UBC Theses and Dissertations

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UBC Theses and Dissertations

A mathematical theory of elastic orthotropic plates in plane strain and axi-symmetric deformations Lin , Yi Han 1987

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A MATHEMATICAL THEORY OF ELASTIC ORTHOTROPIC PLATES IN PLANE STRAIN AND AXI-SYMMETRIC DEFORMATIONS By Y I H A N L I N B.Sc, Fudan University, 1969 M.Sc, The University of British Columbia, 1983 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D e p a r t m e n t of M a t h e m a t i c s I n s t i t u t e o f A p p l i e d M a t h e m a t i c s We accept this thesis as conforming to the required standard 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 June 1987 © Yihan Lin, 1987 4 6 ii I n 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 t h e r e q u i r e m e n t s f o r a n ad-v a n c e d degree at 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 agree t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r refe r e n c e a n d s t u d y . I f u r t h e r agree 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 of 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 m a y b e g r a n t e d b y t h e H e a d o f m y D e p a r t m e n t o r b y h i s o r h e r 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 financial g a i n s h a l l n o t be a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t of M a t h e m a t i c s 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 1956 M a i n M a l l V a n c o u v e r , B r i t i s h C o l u m b i a C a n a d a V6T 1W5 J u n e 1987 iii Abstract We p r e s e n t a n e l a s t i c o r t h o t r o p i c p l a t e t h e o r y i n p l a n e s t r a i n a n d a x i s y m -m e t r i c d e f o r m a t i o n s b y first d e v e l o p i n g t h e i r u n i f o r m a s y m p t o t i c e x p a n s i o n s o f t h e e x a c t s o l u t i o n s f o r t h e b a s i c g o v e r n i n g b o u n d a r y v a l u e p r o b l e m s . T h e n , t h e e s t a b l i s h m e n t o f t h e necessary c o n d i t i o n s f o r d e c a y i n g s t a t e s , b o t h e x p l i c i t l y a n d a s y m p t o t i c a l l y , e n a b l e s us t o d e t e r m i n e t h e o u t e r s o l u t i o n w i t h o u t r e f e r e n c e t o t h e i n n e r s o l u t i o n a n d c l a r i f y t h e pre-cise m e a n i n g of t h e w e l l k n o w n St.Venant's p r i n c i p l e u n d e r t h e c i r c u m s t a n c e s c o n s i d e r e d here. T h e p o s s i b l e e x i s t e n c e of c o r n e r s t r e s s s i n g u l a r i t i e s was e x a m i n e d b y e s t a b l i s h i n g a n d s o l v i n g t h r e e t r a n s c e n d e n t a l g o v e r n i n g e q u a t i o n s . B y d e v e l o p i n g a g e n e r a l i z e d C a u c h y t y p e s i n g u l a r i n t e g r a l e q u a t i o n f o r t h e p l a n e s t r a i n d e f o r m a t i o n a n d a n i n t e g r a l e q u a t i o n of t h e s e c o n d k i n d f o r t h e a x i - s y m m e t r i c d e f o r m a t i o n a n d t a k i n g t h e c o r n e r s t r e s s s i n -g u l a r i t i e s i n t o c o n s i d e r a t i o n , we o b t a i n e d a c c u r a t e n u m e r i c a l s o l u t i o n s f o r a l l c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s w h i c h are needed i n t h e a s y m p t o t i c n e c e s s a r y c o n d i t i o n s f o r de-c a y i n g s t a t e s . F i n a l l y , t h e a c c u r a c y of t h e n u m e r i c a l s o l u t i o n s of c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s a n d t h e e f f i c i e n c y of t h e p l a t e t h e o r y were c o n f i r m e d t h r o u g h t h e a p p l i c a t i o n s of s o l v i n g t w o p h y s i c a l p r o b l e m s a n d c o m p a r i n g w i t h t h e e x i s t i n g r e s u l t s . iv Contents A b s t r a c t H i F i g u r e s v i A c k n o w l e d g e m e n t s v i i 1 I n t r o d u c t i o n 1 1 P L A N E S T R A I N D E F O R M A T I O N 6 2 O u t e r a n d I n n e r s o l u t i o n s 7 2.1 B a s i c E q u a t i o n s a n d B o u n d a r y C o n d i t i o n s 7 2.2 O u t e r S o l u t i o n 10 2.3 I n n e r S o l u t i o n 13 2.4 N u m e r i c a l s o l u t i o n s f o r e i g e n v a l u e s 18 3 N e c e s s a r y C o n d i t i o n s f o r D e c a y i n g S t a t e s 2 1 3.1 N e c e s s a r y C o n d i t i o n s f o r D e c a y i n g S t a t e s 21 3.2 A s y m p t o t i c N e c e s s a r y C o n d i t i o n s f o r D e c a y i n g S t a t e s . . . . 27 4 C o r n e r S t r e s s S i n g u l a r i t i e s 2 9 4.1 S o l u t i o n i n P o l a r C o o r d i n a t e s 30 4.2 E q u a t i o n f o r e i g e n v a l u e s 32 4.3 N u m e r i c a l A n a l y s i s 35 5 C a n o n i c a l P r o b l e m s 3 8 5.1 I n t e g r a l E q u a t i o n f o r S y m m e t r i c C a s e 39 5.2 R e d u c t i o n t o S i n g u l a r I n t e g r a l E q u a t i o n 42 5.3 I n t e g r a l E q u a t i o n f o r A n t i - S y m m e t r i c case 45 5.4 M e t h o d of E i g e n f u n c t i o n E x p a n s i o n s . . 47 5.5 N u m e r i c a l A n a l y s i s 50 G A p p l i c a t i o n s 6 0 6.1 D e t e r m i n a t i o n o f O u t e r S o l u t i o n s 60 6.2 F l e x i b i l i t y C o e f f i c i e n t o f S h e a r e d B l o c k 63 n A X I - S Y M M E T R I C D E F O R M A T I O N 6 6 7 O u t e r a n d I n n e r s o l u t i o n s 6 7 7.1 B a s i c E q u a t i o n s a n d B o u n d a r y C o n d i t i o n s 67 Contents v 7.2 O u t e r S o l u t i o n 70 7.3 I n n e r S o l u t i o n 73 8 N e c e s s a r y C o n d i t i o n s f o r D e c a y i n g S t a t e s 7 8 8.1 N e c e s s a r y C o n d i t i o n s f o r D e c a y i n g S t a t e s 78 8.2 A s y m p t o t i c N e c e s s a r y C o n d i t i o n s f o r D e c a y i n g S t a t e s . . . . 83 0 C o r n e r S t r e s s S i n g u l a r i t i e s 8 6 9.1 M i x e d B o u n d a r y C o n d i t i o n C a s e 86 9.2 D i s p l a c e m e n t B o u n d a r y C o n d i t i o n C a s e 90 1 0 C a n o n i c a l P r o b l e m s 0 4 10.1 F i n i t e H a n k e l T r a n s f o r m M e t h o d 94 10.2 I n t e g r a l E q u a t i o n of t h e S e c o n d K i n d S6 10.3 M e t h o d of e i g e n f u n c t i o n e x p a n s i o n s 98 10.4 N u m e r i c a l A n a l y s i s 100 1 1 A p p l i c a t i o n s 1 1 4 11.1 S i n g u l a r i t y C a u s e d b y C o n c e n t r a t e d F o r c e at C e n t e r . . . . 1 1 4 11.2 O u t e r S o l u t i o n f o r C l a m p e d C i r c u l a r P l a t e 116 1 2 C o n c l u s i o n s 121 R E F E R E N C E S . 1 2 4 A P P E N D I X . . . . . . . . . . . 1 2 7 A D e r i v a t i o n o f k e r n e l i n C h a p t e r 5 1 2 7 B A s y m p t o t i c E x p a n s i o n s i n C h a p t e r 5 1 3 0 B . l S y m m e t r i c C a s e 130 B. 2 A n t i - s y m m e t r i c C a s e 132 C K e r n e l s o f S . I . E s 1 3 3 C. l S y m m e t r i c C a s e 133 C.2 A n t i - s y m m e t r i c case 135 D T h e o r y f o r G e n e r a l i z e d C a n c h y T y p e S . I . E s . . . . . 1 3 6 E O u t e r S o l u t i o n s i n A x i - S y m m e t r i c C a 9 e 1 3 9 E . l A n t i - S y m m e t r i c case 139 E . 2 S y m m e t r i c C a s e 142 F D e r i v a t i o n o f G r e e n ' s F u n c t i o n 1 4 4 vi Figures 2.1 G e o m e t r y of a r e c t a n g u l a r plate 8 -5.1 axz0 i n e x t e n s i o n . . . ~ . 53 5.2 a r z 0 i n b e n d i n g 54 5.3 a r z 0 i n s h e a r i n g 55 5.4 aXQ i n e x t e n s i o n 56 5.5 c r x 0 i n b e n d i n g 57 5.6 aXQ i n s h e a r i n g 58 5.7 C o m p a r i s o n b e t w e e n S.I.E. a n d series s o l u t i o n s 59 6.1 C o m p a r i s o n w i t h u p p e r a n d l o w e r b o u n d s 65 7.1 G e o m e t r y of a c i r c u l a r p l a t e 68 10.1 u 0 i n c a n o n i c a l B V P 1 102 10.2 u0 i n c a n o n i c a l B V P 2 103 10.3 u 0 i n c a n o n i c a l B V P 3 104 10.4 c r r 0 i n c a n o n i c a l B V P 4 105 10.5 c r r 0 i n c a n o n i c a l B V P 5 106 10.6 c r r 0 i n c a n o n i c a l B V P 6 107 10.7 arz0 i n c a n o n i c a l B V P 1 108 10.8 arz0 i n c a n o n i c a l B V P 2 109 10.9 (TrzQ i n c a n o n i c a l B V P 3 110 10.10 arz0 i n c a n o n i c a l B V P 4 . H I 10.11 ar:Q i n c a n o n i c a l B V P 5 112 10.12 arzQ i n c a n o n i c a l B V P 6 113 11.1 C o m p a r i s o n w i t h t h e c l a s s i c a l r e s u l t 120 vii Acknowledgements T h e a u t h o r w i s h e s t o t h a n k D r . F. W a n a n d D r . G. B l u m a n f o r t h e i r a d v i c e a n d h e l p t h r o u g h o u t t h e e x e c u t i o n of t h i s work. A s w e l l , t h e financial s u p p o r t of t h e I. W. K i l l a m M e m o r i a l F e l l o w s h i p is g r a t e f u l l y a c k n o w l e d g e d . T h e a u t h o r expresses h i s s i n c e r e a p p r e c i a t i o n t o t h e I n s t i t u t e o f A p p l i e d M a t h e m a t i c s , t h e D e p a r t m e n t of M a t h e m a t i c s , a n d 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 , w i t h o u t whose v a l u a b l e s u p p o r t he w o u l d n e v e r have c a r r i e d o u t t h i s work. F i n a l l y , t h e a u t h o r i s d e e p l y a p p r e c i a t i v e o f t h e s u p p o r t a n d e n c o u r a g e m e n t o f f e r e d b y h i s wife, Y a q i n , d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s . CHAPTER 1 INTRODUCTION P l a t e s t r u c t u r e s w h i c h have one d i m e n s i o n ( t h i c k n e s s ) m u c h s m a l l e r t h a n t h e o t h e r t w o d i m e n s i o n s ( l e n g t h a n d w i d t h ) are w i d e l y used i n v a r i o u s areas i n i n d u s t r y . T h e b o u n d a r y v a l u e p r o b l e m g o v e r n i n g d e f o r m a t i o n s of t h e 3 d i m e n s i o n a l b o d y of t h e p l a t e does n o t a d m i t a n e x a c t a n a l y t i c s o l u t i o n e x c e p t f o r t h e s i m p l e s t cases. T o m a k e s u c h p r o b l e m s m o r e t r a c t a b l e , t h e y are r e d u c e d t o t h o s e f o r d e f o r m a t i o n s of 2 d i m e n s i o n a l m i d -d l e p l a n e of the p l a t e i n e n g i n e e r i n g p l a t e t h e o r i e s b y a p p l y i n g v a r i o u s a d h o c a p p r o x i m a t e h y p o t h e s e s . I n the case of s m a l l d e f l e c t i o n s f o r t h i n p l a t e s c o m p o s e d of i s o t r o p i c m a t e r i a l , t h e c l a s s i c a l K i r c h h o f f t h i n p l a t e t h e o r y g e n e r a l l y gives s a t i s f a c t o r y r e s u l t s w h e r e a s h i g h o r d e r t h e o r i e s are ne e d e d f o r t h i c k e r p l a t e s e s p e c i a l l y t h o s e m a d e of a n i s o t r o p i c m a t e r i a l i n c o m p l e x l o a d i n g c i r c u m s t a n c e s . A r a t i o n a l a n d s y s t e m a t i c d e r i v a t i o n o f l i n e a r p l a t e t h e o r i e s are k n o w n t o b e p o s s i b l e b y a n a p p l i c a t i o n o f p e r t u r b a t i o n o r i t e r a t i o n meth-ods [0,7,8,29]. W h e n a p p r o p r i a t e l y n o n d i m e n s i o n a l i z e d , t h e g o v e r n i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n s h a v e a s m a l l p a r a m e t e r e ( r a t i o o f t h i c k n e s s t o l e n g t h ) i n t h e c o e f f i c i e n t s . T h e s t r u c t u r e of t h e l i n e a r e q u a t i o n s is of t h e s i n g u l a r p e r t u r b a t i o n t y p e a n d t h e m e t h o d of m a t c h e d a s y m p t o t i c e x p a n s i o n s suggests a d e c o m p o s i t i o n of t h e e x a c t s o l u t i o n o f t h e l i n e a r b o u n d a r y v a l u e p r o b l e m i n t o a n o u t e r s o l u t i o n ( o r i n t e r i o r s o l u t i o n ) a n d a n i n n e r s o l u -t i o n ( o r b o u n d a r y l a y e r s o l u t i o n ) . T h e c l a s s i c a l K i r c h h o f f t h e o r y is now k n o w n t o be t h e l INTRODUCTION / 1.0 2 l e a d i n g t e r m of t h e o u t e r s o l u t i o n f o r l i n e a r e l a s t o s t a t i c s o f i s o t r o p i c p l a t e s . T h e o u t e r s o l u t i o n ( o r i t s a p p r o x i m a t i o n ) w h i c h is s i g n i f i c a n t t h r o u g h o u t t h e p l a t e c a n n o t s a t i s f y g e n e r a l ( a d m i s s i b l e ) b o u n d a r y c o n d i t i o n s . T h e s u p p l e m e n t a r y i n n e r s o l u t i o n ( o r i t s ap-p r o x i m a t i o n ) , w h i c h d e c a y s e x p o n e n t i a l l y a way f r o m t h e b o u n d a r y a n d is s i g n i f i c a n t o n l y • " " " i n t h e b o u n d a r y l a y e r r e g i o n , i s u s u a l l y t e d i o u s a n d / o r d i f f i c u l t t o d e t e r m i n e a n d o f t e n of l i t t l e i n t e r e s t i n d e s i g n . I t is t h e r e f o r e d e s i r a b l e t o o b t a i n t h e o u t e r s o l u t i o n w i t h o u t reference t o t h e i n n e r s o l u t i o n e x c e p t f o r some c a n o n i c a l p r o b l e m s w h i c h are s o l v e d o n c e a n d f o r a l l . R e c e n t l y t h i s was s h o w n t o be p o s s i b l e f o r i s o t r o p i c p l a t e s b y t h e e s t a b l i s h -m e nt of n e c e s s a r y c o n d i t i o n s f o r d e c a y i n g s t a t e s [ l l , 1 2 , 1 3 , 1 4 ] . S o me ne c e s s a r y c o n d i t i o n s have e x a c t , s i m p l e a n d e x p l i c i t f o r m s , b u t o t h e r s c a n o n l y b e e x p r e s s e d a s y m p t o t i c a l l y i n t e r m s of i n t e g r a l s of s o l u t i o n s o f w e l l d e f i n e d c a n o n i c a l p r o b l e m s w h i c h m u s t be s o l v e d n u m e r i c a l l y . T h e s o l u t i o n s of these c a n o n i c a l p r o b l e m s d e p e n d o n l y o n t h e g e o m e t r y a n d m a t e r i a l p r o p e r t y of t h e p l a t e ; once o b t a i n e d , t h e y c a n be u s e d f o r d i f f e r e n t edge d a t a f o r t h e same p l a t e . T h e s e necessary c o n d i t i o n s f o r d e c a y i n g s t a t e s c o r r e s p o n d t o t h e v/ e l l k n o w n St.Venant's p r i n c i p l e w h i c h does n o t a p p l y t o these p r o b l e m s . T h i s t h e s i s i s d e v o t e d t o t h e d e v e l o p m e n t of a c o r r e s p o n d i n g t h e o r y f o r or-t h o t o p i c p l a t e s . P a r t I (Ch.2-Ch.6) d e a l s w i t h p l a n e s t r a i n d e f o r m a t i o n s of C a r t e s i a n or-t h o t o p i c p l a t e s a n d p a r t I I ( C h . 