ) is given by the following expressions: (3) and ( S ^ p ) * 2.C - 2(bp+c^) - ( p * - c f ^ 2 S t i - p ) where b = — 7\ + |x to b = 0 and c which reduce and C= — ^ - 1/4 on the assumption that 7v = We now define a term £K such that TABLE (l) OF RELATIONSHIPS 2,= ft - 1  b ' " 2(5x1-p)* r i Bz-"' S^S^p) 2-r . B s^+^pSii - S p 2 -A * = s f B * - -IF ^ f A - % B s si z 2(S,*q,) ft* =. ~ ' r * _ - ( 3 s, + a) A * - 1 Q * _ 2 s , t q , r * . 8S,2 r 9qS, *3q/-3 S, 5(S,rq ) 3 B * " s ? r * - -lS s? s, 7 - — s T EV tke above tctUe opera.tio.aS -<bAl = t B . . ^ = t g? . ^ A c _ - ^ B ' j - y*B* ! ^ -"Ait, .-Ait, of tWe f0tt°wuag forvw a.re valid; "dy, "d^i C)y2 £y a ^ 3 T A B L E A.I TABLE (II) OF RELATIONSHIPS I,-- X^ U C s^tt)- 2. ( s^ t ) - s t * t U(s4+t) T - 1 1 . 3 S i f t i+ s2 J - 3t kc - ^ , /os t u * si U " s/ s? r,* = ^ - . ^ U s , . t ) - a t t t ) *" " £(s, +t) 2- L, - s , (s l +t) 3 J * " s, + t k + - 1 , * _ zs, +-t L z - ' s,3Cs,+t)3 i s * a -U(s,+t) T*= 1 ^ s,»(s,+t)» 1 * - 8S,1 + 9ts, +3tz l* s, J* ~ s,3 k* - 1 - L*" sr r ? " s? k* = ' 5 t -r * - 3(5tZ-S?) S," I n t k e above t a t k o p e r a W s ^ j - ^ . ^  j* _ _ > K . . K?} BE = -pKi 3 ^ J f . q k l o-f tke ^ roilowtiacj fov-m. are valid; ^y, "by,. c)yz cly^ "dy5 TABLE A . £ - 8 6 -(4) This implies that the displacement f i e l d u K can be expressed as where 1. The components £ K Prom equation (A.4) we have -(_ L LnJdx, = n^ L L»;-z 4)dx, = / * J ^ t ( H + - Z ; ) d x , (5) ( 6 ) (7) - 8 7 -and s / ^ * ( - A , * ^ A a + 4 x 3 L ) 3 A s ) ( 8 ) where f(x,,o,x 3) = f ( L, O, x 3) - f (-L , 0,X 3) whenever the symbol appears. 2 . The derivatives of € K We shall now proceed to evaluate the deriva-tives £K,U and 6 1 , C k . which are required in equa-tion (A.5b). a) The component fe,,,, = )^\jzt (A4-A4*)|| 6>>2 = -/^ [Lr\Cs,+t)(Sa+t)t^(j3i-T3*)]|| ( 9 ) € z z = y z [-3 (J 3 tT3*) 1- yI (L, +L: + J a S J,**)] I -88-b) The € 2 component •4 (10) = -jj.t ( Z 4 -Z 4) ZW/JL [ C 2 + - 2 4 ) - t H A 4 - A 4 * ) ( z 4 - 2 4 ) _ € t > 3 3=^ [A4(s*-p*)-A4 (sf + cf.) c) The £3 component ^ 3 , 1 = j ^ t ( " B , 2^<^ B;, + - 4 X 3 L J 3 B 3 ) ^ 3 , i « * ^ y * j t * ( - 0 2 c j C i + 4 x 3 y 3 C 3 ) - ( - B 1 + 2 ^ B 2 + 4 x 3 y 3 B s ) = ^ [ - A ^ q A * + 4 * 3 ^ 3 - y l (-B, + 2 q ^ i - 4 x 3 y 3 B 3 ) £ 3 ^ = ^ [ - 3 ( - B , t 2 q B 2 H x 3 y 3 B 3 ) + y £ ( - C + ^ C * 4 - 4 x 3 y 3 C 3 ) e 3, 3 = jLxyj, (-A2. +• 2 p A 3 - 4 x 3 y 3 A 4 ) 3 A 3 - 2 A 4 ( p i - Z x 3 ) ^ 4 x 3 y 3 A 5 V % - 4 ^ y a ( A 3-2x 3A 4) - 8 9 -d) The divergence duv 6 = 61,1 €.4,1 f 63,: 2 ( 7 * - £ W W p A 3 - 4 x 3 y 3 A 4 (12) e) The term Grad K (div £ ) Grad.Cdiv e ) = ^ L j z t ( z A 4 - ^ A 4 - B z t2pB 3 - 4 x 3 L