7 - C h . l l ) w i t h a x i - s y m m e t r i c d e f o r m a t i o n s of c y l i n d r i c a l l y o r t h o t r o p i c p l a t e s . T h e b o u n d a r y v a l u e p r o b l e m f o r a g e n e r a l i z e d b i - h a r m o n i c e q u a t i o n i n C a r t e s i a n c o o r d i n a t e s g o v e r n i n g t h e p l a n e s t r a i n d e f o r m a t i o n i s f i r s t f o r m u l a t e d i n Ch.2 a n d t h e n t h e u n i f o r m a s y m p t o t i c e x p a n s i o n of t h e e x a c t s o l u t i o n i s d e v e l o p e d b y f o l l o w i n g t h e g e n e r a l p r o c e d u r e f o r s i n g u l a r l y p e r t u r b e d e q u a t i o n s . T h e d e c a y i n g r a t e s of t h e i n n e r s o l u t i o n s f o r some o r t h o t r o p i c m a t e r i a l s are e v a l u a t e d b y u s i n g a n e f f i c i e n t a l g o r i t h m f o r eigenvalues. I n Ch.3, n e c e s s a r y c o n d i t i o n s f o r d e c a y i n g s t a t e s , u s e f u l f o r d e t e r m i n i n g t h e o u t e r s o l u t i o n , are d e v e l o p e d . W e e s t a b l i s h a l l p o s s i b l e necessary c o n d i t i o n s e x p r e s s e d i n t e r m s of t h e edge d a t a of t h e p l a t e i n a n e x p l i c i t i n t e g r a l f o r m . F o r t h e cases w h e r e e x p l i c i t ones are n o t p o s s i b l e , t h e n u m e r i c a l s o l u t i o n s of t h r e e c a n o n i c a l b o u n d a r y v a l u e INTRODUCTION / 1.0 3 p r o b l e m s are n e e d e d t o e x p r e s s t h e n e c e s s a r y c o n d i t i o n s a s y m p t o t i c a l l y b y a n a p p l i c a t i o n of t h e r e c i p r o c a l t h e o r e m as has b e e n d o n e f o r t h e i s o t r o p i c case. T h e e v a l u a t i o n of a stress s i n g u l a r i t y at a c o r n e r w i t h one s i d e t r a c t i o n f r e e a n d t h e o t h e r s i d e fixed i s t h e first s t e p t o w a r d s t h e a c c u r a t e c o m p u t a t i o n of n u m e r i c a l s o l u t i o n s o f c a n o n i c a l p r o b l e m s . T h e e q u a t i o n g o v e r n i n g t h e s t r e s s s i n g u l a r i t y e x p o n e n t has a s i m p l e f o r m i n t h e i s o t r o p i c case w h e r e a s a r a t h e r i n v o l v e d t r a n s c e n d e n t a l e q u a t i o n has t o b e s o l v e d n u m e r i c a l l y f o r t h e o r t h o t r o p i c case. I n Ch.4, t h e e x i s t e n c e of t h e s o l u t i o n of t h e r e l e v a n t e i g e n v a l u e p r o b l e m f o r s t r e s s s i n g u l a r i t y e x p o n e n t s i s p r o v e d f o r o r t h o t r o p i c p l a t e s a n d i t s u n i q u e n e s s is p r o v e d f o r i s o t r o p i c p l a t e s . A c c u r a t e n u m e r i c a l s o l u t i o n s are o b t a i n e d . T h e c a n o n i c a l p r o b l e m s f o r a s e m i - i n f i n i t e s t r i p c l a m p e d at t h e r o o t a n d s u b j e c t t o u n i t e x t e n s i o n , b e n d i n g a n d s h e a r i n g , r e s p e c t i v e l y , at i n f i n i t y , h a ve been t o p i c s f o r e n g i n e e r s a n d a p p l i e d m a t h e m a t i a n s f o r s e v e r a l decades [2]. It has b e e n c l e a r f r o m o u r e x p e r i e n c e as w e l l as [35] a n d [3] t h a t t h e f o r m of t h e s o l u t i o n d e v e l o p e d i n Ch.2 is n o t e f f i c i e n t f o r stresses at t h e r o o t w h e r e d e c a y i n g t e r m s are s i g n i f i c a n t . W i t h t h e h e l p of a g e n e r a l i z e d o r t h o g o n a l i t y c o n d i t i o n , we are a b l e t o d e r i v e i n Ch.5 a n i n t e g r a l e q u a t i o n of t h e first k i n d a n d a r e g u l a r i z a t i o n t e c h n i q u e is a p p l i e d t o t h i s e q u a t i o n t o o b t a i n a s m o o t h s o l u t i o n . O n t h e o t h e r h a n d , F o u r i e r s i n e ( o r cosine) t r a n s f o r m s c o m b i n e d w i t h a s y m p t o t i c e x p a n s i o n s f o r a lar g e p a r a m e t e r g i v e us a s i n g u l a r i n t e g r a l e q u a t i o n of t h e first k i n d w h ose s i n g u l a r i t y is a n a l y s e d b y t h e m e t h o d d e v e l o p e d i n [24]. T h i s s e c o n d m e t h o d , w h i c h m a y b e r e g a r d e d as a s u b s t a n t i a l i m p r o v e m e n t over those i n [2] a n d [16], p r o d u c e s s t a b l e c o n v e r g e n t n u m e r i c a l s o l u t i o n s a n d is t h e r e f o r e p r o v e d t o be m u c h s u p e r i o r t o t h e first m e t h o d . O u r F O R T R A N p r o g r a m s f o r t h e i m p l e m e n t a t i o n of these m e t h o d s c a n d e a l w i t h b o t h o r t h o t r o p i c a n d i s o t r o p i c cases a n d t h u s s o l v e t h e c a n o n i c a l p r o b l e m s c o m p l e t e l y . N u m e r i c a l s o l u t i o n s f o r these c a n o n i c a l p r o b l e m s i n g r a p h i c a l f o r m a r e g i v e n i n Ch.5. A s a n a p p l i c a t i o n of t h e t h e o r y a n d a l s o a check of t h e a c c u r a c y of n u m e r i c a l s o l u t i o n s i n Ch.5, we o b t a i n i n Ch.6 t h e f l e x i b i l i t y c o e f f i c i e n t of a s h e a r e d b l o c k . O u r r e s u l t lies b e t w e e n t h e u p p e r a n d l o w e r b o u n d c u r v e s a c c u r a t e l y set i n [25] a n d s h ows t h a t t h e u p p e r b o u n d c u r v e is m o r e a c c u r a t e t h a n t h e l o w e r b o u n d cu r v e . INTRODUCTION / 1.0 4 P a r t I I o f t h i s t h e s i s c o n t a i n s a p a r a l l e l d e v e l o p m e n t f o r a x i s y m m e t r i c de-f o r m a t i o n s w h e r e we d e a l w i t h t h e g e n e r a l i z e d b i - h a r m o n i c e q u a t i o n i n p o l a r c o o r d i n a t e s . T h e o u t e r s o l u t i o n is w o r k e d o u t f o r t h e first t i m e i n Ch.7. W h e n s p e c i a l i s e d t o i s o t r o p i c p l a t e s , i t is c o n s i s t e n t w i t h t h e w e l l k n o w n L e v y s o l u t i o n [12,13]. T h e c o m b i n a t i o n of t h e o u t e r s o l u t i o n a n d t h e i n n e r s o l u t i o n , w h ose d e c a y i n g n a t u r e is g i v e n b y B e s s e l f u n c t i o n s i n series f o r m , p r o v i d e s t h e u n i f o r m a s y m p t o t i c e x p a n s i o n of t h e e x a c t s o l u t i o n . S i x c a n o n i c a l p r o b l e m s f o r m u l a t e d f o r c i r c u l a r p l a t e s are s t u d i e d i n Ch.8 t o e s t a b l i s h n e e d e d a s y m p t o t i c necessary c o n d i t i o n s f o r d e c a y i n g s t a t e s i n a d d i t i o n t o t h e e x p l i c i t ones. Ch.9 gives t w o t r a n s c e n d e n t a l e q u a t i o n s g o v e r n i n g t h e s t r e s s s i n g u l a r i t y e x p o n e n t s i n t h e c l a m p e d a n d m i x e d edge b o u n d a r y c o n d i t i o n cases r e s p e c t i v e l y . T h e s i m p l e c o u n t e r p a r t s o f t h e a b o v e e q u a t i o n s i n t h e i s o t r o p i c case e n a b l e us t o c o n f i r m t h e n o n e x i s t e n c e of t h e s o l u t i o n of t h e r e l e v a n t e i g e n v a l u e p r o b l e m f o r s t r e s s s i n g u l a r i t y f o r a m i x e d edge b o u n d a r y c o n d i t i o n a n d i t s e x i s t e n c e a n d u n i q u e n e s s f o r a fixed edge b o u n d a r y c o n d i t i o n . T h e c o r r e s p o n d -i n g c o n c l u s i o n i n t h e o r t h o t r o p i c case i s v e r i f i e d n u m e r i c a l l y . Ch.10 g i v e s a d i s c u s s i o n of t h e s o l u t i o n f o r t h e s i x c a n o n i c a l p r o b l e m s . F i r s t , t h e stress c o m p o n e n t az1 w h i c h is o b s e r v e d t o o b e y t h e same d i f f e r e n t i a l e q u a t i o n as t h e s t r e s s f u n c t i o n a n d s a t i s f i e s homoge-neous b o u n d a r y c o n d i t i o n s at b o t h t h e t o p a n d b o t t o m sur f a c e s , i s c hosen t o b e t h e b a s i c u n k n o w n v a r i a b l e f o r o u r p r o b l e m . T h e n <72 is e x p r e s s e d i n t e r m s of edge d i s p l a c e m e n t a n d s t r e s s i n a n i n t e g r a l f o r m by a n a p p l i c a t i o n of finite H a n k e l t r a n s f o r m . F i n a l l y , an i n t e g r a l e q u a t i o n of t h e s e c o n d k i n d is d e r i v e d f o r e i t h e r edge d i s p l a c e m e n t uQ o r edge stres s arQ w i t h o u t u s i n g B e s s e l f u n c t i o n s . O u r F O R T R A N p r o g r a m s f o r t h e i m p l e m e n -t a t i o n of t h i s s o l u t i o n p r o c e d u r e h a n d l e b o t h t h e o r t h o t r o p i c case a n d t h e i s o t r o p i c case f o r w h i c h t h e s o l u t i o n s f o r c a n o n i c a l p r o b l e m s have n o t b e e n k n o w n so f a r . S i m i l a r t o p a r t I, we d e v e l o p a n a l t e r n a t i v e m e t h o d of s o l u t i o n b a s e d o n g e n e r a l i s e d o r t h o g o n a l i t y c o n d i t i o n s a n d a p p l i c a t i o n of t h e r e g u l a r i z a t i o n t e c h n i q u e t o a n i n t e g r a l e q u a t i o n . Ch.10 gives g r a p h s of n u m e r i c a l s o l u t i o n s of c a n o n i c a l p r o b l e m s . I n C h . l l , a n a p p l i c a t i o n t o a c l a m p e d c y l i n d r i c a l l y o r t h o t r o p i c c i r c u l a r p l a t e l o a d e d b y a c o n c e n t r a t e d f o r c e at i t s c e n t e r INTRODUCTION / 1.0 5 shows r e m a r k a b l e c o i n c i d e n c e b e t w e e n o u r r e s u l t s a n d t h o s e b y c l a s s i c a l p l a t e t h e o r y a way f r o m t h e p l a t e edge a n d t h e center. F i n a l l y , we g i v e i n Ch . 1 2 a s u m m a r y o f t h e m a i n c o n t r i b u t i o n s o f t h i s t h e s i s a n d a b r i e f d e s c r i p t i o n o f f u r t h e r work. PLANE STRAIN DEFORMATION OF ELASTIC ORTHOTROPIC PLATES / 1.0 6 PART I PLANE STRAIN DEFORMATION OF ELASTIC ORTHOTROPIC PLATES OUTER AND INNER SOLUTIONS / 2.1 7 CHAPTER 2 OUTER AND INNER SOLUTIONS 2.1 BASIC EQUATIONS AND BOUNDARY CONDITIONS T h e first p a r t of t h i s t h e s i s is d e v o t e d t o t h e p l a n e s t r a i n p r o b l e m of a n o r t h o t r o p i c p l a t e whose g e o m e t r y is d e p i c t e d i n Fig.2.1. T h e c o n s t a n t t h i c k n e s s of t h e p l a t e is 2h, i t s w i d t h is 21, t h e x—, y — a x e s are i n t h e m i d p l a n e of t h e p l a t e a n d t h e z—axis is n o r m a l t o t h e m i d p l a n e . A l l p h y s i c a l q u a n t i t i e s are i n d e p e n d e n t of t h e y c o o r d i n a t e , i n c l u d i n g m a t e r i a l p r o p e r t y , p l a t e geometry, l o a d s a n d b o u n d a r y c o n d i t i o n s . T h e b a s i c e q u a t i o n s g o v e r n i n g d e f o r m a t i o n s of a n e l a s t i c o r t h o t r o p i c p l a t e are w e l l k n o w n , e.g.[19]. B y s u b t r a c t i n g off a p p r o p r i a t e p a r t i c u l a r s o l u t i o n s , b o d y fo r c e s c a n a l w a y s be r e m o v e d a n d hence a s s u m e d t o b e absent. F o r p l a n e s t r a i n d e f o r m a t i o n s , t h e r e l e v a n t e q u i l i b r i u m e q u a t i o n s are dax dcrxz — - H — = 0, dx dz fn i\ daxz daz _ Q ^ ' ' dx dz O r t h o t r o p i c m a t e r i a l s w i t h t h r e e p r i n c i p a l axes p a r a l l e l t o t h e x, y, z axes r e s p e c t i v e l y h ave 9 m a t e r i a l c o n s t a n t s i n g e n e r a l . T h e n u m b e r of m a t e r i a l c o n s t a n t s i s r e d u c e d t o 4 f o r p l a n e OUTER AND INNER SOLUTIONS / 2.1 8 F I G U R E 2.1 geometry of a rectangular plate. d e f o r m a t i o n cases as s h o w n i n t h e f o l l o w i n g c o n s t i t u t i v e e q u a t i o n s : / " a <*i2 0 W M I l l E * ~vnlE2 0 \ ( o x \ ez = a 1 2 a 2 2 0 a , = - « / 2 1 / ^ l / £ 2 0 a , (2.2) where f 1 2 , i / 2 l > -^2 a n <^ ^ * a r e p o s i t i v e m a t e r i a l c o n s t a n t s w i t h un/E2 = fzi/Ei. F o r p l a n e s t r e s s case, f 1 2 a n d f 2 1 are Poisson's r a t i o ; Elt E2 a n d G a r e Young's a n d s h e a r m o d u l i r e s p e c t i v e l y . F o r p l a n e s t r a i n case, t h e c o r r e s p o n d i n g m a t e r i a l c o n s t a n t s a r e k n o w n f u n c t i o n s o f P o i s s o n ' s r a t i o , Young's a n d s h e a r m o d u l i . F o r i s o t r o p i c m a t e r i a l P o i s s o n ' s r a t i o c o u l d be n e g a t i v e i n t h e o r y w h e r e a s i s o b s e r v e d t o b e p o s i t i v e i n p r a c t i c e . ax,oxy<rxz are s t r e s s c o m p o n e n t s a n d c x , e , , 7 X 2 are s t r a i n c o m p o n e n t s . T h e s t r a i n d i s p l a c e m e n t e q u a t i o n s f o r s m a l l d i s p l a c e m e n t s OUTER AND INNER SOLUTIONS / 2.1 9 i m p l y t h e c o m p a t i b i l i t y e q u a t i o n d2-ir, d2er d2er , x z — 2 + x (2 41 dxdz dx2 dz2 1 j w h e r e u a n d w are d i s p l a c e m e n t s i n t h e x a n d z d i r e c t i o n s r e s p e c t i v e l y . T h e e q u i l i b r i u m e q u a t i o n s (2.1) c a n b e s a t i s f i e d b y e x p r e s s i n g t h e s t r e s s c o m p o n e n t s i n t e r m s of a s t r e s s f u n c t i o n <f> as f o l l o w s : _ d2<f> _ d2<f> _ d24> °x~ dz2' a * ~ dx2' dxdz' ( 2 , 5 ) Eqs.(2.5) a n d (2.2) t u r n eq.(2.4) i n t o ^txxzx + (2o12 + aM)<f>zxzz + all<t>zzzz = 0 (2-6) where s u b s c r i p t s d e n o t e s d e r i v a t i v e s . U s e of t h e s c a l i n g x = ax, a = (a22/an)1/i=:(El/E2)1/4 t r a n s f o r m s (2.6) i n t o w h ere teiii + (2 + P)<f>iiz2 + 4>zzzz = 0 (2.7) 2 a 1 2 + a 3 3 \fExE2  P = — , 2 = — — 2(1 + V/VTO^I ) V v ^ I T a ^ G K V l~ 2 i ) (2.8) = E/G — 2(1 -f v) w i t h E = \ZEiE2, v = sjvnv2x. N o t e t h a t we have p = 0 f o r a n i s o t r o p i c m a t e r i a l a n d g e n e r a l l y p > — 4 due t o v < 1 b y p o s i t i v e - d e f i n i t e n e s s of t h e m a t r i x i n eq.(2.2), i.e. p o s i t i v e - d e f i n i t e n e s s of t h e s t r a i n e n e r g y of t h e p l a t e . S i n c e t h e case p = 0 has been d i s c u s s e d i n l i t e r a t u r e , we assume, b a s e d o n o u r o b s e r v a t i o n s o n t h e d a t a f o r o r t h o t r o p i c m a t e r i a l s , p > 0 f o r t h e d e v e l o p m e n t of s o l u t i o n s of eq.(2.7). T h u s t h e d i f f e r e n t i a l e q u a t i o n i s e l l i p t i c . T h e t o p a n d b o t t o m s u r f a c e s of t h e p l a t e are t r a c t i o n free, i.e. at z = ±h: a X i = 0, trt = 0, (2.9) OUTER AND INNER SOLUTIONS / 2.2 10 o r at z = ±h: 4>iz=0t 4>iz = 0, (2.10) w h i l e at each s i d e of t h e p l a t e , i.e. at x = ±/, t h e b o u n d a r y c o n d i t i o n s are one of t h e f o l l o w i n g c o m b i n a t i o n s : (1) ax = ax, <rxz = &xg] (2) £rx = a x , u; = u>; (3) u = u, aXi = axz; (4) u = u, w = w where t h e b a r r e d q u a n t i t i e s are g i v e n f u n c t i o n s of z. I n w h a t f o l l o w s , we g i v e t h e asymp-t o t i c e x p a n s i o n s of s o l u t i o n s f o r t h e b o u n d a r y v a l u e p r o b l e m s d e f i n e d above. 2.2 OUTER SOLUTION T h e o u t e r s o l u t i o n ( o r i n t e r i o r s o l u t i o n ) i s t h e s o l u t i o n of (2.7) s i g n i f i c a n t t h r o u g h o u t t h e r e g i o n . T h e c o m b i n a t i o n of t h e o u t e r s o l u t i o n a n d t h e i n n e r ( o r b o u n d a r y la y e r ) s o l u t i o n gives t h e c o m p l e t e s o l u t i o n of t h e b o u n d a r y v a l u e p r o b l e m [ 6 ] . I t m a y be e x p e c t e d t h a t t h e d i f f e r e n c e i n t h e i n t e r i o r r e g i o n between t h e o u t e r s o l u t i o n a n d s o l u t i o n s p r o d u c e d by v a r i o u s r e a s o n a b l e e n g i n e e r i n g p l a t e t h e o r i e s is s m a l l f o r t h i n p l a t e s w i t h s l o w l y v a r y i n g p r o p e r t i e s a n d loads. T o find t h e o u t e r s o l u t i o n of eq.(2.7), we b e g i n w i t h new s c a l i n g s t = z/h, s = x/l w h i c h change (2.7) i n t o e V . „ . + (2 + P)e 2*..« + 4>tm = 0 (2.11) w h e r e € = ah/l<&l (2.12) OUTER AND INNER SOLUTIONS / 2.2 11 is a s s u m e d f o r p l a t e s . I n t h e p l a t e i n t e r i o r , t h e s o l u t i o n is e x p e c t e d t o change s l o w l y . W e m a y t a k e t h e s o l u t i o n t h e r e , i.e., t h e o u t e r s o l u t i o n , i n t h e f o r m 4> = 0o(*. 0 + 0 + € 40 2(«, <) + •••• (2.13) Eq.(2.11) t h e n gives a series of e q u a t i o n s 4>Qttu = 0| tutu = ~{2 + P)4>o„tt, (2.14) fautt = -(2 + V)4>u*u ~ ^o««««i etc.. C o r r e s p o n d i n g l y , t h e b o u n d a r y c o n d i t i o n s (2.10) at t = ± 1 b e c o m e <t>o»t = 0, 4>q„ = 0, <f>ut = 0, 4U, = 0 i (2.15) 2^«e = °> 2^»« = 0, etc. f r o m w h i c h t h e s o l u t i o n <f>Q c a n be e x p r e s s e d as = Soo(s) + *oi(*)< + s02(s)t2 + s 03(s)< 3 (2.16) w i t h «oi(s) + 2*o2(«) + 3s{,3(a) =0 *oi(«) - 2 s o 2 ( s ) + 3s{, 3(s) = 0 «oo(a) + *oi(«) + *02(«) + = 0 «OO(*)-«OI(«) + *S2(«)-«M(*) = 0 where p r i m e s s t a n d f o r d e r i v a t i v e s . B y n e g l e c t i n g l i n e a r f u n c t i o n s i n s a n d t w h i c h g i v e n o c o n t r i b u t i o n s t o t h e s t r e s s c o m p o n e n t s , we have <f>0 = Axa{t - t 3 / 3 ) + A2t2 + A 3 t 3 (2.17) w h e r e A i t A2 a n d A$ are c o n s t a n t s . W i t h (2.17), t h e r i g h t h a n d s i d e of t h e s e c o n d e q u a t i o n of (2.14) t u r n s o u t t o be z e r o a n d we a r r i v e at 4>x = Bxs{t-t*lZ) + B2t2 + B3t3 (2.18) OUTER AND INNER SOLUTIONS / 2.2 12 whose f o r m i s s i m i l a r t o <f>0. T h e c o n t i n u a t i o n o f t h e a b o v e p r o c e d u r e g i v e s <t>2l<f>$, e t c . i n a s i m i l a r f o r m a n d t h u s c o n f i r m s t h a t <f> =s(t - *3/3)(A! + Bx<? + ( V + • • •) + t2(A2 + B2e2 + C 2 e 4 + - ) + ts(Az + Bie2 + C3ei + ---) =s{t-t3/3)A + t2B + tzC o r x Nz2 M 0 z 3 3Q . z\ , x where N,M0 a n d Q are c o n s t a n t s o f i n t e g r a t i o n . It c a n now be v e r i f i e d b y a d i r e c t s u b s t i t u t i o n t h a t (2.19) is a n e x a c t s o l u t i o n of eq.(2.11) f o r a n y v a l u e of e t h o u g h i t was o r i g i n a l l y d e r i v e d b y w a y of a p e r t u r b a t i o n e x p a n s i o n i n p o w e r s of s m a l l e. T h e c o r r e s p o n d i n g e x p r e s s i o n s f o r t h e s t r e s s c o m p o n e n t s are : _ N 3 M 0 3 Q az = 0, (2.20) It is not d i f f i c u l t t o see t h a t t h e o u t e r s o l u t i o n i s i d e n t i c a l w i t h t h o s e o b t a i n e d f r o m ... . K i r c h h o f f ' s t h e o r y f o r p r o b l e m s of p l a n e s t r a i n d e f o r m a t i o n s . F r o m (2.3), we get t h e d i s p l a c e m e n t e x p r e s s i o n s N x M0 3xz U = T 2 E X + HFWi Q ,3x2z u^zz z3, _ Nu2lz 3M 0 u2xz2 x2  W ~ h 2EX Ah3 1 Ex Ex} Q ,3h2x x3 3i/2iZ22x where t h e new c o n s t a n t s of i n t e g r a t i o n u, c, d c h a r a c t e r i z e r i g i d b o d y m o t i o n s . T h e deter-m i n a t i o n of these c o n s t a n t s are n o t t r i v i a l w h e n d i s p l a c e m e n t edge b o u n d a r y c o n d i t i o n s OUTER AND INNER SOLUTIONS / 2.3 13 are p r e s c r i b e d . W i t h i t s s i m p l e d e p e n d e n c e o n z, t h e o u t e r s o l u t i o n c a n n o t s a t i s f y gen-e r a l a d m i s s a b l e edge b o u n d a i y c o n d i t i o n s . A s a n e x a c t s o l u t i o n o f t h e g o v e r n i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n , i t i s i n c o m p l e t e . T h e m i s s i n g s o l u t i o n c o m p o n e n t s , c a l l e d t h e i n n e r s o l u t i o n , t u r n o u t t o be s i g n i f i c a n t o n l y i n t h e b o u n d a r y l a y e r r e g i o n . I n g e n e r a l , t h e de-t e r m i n a t i o n o f t h e o u t e r s o l u t i o n is c o u p l e d w i t h t h a t o f i n n e r s o l u t i o n . N o t e t h a t we have e x c l u d e d o u t e r s o l u t i o n c o m p o n e n t s f r o m t h e i n n e r s o l u t i o n t o d e f i n e t h e i n n e r s o l u t i o n as a b o u n d a r y l a y e r s o l u t i o n . 2.3 INNER SOLUTION G i v e n t h e e x p a n s i o n (2.13) f o r t h e o u t e r s o l u t i o n , t h e s u p p l e m e n t a r y i n n e r s o l u t i o n m u s t c h a n g e s i g n i f i c a n t l y i n t h e b o u n d a r y l a y e r r e g i o n , a n d a d i f f e r e n t s c a l i n g f r o m t h a t f o r t h e o u t e r s o l u t i o n w i l l have t o be used. I n o r d e r t o have a f o u r t h o r d e r d i f f e r e n t i a l e q u a t i o n i n a v a r i a b l e a l o n g t h e x d i r e c t i o n t o s a t i s f y t h e edge b o u n d a r y c o n d i t i o n s , we set s = er. T h e n , eq.(2.11) a n d t h e t o p a n d b o t t o m s u r f a c e b o u n d a r y c o n d i t i o n s b e c o m e <t>rrrr + (2 + p)<f>rrU + <fiuu=0 (2.22) w i t h 4>rl = 0 , 4>rr = 0 at t = ± 1 . (2.23) T h e a s y m p t o t i c e x p a n s i o n s of t h e i n n e r s o l u t i o n are g e n e r a l l y of t h e f o r m 4> = /*o(e)0o(r, 0 + M i ( e ) h ( r , 0 + M 2 (<0&>(r, t) + • • • (2.24) w h e r e Ho(e), e t c . are t o be d e t e r m i n e d . S i n c e c does n o t a p p e a r i n (2.22) a n d (2.23), t h e s u b s t i t u t i o n of (2.24) i n t o (2.22) a n d (2.23) r e v e a l s t h a t t h e <f>i(r,t) are n o t c o u p l e d t o each o t h e r a n d s a t i s f y t h e s a me e q u a t i o n a n d b o u n d a r y c o n d i t i o n s as (2.22) a n d (2.23). I n OUTER AND INNER SOLUTIONS / 2.3 14 o t h e r words, t h e <f>i{r,t) are c o m p o s e d o f t h e e i g e n f u n c t i o n s fa(r,t) s a t i s f y i n g (2.22) a n d (2 .23). T h u s , we have & ( r,0 = X > a / a ( r , 0 (2.25) a w h e r e A < a are c o n s t a n t s . A s u b s t i t u t i o n o f (2.25) i n t o (2.24) gives t h e e x p r e s s i o n f o r i n n e r s o l u t i o n * = X > ( < O * . - ( M ) = E M O £ 4 « / « * ' ° (2 26) a i a where .Aft are c o n s t a n t s . T o find t h e e i g e n f u n c t i o n s , we c o n s i d e r a s e p a r a t i o n o f v a r i a b l e s <f>(r,t) = R(r)T(t) (2.27) w h i c h t u r n s (2.22) i n t o - ^ r - r ( < ) + (2 + p ) - ^ ^ - + ^ ( r j - ^ j - - 0. (2.28) Eq.(2.28) m a y b e r e g a r d e d as a n o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r R(r) w i t h c o n s t a n t c o e f f i c i e n t s (t is a p a r a m e t e r here). T h e r e f o r e , i t s s o l u t i o n is c o m p o s e d of ear w h e r e a .must be a c o n s t a n t i n o r d e r f o r R(r) t o be a f u n c t i o n of r o n l y . N o w (2.28) c a n be w r i t t e n as Tmt+a2(2 + p)Tu+a4T = 0 whose s o l u t i o n is c o m p o s e d of e11 w i t h a a n d 7 r e l a t e d b y 7 4 + (2 + p)a V + a 4 = 0 o r f = -k2a2, -a2/k2 where k2 = 1 + p/2 + > / ( l + p / 2 ) 2 - 1 > 1 (2.29) f o r p > 0. I n a d d i t i o n t o ear, R(r) m a y al s o have s o l u t i o n s of t h e f o r m reQr, r2ear, r 3 e o r OUTER AND INNER SOLUTIONS / 2.3 15 i f a is a r e p e a t e d r o o t of t h e c h a r a c t e r i s t i c e q u a t i o n of (2.28). A d i r e c t s u b s t i t u t i o n r e v e a l s t h a t t h i s is p o s s i b l e o n l y w h e n a = 0. T h e c o r r e s p o n d i n g s o l u t i o n f o r <f> i n t h a t case i s i d e n t i c a l w i t h t h e o u t e r s o l u t i o n o b t a i n e d before. I n o t h e r words, we h a v e c o n f i r m e d t h a t t h e o u t e r s o l u t i o n is c o m p o s e d of e i g e n f u n c t i o n s f o r t h e v a n i s h i n g e i g e n v a l u e of t h e r e l e v a n t e i g e n v a l u e p r o b l e m . I n o r d e r t o s a t i s f y b o u n d a r y c o n d i t i o n s o n t h e s u r f a c e s o f t h e p l a t e , t h e c o n s t a n t s Ci,C2,C3,C^ i n t h e g e n e r a l s o l u t i o n 4> = ear{Cx cos(fcar) + C2 s i n ( f c a i ) + C 3 cos(ar/Jfc) + C 4 sin(ai/Jfc)} m u s t be chosen so t h a t / c o s ( * a ) c o s ( a / A ) \ ( C A _ /()\ \k2sm(ka) sin(a/Jfc) / \C3) ~\0J ^-dUJ a n d / sin(A;a) «n(a/*) \ (C2\ _ (0\ \k2cos{ka) c o s ( a / f c ) J \ c j ~ \0J ' Eq.(2.30) r e q u i r e s t h e e i g e n v a l u e p a r a m e t e r a t o s a t i s f y cos(fca) sin(a/A;) - k2 sin(A;a) cos(a/A;) = 0 (2.32) w i t h e i g e n f u n c t i o n s of t h e f o r m ... a kaz , t . a z . <pQ[z) = cos — cos — c o s ( K a ) cos -—. (2.33) k n kh S i m i l a r l y , eq.(2.31) r e q u i r e s t h e ei g e n v a l u e t o s a t i s f y sin(fc/?) cos(/9/fe) - k2 cos(k(3) am{0/k) = 0 (2.34) w i t h e i g e n f u n c t i o n s , / v . P . Pkz . . 3z , 4>Jz) = s in ^  s m sm(pk) s in — (2.35) k ft kh w h e r e a has been r e p l a c e d b y /? t o i n d i c a t e a d i f f e r e n t t y p e o f eigenvalues. T h e e x a c t s o l u t i o n i s a s y m p t o t i c t o a s u m of t h e o u t e r s o l u t i o n a n d t h e i n n e r s o l u t i o n w h i c h d e c a y s t o z e r o away f r o m t h e b o u n d a r i e s . W e have seen t h a t t h e OUTER AND INNER SOLUTIONS / 2.3 16 o u t e r s o l u t i o n i s a l s o t h e s o l u t i o n c o r r e s p o n d i n g t o t h e z e r o eigenvalue. T h e r e f o r e , i n t h e p l a n e s t r a i n p r o b l e m , t h e u n i f o r m a s y m p t o t i c e x p a n s i o n o f t h e e x a c t s o l u t i o n is o f t h e s ame f o r m as t h e i r e i g e n f u n c t i o n e x p a n s i o n s . T h e u n i f o r m a s y m p t o t i c e x p a n s i o n f o r <f> as e - » 0 is + £ A a e ^ ~ ( , ) + £ A ^ - ^ ( z ) ( 2 3 6 ) a 0 + £ Bae*+*M<4a{z) + £ B ^ + ' M ' M z ) « P w h e r e a,P,<t>a,<f>p are d e f i n e d b y eqs.(2.32),(2.34),(2.33) a n d (2.35) r e s p e c t i v e l y . T h e s i g n s of t h e r e a l p a r t s of a a n d p are so c hosen t h a t i n n e r s o l u t i o n c o m p o n e n t s d e c a y away f r o m b o u n d a r i e s . Nt M 0 , Q, A a , Apt Ba a n d Bp are u n k n o w n c o n s t a n t s w h i c h m a y b e d e t e r m i n e d b y m a t c h i n g t h e s o l u t i o n w i t h t h e edge c o n d i t i o n s . We w i l l have m o r e d i s c u s s i o n s o n t h i s l a t e r . A c c o r d i n g l y , we have t h e f o l l o w i n g u n i f o r m a s y m p t o t i c e x p a n s i o n s f o r stresses a n d d i s p l a c e m e n t s : _N_ 3 M o 3<? °x~2h + 2 h * Z + 2 h * X Z + p { E ^ « 2 « a ( I - / l ) / e ^ « w + E ^ v ( W I ) / e ^ w a fi ^ = ^ { £ ^ " 2 E A ( 1 - X / 0 A < M * ) + E V ^ ( 1 - ' / L ) / E M * ) a a fi + £ BQah°^*M<<l>a(z) + £ Bt,pVll+*M<+p(z)} 0 + ^ 2 {" £ i t a ^ e - ^ - ^ a t * ) + E V 2 ^ ( 1 - X / l , / 4 ^ W a 0 + E ^ V ^ A ^ , ) - E ^ V ( l + ' / 0 / V M / | W } OUTER AND INNER SOLUTIONS / 2.3 17 N x MQZXZ Q .Zx2z i/nz3 z3. , + ^ {-E^«o(1"*/0/e«-w -Ev^" a / 0 Sw a n d N u2\Z ZMQ.VOIZ1 x2. Q , 3 / i 2 z z 3 3 r / 2 1 z z 2 a fi w h e r e / v f 2 or af c z 1 a z ^xal^J = c o s T c o s —r— + 7? c o s f f c a ) COS — k h k2 kh t \ L.2 • P • P k z i 1 • fUa\ • Pz <7z/3lz) — —k s in - s in + ^ sm{k(J) s in — , . . a . fcaz 1 . a z axza\z) — k cos — s in — — cos(fca) s in — fZ tit K Kft . , , . p kfiz 1 . pzy °~xzfs{z) = k s in - cos — - - sm{kp) cos — } /• •> / i 2 \ a kOLZ i 1 x /•. x u q ( 2 J = — ( " + V ) C O S T C O S ~~i \To + u) cos(«a) cos — k n k£ kh up{z) = -{k s in - s in — + + sin(fc/9) s m — * v ,, E l 7 . a . A;az 1 . i? 1 . ,, . , a z u> a(z) = -k[v - g + k ) cos - s m — + -{u - — + cos{ka) s in — / x , / E ,2^ • P k8z !/ E L . 5 s M 2 ) = k(u - £ + k ) s in - cos — - -(«, - - + ^ ) s i n ( * f l cos — . F r o m t h e above we see t h a t t h e i n n e r s o l u t i o n is i n a c o m p l i c a t e d series f o r m a n d i t s d e t e r m i n a t i o n w i l l n o t be easy. O n t h e o t h e r h a n d , t h e i n n e r s o l u t i o n i s c o n f i n e d t o b o u n d a r y l a y e r r e g i o n s . W h e n i t is n o t needed i n a p a r t i c u l a r p r o b l e m , i s i t p o s s i b l e t o d e t e r m i n e t h e o u t e r s o l u t i o n w i t h o u t reference t o i n n e r s o l u t i o n ? F o r p l a n e s t r a i n p r o b l e m s , t h i s c a n be a c h i e v e d b y t h e e s t a b l i s h m e n t of necessary c o n d i t i o n s f o r d e c a y i n g s t a t e s d i s c u s s e d i n C h . 3 . OUTER AND INNER SOLUTIONS / 2.4 18 2.4 NUMERICAL SOLUTIONS FOR EIGENVALUES H e r e we d e v e l o p a n ef f i c i e n t a l g o r i t h m t o e v a l u a t e e i g e n v a l u e s a f r o m cos(fca) sin(a/A:) - c 2 sin(A:a) cos(a/A:) = 0 (2.37) w h e r e A; > 1. Eq.(2.37) b e c omes (2.32) f o r c = k a n d (2.34) f o r c = \jk. T h e l e a d i n g t e r m of t h e T a y l o r e x p a n s i o n of (2.37) at k = 1 gives t h e w e l l k n o w n e q u a t i o n 2 a ± s i n 2 a = 0 fo r t h e i s o t r o p i c case. W e assume cos(fca) ^  0 a n d c o s ( a / f c ) ^  0, t h e n eq.(2.37) c a n b e w r i t t e n as tan(a/A;) = c 2 t a n ( f c a ) . D e n o t i n g kot = /?, we h a v e tan(/?/fc 2) = c 2 t a n / ? . (2.38) We m a y r e s t r i c t P > 0 a n d s e a r c h n u m e r i c a l l y f o r r e a l s o l u t i o n s i n each i n t e r v a l [nir — n/2, nn + n/2] f o r n=0,l,2, e t c . P a p p r o a c h s mr as k —• oo. If a is a c o m p l e x s o l u t i o n o f (2.37), so are —a, a, —a. H e r e , a b a r d e n o t e s a co m p l e x c o n j u g a t e . T h e r e f o r e , we m a y set a = s + it(s > 0, t > 0). A s e p a r a t i o n of t h e r e a l a n d i m a g i n a r y p a r t s of eq.(2.37) gives s HAt) tan£ Hz(t) o r - f o r c = k > 1 a n d 2 s _ Hx{t)Hz{t) _ 2  t a n fc " H2(t)HA(t) ~ f W' t a n k s - H 2 w m ~ 9 [ t ) 2 s _ H2{t)H&)  i a n k Hx(t)Hz(t)' t a n S k ~ Hx{t)HA{t) f o r c=l/k. I n (2.40) a n d (2.41), Hi(t) are d e f i n e d b y Hx{t) = k2 t a n h kt - t a n h ^, H2{t) = k2 t a n h ^  - t a n h kt, J73(<) = A:2 - t a n h kt t a n h |, # 4 ( 0 = 1 - k2 t a n h ibi t a n h ^ . (2.40) (2.41) (2.42) OUTER AND INNER SOLUTIONS / 2.4 19 I t i s o b v i o u s t h a t Hi(t) > 0, H$(t) > 0 an d , s i n c e H2[t) I S a n i n c r e a s i n g f u n c t i o n , H2(t) > H2(0) = 0. S i n c e H^(t) decreases f r o m 1 t o 1 — A;2 < 0 a n d m u s t b e p o s i t i v e a c c o r d i n g t o (2.40), t is r e s t r i c t e d t o 0 < t < tQ w h e r e tQ is t h e u n i q u e r o o t o f Hi{t0) = 0, o r 1 - k? t a n h kt0 t a n h ^  = 0. (2.43) T h a t i s , we o n l y n e e d t o s e a r c h f o r s o l u t i o n s i n t h e i n t e r v a l ( 0 , * 0 ) - Eq.(2.40) m a y be w r i t t e n as t a n ^  = ±/(i), tan(fcs) = ±g(t) (2.44) o r — = ±Arctan/(i) + mir, k (2.45) ks = ±Arctanj(r) + nir. A c c o r d i n g t o (2.39), t h e s i g n s i n (2.44) are e i t h e r s i m u l t a n e o u s l y p o s i t i v e o r s i m u l t a n e o u s l y n e g a t i v e . T h u s , we ha v e i n t h e f i r s t case: s = A : A r c t a n / ( r ) + mkir, (m = 0,1,2,...) A : A r c t a n / ( t ) + mkn — - ^ A r c t a n ^ ( f ) + nir/k. ( n = 0,1,2,...) k F o r a g i v e n i n t e g e r n, we o n l y have t o s e a r c h f o r s o l u t i o n s i n 0 < t < t0 f o r t h e i n t e g e r m s a t i s f y i n g n/k2 — 1/2 < m < (n + \/2)/k2. I n t h e s e c o n d case, we have s = — & A r c t a n / ( < ) + mkn, (m=l,2,.. i) — A;Arctan/(2) + mkir = — ^ A r c t a n ( r ) + nn/k. (n = 1,2,...) F o r a g i v e n i n t e g e r n, we o n l y have t o s e a r c h f o r s o l u t i o n s i n 0 < t < t0 f o r t h e i n t e g e r m s a t i s f y i n g (n — 1/2)/k 2 < m < n/k2 + 1/2. T h e above a l g o r i t h m has been i m p l e m e n t e d a n d gives t h e f o l l o w i n g n u m e r i c a l r e s u l t s : OUTER AND INNER SOLUTIONS / 2.4 20 m a t e r i a l p i n e w o o d 1 p i n e w o o d 2 p l y w o o d 1 p l y w o o d 2 i s o t r o p i c Ex 100000 4200 120000 64400 E2 4200 100000 64400 120000 G 7500 7500 7200 7200 "12 0.01 0.238 0.044 0.082 l>2i 0.238 0.01 0.082 0.044 a 2.2090 0.4527 1.1684 0.8559 1 P 0.6349 0.6349 10.0895 10.0895 0 k 1.4749 1.4749 3.4650 3.4650 1 2.2557 +1'0.8690 2.2557 +10.8690 0.9132 0.9132 2.106+ i l . 1 2 5 A i 3.4940 3.4940 1.2998 1.2998 3.749 +1'1.384 Re{X0}/a - 1.0211 4.9828 0.7816 1.0669 2.106 Re{\i)[a 1.5817 7.7181 1.1125 1.5186 3.749 Here, t h e u n i t f o r e l a s t i c i t y m o d u l u s is kg/cm2. XQ i n t h e t a b l e is t h e e i g e n v a l u e w i t h the m i n i m u m p o s i t i v e r e a l p a r t f o r eq.(2.32) a n d A j is t h e c o r r e s p o n d i n g e i g e n v a l u e f o r eq.