j 3B 4) Grad* ( c k v [ A i - ^ H ; + 4 p A 3 - 4 x 3 y 3 A 4 - y | ( B l - 2 A ; + 4 p ^ -4X3LJ3BJ G r a d 3 ( d i v e) = u u z hH - 2 2 f f +3A3 ~ 2 A , ( p t £ x 3 ) r4x3^ A * (13) 3. The formulation of the terms lOk We can now determine the expressions by applying the r e s u l t s of the previous section (appendix A.2) to equation (A.5b). The re s u l t i n g terms are £ | A L J J T 3 + T 3 * ) - § / * y * t ( A 4 f 3 p B 3 - Z A ; - 4 x 3 L j 3 B 4 ) uu 7 - *(2 4 +2*)- f ( A z - a 2 4 ^ 4 P A 3 - 4 x 3 y , A 4 ) + f y ! ( A 4 r 3 P B 3 - a A ; - 4 x 3 y 3 B 4 ) - 9 0 -a n d ( 1 4 ) F o r t h e s a k e o f b r e v i t y , i t i s a d v a n t a g e o u s t o d e f i n e f o u r new e x p r e s s i o n s , v i z . : F, = J3 + J3* F t - A 4 - M 4 +-3pB 3-4^y3B 4 F 3 -- Z(\z - Z l - p A 3 ^-4x 3y 3A 4 F 4- 3A3 + 2A 4(y s-3x 3)~4x 3y 3 A 5 +Z(2s-2?) 4 . T h e a c t u a l d i s p l a c e m e n t f i e l d s (15) B y s u b s t i t u t i n g t h e e x p r e s s i o n s f o r u->\ , <*->t a n d t o 3 i n t o e q u a t i o n (A .5a) we o b t a i n t h e t h r e e d i s -p l a c e m e n t s u , = - ^ r D u , ( x 3 ) y a ( F , . t i J d x 3 D IZTT a n d U 3 = u,Cxs)(F31 y£ Fe)dx3 I u, (x3) y* F 4 dx 3 J d (16) -91-which can be evaluated very easily with the aid of an electronic computer (an I.B.M. 7040 was used in our case). -92-B. THE STRESS CHANGES From equation (2.17), the changes i n stress are given by We s h a l l now evaluate the derivatives L/JK,K 5 uj£,j and ^Jjii. which are required to formulate these equa-tions e x p l i c i t l y . 1. The derivatives of oU*. \ a) The tOx component 3 ( T 4 + X 4 * ) - A 4 + 2 At - 3 p B 3 ^ - 4 x 3 y 3 B * f t z ( B 4 - £ B 4 * i - 3 p C 3 - 4 X 3 L J 3 C 4 ) cu.,* ^ -ZJA T3 +• J 3 * - y^ 2 ( Ka r k3*) (2) | [ ( A 4 - 2 A ; + 3 p B 3 H x 3 y 3 B 4 ) - y f ( B 4 - ^ B 4 * + 3 p C 3 - 4 x 3 y 3 C -uo,,3 « 2 / X L J z [ p K 3 - ^ K t - | ( - A 5 ^ 2 A f f + f 3 B 3 - B 4 ( 3 p + 4* 3) + 4 X 3 L J 3 B S ) J - 9 3 -b) The component W i , , = |/At ZBa-Aj-pB 3 ^ 4x3y3B4 + yUB4-2.Bl+3pC3-4x3ij3C4) 4* o j i j 2 . = - -|^<y* [2.Bz-A4-pB3t4x3y3B4_ -- 2 ( A 4 - 2 A 4 % 3 p B 3 - 4 x 3 y 3 B 4) + y i ( & * - 2 * 3 p C 3 - 4*9 ysC*) |x [2 5 -3A3 + A4(p+-4x3) - 4 x 3 y 3 A s + yl(2Aj -A5 + 3B3 - B4(3pi-4x3) + 4x3y3B<f) c) The U J 3 component cu 3 l l = | / x y 2 t [ 3 B 3 + Z B 4 ( y 3 - 3 x 3 ) - 4 x 3 y 3 B 5 - 2 A s +2.As 3A3^ 2A4(y3 -3x3)-4x3iy3 A5 + 2(25 -Zj) (4) y* (3B3 t-2B4(y3 -3x3)-4x3y3Bs f 2(A<?