(2.34). T h e d e c a y i n g r a t e d e f i n e d b y A i n e~Xxlh i s e q u a l t o e i t h e r i ? e { A 0 } / a o r Re{\i)/a. A s is w e l l known[17], t h e d e c a y i n g r a t e f o r a n o r t h o t r o p i c m a t e r i a l m a y b e l a r g e r o r s m a l l e r t h a n t h a t f o r t h e i s o t r o p i c case. NECESSARY CONDITIONS FOR DECAYING STATES / S.l 21 CHAPTER 3 NECESSARY CONDITIONS FOR DECAYING STATES A n e l a s t o s t a t i c s t a t e s u c h as a n i n n e r s o l u t i o n i n o u r p r o b l e m , w h i c h d e c a y s e x p o n e n t i a l l y t o z e r o away f r o m b o u n d a r i e s , i s c a l l e d a d e c a y i n g s t a t e . T h e r e s u l t s of the l a s t c h a p t e r show t h a t t h e d i f f e r e n c e b e t w e e n t h e e x a c t s o l u t i o n a n d t h e o u t e r s o l u t i o n is a d e c a y i n g s t a t e . We k n o w f r o m [11] t h a t t h e o u t e r s o l u t i o n i n t h e i s o t r o p i c case m a y be o b t a i n e d w i t h o u t a n y reference t o t h e i n n e r s o l u t i o n b y e s t a b l i s h i n g e n o u g h n e c e s s a r y c o n d i t i o n s f o r d e c a y i n g s t a t e s . T h i s also a p p l i e s t o t h e o r t h o t r o p i c case. T h e n e c e s s a r y c o n d i t i o n s f o r d e c a y i n g s t a t e s w i l l be d e d u c e d i n t h i s c h a p t e r a n d t h e y are of t w o t y p e s . T h e f i r s t t y p e has a n e x a c t , e x p l i c i t f o r m d e r i v e d b y i n t e g r a t i n g t h e b a s i c e q u a t i o n s a n d u s i n g t h e f a c t t h a t t h e a s y m p t o t i c e x p a n s i o n of t h e i n n e r s o l u t i o n is a l s o t h e e x a c t e x p r e s s i o n of t h e d e c a y i n g s o l u t i o n . T h e s e c o n d t y p e is o b t a i n e d b y a p p l y i n g t h e r e c i p r o c a l t h e o r e m as done i n [11] f o r i s o t r o p i c m a t e r i a l s . T h e l a t t e r t y p e i s of a s y m p t o t i c n a t u r e ; t h e e r r o r i n v o l v e d is e x p o n e n t i a l l y s m a l l as e —• 0. M o r e o v e r , it is e x p r e s s e d i n t e r m s of t h e s o l u t i o n s o f t h r e e c a n o n i c a l p r o b l e m s d e f i n e d i n t h i s c h a p t e r w h i c h c a n o n l y be o b t a i n e d n u m e r i c a l l y . T h e c o n t e n t s of t h i s c h a p t e r , t o g e t h e r w i t h c h a p t e r s 2,7 a n d 8, have b e e n p u b l i s h e d i n [20]. .3.1 NECESSARY CONDITIONS FOR DECAYING STATES NECESSARY CONDITIONS FOR DECAYING STATES / 3.1 22 We b e g i n w i t h t h e d e f i n i t i o n o f t h r e e i n t e g r a l s w h i c h w i l l be c a l l e d m e m b r a n e stres s r e s u l t a n t , m o m e n t r e s u l t a n t a n d t r a n s v e r s e s h e a r r e s u l t a n t r e s p e c t i v e l y : axdz, M x = zaxdz, Qx = aX2dz. (3.1) -h J-h J-h I n t e g r a t i o n of eq.(2.1) w i t h r e s p e c t t o z a n d use of t h e b o u n d a r y c o n d i t i o n (2.9) y i e l d ^ = 0 , ^ = 0 dx dx o r Nx = const., Qx = const.. (3.2) B y m u l t i p l y i n g t h e f i r s t e q u a t i o n of (2.1) b y z a n d i n t e g r a t i n g w.r.t. z, we o b t a i n dMx_ fh da„ h fh ~~dx~ J h ~dz~ ~za**\-h + J ***** "* o r Mx = Qxx + const.. (3.3) For a d e c a y i n g s t a t e , t h e c o n s t a n t s i n (3.2) a n d (3.3) v a n i s h . I n f a c t , we have seen i n Ch.2 t h a t d e c a y i n g s t a t e s are c o m p o s e d of e i g e n f u n c t i o n s a s s o c i a t e d w i t h non-zero eigenvalues. E a c h e i g e n f u n c t i o n has t h e f o r m <f> = X{x)Z{z) = tXxZ{z) (3.4) where A ^  0 a n d Z{z) s a t i s f i e s t h e b o u n d a r y c o n d i t i o n s at t h e t o p a n d b o t t o m surfaces: Z{h) = Z{-h) = 0, Z'{h) = Z'(-h) = 0. (3.4a) F o r s i m p l i c i t y , we use one g e n e r a l t e r m t o s t a n d f o r t h e l i n e a r c o m b i n a t i o n of t h e eigen-f u n c t i o n s , i.e. t h e d e c a y i n g s t a t e s , i n t h i s s e c t i o n . Now, we have f o r a d e c a y i n g s t a t e Nx = £ a x d z = £ ^ d z =^z\h-h = X{x)Z'{z)th = 0, Mx = J" z*xdz = j h Z ^ d z = (z^-<t>)\h_h = X(x)(zZ'{z)- Z(z))\h_h = 0. NECESSARY CONDITIONS FOR DECAYING STATES / 3.1 23 f axdz = 0, (3.5) J-h I zoxdz = 0, (3.6) J-h [ axgdz = 0. (3.7) J-h T h u s , we h a v e c o n f i r m e d f o r a d e c a y i n g s t a t e necessary condition 1 necessary condition 2 necessary condition 3 -h A s a n a p p l i c a t i o n o f these n e c e s s a r y c o n d i t i o n s , we e x p r e s s t h e c o n s t a n t s of i n t e g r a t i o n i n t h e o u t e r s o l u t i o n , i.e. N, Q, MQ i n t e r m s of t h e stre s s i n t e g r a l s . A c c o r d i n g t o C h . 2 , f o r a g e n e r a l p l a n e s t r a i n s t a t e , t h e d i f f e r e n c e b e t w e e n t h e e x a c t s o l u t i o n a n d t h e o u t e r s o l u t i o n is n e c e s s a r i l y a d e c a y i n g s t a t e . A p p l i c a t i o n o f t h e n e c e s s a r y c o n d i t i o n s g i v e s >h [ax - ax)dz = 0, —h h 0> / z{ax - ax)dz = 0, J-h rh / (cr„-a2,)«fe = 0 J-h where (TX,(TXZ s t a n d f o r t h e v a r i o u s st r e s s c o m p o n e n t s of t h e e x a c t s o l u t i o n a n d CTX,<JXZ f o r the c o r r e s p o n d i n g st r e s s c o m p o n e n t s of t h e o u t e r s o l u t i o n . B y u s i n g e x p r e s s i o n s f o r t h e o u t e r s o l u t i o n i n C h . 2 , we have •ft Srr _ o r H . N 3 M „ 3 Q . . , n  H{ ° * - U - 2 l S Z - 2 ¥ X z ) d z = 0> H , N 3 M 0 3Q n _ H Z ^ * - U - 2 l S Z - 2 V X z ) d Z = 0> {<rxz--A%(h2-z2))dz = 0 / N = j ha*dzi Q = f °xzdz, J-h fh MQ + Qx — I zuxdz J-h f o r any v a l u e o f x, i n p a r t i c u l a r at t h e b o u n d a r y . T h i s gives a c l e a r p h y s i c a l m e a n i n g f o r A r, Q a n d M c NECESSARY CONDITIONS FOR DECAYING STATES / 3.1 24 T h r e e o t h e r necessary c o n d i t i o n s a r e r e l a t e d t o d i s p l a c e m e n t s a n d stresses. I n t e g r a t i o n of t h e first e q u a t i o n o f (2.3) du — = €x = anax + al2(Tz ox across t h e t h i c k n e s s g i v e s ^ rh rh dx j_u ( / udz) = a n N x + a12 f <rzdz J-h J-h = - f li2 / J-h fH d<J*>A = °12 / — d z J-h 9x z-r-dz -h >h -h where t h e ne c e s s a r y c o n d i t i o n Nx = 0, t h e b o u n d a r y c o n d i t i o n az = 0 at z = ±h a n d eq.(2.1) have been used. I n t e g r a t i o n w.r.t. x gives ?h /udz — al2 / zaxzdz = c o n s t . -h J-h w h e r e t h e c o n s t a n t c a n be p r o v e d t o be z e r o f o r d e c a y i n g s t a t e s . I n f a c t , f o r d e c a y i n g s t a t e s we have jj- = ex = ax/Ex - azul2/E2 = eXxZ"(z)/E1 - ul2X2eXxZ(z)/E2 o r u = {Z"(z)/{XE1) - u12XZ(z)/E2}eXx where (3.4) has been used. It f o l l o w s t h a t u - a12zaX2 = {Z"(z)/(XE1) - ul2XZ{z)/E2 - ul2XzZ'(z)/E2}eXx, a n d , b y use of (3.4a), / ( u - al2zcXi)dz = Z'{z)l{XEx)\h_h - vl2XzZ(z)/E2\h_h = 0 J-h w h i c h gives f o r d e c a y i n g s t a t e s n e c e s s a r y c o n d i t i o n 4 NECESSARY CONDITIONS FOR DECAYING STATES / 3.1 25 rh .. i-h -h where J udz + J zaxzdz = 0 (3.8) v _ vx2 _ u2l _ ^/ul2u2l _ E - E ^ - E ; - T^IT - "°12- ( 3 - 9 ) Another necessary condition may be derived by multiplying the first equation of (2.3) by z and integrating across the thickness: f zudz = a,\\Mx + a12 f zcr.dz ox J^h J-h fh z2 daz J fh z2 daXi = -AI2J_hJ-dz-DZ=ANJ_hJ-dx-dz where Mx = 0, az = 0 at z = ±h and an equilibrium equation have been used. Integration w.r.t. x gives fh fh z2 I zudz — a12 I —o~ x zdz = const. J-h J-h 2 where the constant can be shown to be zero for decaying states by a similar argument. In fact, for decaying states, zu - al2z2axz/2 = ex*{z/{\El)Z"{z) - ul2\zZ{z)/E2 - unXz2 Z\z)/{2E2)} and j \ z u - a12jaTZ)dz = eXx{[zZ'(z) - Z{z))/{\E{) - ul2Xz2Z(z)/(2E2)}\h_h = 0 which gives for decaying states necessary condition 5 rh v ,h z2 J_ zudz +EJ_ J ^ z d z = °- (3-10) .  fc ,2 zudz + - J To find one more necessary condition, we multiply the third equation of (2.3) NECESSARY CONDITIONS FOR DECAYING STATES / 3.1 26 b y z a n d i n t e g r a t e w.r.t. z t o get *h rh a.. i fh d f 2 i r 2 ^ 1 r 2 • T x j . h z w d z + j _ h z T z d z = G ) _ h z a " d z K Z33(T h 3 5* -h 3 ax O n t h e o t h e r h a n d , t h e d i r e c t i n t e g r a t i o n of t h e same e q u a t i o n gives • /> rh - h i - h i : ^-dz (3.11) d r J r duJ QX N , , It is n o t d i f f i c u l t t o see t h a t z2—dz — 2 2 u | \ — / 2;zu<£z —dz -2 1 zudz •k dz y _ A =-h2TXS-hwdz+^S-hz2<Txzdz o r /•n 2 9 U j l2d fh , v fh z3daXj / z2—dz = -h2 — wdz + — ——*dz. (3.13) dz dx J_h E J_h 3 dx K J S u b s t i t u t i n g (3.13) i n t o (3.11) a n d i n t e g r a t i n g across t h e t h i c k n e s s y i e l d /h 1 v fh jz3 {h? - z2)wdz + ( - - — ) J —vxdz = c o n s t . where the c o n s t a n t c a n be s h o w n t o be zero f o r d e c a y i n g s t a t e s . I n f a c t , f o r t h e g e n e r a l t e r m of a d e c a y i n g s o l u t i o n , B y u s i n g i n t e g r a t i o n b y p a r t s a n d t h e b o u n d a r y c o n d i t i o n s o n t h e p l a t e s u r f a c e s , we get J\(k2 - z2)w + ( I - ^ jvjdz = ( | - ±)eXx 2zZ(z)dz + 0 + ( ^ ~ ^ ) e A l / ' i , 2 2 Z ( 2 M 2 = 0 NECESSARY CONDITIONS FOR DECAYING STATES / 3.2 27 w h i c h m a y be r e w r i t t e n as n e c e s s a r y c o n d i t i o n 6 f\h* - z*)wdz y* ^axdz = 0. (3.14) T h e s e s i x e x p l i c i t n e c e s s a r y c o n d i t i o n s a r e c o n s i s t e n t w i t h t h o s e i n t h e i s o t r o p i c case w h i c h were o b t a i n e d b y an a p p l i c a t i o n of t h e r e c i p r o c a l t h e o r e m a n d p r o v e d t o b e c o r r e c t i n a n a s y m p t o t i c sense as e —» 0. We have r e m o v e d t h e a s y m p t o t i c i t y r e s t r i c t i o n . 3.2 ASYMPTOTIC NECESSARY CONDITIONS FOR DECAYING STATES We are u n a b l e t o get e x p l i c i t n e c e s sary c o n d i t i o n s e x p r e s s e d i n t e r m s of t h e i n t e g r a l s of d i s p l a c e m e n t s . O n t h e o t h e r h a n d , we k n o w f r o m [11,12] t h a t , b y a n a p p l i c a t i o n of t h e r e c i p r o c a l t h e o r e m t o a d e c a y i n g s t a t e a n d d i f f e r e n t r e g u l a r s t a t e s a s s o c i a t e d w i t h c e r t a i n c a n o n i c a l p r o b l e m s , we m a y d e v e l o p necessary c o n d i t i o n s f o r d i s p l a c e m e n t edge - -data u p to a n e r r o r e x p o n e n t i a l l y s m a l l as" e —• 0. I n o t h e r words, t h e s e necessary c o n d i t i o n s are c o r r e c t a s y m p t o t i c a l l y . T h e d e c a y i n g s t a t e s c o n s i d e r e d here are f o r d i s p l a c e m e n t edge d a t a u = u, w = w at x — — I a n d o t h e r b o u n d a r y c o n d i t i o n s at x — /. T h e c a n o n i c a l p r o b l e m s we need are f o r t h e d i s p l a c e m e n t b o u n d a r y c o n d i t i o n s u = 0, w = 0 at x = — / a n d have t h e f o l l o w i n g s p e c i f i e d o u t e r s o l u t i o n s r e s p e c t i v e l y : C B V P 1 : N = 1, M 0 = 0, Q = 0; C B V P 2 : N = 0, A f 0 = 1, Q = 0; (3.15) C B V P 3 : N = 0, M0 = I, Q = 1. N o t e t h a t M 0 i n C B V P 3 has b e e n so chosen t h a t M x = 0 at x = — I. I n p r i n c i p l e , we c a n choose a n y l i n e a r c o m b i n a t i o n of o u t e r s o l u t i o n s i n (3.15) t o d e f i n e c a n o n i c a l p r o b l e m s . T h e r e c i p r o c a l t h e o r e m says t h a t t h e w o r k on t h e d i s p l a c e m e n t s of s t a t e 1 done by t h e forces of s t a t e 2 is e q u a l t o t h e w o r k o n t h e d i s p l a c e m e n t s of s t a t e 2 d o n e by t h e forces of NECESSARY CONDITIONS FOR DECAYING STATES / 3.2 28 s t a t e 1. W e c hoose s t a t e 1 t o b e t h e d e c a y i n g s t a t e i n d u c e d b y t h e ( r e s i d u a l ) edge d a t a a n d s t a t e 2 t o be t h e s o l u t i o n of a c a n o n i c a l b o u n d a r y v a l u e p r o b l e m , a n d a p p l y t h e r e c i p r o c a l t h e o r e m t o t h e s t r i p b o u n d e d b y x = — I a n d x = 0. S i n c e t h e r e i s n o c o n t r i b u t i o n f r o m t h e t r a c t i o n f r e e t o p a n d b o t t o m surfaces, we o b t a i n w h e r e oxi,ozzi are stresses at x = —/ i n t h e c a n o n i c a l p r o b l e m i a n d t h e r i g h t h a n d s i d e r e p r e s e n t s c o n t r i b u t i o n s f r o m t h e c r o s s s e c t i o n x = 0. T h e c o n s t a n t A 0 s t a n d s f o r t h e least p o s i t i v e r e a l p a r t of t h e eigenvalues f o r a i n (2.32) a n d /? i n (2.34). B y n e g l e c t i n g the e x p o n e n t i a l l y s m a l l e r r o r t e r m as e —• 0, we h ave f o r d e c a y i n g s t a t e s a s s o c i a t e d w i t h d i s p l a c e m e n t edge d a t a - a s y m p t o t i c n e c e s s a r y c o n d i t i o n s N o t e t h a t i n the d e f i n i t i o n of c a n o n i c a l p r o b l e m s , t h e b o u n d a r y c o n d i t i o n s at x = I have not b e en g i v e n s i n c e t h e y o n l y have an e x p o n e n t i a l l y s m a l l i n f l u e n c e o n t h e a b o v e n e c e s s a r y c o n d i t i o n s . I n o t h e r words, we ma y r e g a r d c a n o n i c a l p r o b l e m s as s e m i - i n f i n i t e s t r i p s c l a m p e d at t h e end a n d w i t h a s p e c i f i e d a p p l i e d l o a d of a x i a l f o r c e , b e n d i n g m o m e n t a n d s h e a r i n g f o r c e at i n f i n i t y . A c c u r a t e n u m e r i c a l s o l u t i o n s f o r t h e t h r e e c a n o n i c a l p r o b l e m s w h i c h are u s e d i n (3.17) have been a c h i e v e d b y first e x t r a c t i n g t h e c o r n e r s t r e s s s i n g u l a r i t y d i s c u s s e d i n t h e n e x t c h a p t e r . T h e a c t u a l s o l u t i o n s of t h e c a n o n i c a l p r o b l e m s w i l l be d i s c u s s e d i n C h . 5 . T h e a p p l i c a t i o n s of t h e n e c e s s a r y c o n d i t i o n s d e r i v e d i n t h i s c h a p t e r w i l l be p r e s e n t e d i n Ch.6. (3.16) (» = 1,2,3). (3.17) CORNER STRESS SINGULARITIES / 4.0 29 CHAPTER 4 CORNER STRESS SINGULARITIES I n s e c t i o n 3.2, i t was s h o w n t h a t s o l u t i o n s of c e r t a i n c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s f o r c a n t i l e v e r e d s e m i - i n f i n i t e s t r i p s a r e n e e d e d f o r t h e a p p l i c a t i o n s of n e c e s s a r y c o n d i t i o n s f o r a d e c a y i n g s t a t e i n v o l v i n g d i s p l a c e m e n t b o u n d a r y c o n d i t i o n s . I n t h i s a n d n e x t c h a p t e r s we w i l l s o l v e t h e b o u n d a r y v a l u e p r o b l e m s of a f o u r t h o r d e r , l i n e a r , c o n s t a n t c o e f f i c i e n t p a r t i a l d i f f e r e n t i a l e q u a t i o n i n a n u n b o u n d e d , n o n s m o o t h d o m a i n . It i s w e l l k n o w n i n t h e t h e o r y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s t h a t t h e s o l u t i o n o f t h e b o u n d a r y v a l u e p r o b l e m is c o r r e s p o n d i n g l y s m o o t h i f t h e c o e f f i c i e n t s o f t h e e q u a t i o n a n d b o u n d a r y o p e r a t o r s , t h e i r r i g h t h a n d sides, a n d t h e b o u n d a r y of t h e d o m a i n are s u f f i c i e n t l y s m o o t h . O n t h e o t h e r h a n d , t h e l a c k of t h e b o u n d a r y s m o o t h n e s s m a y l e a d t o t h e o c c u r r e n c e of s i n g u l a r i t i e s o f t h e s o l u t i o n i n t h e n e i g h b o u r h o o d o f n o n - r e g u l a r p o i n t s (e.g. c o r n e r s ) of t h e b o u n d a r y . A t h o r o u g h s u r v e y of t h i s t h e o r y i s g i v e n i n [18]. T h e e x i s t e n c e of a st r e s s s i n g u l a r i t y at t h e v e r t e x o f a r i g h t i s o t r o p i c c o r n e r w i t h one s i d e t r a c t i o n free a n d t h e o t h e r s i d e fixed i s w e l l k n o w n [38]. W e pr o v e i n t h i s c h a p t e r t h a t t h i s i s a l s o t r u e f o r a n o r t h o t r o p i c r i g h t c o r n e r w i t h t h e same b o u n d a r y c o n d i t i o n s even t h o u g h t h e g o v e r n i n g e q u a t i o n i s m o r e c o m p l i c a t e d t h a n f o r t h e i s o t r o p i c case. F u r t h e r m o r e , we f o u n d t h a t t h e str e s s s i n g u l a r i t y e x p o n e n t r e m a i n s t h e same i f t h e o r t h o g o n a l m a t e r i a l p r i n c i p a l axes are s u b j e c t t o a r o t a t i o n b y 90 degrees, o r a re e x c h a n g e d . W i t h a c o r n e r s t r e s s CORNER STRESS SINGULARITIES / 4.1 30 s i n g u l a r i t y a c c u r a t e l y e v a l u a t e d a n d e x t r a c t e d , t h e r e s i d u a l stresses at t h e fixed e n d of t h e c a n t i l e v e r e d s e m i - i n f i n i t e s t r i p of t h e c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s b e c o m e b o u n d e d a n d c a n be c o m p u t e d a c c u r a t e l y as s o l u t i o n s of i n t e g r a l e q u a t i o n s d i s c u s s e d i n c h a p t e r 5. A F O R T R A N p r o g r a m has been d e v e l o p e d t o get a c c u r a t e n u m e r i c a l r e s u l t s f o r s t r e s s s i n g u l a r i t y e x p o n e n t s . 4.1 SOLUTION IN POLAR COORDINATES We b e g i n w i t h eq.(2.7) w h i c h c a n b e w r i t t e n as 'd£ 2 ^ k2dz2)Kdx2 + dz fd2 1 d \ f d 2 t 2 d 2 w n • % o r where — + k2 — di2 dz T h e e q u a t i o n is d e f i n e d i n a r i g h t c o r n e r x > 0, z > 0 w i t h b o u n d a r y c o n d i t i o n s °xz = cr 2 = 0, at 2 = 0, (4.4) u = 0, w = 0, at x = 0. (4.5) W i t h t h e p o l a r c o o r d i n a t e s eq.(4.2) becomes i = pcos(p, kz = psintp, (4.6) , d2 1 d 1 0 2 x ^ n ^ + pd-p + 7 2 ^ ^ 0 { A 1 ) w h i c h h as t h e t r i v i a l s o l u t i o n a n d s o l u t i o n s f r o m s e p a r a t i o n of v a r i a b l e s $ = px cos(Ay?), px sin(Ay?) (4.8) CORNER STRESS SINGULARITIES / 4.1 31 w h e r e A i s a n a r b i t r a r y c o n s t a n t . Eq.(4.8) c a n now b e w r i t t e n as + h>)4> = ( A o s f A ^ y sin(Av?),0} (4.9) w h e r e a c o n s t a n t f a c t o r has b e en d r o p p e d s i n c e we are o n l y i n t e r e s t e d i n t h e f o r m o f t h e s o l u t i o n s . W i t h t h e p o l a r c o o r d i n a t e s ki = rcos9, 2 = r s i n 0 , (4-10) eq.(4.9) becomes ^dr2 + rd? + 7>dT>)4, = { A o s ( A p ) , / s i n ( A y > ) , 0 } . (4.11) S i n c e t h e c o o r d i n a t e s ( r , 9) a n d (/?, <p) are r e l a t e d b y p = r\/k2 s i n 2 9+1/k2 c o s 2 9, t a n y ? = fc2tan0, (4.12) t h e r i g h t h a n d s i d e of (4.11) m a y be w r i t t e n as pxcos(X<p) = r Ap£ cos(Aw0), px sin(A^>) = rx px &'m(\u>o) where = A; s i n 2 0 + l / f c 2 c o s 2 9, UQ = A r c t a n ( A ; 2 t a n 9). S o l u t i o n s of eq.(4.11) f o r t h e case of a z e r o r i g h t h a n d s i d e are of t h e t y p e (4.13) (4.14) r ^ c o s ( / z ^ ) , r^sinf ^ i f l ) w h e r e \i is a n a r b i t r a r y c o n s t a n t . A p a r t i c u l a r s o l u t i o n o f eq.(4.11) f o r t h e r i g h t h a n d s i d e /? Acos(A^?) is of t h e t y p e w here f\(6) is g o v e r n e d b y t h e e q u a t i o n f'{{9) + (A + 2 ) 2 / A ( 0 ) = px0 c o s ( A ^ ) . (4.15) CORNER STRESS SLVGULARITIES / 4.2 32 Eq.(4.15) has a p a r t i c u l a r s o l u t i o n •e S i m i l a r l y , f o r eq.(4.11) w i t h r i g h t h a n d s i d e pXBin(X<p)} we h a v e t h e s o l u t i o n w h e r e i r° 9x(0) = j P<> s m ( A ^ ) s i n ( A + 2){6- (4.17) (4.18) 'o T h e g e n e r a l s o l u t i o n f o r eq.(4.1) i s t h e r e f o r e <f>(r, 9) = r A + 2 { C ! cos(A + 2)9 + C2 s i n ( A + 2)9 + C 3 / A ( 0 ) + CAgx{9)} = rX+2Gx(9) w h e r e Cl,C2,C2,Ci a n d A have t o be d e t e r m i n e d f r o m t h e b o u n d a r y c o n d i t i o n s . F o r a n i s o t r o p i c m a t e r i a l , we have k = 1, Prj, = 1, = tp. I n t h a t case, f\(0) r e d u c e s t o a l i n e a r c o m b i n a t i o n of c o s A 0 a n d cos(A + 2)9, a n d gx(9) t o t h a t of s i n A0 a n d s i n ( A + 2)9. T h e y are c o n s i s t e n t w i t h [38]. 4.2 EQUATION FOR EIGENVALUES S i n c e t h e c o o r d i n a t e s (x,z) a n d ( r , 9) are r e l a t e d b y z = 7 r c o s 0 , z = r s i n 0 , (4.19) we have d k. „ d s i n 0 d . — = - ( c o s (9 ) dx a dr r d9 d . d cos 9 d — = smO- + —. dz dr r d9 T h e y give e x p r e s s i o n s f o r t h e stresses i n t e r m s of ( r , 6) c o o r d i n a t e s as f o l l o w s k2 . 2.d2 s i n 29 d2 s i n 2 9 d2 s i n 2 9 d s i n 20 d , . °> = ^{C°S e'dV2 ~ ~r~d9dr + -^~oT2 + ~r~d~r + ~ 0 * > * ' _ k sin29 d2 sin29 d2 c o s 2 0 d 2 _ s i n 2 0 d c o s 2 0 d (420) * " ~ ~a{~~T~d^ ~ 2 r 2 W + r ~dr~d9 ~ 2r dr ~ ~ 7 2 " d 0 ^ ' -f,;2ad2 cos29d2_ s i n 29 d2 cos2 9 d s i n 20 d Wr2+~~r2~d92 + r drd9 + r d~r ~ r 2 de**' X CORNER STRESS SINGULARITIES / 4.2 33 T h u s t h e b o u n d a r y c o n d i t i o n cz = 0 at 0 = 0 gives dr2 o r f r o m (4.18) Ci = 0 . O n t h e o t h e r h a n d , t h e b o u n d a r y c o n d i t i o n o~xz = 0 at 0 = 0 y i e l d s r <?r<90 r 2 86 o r f r o m (4.18) G' A(0) = 0 w h i c h r e s u l t s i n C 2 = 0 where / A ( 0 ) = <7A(0) = 0 h a v e b e e n used. W i t h Cx = C 2 = 0 eq.(4.18) b e c o m e s <P(r,6) = rx+2{C3fx(e) + C4gx(6)}. (4.21) T h e b o u n d a r y c o n d i t i o n w = 0 at t h e edge a n d t h e s e c o n d e q u a t i o n of (2.3) g i v e "21 1 wh i c h , u p o n t h e s u b s t i t u t i o n o f e x p r e s s i o n s o f ax a n d az i n (4.20) e v a l u a t e d at 6 = ir/2, y i e l d s a n e q u a t i o n f o r C 3 a n d C 4 : C 3{/A(V2) + /A(T/2)7A> + C 4 { ^ ( i r / 2 ) + 9^/2)lx) = 0 (4.22) where 7 A = (A + 2 ) { l - ( A + l ) i V * 2 } . O n t h e o t h e r h a n d , d i f f e r e n t i a t i o n w i t h r e s p e c t t o z of t h e t h i r d e q u a t i o n of (2.3), use of t h e b o u n d a r y c o n d i t i o n u = 0 at t h e edge a n d t h e s e c o n d e q u a t i o n o f (2.3) g i v e + " E T ^ r = l 4 - 2 3 ) E, dx E<> dx G dz CORNER STRESS SINGULARITIES / 4.2 34 w h i c h is e v a l u a t e d at 6 = n/2 a n d r e s u l t s i n a n o t h e r e q u a t i o n f o r C 3 a n d C 4 : C 3 { / " V / 2 ) + + CM'(*/2) + </x(*/2)8x} = 0 (4.24) w h e r e 6X = 3 A + 4 + A ( A + 1){E/G - u)/k2 F o r C 3 a n d C 4 n o t t o b o t h be zero, t h e d e t e r m i n a n t of (4.22) a n d (4.24) m u s t v a n i s h : /A( */2) + 7A/A(*/2) fi{*/2) + 7A<7A(*/2) / r (*/2) + 5A/AU/2) < ( T / 2 ) + 5 A ^ A ( i r / 2 ) = 0. T h i s gives a n e q u a t i o n f o r t h e e i g e n v a l u e A k2X~2 - kx{\ + \)kx~2 / p $ w n ( A + 2 ) ( f f/2 -V)cosA ( ir/2-a^)<ty /.TT/2 + fc2(A + l)kx / cos ( A + 2)(7r/2 - V) s i n A(TT/2 - u^dip Jo + fcxA:2(A + l ) 2 / 2 / / 4 4 s i n ( A + 2 ) ( ^ - t f ) s i n A ( a ^ - aty)<ty<ty = 0 «/ 0 JO where fcx = 1 + I//A;2, Ar2 = \ + {u - E/G)/k2 are m a t e r i a l c o n s t a n t s . F o r a n i s o t r o p i c m a t e r i a l , we have a = 1, A; = 1, /ty = 1, = \p,ki — I + u, k2 = - ( 1 + y ) a n d eq.(4.25) re d u c e s t o s m 2 ( A + 1 ) - - T + * - — = 0 (4.25) (4.26) (4.27) w h i c h is e x a c t l y t h e same as i n [38 ] . We m e n t i o n i n p a s s i n g t h a t w i t h A d e t e r m i n e d b y (4.25), t h e r a t i o o f s t r e s s i n t e n s i t y f a c t o r , <rx{r,*/2) CORNER STRESS SINGULARITIES / 4.3 35 m a y be e v a l u a t e d u s i n g t h e e x p r e s s i o n s *,(r , ir/2) = (A + 1)(A + 2 ) r AG A(*/2) = ^(AJr*^, k (4.28) *«(r,ir/2) = - ( A + l)r AG 'A(7r/2) = J f 2 ( A ) r A C 0 a w h e r e K^X) = (A + 1)(A + 2) 2Jfc A / s i n ( A + 2)(TT/2 - V) s i n A(JT/2 - u^dif*, Jo K2{X) = £(A + 1)(A + 2 ) 2 { f c A J* 4 cos(A + 2)(ir/2 - V) s i n A(rr/2 - u^dxfj (4.29)} r*/2 /-*/2 , . + *!(A + 1) / / p jp j c o s ( A c ^ ) s i n ( A w ^ ) s i n ( A + 2)(<f> - ip)d<f>di>} Jo Jo a n d CQ is a c o n s t a n t f a c t o r . 4.3 NUMERICAL ANALYSIS F o r an i s o t r o p i c m a t e r i a l we c a n show th e e x i s t e n c e a n d t h e u n i q u e n e s s of a co r n e r s t r e s s s i n g u l a r i t y . F i r s t , we r e s t r i c t R e { A } > — 1 for t h e finiteness of s t r a i n e n e r g y a n d R e { A } < 0 f o r a stres s s i n g u l a r i t y . D e n o t i n g t h e e q u a t i o n (4.27) b y F ( A ) = 0, we see f r o m t h e f a c t f ' - 1 ' = ( 1 + , K 3 - y ) < ° - f ( ° ) = ( 1 + , ) ( 3 - t - ) > ° a r e a l s o l u t i o n i n i n t e r v a l — 1 < A < 0 e x i s t s . T o p r o v e t h a t t h e r e is o n l y one s o l u t i o n w i t h — 1 < Re { A } < 0, we w r i t e 1 + A = a + ib (0 < a < 1), a n d c o n s i d e r a r e c t a n g u l a r c o n t o u r i n t h e c o m p l e x A p l a n e c o n s i s t i n g of t h e segments of s t r a i g h t l i n e s : R e { A } = - l , R e { A } = 0, I m { A } = ±H w h e r e H is a n a r b i t r a r i l y l a r g e p o s i t i v e n u m ber. We t h e n s t u d y t h e i m a g e of t h i s c o n t o u r m a p p e d b y t h e a n a l y t i c f u n c t i o n " z = rW = - J!-1l+\ + ( 1 + ? ( a 2 - t ; ) - i — c - h * v ' 2(1 + v) 3-1/ 2 2(l + i/)o6 1 . .... + t f- - sin air sinn bir\ 3 — i/ 2 CORNER STRESS SINGULARITIES / 4.3 36 w h i c h h a s t h e p r o p e r t y o f Im{Z} > 0 f o r 6 > 0 a n d Im{Z} < 0 f o r b < 0. A s a p o i n t A moves a l o n g t h e c o n t o u r i n t h e u p p e r h a l f A p l a n e (6 > 0) f r o m A = 0 t o A = — 1 , t h e c o r r e s p o n d i n g p o i n t Z moves a l o n g a c u r v e i n t h e u p p e r h a l f Z p l a n e f r o m F(0) > 0 t o F{— 1) < 0 (Z is o n t h e n e g a t i v e r e a l a x i s as A i s o n t h e l i n e R e { A } = — 1 ) . T h e n , as A co n t i n u e s t o m o v e a l o n g t h e c o n t o u r i n t h e l o w e r h a l f A p l a n e (6 < 0) b a c k t o A = 0, Z c o n t i n u e s t o move a l o n g a c u r v e i n t h e l o w e r h a l f Z p l a n e b a c k t o F(0). A c c o r d i n g t o t h e A r g u m e n t P r i n c i p l e , we t h u s p r o v e t h a t t h e r e i s o n l y one z e r o f o r t h e a n a l y t i c f u n c t i o n Z = F(X) i n t h e r e g i o n - 1 < R e { A } < 1. T h a t i s , we need o n l y t o s e a r c h f o r a r e a l s o l u t i o n b e t w e e n — 1 a n d 0. F o r a n o r t h o t r o p i c m a t e r i a l , we c a n e a s i l y show t h e e x i s t e n c e o f a st r e s s s i n g u l a r i t y , i.e. a r e a l s o l u t i o n o f A betw e e n — 1 a n d 0. D e n o t i n g eq.(4.25) b y F(X) = 0, we t h e n see f r o m t h e f a c t F{0) = -u/k4 < 0, F ( - l ) = 1/Jfc4 > 0 the e x i s t e n c e of a r e a l s o l u t i o n b e t w e e n — 1 a n d 0. A c o m p l e x s o l u t i o n w i t h — 1 < R e { A } < 0 seems t o be p h y s i c a l l y u n r e a l i s t i c s i n c e a n i m a g i n a r y p a r t o f A w o u l d r e s u l t i n a s o l u t i o n o s c i l l a t i n g w i t h an i n f i n i t e l y l a r g e f r e q u e n c y as r —• 0. T h u s we r e s t r i c t o u r s e l v e s t o t h e r e a l s o l u t i o n — 1 < A < 0. A F O R T R A N p r o g r a m h a s been d e v e l o p e d t o find r e a l s o l u t i o n s of (4.25) b e t w e e n — 1 a n d 0 by a s t a n d a r d s u b r o u t i n e Z E R O l u s i n g a c o m b i n a t i o n of b i s e c t i o n a n d Newton's m e t h o d t o solve a n o n l i n e a r e q u a t i o n . A f t e r one s o l u t i o n A 0 i s f o u n d , t h e i n t e r v a l (—1, A 0 — e) is s e a r c h e d t o m a k e sure t h e r e is no s m a l l e r s o l u t i o n . T h e d o u b l e i n t e g r a l i n F[X) i s c o n v e r t e d t o s i n g l e i n t e g r a l s w h i c h are c o m p u t e d b y t h e s t a n d a r d s u b r o u t i n e D Q U A N K u s i n g Simpson's r u l e w i t h a r e f i n e d e r r o r c o n t r o l . N u m e r i c a l r e s u l t s have b e e n c h e c k e d i n t w o ways: F i r s t , t h e p r o g r a m is r u n f o r a n i s o t r o p i c m a t e r i a l a n d t h e r o o t f o u n d is s u b s t i t u t e d i n t o eq.(4.27), a s i m p l e e q u a t i o n f o r t h e i s o t r o p i c case. S e c o n d l y , c o m p a r i s o n is m a d e b e t w e e n t h e s o l u t i o n f o u n d here a n d t h e one f o u n d b y t h e s i n g u l a r i n t e g r a l e q u a t i o n m e t h o d w h i c h w i l l be d i s c u s s e d l a t e r . T h e agreement i s e x t r e m e l y g o o d . CORNER STRESS SINGULARITIES / 4.3 37 F o r a m a t e r i a l w i t h EX = 1 0 5 , E2 = 4200, G = 7500, vx2 = 0.01, b o t h m e t h o d s g i v e A = -0.07624379. It is c l e a r t h a t t h e eq.(4.25) as w e l l as i t s s o l u t i o n A r e m a i n u n c h a n g e d i f t h e o r t h o g o n a l m a t e r i a l p r i n c i p a l axes are s u b j e c t t o a r o t a t i o n b y 90 degrees, o r are i n t e r c h a n g e d . S o me r e s u l t s a r e l i s t e d b e l ow f o r f u t u r e reference. T h e u n i t f o r e l a s t i c i t y m o d u l u s i s kg/cm2. m a t e r i a l E2 G 1^2 2^1 A i s o t r o p i c 0.3 -0.24165 i s o t r o p i c 0.5 -0.31002 p l y w o o d 1 1.2 x 1 0 5 6.44 x 1 0 4 7200 0.044 0.082 -0.10704 p l y w o o d 2 6.44 x 1 0 4 1.2 x 1 0 5 7200 0.082 0.044 -0.10704 p i n e w o o d 1 10 5 4200 7500 0.01 0.238 -0.07624 p i n e w o o d 2 4200 1 0 5 7500 0.238 0.01 -0.07624 CANONICAL PROBLEMS / 5.0 38 CHAPTER 5 CANONICAL PROBLEMS T h i s c h a p t e r gives d e t a i l e d d i s c u s s i o n s o n m e t h o d s f o r s o l v i n g t h e c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s d e f i n e d i n c h a p t e r 3 i n c l u d i n g t h e n u m e r i c a l r e s u l t s . A s men-t i o n e d i n t h e I n t r o d u c t i o n , t h e p r o b l e m f o r stresses at t h e r o o t of a s e m i - i n f i n i t e i s o t r o p i c s t r i p u n d e r e x t e n s i o n , b e n d i n g a n d s h e a r i n g has a l o n g h i s t o r y a n d s t i l l needs f u r t h e r work. T h e r e s u l t s i n [35] a n d [3] show t h a t a series s o l u t i o n at t h e r o o t x = 0 converges slowly[9] a n d p a y i n g no s p e c i a l a t t e n t i o n t o t h e c o r n e r s t r e s s s i n g u l a r i t y m a y a c t u a l l y re-s u l t i n d i v e r g e n c e i n p r a c t i c e . B y u s i n g t h e m e t h o d of p r o j e c t i o n t o d e a l w i t h t h e c o r n e r stress s i n g u l a r i t y , c o n v e r g e n c e was a c h i e v e d i n [10] i n t h e sense of t h e L2— n o r m w i t h t h e d i s a d v a n t a g e of c o n s t r u c t i n g a n d s o l v i n g a l a r g e set of e q u a t i o n s d u e t o s l o w convergence. W i t h t h e h e l p of a g e n e r a l i z e d o r t h o g o n a l i t y c o n d i t i o n , we are a b l e t o r e d u c e t h e c a n o n i c a l p r o b l e m s t o s o l v i n g a n i n t e g r a l e q u a t i o n o f t h e first k i n d w i t h a k e r n e l i n a series f o r m . T h e i n t e g r a l e q u a t i o n i s t h e n s o l v e d b y a p p l y i n g a r e g u l a r i z a t i o n t e c h n i q u e d e v e l o p e d i n [28,32,33]. C o n v e r g e n c e i s a c h i e v e d w i t h a n a p p r o p r i a t e c h o i c e of a p a r a m e t e r 5 i n t h e reg-u l a r i z a t i o n p r o c e d u r e . E f f o r t s have a l s o b e e n m a d e t o get p o i n t w i s e c o n v e r g e n t s o l u t i o n s f o r a s e m i - i n f i n i t e i s o t r o p i c s t r i p b y i n t e g r a l t r a n s f o r m m e t h o d s [36,16,2]. I n p a r t i c u l a r , [36] c o n s i d e r s a p a r t i c u l a r p r o b l e m w i t h t h e e n d f i x e d a n d a p a i r of c o n c e n t r a t e d n o r m a l fo r c e s a p p l i e d at t h e u p p e r a n d l o w e r s u r f a c e s , b u t c o u l d n o t r e d u c e t h e d e r i v e d i n t e g r a l CANONICAL PROBLEMS / 5.1 39 e q u a t i o n t o one of C a u c h y t y p e . O n t h e o t h e r h a n d , [16] s o l v e s t h e p r o b l e m of u n i t ex-t e n s i o n a n d o b t a i n s a g e n e r a l i z e d C a u c h y t y p e s i n g u l a r i n t e g r a l e q u a t i o n b y a m e t h o d i n w h i c h an i n t e g r a l does n o t e x i s t a c c o r d i n g t o [2]. A l s o i n [2], t h e i s o t r o p i c p r o b l e m s are r e d u c e d t o a s y s t e m of t w o s i n g u l a r C a u c h y t y p e i n t e g r a l e q u a t i o n s . We e m p l o y here a l s o t h e i n t e g r a l t r a n s f o r m m e t h o d b u t now d e r i v e a s i n g l e g e n e r a l i z e d C a u c h y - t y p e s i n g u l a r i n t e g r a l e q u a t i o n f o r a s e m i - i n f i n i t e o r t h o t r o p i c s t r i p i n e x t e n s i o n , b e n d i n g a n d flexure, r e s p e c t i v e l y . T h e e q u a t i o n s o b t a i n e d are s l i g h t l y m o r e g e n e r a l t h a n those s t u d i e d i n [4,5] a n d are a n a l y z e d i n a s i m i l a r way. T h e c o m p u t e r p r o g r a m s we d e v e l o p e d f o r t h e g e n e r a l o r t h o t r o p i c case c a n a l s o d e a l w i t h t h e i s o t r o p i c case, a n d a p p e a r t o be s u p e r i o r t o t h o s e i n [36,16,2]. O u r e x p e r i e n c e shows t h a t i t is easy t o get a s t a b l e c o n v e r g e n t s o l u t i o n b y t h e i n t e g r a l t r a n s f o r m m e t h o d . A s i t is s u p e r i o r t o m e t h o d s u s i n g series e x p a n s i o n s , we w i l l first d i s c u s s t h e i n t e g r a l t r a n s f o r m m e t h o d below. 