-A?) cu3,3 - j jj^ji -A4 + 2 ^ ? + 4x*y3Afe+- 2, ("2s-22)] -94-2. The grouping of s p e c i f i c terms a) The divergence :LV <JO = CO.,i r LOz,z -f- u j 3 j 3 ± j j L y l ' ( Z A + - A * -2x 3A 5) (5) b) The symmetric sums 3 ( J 3 f J 3 * ) - 3 y i ( K 3 - r K 3 ) - a t y * ( B 4 - 2 B 4 +-3pC3 - T x 3 u ) 3 C f -t(6B z-5A 4+8x 3y 3B 4+A4) ^ 3 , 1 t w „ 3 = | ^ 3(pK 3 - c^k3) • - 4 t(Af + 2,X 3y 3 Bs ~ B 4(x 3 (6) t- A 4(x 3 r 5y 3) - 8x3y3As r 4 y ! (A* 5 + 2 x 3 y 3 B 5 - B 4 ( x 3 + a y 3 ) ) 3. The actual stress changes Substituting the equations of appendix (B.2) above into equation (B.l) we obtain the six components - 9 5 -of the stress change ^ " &njA ^ f 2-LO'^) °l *3 ^ Z Z ~ 8rr I u»^**) 0 0 + ^ ^ f c ) °lx3 Jd T33 - 5 - fDu,(x3) (dUv u> f 2 u J M ) c(x3 i r D t>z = j u,(x3) ( UJ,, 2 +-UJ2,,) dx3 (7) 8 T T J a Analytical solutions can easily be obtained for each of these expressions with the aid of an electronic computer (an I.B.M. 7040 was used to perform this task). -96-C. THE MAXIMUM SHEAR STRESS Having evaluated equations (B.7) at any given point we may calculate the p r i n c i p a l stresses and hence the maximum shear stress at that point. Equa-t i o n (2.20), Tdj - t (J'cj | = O y i e l d s a cubic equation of the form . - T 3 - i - ©, T 2 , - © 2 . T +• © 3 - O ( i ) (2) where © , , © * and © 3 are invariants of the stress tensor and are described by the following equations: T H +• Tz2. +• T 3 3 © 2 = T T * <-T*.T3 +- T 3 T , Tz.3 T, T21 T2.3 T 33 T3 I T 3 * T n «7- W U5L T; *2. -97-and © 3 = T, T i T 3 T r, Tz 3 (3) Here T , ^ and T 3 a r e t h e t h r e e r o o t s o f the c u b i c e q u a t i o n such t h a t T 3 > Tj, > T| . The maximum shear s t r e s s i s g i v e n by F o r the reas o n s mentioned i n Chapter 2.7 no r e f e r e n c e w i l l be made t o the d i r e c t i o n s o f maximum sh e a r s t r e s s . However, i t i s i m p o r t a n t t o n o t e t h a t when an i n i t i a l s t r e s s d i s t r i b u t i o n L12. e x i s t s i n the medium, i t must be added t o the s t r e s s change g i v e n by e q u a t i o n (B.7d). Thus i n our case a l l . t he T12. terms i n equa-t i o n s (C.3) a r e o f the form s ( 5 ) -DE-REFERENCES Anderson, E. M., Dynamics o f F a u l t i n g , 2nd ed., O l i v e r and Boyd, E d i n b u r g h , 1951. B a d g l e y , P. C , S t r u c t u r a l Methods f o r t h e E x p l o r a t i o n G e o l o g i s t , H a r p e r and B r o s . , New Y o r k , 1959. Ben-Menahem, A. and Tokso, M. N., Source Mechanism from S p e c t r a o f L o n g - p e r i o d S e i s m i c S u r f a c e Waves, J . Geophys. Res., 6j_, 1943-1955, 1962. Ben-Menahem, A. and Tokso, M. N., Source Mechanism f o r Spectrums o f L o n g - p e r i o d S u r f a c e Waves, J . Geophys. Res., 68, 5207-5222, 1963. B e n i o f f , H., Mechanism o f Earthquake G e n e r a t i o n , B u l l . 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