5.1 INTEGRAL EQUATION FOR SYMMETRIC CASE C a n o n i c a l b o u n d a r y v a l u e p r o b l e m 1, o r C B V P l f o r s h o r t , is t h e e x t e n s i o n p r o b l e m d e f i n e d b y eq.(2.7) <f>xxix + (2 + V)4>xxzz + 4>zzzz = 0 f o r 0 < x < oo, —h<z<h w i t h b o u n d a r y c o n d i t i o n s az = axz = 0 , at z — ±h u = 0, w = 0, at x = 0 an d u n i t t r a c t i o n f o r c e at x = oo. We s u b t r a c t off t h e f o l l o w i n g p a r t i c u l a r ( o u t e r ) s o l u t i o n : a* ~ oh* °xz = °' °z~ °' <5-2> w h e r e u a n d c w i l l b e set t o z e r o f o r s y m m e t r y w i t h r e s p e c t t o t h e z - a x i s . T h e b o u n d a r y v a l u e p r o b l e m is t h e n r e d u c e d t o one f o r a d e c a y i n g s t a t e d e f i n e d b y t h e s a me e q u a t i o n , CANONICAL PROBLEMS / 5.1 40 t h e s a me b o u n d a r y c o n d i t i o n s at z = ±h, b u t d i f f e r e n t b o u n d a r y c o n d i t i o n s at x = 0 a n d x = oo: (5.3) u = u 0 = - d , w = u>0 = - - ^ r a< x = 0, °"D O'xi, <x2, u, tu —* 0 e x p o n e n t i a l l y . T o s o l v e t h e r e d u c e d p r o b l e m , we a p p l y t h e F o u r i e r c o s i n e t r a n s f o r m f°° $ ( s , z ) = / 4>(xy z) cos(xs)dx (5.4) •Jo t o t h e b a s i c e q u a t i o n a n d use poo / 4>ii cos(sx)dx = -<t>x{0} z) - s 2 $ ( s , z ) , Jo / txiii c o s ( s x ) d x = - ^ i i ( 0 » + s2<f>x(0, z) + s 4 $ ( s , z ) 0^ (5.5) t o o b t a i n w h e r e $ z z z z - s 2 ( 2 + p)$zz + s 4 $ = <j(s, z ) (5.6) <7(s,z) = fe(0,z) - s 2 ^ ( 0 , z ) + (2 + p)4>izz{0,z). (5.7) T h e F o u r i e r s i n e t r a n s f o r m t r a n s f o r m s t h e b o u n d a r y c o n d i t i o n axz(xtk) = 0 t o $z{s,h) = 0. (5.8) H e r e we need o n l y t o b e c o n c e r n e d w i t h t h e b o u n d a r y c o n d i t i o n at z = h because o f t h e s y m m e t r y . B y a F o u r i e r c o s i n e t r a n s f o r m , t h e b o u n d a r y c o n d i t i o n <rz(x,h) = 0 becomes - s 2 $ ( s , / 0 - ^ ( o , / 0 = o o r $ ( s , A ) = 0 (5.9) fo r a d e c a y i n g s t a t e <p(x, z) i n a series f o r m of t h e t y p e X(x)Z(z) w i t h Z(k) = Z'[h) = 0. T h e f o l l o w i n g r e l a t i o n s f o r a d e c a y i n g s t a t e , <t>x(°,z) = -a f vzzo{y)dy, Jh <t>xzz(0,z) = -a±axz0(z), (5.10) <t>ixi{0,z) = -aE-j-2Mz) + a (§ - « 0 ^ « o ( « ) CANONICAL PROBLEMS / 5.1 41 a l l o w us t o e x p r e s s g(s} z) i n t e r m s of u 0 ( z ) a n d a X 2 0 ( z ) : d d2 tz g(s,z) = av—axz0{z)-aE-j^u0{z)+as2j^ axz0{y)dy (5.11) w h e r e UQ(Z),(TX:Q(Z) are t h e v a l u e s o f u ( z ) a n d 0xz(z) at x = 0. T h e b o u n d a r y v a l u e p r o b l e m d e f i n e d b y (5.6), (5.8) a n d (5.9) has s o l u t i o n ${s,z)= [ K{s,z,y)g{s,y)dy (5.12) J o where K(s,z,y) is d e r i v e d i n a p p e n d i x A a n d l i s t e d as f o l l o w s : K ( s z v ) _ _ J L_ I O^k) + kHT^k) + T^l/k)} + k'O^l/k), 0 < y < z; { ' , V ) ~ k* - 1 s 3 A , 1 02(k) + k2{T2(k) + T2(l/k)} + k*02{l/k), z<y<h, (5.13) w i t h k d e n n e d i n (2.29) a n d A , = cosh(skh) s'mh(sh/k) — k2 8mh(skh)cosh(sh/k), Oi(k) = sin h ( s / i / f c ) cosh(sA:y) sinh(sA;(z — h)), 02(k) = s'mh(sh/k) cosh(sfcz) sinh(sA:(y — h)), Ti(k) = cosh(sA:t/){cosh(s/i/A:) cosh(sA:(z — h)) — cosh(sz/k)}, T2(k) = cosh(sA;z){cosh(s/i/A;) cosh(sfc(y — h)) — cosh(syfk)}. S u b s t i t u t i n g (5.11) i n t o (5.12) a n d i n t e g r a t i n g b y p a r t s g i v e us ${s,z) = -aE J K{s,z,y)d^-dy + J K{s,z,y){au?^ + as2 J\xz0dy}dy „ [ h d K d u Q , f h f dK * f* r r t —\ i—\ = a E T~-£~dy- {av— + as / h{a,zty)dy}tTxt(i[y)dy. Jo °y °y Jo °y Jo (5.14) A n i n v e r s e F o u r i e r c o s i n e t r a n s f o r m t h e n y i e l d s 2 f°° 4>(x,z) =— I c o s ( z s ) $ ( s , z)ds * Jo = -aE f J ( x , « f y ) ^ r f y (5.15) n Jo ay 2 fh - - / {aul(i, z, y) + aJ{i, z, y)}<rxz0(y)dy x Jo CANONICAL PROBLEMS / 5.2 42 w h e r e I[x,z,y) = I cos[xs)——ds Jo oy f°° i fy J(x,z,y) = I cos(xs)s I K(s,z,y)dyda Jo Jo (5.16) '0 0^We i n s e r t (5.15) i n t o t h e s t r a i n d i s p l a c e m e n t r e l a t i o n = ^/E2 - M E i ) ° x = D4>{x,z) (5.17) w h e r e „ 1 d2 v d2 1 „ v n , a n d let x —• 0 to get t h e needed i n t e g r a l e q u a t i o n f o r t h e u n k n o w n <TXZQ: § j\au{DI), + a(DJ)0}axzQ(y)dy = + laE j \ D l ) 0 ^ M d y ( 5.19) where ( ) 0 d e n o t e s f o r t h e v a l u e of ( ) as x -* 0. F o r t h e C B V P l w i t h d a t a (5.3), t h e i n t e g r a l e q u a t i o n b e c omes 2- J {au(DI)0 + a(DJ)0}axz0{y)dy = - J j ^ . (5.20) F r o m we have <TX0(Z) = l i m <f>zz{x,z) x—*u = «aEh fe)0~3y~ V (5-21) 2 fh d2I d2J 2 5 2 / d 2 J <rxo(z) = -- / )o + 4 ^ - r ) o } ^ o ( y ) d y 7T yo &z~ &z f o r C B V P l . 5.2 REDUCTION TO SINGULAR INTEGRAL EQUATION F o r l a t e r references, we first w r i t e d o w n t h e f o l l o w i n g e x p r e s s i o n s f o r ^  a n d JQ K(s,z,y)dy d e r i v e d f r o m (5.13): ^ / ^ i W + Q i W + f c 2 { A ( l A ) + Q l ( l A ) } , 0 < y < z ; , . 3 y ~ fc4 - 1 s 2 A , 1 P2(k) + Q2(k) + k2{P2(l/k) + Q2(l/k)}t z<y<h, ^ £ £ J CANONICAL PROBLEMS / 5.2 43 a n d T K(s z v)dv - — — i + *2«i(V*) + (k) + k^il/k), 0 < y < z; Jo 1 ' ' V ) V " A 4 - 1 s 4 A , 1 P 2(Ar) + k2Q2(l/k) + k*Q2(k) + k*P2(l/k), z<y<ht --• (5.23) w h e r e P\{k) = sinh(s/i/A;) s i n h ( s f c y ) sinh(sA:(z — h)), P2(k) = smh[sh/k) cosh(3A;z) cosh(sA:(y — A ) ) , Qi(k) — sinh(sy/A:){cosh(sA:/i) c o s h ( s ( z — h)/k) — cosh(sArz)}, Q2(k) = cosh(skh) cosh(sz/k) s i n h ( s ( y — h)/k) — cosh(sfcz) sinh(sy/A:). We have d r o p p e d a t e r m of 1 / s 4 i n fQy K(s,zty)dy f o r z < y < h. B y eq.(5.16), t h i s t e r m gives ris e t o a t e r m i n J of t h e f o r m f0°° cos(xs)/s2ds g i v i n g no c o n t r i b u t i o n t o i n t e g r a l e q u a t i o n (5.20) a n d axo(z) b e c a u s e of t h e i d e n t i t y d2 / ' c o c o s ( x s ) J d r°° - s i n ( x s ) , d f°° - s i n x , ^ 2 / —-2-^ ds = / ds = I F / d x = °-oxz JQ dx JQ S ox JQ x N o t e t h a t i n o r d e r t o s i m p l i f y t h e s t a t e m e n t we have used a n i n t e g r a l w h i c h is d i v e r g e n t at s — 0. T h i s is a l s o t h e case f o r t h e i n t e g r a l s w i l l b e d e f i n e d below. S i n c e t h e y o c c u r as a s u m i n (5.23), t h e final r e s u l t does not have a n y d i v e r g e n c e at s = 0. B y d e f i n i n g a n d u s i n g A.(l/*) = ~A.{k) we m a y e x p r e s s 7 ( x , z , y) a n d J ( x , z , y) d e f i n e d i n (5.16) as f o l l o w s : 1 { i , V ) - _ * L . { A W - - ( A ( l / * ) - QiWh 0 < y < z; k * - l \ P2(k) - Q2(l/k) - {P2(l/k) - Q2(k)}, z<y<h, CANONICAL PROBLEMS / 5.2 44 a n d J [ X > Z > y ) A 4 - 1 \ P2(k) - Q2{l/k) - k*{P2{l/k) - Q2{k)}, z<y<h. We m a y now s p l i t each i n f i n i t e i n t e g r a l i n t h e e x p r e s s i o n s f o r I ( x , z, y) a n d J ( x , z , y) i n t o t w o p a r t s : / = / + / Jo Jo JN w h e r e TV is a la r g e p o s i t i v e n u m b e r t o b e chosen l a t e r i n n u m e r i c a l c o m p u t a t i o n s . T h e s e c o n d p a r t , f^t has a n a s y m p t o t i c e x p a n s i o n f o r l a r g e s (see a p p e n d i x B ) ; t h e r e m a i n d e r of i t s l e a d i n g t e r m a p p r o x i m a t i o n i s 0{e~Nhlk). T h i s l e a d i n g t e r m m a y be f u r t h e r s p l i t i n t o t w o p a r t s . O n e c o n t r i b u t e s t o t h e s i n g u l a r k e r n e l a n d t h e o t h e r is c o m b i n e d w i t h t h e rN first p a r t , J Q , c o n t r i b u t i n g t o t h e r e g u l a r k e r n e l o f t h e i n t e g r a l e q u a t i o n . I n t h i s way, we have (DI)0 = (DI)R + (DI)e + 0(e-Nh'k), (5.24) ( Z ? J ) 0 = (DJ)R + (DJ), + 0{e-Nhlk) where {DI)R){DJ)R are c o m p o n e n t s of t h e r e g u l a r k e r n e l a n d {DI)t,(DJ)s c o m p o n e n t s of t h e s i n g u l a r k e r n e l (see a p p e n d i x C ) . S u b s t i t u t i o n o f (5.24) i n t o (5.20) gives - / {av(DI)8 + a(DJ)t}o-Tz0(y)dy+l f {au{DI)N + a(DJ)N}axz0{y)dy K Jo * Jo o r * Jo i ^1 2^ ^3 4^ ^5 i / \ j {j—z + y-^Tz + 2h-y-z + h-z + k2{h-y) + k?{h - z) + h - y ) < T * M V + j KN(z, y)axz,{y)dy = + 0{e~Nhlk) 10 (5.25) w h e r e t h e c o n s t a n t s A,- a n d the r e g u l a r k e r n e l KN{z, y) are g i v e n i n a p p e n d i x C. Eq.(5.25) i s a so c a l l e d g e n e r a l i z e d C a u c h y - t y p e s i n g u l a r i n t e g r a l e q u a t i o n f r o m w h i c h t h e s i n g u l a r i t y of s o l u t i o n <TXZQ(Z) c a n b e a n a l y s e d . I n s p e c i a l i z i n g t h e g e n e r a l t h e o r y i n t r o d u c e d i n a p p e n d i x CANONICAL PROBLEMS / 5.3 45 D, eq.(5.25) is o b t a i n e d f r o m eq.(D.l) b y s e t t i n g a = 0, b = h, t = y, x = z, J = 1, S = 3, CQ = \ X , Ci — A 2, kx = 1, $i = JT, dx = — A 3 , /ij = 1, u>x — 0, <*2 = -<V*2, ^2 = V^2. w2 = Of dz = — A 5 , h$ = A;2, w 3 = 0. T h e r e f o r e , t h e s i n g u l a r i t y e x p o n e n t (3 at z = A is g o v e r n e d b y F(/3) = A x C O S ( T T / ? ) - A 3 - A 4 J f c 2 / ? - 2 - \hk~2? = 0. (5.26) A f t e r c a n c e l l i n g off t h e c o m m o n f a c t o r a i n A,-, t h e eq.(5.26) r e m a i n s t h e same i f t h e o r t h o g -o n a l m a t e r i a l p r i n c i p a l axes are s u b j e c t t o a r o t a t i o n b y 90 degrees, o r a re i n t e r c h a n g e d . M o r e o v e r , b y u s i n g L ' H o p i t a l ' s r u l e t o e v a l u a t e t h e l i m i t s as k —» 1 a n d n o t i c i n g (3 = — A , we f o u n d t h a t eq.(5.26) is r e d u c e d t o t h e one f o r i s o t r o p i c m a t e r i a l , i.e. eq.(4.27). S i n c e F\p) < 0 f o r 0 < p < 1 a n d an u n i q u e r e a l s o l u t i o n e x i s t s i n 0 < P < 1. A s p o i n t e d o u t i n Sec.4.3, t h e agreement of. the s o l u t i o n of (5.26) w i t h t h a t of Ch.4 i s r e m a r k a b l e . 5.3 INTEGRAL EQUATION FOR ANTI-SYMMETRIC CASE T h e o n l y d i f f e r e n c e s b e t w e e n C B V P l a n d C B V P 2 o r C B V P 3 are t h e b o u n d -a r y c o n d i t i o n s at x = oo, w h i c h now c o r r e s p o n d t o a u n i t b e n d i n g a n d a u n i t t r a n s v e r s e s h e a r i n g f o r c e , r e s p e c t i v e l y . B y s u b t r a c t i n g off t h e f o l l o w i n g p a r t i c u l a r ( o u t e r ) s o l u t i o n 3e n °x = ^ » °xz = °. °z = °> 3xz A 3 , 2 (5-27) CANONICAL PROBLEMS / 5.3 46 t h e s o l u t i o n f o r t h e m o d i f i e d C B V P 2 is o n l y a d e c a y i n g s t a t e w i t h t h e f o l l o w i n g d i s p l a c e -m e n t b o u n d a r y c o n d i t i o n s at x = 0 U = UQ = wz Zv2xz2 (5.28) w h e r e d has been set t o z e r o f o r a n t i - s y m m e t r y w i t h r e s p e c t t o t h e x - a x i s . S i m i l a r l y , b y s u b t r a c t i n g off t h e f o l l o w i n g p a r t i c u l a r ( o u t e r ) s o l u t i o n 3 x z 1 3 x 2 z 4h*[ EX + U2XZZ z 3 --)-G] 1 Zh?x 4h^ G X 3 Zv2Xxz Ei Ei U = -Ah^- T + "EY ~ c > ~ u z + d> (5-29) , 3 / i 2 x x 3 Si/2Xxz2. W = 4 & { - G - - E - x - - r ) + U , X + C' the s o l u t i o n f o r t h e m o d i f i e d C B V P 3 a l s o b e c omes a d e c a y i n g s t a t e w i t h t h e f o l l o w i n g d i s p l a c e m e n t b o u n d a r y c o n d i t i o n s at x = 0 1 / 1 "21\ 3 4 h G Ex (5.30) W = WQ = — C where d has a l s o b e en set t o z e r o f o r a n t i - s y m m e t r y . F o l l o w i n g t h e same d e d u c t i o n as i n C B V P l , one a r r i v e s at eq.(5.13) a g a i n b u t now w i t h A , a n d Ox,02iTx,T2 g i v e n b y t h e f o l l o w i n g e x p r e s s i o n s : Ox(k) = cosh(shfk) s i n h ( s f c y ) s i n h ( s / ; ( z — h)), 02(k) = cosh(sh/k) s i n h ( s f c z ) s i n h ( s f c ( y — h)), Tx(k) = si n h ( s A : y ) { s i n h ( s / i / * ) cosh(sfc(/i - z ) ) - s i n h ( s z / f c ) } , (5.31) T2(k) = sinb(sA:z){sinh(s/i/fc) cosh(sA:(/i — y)) — s i n h ( s y / f c ) } , A „ = smh(skh) cosh(s h/k) — & 2cosh(sA:/i)sinh(s/i/A:). T h e n , we c a r r y o n as i n C B V P l l e a d i n g t o t h e same f o r m f o r t h e i n t e g r a l e q u a t i o n s , i.e. (5.15), (5.16), (5.18) a n d (5.19). Eqs.(5.22) a n d (5.23 ) a r e still v a l i d w i t h d i f f e r e n t CANONICAL PROBLEMS / 5.4 47 e x p r e s s i o n s f o r P j , F 2> Qi> Q2' P\{k) = cosh(shfk) cosh(sfcy) sinh(sA:(z — h))} P2{k) = cosh(sh/k) sinh(sA:z) cosh(sA:(y — h))t (5.32) Qi(k) = cosh(sy/A:){sinh(sA;Ji) c o s h ( s ( / i — z)/k) — sinh(sA:z)}, Q2(k) = s i n h ( s z / f c ) sinh(sA:/i) s i n h ( s ( y — h)/k) — sin h ( s f c z ) ccsh(sy/Jfe). T h e c o r r e s p o n d i n g e x p r e s s i o n s n e e d e d f o r t h e s i n g u l a r i n t e g r a l e q u a t i o n i n each case a r e l i s t e d i n a p p e n d i x B, Sec. B.2. W e l i s t h e r e o n l y t h e s i n g u l a r i n t e g r a l e q u a t i o n f o r b o t h C B V P 2 a n d C B V P 3 : ^ f^f A i A 2 A3 A4 A5 . / \ j v Jo {y~^~z + y~+~z + 2h-y-z + h-z + k2{h-y) + k2(h-z) + h - y } ( T x z M V + f KN(z, y)axzo{y)dy = F(z) + 0{e-Nhlh) Jo (5.33) where A,- a n d KN(z,y) f o r t h e a n t i - s y m m e t r i c case are i l l u s t r a t e d i n a p p e n d i x C Sec.C.2. T h e r i g h t h a n d s i d e is 3uz f o r C B V P 2 a n d f o r C B V P 3 . F { Z ) = 4 ^ ( l " v ) f(DI)oy2dy (5.35) 5.4 METHOD OF EIGENFUNCTION EXPANSIONS So f a r we have d i s c u s s e d t h e F o u r i e r t r a n s f o r m m e t h o d f o r s o l v i n g c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s . E f f o r t s h ave also b e e n m a d e t o use a n e i g e n f u n c t i o n e x p a n s i o n m e t h o d w h i c h h a s p r o v e d t o b e less e f f e c t i v e . A s we have seen i n Ch.2, t h e d e c a y i n g s t a t e <f>D = 4>- tout*- = fanner m a y be w r i t t e n as fc> = £4fc«°*'/('*,/*M (5.36) k CANONICAL PROBLEMS / 5.4 48 w h e r e fk(z) is e i t h e r <}>a i n (2.33) o r $p i n (2.35) s a t i s f y i n g e q u a t i o n k4fr+{2+P)aih2fi:+aifk=o a n d t h e b o u n d a r y c o n d i t i o n s o n t h e faces o f t h e p l a t e : fk(±h) = f'k(±h) = 0. w h e r e t h e p r i m e d e n o t e s JJ. T h e g e n e r a l i z e d o r t h o g o n a l i t y c o n d i t i o n f o r t h e series i n (5.36) is d e r i v e d b y c o n s i d e r i n g >h -h w h i c h , b y i n t e g r a t i o n b y p a r t s a n d t h e b o u n d a r y c o n d i t i o n s o n t h e s u r f a c e s , b e c o m e s >h -h S i m i l a r l y , we also have -h -h S u b t r a c t i n g (5.38) f r o m (5.37) gi v e s t h e needed g e n e r a l i z e d o r t h o g o n a l i t y c o n d i t i o n >h -h f o r I ^  k. T o have an e x p l i c i t e x p r e s s i o n f o r AK i n (5.36), we c o n s i d e r t h e s u m /* { f c V r + (2 + p)a2h2f'k' + o'Mftdz = 0 J-h "? r < f c 4 t f / f - < 2 + P)«kh*fkfl + °Ukfi}** = 0. (5.37) J-h 4 fH{b'fkfl " (2 + vWih?fhf{ + aUkfl)dz = 0. (5.38) J-h fH infill ~ <*l*2flfk}dz = 0 (5.39) J-h £ M fH {tffifi - a2a2k ftfk)dz. (5.40) B y a p p l i c a t i o n o f (5.39) i t is e q u a l t o Akck w i t h >h -h T h u s ck= fk{h*fp-atfi)dz. (5.41) J-h AK = - f {*4 (£>//')# - ajQT A)*r (5.42) c* J-h , , CANONICAL PROBLEMS / 5.4 49 w h e r e t h e o r d e r of t h e i n t e g r a t i o n a n d s u m m a t i o n have b e e n i n t e r c h a n g e d . T h e s u b s t i t u -t i o n of £ > / / ' = < W z ) (5.43) a n d £ Atajfi = h2{Ew'm(z) + v*xm{z)) (5.44) l i n t o (5.42) gives Ck J-h w h e r e s u b s c r i p t s D a n d 0 rep r e s e n t t h e d e c a y i n g s t a t e a n d t h e e v a l u a t i o n at x = 0, res p e c t i v e l y . A n i n t e g r a l e q u a t i o n m a y now b e o b t a i n e d b y e s t a b l i s h i n g a r e l a t i o n at x = 0: rh rh / K{z,y)axDQ{y)dy = um{z) + / Kl{ziy)w'D0(y)dy (5.46) J-h J-h Ak = ^ f h {(h2fk' - va2fk)axDQ{z) - a2Efkw'm{z)}dz (5.45) w here K(z, y) = h 2 1 £ Uh2fl\y) - valMyKufc), Kx{z,y) = Eh2Y,<*l/cifl{y)ui{z)> (5.47) Eq.(5.46) is a n i n t e g r a l e q u a t i o n of t h e first k i n d f o r o-x£>o(z) f r o m w h i c h Af. c a n be o b t a i n e d b y eq.(5.45). T h u s t h e p r o b l e m i s s o l v e d . N o t e t h a t uD0(z) a n d W£>0(z) are k n o w n u p t o some c o n s t a n t s ( r i g i d b o d y m o t i o n ) w h i c h have t o b e d e t e r m i n e d s i m u l t a n e o u s l y f r o m (TXDQ(Z) b y u s i n g t h e n e c e s s a r y c o n d i t i o n s d e v e l o p e d i n Ch.3. W e are u n a b l e t o c o n v e r t eq.(5.46) t o a s i n g u l a r i n t e g r a l e q u a t i o n as we have d o n e t h r o u g h t h e F o u r i e r t r a n s f o r m m e t h o d ; we do n o t h a v e s i m i l a r a s y m p t o t i c e x p r e s s i o n s i n t h e p r e s e n t a p p r o a c h . B y ke e p i n g a finite n u m b e r t e r m s of t h e i n f i n i t e series i n t h e k e r n e l , we get a F r e d h o l m i n t e g r a l e q u a t i o n of t h e first k i n d whose i l l - p o s e d n e s s is w e l l k n o w n , e.g. [34]. T h e n , t h e t e c h n i q u e of r e g u l a r i z a t i o n d e v e l o p e d i n [28,32,33] r e d u c e s t h e p r o b l e m t o s o l v i n g a s t a b l e m i n i m i z a t i o n p r o b l e m w i t h a p a r a m e t e r a(<5), w h i c h is a f u n c t i o n of t h e e r r o r l e v e l 6 of t h e r i g h t h a n d s i d e CANONICAL PROBLEMS / 5.5 50 of t h e i n t e g r a l e q u a t i o n . G e n e r a l l y a is c h osen b y the so c a l l e d d i s c r e p a n c y p r i n c i p l e [ 2 3 , 1 5 ] . T h e d i f f i c u l t y i n o u r p r o b l e m i s t h a t t h e e r r o r l e v e l 6 a s s o c i a t e d w i t h t h e r e m a i n d e r of t h e series i n t h e k e r n e l is u n k n o w n . I n t h i s t h e s i s 8 i s c h o s e n t o s m o o t h t h e s o l u t i o n . I n t h i s w a y we o b t a i n e d a s o l u t i o n w h i c h agrees w i t h t h a t f r o m t h e F o u r i e r t r a n s f o r m m e t h o d . 5.5 NUMERICAL ANALYSIS F O R T R A N p r o g r a m s f o r n u m e r i c a l s o l u t i o n s of t h r e e c a n o n i c a l p r o b l e m s d e f i n e d i n t h i s c h a p t e r h a v e b e e n d e v e l o p e d a n d i m p l e m e n t e d o n t h e A m d a h l 5850 i n t h e U B C C o m p u t i n g C e n t e r . P r o g r a m S X Z T T gives n u m e r i c a l s o l u t i o n s of crxz0 a n d axQ f o r C B V P 1 b y F o u r i e r T r a n s f o r m m e t h o d s whereas p r o g r a m s S X Z B T a n d S X Z F T are f o r t h o s e of C B V P 2 a n d C B V P 3 r e s p e c t i v e l y . T h e c o l l o c a t i o n m e t h o d , as d e s c r i b e d i n [1], is u s e d t o d i s c r e t i z e t h e i n t e g r a l e q u a t i o n . B a s e d o n t h e s i n g u l a r i t y e x p o n e n t 7 o b t a i n e d i n Ch.4, c o r r e s p o n d i n g J a c o b i p o l y n o m i a l s are chosen t o r e p r e s e n t c r x ; 0 [ 4 ] . N u m e r i c a l i n t e g r a t i o n s are p e r f o r m e d b y G a u s s i a n q u a d r a t u r e whose w e i g h t s a n d r o o t s are p r o d u c e d b y a s u b r o u t i n e n a m e d G a u s s b a s e d o n [31]. S y m m e t r y of a X 2 0 has b e e n u s e d t o r e d u c e t h e n u m b e r of a l g e b r a i c e q u a t i o n s . T h e p r o g r a m s d e v e l o p e d f o r t h e g e n e r a l o r t h o t r o p i c case c a n also be u s e d f o r t h e i s o t r o p i c case b y s e t t i n g t h e n u m e r i c a l d a t a s u c h t h a t Ei —> E2, G = 2(1 + uX2) R a p i d , s t a b l e c o n v e r g e n c e is o b s e r v e d i n p r a c t i c e . N u m e r i c a l s o l u t i o n s h a v e been o b t a i n e d f o r t h e i s o t r o p i c case h a v i n g ^ = 20000.00001, .02 = 20000, G = 7500, v = 0.33333 w h i c h c o r r e s p o n d s t o a v a l u e of a = 1/4 f o r Poisson's r a t i o . A close a g reement w i t h t h e r e s u l t s i n [10] i s a c h i e v e d . A d d i t i o n a l n u m e r i c a l r e s u l t s have b e e n o b t a i n e d f o r s i x o t h e r CANONICAL PROBLEMS / 5.5 51 cases as g i v e n below: m a t e r i a l Ei E2 G 1^2 p i n e w o o d 1 100000 4200 7500 0.01 p l y w o o d 1 120000 64400 7200 0.044 p i n e w o o d 2 4200 100000 7500 0.238 p l y w o o d 2 64400 120000 7200 0.082 i s o t r o p i c 1 15732 15732 7500 0.0488 i s o t r o p i c 2 15265 15265 7200 0.06 N o t e t h a t t h e u n i t f o r e l a s t i c i t y m o d u l u s is kg/cm2 a n d h = 1cm f o r a l l t h e cases. T h e v a l u e of v f o r i s o t r o p i c 1 ( i s o t r o p i c 2) is e q u a l t o y/v 12^21 for p i n e w o o d 1 o r 2 ( p l y w o o d 1 or 2). T h e g r a p h s o f axz0 a n d a x 0 f o r t h r e e C B V P s are s h o w n i n F i g . 5.1 t o F i g . 5.6. A l l figures a n d d a t a r e f e r t o t h e c o m p l e t e s o l u t i o n s o f u n i t e x t e n s i o n , b e n d i n g a n d s h e a r i n g . T h e p a r a m e t e r N chosen t o s p l i t t h e i n f i n i t e i n t e g r a l s is t a k e n t o be 50 a n d a l a r g e r N does not i m p r o v e t h e r e s u l t s s i g n i f i c a n t l y . A l l t h e n u m e r i c a l r e s u l t s d i s p l a y e d o n t h e g r a p h s a r e o b t a i n e d b y s o l v i n g a s y s t e m of 8 l i n e a r e q u a t i o n s w h i c h g e n e r a l l y g i v e i n d i s t i n g u i s h a b l e figures f r o m a s y s t e m of 10 e q u a t i o n s . T h e m e t h o d is p r o v e d t o b e ef f i c i e n t a n d i n e x p e n s i v e . I n Fig.5.1 a n d Fig.5.2, t h e l a r g e s t m a x i m u m she a r s t r e s s i n e x t e n s i o n a n d b e n d i n g i s g i v e n b y p i n e w o o d 2 w h i c h h a s t h e s m a l l e s t E1. I n Fig.5.3, t h e figures f o r p i n e w o o d 1 a n d 2, p l y w o o d 1 a n d 2, i s o t r o p i c 1 a n d 2 are a l m o s t i n d i s t i n g u i s h a b l e , r e s p e c t i v e l y . T h a t i s , CTXZQ i n s h e a r i n g i s n o t s e n s i t i v e t o t h e d i r e c t i o n o f o r t h o t r o p y . T h e p i n e w o o d 1 a n d 2 have the s m a l l e s t m a x i m u m axzQ i n s h e a r i n g . T h e figures f o r p i n e w o o d 1 a n d 2, p l y - w o o d 1 an d 2 are i n d i s t i n g u i s h a b l e i n Fig.5.4. T h a t m e a n s t h a t <7 X 0 i n t h e u n i t e x t e n s i o n i s n o t s e n s i t i v e t o t h e d i r e c t i o n of o r t h o t r o p y . I n Fig.5.5, a l m o s t a l l figures are i n d i s t i n g u i s h a b l e . I n o t h e r words, t h e S t . V e n a n t s o l u t i o n f o r aXQ i n b e n d i n g i s q u i t e a c c u r a t e . I n Fig.5.6, 0-xo f o r i s o t r o p i c 1 a n d 2 i n s h e a r i n g i s i n d i s t i n g u i s h a b l e , a n d p l y w o o d 1 h a s t h e l a r g e s t m a x i m u m a l 0 - Fig.5.7 c o m p a r e s axz0 i n u n i t e x t e n s i o n b y t h e F o u r i e r t r a n s f o r m m e t h o d w i t h t h o s e b y t h e series e x p a n s i o n m e t h o d f o r p l y w o o d 1. T h e c u r v e D E L 0 r e p r e s e n t i n g t h e CANONICAL PROBLEMS / 5.5 52 n o n r e g u l a r i z a t i o n r e s u l t (6 = 0) i s o s c i l l a t i n g a n d f a r f r o m t h e r e a l s o l u t i o n . B y c h o o s i n g 6 p r o p e r l y , we see t h a t t h e r e g u l a r i z e d s o l u t i o n s ( D E L I f o r 5 = 0.01, D E L 2 f o r 6 = 0.005, D E L 3 f o r 6 — 0.00175) a p p r o a c h t h e s o l u t i o n o b t a i n e d b y t h e t r a n s f o r m m e t h o d . CANONICAL PROBLEMS / 5.5 53 O.OO-i -0.01--0.02--0.03--0.04 -0.05--0.06--0.07--0.08--0.09-Legend p i n e w o o d 1 p l y - w o o d 1 p i n e w o o d 2  p l y - w o o d 2  i s o t r o p i c 1 i s o t r o p j c _ 2 _ 0 0.2 — I — 0.4 0.6 \ > 0.8 \ \ \ \ \ FIGURE 6.1 <TXZQ in extension. CANONICAL PROBLEMS / 5.5 0 . 0 5 -1 0 . 0 0 - 0 . 0 5 -axzQ FIGURE 5.2 axz0 in bending. CANONICAL PROBLEMS / 5.5 0.7 0.1 i 1 1 1 1 0 0.2 0.4 0.6 0.8 1 Z FIGURE 5.3 axz0 In shearing. CANONICAL PROBLEMS / 5.5 56 0.56 0.55-0.54 0.53 0.52-0.51-0.50-0.49 7 0 Legend pine wood 1 p)y-yfood 1 pine wood 2  ply-wood 2  isotropic 1 isotropjc_2_ 0.2 T T 0.4 0.6 Z 0.8 FIGURE 5.4 drf in extension. CANONICAL PROBLEMS / 5.5 Z FIGURE 5.5 in bending. CANONICAL PROBLEMS / 5.5 7-1 Z FIGURE 5.6 a x 0 in shearing. CANONICAL PROBLEMS / 5.5 Z FIGURE 5.7 Comparison between S.I.E.and series solutions. APPLICATIONS / 6.1 60 CHAPTER 6 APPLICATIONS I n t h i s c h a p t e r , we c o n s i d e r some t y p i c a l b o u n d a r y v a l u e p r o b l e m s a n d show how t h e i r o u t e r ( o r p l a t e t h e o r y ) s o l u t i o n s c a n be d e t e r m i n e d b y u s i n g t h e n e c e s s a r y c o n d i t i o n s f o r d e c a y i n g s t a t e s d e v e l o p e d i n Ch.3. T h e n , t h e p r o b l e m of a s h e a r e d b l o c k is s o l v e d i n t h i s way t o i l l u s t r a t e b o t h t h e e f f i c i e n c y o f t h e t h e o r y a n d t h e a c c u r a c y of n u m e r i c a l r e s u l t s o b t a i n e d i n Ch.5. 6.1 DETERMINATION OF OUTER SOLUTIONS F o r a g i v e n b o u n d a r y v a l u e p r o b l e m f o r a s t r i p i n p l a n e s t r a i n d e f o r m a t i o n s h o w n i n Fig.2.1, t h e s o l u t i o n i s g e n e r a l l y n o t a d e c a y i n g s t a t e . However, t h e d i f f e r e n c e b etween t h e e x a c t s o l u t i o n a n d t h e o u t e r s o l u t i o n i s e x p e c t e d t o b e a d e c a y i n g s t a t e . T h e n e c e s s a r y c o n d i t i o n s i n Ch.3 c a n t h e n be a p p l i e d t o t h i s d e c a y i n g s t a t e a n d t h e r e b y d e t e r m i n e t h e o u t e r s o l u t i o n . T h e d i s c u s s i o n of t y p i c a l b o u n d a r y v a l u e p r o b l e m s g i v e n below shows how t h e m e t h o d works. E x a m p l e 1. Stresses axi <rxz are g i v e n at b o t h ends. T h e s a t i s f a c t i o n o f t h e n e c e s s a r y c o n d i t i o n s (3.5),(3.6),(3.7) at e i t h e r end, say x = —I, b y t h e d i f f e r e n c e b e t w e e n t h e e x a c t a n d o u t e r s o l u t i o n s APPLICATIONS / 6.1 61 a n d gives N= f Gxdz (6.3) J-h Q = / axtdz (6.4) M 0 + Q z = / oxzdz (6.5) w h e r e cx,axz are g i v e n stresses at z = —/. O n t h e o t h e r h a n d , n o r e s t r i c t i o n h a s b e e n i m p o s e d o n d i s p l a c e m e n t s b y t h e b o u n d a r y c o n d i t i o n s . T h u s , t h e o u t e r s o l u t i o n i s d e t e r -m i n e d u p t o a r i g i d b o d y m o t i o n as e x p e c t e d . N o t e t h a t t h e s t r e s s d a t a at z = / a n d x = —I s h o u l d be so p r e s c r i b e d t h a t t h e s t r i p i s i n o v e r a l l e q u i l i b r i u m . E x a m p l e 2. S t r e s s ax a n d d i s p l a c e m e n t w are g i v e n at b o t h ends. N c a n be o b t a i n e d f r o m (6.3) at e i t h e r i = - I o r i = I w h e r e t h e edge d a t a m u s t have t h e same r e s u l t a n t s . Eq.(6.5) at b o t h x — I a n d x = —I solves Q a n d M 0 . T h e s a t i s f a c t i o n of th e n e c e s s a r y c o n d i t i o n (3.14) at b o t h ends b y t h e d i f f e r e n c e b e t w e e n t h e e x a c t a n d o u t e r s o l u t i o n s , (6.1) a n d w _ t / 2 i ^ 1 _ 3 M 0 u2lz2 _ z* h1 2EX' Ah? K Ex Ex> Q ,3h2x z 3 3u2ixz2^ rr( —7z " T ; — 1 — ux — c, 4 / i 3 V G Ei Ei (6.6) gives us t h e f o l l o w i n g t w o e q u a t i o n s f o r u a n d c: >h „3 j[h2 - z2)wdz+(^ ~^)J h J*,d* = A f 0 { - ^ + " f ) y } + ^ ( « * + c) (6-7) 2 1 2 i / . A 2 , Ah3 Oil- - - — \ + Q " G " E> 5 " ZEX' w h e r e z = —/, / r e s p e c t i v e l y a n d w,ax are t h e c o r r e s p o n d i n g g i v e n d a t a . S i n c e t h e b o u n d -a r y c o n d i t i o n s i m p o s e n o r e s t r i c t i o n s o n u , we m a y set t h e r i g i d b o d y t r a n s l a t i o n e q u a l t o ze r o so t h a t d = 0 a n d t h u s d e t e r m i n e t h e o u t e r s o l u t i o n c o m p l e t e l y . APPLICATIONS / 6.1 62 E x a m p l e 3. S t r e s s axz, d i s p l a c e m e n t u are g i v e n at b o t h ends. Q i s o b t a i n e d f r o m (6.4). T h e s a t i s f a c t i o n o f t h e n e c e s s a r y c o n d i t i o n s (3.8) a n d (3.10) at b o t h e n d s b y t h e d i f f e r e n c e b e t w e e n t h e e x a c t a n d o u t e r s o l u t i o n s , (6.2) a n d N x MQSxz Q ,3x2z i/2lz3 2 3 . , „ gives us t h e f o l l o w i n g f o u r e q u a t i o n s t o s o l v e f o r M 0 , N, d a n d u: fh v fh z2 _ x 2h? I zudz + — I —axzdz ——MQ — w J-h E J_h 2 x z Ex " 3 x 2 ,2v l K h 2 . + Q { 2 - E l + { E - G ) T 0 } ^ where x — —1,1 r e s p e c t i v e l y a n d u , axz are t h e c o r r e s p o n d i n g edge d a t a . T h e r i g i d b o d y t r a n s l a t i o n c is free t o b e set t o z e r o s i n c e no r e s t r i c t i o n h a s b e e n i m p o s e d o n d i s p l a c e m e n t w. T h e o u t e r s o l u t i o n is t h u s t o t a l l y d e t e r m i n e d . E x a m p l e 4. d i s p l a c e m e n t s u a n d w are g i v e n at b o t h ends. T h e s a t i s f a c t i o n o f t h e t h r e e a s y m p t o t i c n e c e s s a r y c o n d i t i o n s (3.17) at b o t h ends b y t h e d i f f e r e n c e b e t w e e n t h e e x a c t a n d o u t e r s o l u t i o n s , (6.6) a n d (6.8), gives us t h e f o l l o w i n g s i x e q u a t i o n s t o s o l v e f o r N, M0, Q, C , d a n d LJ: I. I {axXu + axzXw)dz = \ { ^ r " IfM^ n} + d» ( 6 1 1 ) h i- - - \ • MQ. 3x 3i/2x - , ^(a x 2u + axz2w)dz = - ^ 3 - { — - J ^ M 2 2 i ~ " , Q , 3 x 2 l i / 2 i l . r v 3 i / 2 1 x > v + J?{WX + 4{E-X ~ G ) N " ~ 1 ^ T M 2 2 } ' ( 6- 1 2 ) H ,~ ~ sj M)r 3 i / 2 1 - 3 x 2 Q f\.v2x 1 ~ + ~ S F < - 7 i , M * -w^r^tr dN* l,3h2x x 3 , 3v2xx - . + - < l - G - - T l ) - i E T M 3 2 ) + a Z + C ( 6 ' 1 3 ) w h e r e x = —1,1 a n d u , w are t h e c o r r e s p o n d i n g edge d a t a . tfxi>^*zi>^x2>^*z2i 0 * 3 1 ^ * 2 3 are s o l u t i o n s of C B V P l , 2 , 3 , i.e. f o r u n i t e x t e n s i o n , b e n d i n g a n d t r a n s v e r s e s h e a r i n g ( o r APPLICATIONS I 6.2 63 flexure), r e s p e c t i v e l y . C o n s t a n t s M i l y M 2 2 , M 3 2 , ./V23 a n d J V 3 3 are d e f i n e d b y Mu = / oxz\zdz, M 2 2 = / vxg2z2dz, J-h J-h M32 = / °xz3z2dz> -#23 = / ^x2« 3 r f zi ( 6 - 1 4 ) J-h J-h fk 3 ^33 = / °"*32 O n c e t h e c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s a r e s o l v e d a n d t h e a b o v e five c o n s t a n t s a r e e v a l u a t e d f o r a g i v e n m a t e r i a l , t h e o u t e r s o l u t i o n s of t h e p l a n e s t r a i n p r o b l e m f o r d i f f e r e n t sets of p r e s c r i b e d d i s p l a c e m e n t edge d a t a c a n be o b t a i n e d w i t h o u t a n y r e f e r e n c e t o t h e i n n e r s o l u t i o n s . We l i s t t hese c o n s t a n t s b e l o w f o r t h e m a t e r i a l s c o m p u t e d i n c h a p t e r 5. m a t e r i a l M n M 2 2 J V 2 3 M32 N*3 p i n e w o o d 1 -0.005975 -0.005276 0.6024 0.2685 -0.1884 p l y w o o d 1 -0.007584 -0.006534 0.6003 0.3041 -0.2258 p i n e w o o d 2 -0.02914 -0.02573 0.6024 0.2685 -0.03861 p l y w o o d 2 -0.01035 -0.008921 0.6003 0.3041 -0.1654 i s o t r o p i c 1 -0.01436 -0.01276 0.6030 0.2603 -0.07205 i s o t r o p i c 2 -0.01757 -0.01562 0.6037 0.2603 -0.07205 6.2 FLEXIBILITY COEFFICIENT OF SHEARED BLOCK T h e flexibility c o e f f i c i e n t o f a t r a n s v e r s e l y s h e a r e d b l o c k was i n v e s t i g a t e d i n p a p e r s [25,26,27] w h e r e u p p e r a n d l o w e r b o u n d s f o r t h e flexibility c o e f f i c i e n t were g i v e n i n t e r m s of a p p r o p r i a t e a p p r o x i m a t i o n s f o r t h e p o t e n t i a l a n d c o m p l e m e n t a r y e n e r g y ex-p r e s s i o n s a n d e v a l u a t e d o n t h e b a s i s o f v a r i a t i o n a l p r o b l e m s w h i c h i n v o l v e s o l u t i o n s of b o u n d a r y v a l u e p r o b l e m s . T h e o u t e r s o l u t i o n d i s c u s s e d i n t h i s t h e s i s gives a p l a t e t h e o r y s o l u t i o n f o r t h e same p r o b l e m . F o r s u f f i c i e n t l y t h i n p l a t e s , t h i s o u t e r s o l u t i o n agrees w i t h t h e e x a c t s o l u t i o n u p t o e x p o n e n t i a l s m a l l t e r m s . T h e o r t h o t r o p i c b l o c k , w h i c h i s i n p l a n e APPLICATIONS / 6.2 64 s t r a i n d e f o r m a t i o n a n d free o f t r a c t i o n at t h e t o p a n d b o t t o m su r f a c e s z = ±h, is s u b j e c t t o p r e s c r i b e d d i s p l a c e m e n t s at x = ±/: u = 0, to = ±WN w h e r e w0 i s a c o n s t a n t . T h e u n k n o w n r e s u l t a n t a p p l i e d f o r c e i s g i v e n b y •h -h a n d t h e f l e x i b i l i t y c o e f f i c i e n t C i s g i v e n b y Q J-h (6.15) ~Q' T h e a p p l i c a t i o n o f t h e a n t i - s y m m e t r y of t h e p r o b l e m t o eqs.(6.11)- (6.13) y i e l d s c _ r v 0 _ 1 Sh?l 2/ 3 u2l 1 -+ -~Er-M22 - (-=- - -^)N3Z - — - M 3 2 } . (6.16) (6.17) F o r t h e b l o c k m a d e of p i n e w o o d 1, a c u r v e f o r ^  ( C 0 = 2£hs) i s c o m p u t e d b y use of (6.17) a n d s h o w n i n Fig.6.1. O u r s o l u t i o n l i e s b e t w e e n t h e l o w e r a n d u p p e r b o u n d s o b t a i n e d i n [25] a n d is i n d i s t i n g u i s h a b l e f r o m t h e u p p e r b o u n d o n t h e g r a p h . S o me t y p i c a l r e s u l t s are g i v e n b e l o w : V cu/co Cm/Co CU2/C0 0.05 1.039575 1.039664 1.039693 0.55 5.573960 5.628369 5.630537 1.05 16.85403 17.20965 17.22431 1.95 50.60456 52.80735 52.90249 w h e r e C^/CQ d e n o t e s t h e l o w e r b o u n d , CN2/C0 t h e u p p e r b o u n d i n [25] a n d CM/C0 i s o u r c o m p u t e d r e s u l t . N o t e t h a t o u r r e s u l t s are i n g o o d agreement w i t h t h e u p p e r b o u n d e v e n f o r a r a t h e r t h i c k b l o c k . APPLICATIONS / 6.2 AXISYMMETRIC DEFORMATION OF ELASTIC ORTHOTROPIC PLATES / II.O 66 PART II AXISYMMETRIC DEFORMATION OF ELASTIC ORTHOTROPIC PLATES OUTER AND INNER SOLUTIONS / 7.1 67 CHAPTER 7 OUTER AND INNER SOLUTIONS 7.1 BASIC EQUATIONS AND BOUNDARY CONDITIONS T h e p l a t e c o n s i d e r e d i n p a r t II as s h o w n i n c y l i n d r i c a l c o o r d i n a t e s ( r , z) i n Fig.7.1 is of t h i c k n e s s 2h a n d r a d i u s r 0. T h e p l a t e is m a d e of a c y l i n d r i c a l l y o r t h o t r o p i c m a t e r i a l h a v i n g t h r e e p r i n c i p a l axes c o i n c i d e n t w i t h t h e r a d i a l , c i r c u m f e r e n t i a l a n d a x i a l d i r e c t i o n s , r e s p e c t i v e l y . We f u r t h e r assume t h e t r a n s v e r s e i s o t r o p y of t h e m a t e r i a l , i.e., i s o t r o p y i n t h e p l a n e p e r p e n d i c u l a r t o t h e z a x i s . T h e n u m b e r of i n d e p e n d e n t m a t e r i a l c o n s t a n t s is t h u s r e d u c e d f r o m 9 t o 5 as s h o w n i n t h e f o l l o w i n g s t r e s s - s t r a i n e q u a t i o n s [ 1 9 j : er — anar + anae + a13crz e0 = al2(Tr + an<xe + al3az (7.1) Irz = *U°rz 1r6 = 2(°11 ~ °12)<7r^ w h e r e r, 9, z are t h e r a d i a l , c i r c u m f e r e n t i a l a n d a x i a l c o o r d i n a t e s r e s p e c t i v e l y . T h e m a t e r i a l OUTER AND INNER SOLUTIONS / 7.1 68 r=r„ F I G U R E 7.1 geometry of a circular plate. c o n s t a n t s a,y are c o n n e c t e d w i t h t h e t e c h n i c a l c o n s t a n t s b y t h e f o l l o w i n g e q u a t i o n s : 1 _ -v 1 -v' a13 — ~^J- a44 = r , G' 2(«n ~ « w ) = = £ (7.2a, 6, c) I n t e r m s o f t h e c i r c u m f e r e n t i a l , r a d i a l a n d a x i a l d i s p l a c e m e n t s v = 0,u(r,z) a n d w(r,z), t h e s t r a i n - d i s p l a c e m e n t r e l a t i o n s f o r a x i - s y m m e t r i c d e f o r m a t i o n s a r e : d u U dw dr1 €*"7I € i~5 2' 3u 3tu (7.3a, 6) T h e y i m p l y t w o c o m p a t i b i l i t y e q u a t i o n s dz2 dr2 drdz = 0. (7.4) OUTER AND INNER SOLUTIONS / 7.1 T h e e q u i l i b r i u m e q u a t i o n s w i t h n o b o d y f o r c e ( w i t h o u t loss o f g e n e r a l i t y ) dar darz a. — aa + + — = 0, or dz r doTZ da, arz dr dz r T h e y c a n be s a t i s f i e d b y a str e s s f u n c t i o n <f>(r} z) w i t h d , d2 b d d 2  a r = -rz{dV2 + rd-r + adT2)^ d / L d 2 id a 2 C» = -Tz{hor2 + ro-r + ad?)4,> 9 t A J ° 2 M where a n d &r6 = Vz0 = 0 arz = +a—2)<}>, r ~ dr2 + rdr a = o 1 3 ( a n - a12)/AD b = { a 1 3 ( o n + a 4 4 ) - a12aI3}/AD c = { a 1 3 ( a n - a 1 2 ) + o 1 1 o 4 4 } / A £ ) d = [a2n - a\2)/AD AD = Olla33 — a13-T h e stress f u n c t i o n <f> is r e q u i r e d b y t h e c o m p a t i b i l i t y e q u a t i o n s t o s a t i s f y d2 d4 ( A r A r + (o + c ) A r ^ + dj^)4> = 0 o r w h e r e 5 = a + c + y ^ a + c ) 2 - ^ 1 / 2 2 d ,q + c - v/(° + c ) 2 - 4 d , i / o OUTER AND INNER SOLUTIONS / 7.2 70 We have A p > 0 a n d d > 0 f r o m t h e p o s i t i v e d e f i n i t e n e s s o f t h e s t r a i n energy. T h e t o p a n d b o t t o m s u r f a c e s o f t h e p l a t e a r e free o f t r a c t i o n s , so t h a t az = arz = 0, (at z = ±h). (7.10) Eq.(7.10) c a n b e e x p r e s s e d i n t e r m s of <p a n d i t s d e r i v a t i v e s . A l o n g t h e c y l i n d r i c a l edge r = r 0, we h a v e one of t h e f o l l o w i n g b o u n d a r y c o n d i t i o n s : (1) <7R = a r , cTZ=aTZ] (2) ar = a r , w = to; (7.11) (3) u = u , arz=aTZ\ (4) u = u, w — w. T o g e t h e r w i t h t h e r e q u i r e m e n t of b o u n d e d stresses a n d d i s p l a c e m e n t s t h r o u g h o u t t h e p l a t e , t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n (7.7) a n d t h e b o u n d a r y c o n d i t i o n s (7.10) a n d one of (7.11) define a b o u n d a r y v a l u e p r o b l e m . A n a s y m p t o t i c s o l u t i o n of t h e p r o b l e m w i l l be d i s c u s s e d i n t h e n e x t s e c t i o n . 7.2 OUTER SOLUTION We f o l l o w t h e i d e a d e v e l o p e d i n s e c t i o n 2.2 t o w o r k o u t t h e o u t e r s o l u t i o n . T h e change of v a r i a b l e s r = r 0x, z = ht (7.12) t u r n s t h e b a s i c e q u a t i o n (7.7) i n t o ( e 4 A x A x + e 2 ( a + c ) A , | i + d^)<f> = 0 (7.13) :8t2 dt* w h e r e W i t h (7.6) a n d (7.12), t h e b o u n d a r y c o n d i t i o n s (7.10) b e c o m e ^ ( c € 2 A x + dj^)+ = 0, (at t = ±l) ±(e2Ax + a^)<p = 0. (at t = ± 1 ) (7.15) OUTER AND INNER SOLUTIONS / 7.2 71 T h e n , i t is easy t o see t h a t t h e c o e f f i c i e n t s o f t h e o u t e r e x p a n s i o n 4> = M * . 0 + ^ l f o 0 + e V 2 (*. 0 + e*4>(X, 0 + • • • (7.16) are d e t e r m i n e d b y a series of b o u n d a r y v a l u e p r o b l e m s : { 4>ottu = °» d<Pottt = 0 , at t = ± 1 ; (7.17) % x « = °» a t* = ± J > { d<f>uttt + ( a + c) Ax4>ou = 0, <ty 1 M, + c A x ^ o t = 0, a t t = ± l ; (7.18) a<f>ixtt + &x<f>0x - 0, at * = ± 1 , ( ^ 2 M K + ( A + c)&x4>ut + A x Ax<f>Q = 0, < t y 2 m + cAx<f>u = 0 , at t = ± 1 ; (7.19) afaxtt + A x ^ l x = 0, at t = ± 1 , f + ( a + c)Ax4>2u + AxAx<t>i = 0, < d03«e + cA x<£ 2 t = 0 , at t = ± 1; (7.20) V a4>3xlt + &x<t>2x = °. at t - ± 1 , etc. w h e r e ( ) ( = A p p e n d i x E gives d e t a i l e d d e r i v a t i o n s o f t h e t e r m s { # y ( x , t ) } . R e m a r k a b l y , t h e series (7.16) i t s e l f s u m s t o give a n e x p l i c i t o u t e r s o l u t i o n f o r <f> w i t h t h e c o r r e s p o n d i n g st r e s s a n d d i s p l a c e m e n t e x p r e s s i o n s g i v e n b y aT — AvrA + BcrrB + C<rrG + DarD + EarE, 0$ = A<jQA + Bags + Cage + DffffD + ECTQEI o, = 0, arz = CarzC, u = AuA + BUB + C'uc + ^ u £ > + EuEi w = AwA + Bwg + CWQ + Dwj) + Eu>£ + wQ (7.21) OUTER AND INNER SOLUTIONS / 7.2 72 w h e r e A, B, C, D, E a n d w0 are u n k n o w n c o n s t a n t s a n d °tA = Z / R 2 > °OA = -z/r2, arB = z, a0B = z, °rD = !/ r 2» V0D = - l / r 2 » arE = !) a6E = !> Ore = ^ ~ ^ CZ2 + Ub + l)d~ 2ac}zlnr, ( 6 - l)cz» 3 r °rzC - (d- ac){h2 - z2)/r, a0C = 0 + {(6 + l ) a ' - 2 a c } z l n r + ( 6 - l ) ( f e , = ( o 1 2 - o n)«/r, «2? = (<*12 + O n ) r z , « D = ( ai2 - a n ) / r > = (<*i2 + O n ) r , (1 - 6 ) c z 3 «c = ( ai2 - a n ) — ~ 1" ( ai2 + a n){(& + l)d — 2ac}zrlnr + an(b — l)drz. O T (7.22) W i t h t h e o u t e r s o l u t i o n o f az = 0, t w o i n t e g r a l s of eq.(7.3), w(r,z)= a 1 3 ( c r r + <r$)dz + u>(r,0) Jo a n d w(r, z) = ^  ( a 4 4 < r r 2 - | ^ ) d r + w ( r 0 , 2 ) c a n be c o m b i n e d t o g i v e t h e f o l l o w i n g e x p r e s s i o n f o r o u t e r s o l u t i o n w(r, z): {au°rz - fa)*=odr + JQ ais(ar + °e)dz (7-23) OUTER AND INNER SOLUTIONS / 7.3 73 w h e r e wQ = t o ( r 0 , O ) . F r o m (7.23), v/e get f o r t h e l a s t e q u a t i o n i n (7.21) u>B = -(012 + O i i X * - 2 ~ rl)/2 + a^z2, XVjy = 0 U>E — 2 a 1 3 z , (7.24) u>C = au(d ~ a c ) ^ 2 In — - ( a 1 2 + a n ) { ( 6 + l ) d - 2 a c } ( r 2 m r - r j j l n r 0 ) / 2 + ai3-z 2{[(& + " 2ac] In r + (6 - l ) d / 2 } 2 2 + ^ { ( a 1 2 + a n ) [ ( 6 -f l ) d - 2ac] - 2 a u ( 6 - l ) d } . T h e a b o v e o u t e r s o l u t i o n f o r a n o r t h o t r o p i c c i r c u l a r p l a t e i s o b t a i n e d here f o r t h e f i r s t t i m e . It is a n e x a c t s o l u t i o n of t h e g o v e r n i n g e q u a t i o n s f o r t h r e e d i m e n s i o n a l e l a s t i c i t y t h e o r y f o r a n y e ( t h o u g h d e r i v e d o r i g i n a l l y b y a p e r t u r b a t i o n e x p a n s i o n f o r s m a l l c) a n d r e d u c e s t o t h e L e v y s o l u t i o n i n [12,13] f o r t h e i s o t r o p i c case. g o v e r n i n g t h e i n n e r o r d e c a y i n g s o l u t i o n s h o u l d be e s s e n t i a l l y o f f o u r t h o r d e r . Hence, we r e s c a l e t h e eq.(7.13) t o e l i m i n a t e e 4 i n t h e c o e f f i c i e n t of t h e h i g h e s t o r d e r d e r i v a t i v e b y s e t t i n g x = es (or s = £). Eq.(7.13) a n d b o u n d a r y c o n d i t i o n (7.15) t h u s b e c o m e 7.3 INNER SOLUTION I n o r d e r t o s a t i s f y t h e edge b o u n d a r y c o n d i t i o n s , t h e d i f f e r e n t i a l e q u a t i o n (7.25) * ( c A . + d ^ = 0 ( a * < = ± 1 ) + « * ) , = 0 (at t = ±l) (7.26) w h e r e d2 1 d OUTER AND INNER SOLUTIONS / 7.3 74 T h e a s y m p t o t i c e x p a n s i o n f o r t h e i n n e r s o l u t i o n is g e n e r a l l y o f t h e f o r m <t> = /*o(«)0o(*.O + M O M * . 0 + • • • (7.27) w h e r e Po(e), / ^ ( e ) , e t c . a r e t o b e d e t e r m i n e d . B y s u b s t i t u t i n g (7.27) i n t o (7.25) a n d (7.26), we f o u n d t h a t each $,(s, t) s a t i s f i e s t h e s a me e q u a t i o n a n d b o u n d a r y c o n d i t i o n s as (7.25) a n d (7.26) s i n c e e does n o t a p p e a r e x p l i c i t l y i n t h e e q u a t i o n a n d b o u n d a r y c o n d i t i o n s . I n o t h e r words, e a c h <f>i(s}t) is c o m p o s e d of t h e e i g e n f u n c t i o n s <j>a s a t i s f y i n g (7.25) a n d (7.26), i.e. & ( M ) = X > , a < * a O M ) (7.28) a w h e r e A;a are u n k n o w n c o n s t a n t s a n d i = 0 , l , 2 , T h e n , we have f r o m (7.27) a n d (7.28) ' _ ' _ ° (7.29) a i a w h e r e AQ are u n k n o w n c o n s t a n t s . T h e e x p r e s s i o n (7.29) shows t h a t t h e a s y m p t o t i c ex-p a n s i o n of t h e i n n e r s o l u t i o n m a y b e w r i t t e n as a n e i g e n f u n c t i o n e x p a n s i o n w i t h o u t t h e e x p l i c i t a p p e a r a n c e of e. T o find e i g e n f u n c t i o n s c o r r e s p o n d i n g t o t h e d e c a y i n g s t a t e , we r e w r i t e (7.25) as w i t h , d2 Id Id2 $ = }T2 + - 7 r + - 2 ^ 2 )<*>• (7-31) dsz sds sidt1 We m a y so l v e (7.30) b y s e p a r a t i o n o f v a r i a b l e s t o o b t a i n $ = {0, I0{Xs)cos(slXt), IQ{X8)Bia{al\t)} (7-32) w h e r e f o r a d e c a y i n g s t a t e , i o ( A s ) is a m o d i f i e d B e s s e l f u n c t i o n a n d A i s a n o n z e r o eigen-v a l u e t o be d e t e r m i n e d . A n o t h e r set of s o l u t i o n s w i t h /Q(^ s) r e p l a c e d b y K0(Xs) is a l s o OUTER AND INNER SOLUTIONS / 7.3 75 a d r n i s s a b l e . W i t h (7.32), we s o l v e (7.31) f o r <f> b y s e p a r a t i o n o f v a r i a b l e s a n d t h e s o l u t i o n is a l i n e a r c o m b i n a t i o n of { / 0 ( A s ) c o s ( s y A i ) , / 0(A«) s i n ( s y A * ) } ( j = l , 2 ) . T h e n , t h e e i g e n f u n c t i o n s c a n be o b t a i n e d b y r e q u i r i n g I0(Xs){A c o s ^ A * ) + B s i n ( s i A t ) + C c o s ( s 2 A i ) + Z > s i n ( s 2 A t ) } (7.33) t o s a t i s f y t h e b o u n d a r y c o n d i t i o n s (7.26). F o r t h e a n t i - s y m m e t r i c ( w i t h r e s p e c t t o t h e m i d p l a n e z = 0) s t r e s s a n d d i s p l a c e m e n t case, we have B = D = 0 a n d / s 1 ( d s 2 - c ) s i n ( s 1 A ) s 2(ds 2> - c ) s i n ( s 2 A ) \ (A\ _ (o\ . . \ (1 - o s f t c o s f o A ) (1 - a s ^ ) c o s ( s 2 A ) ) \C) ~ \0) ' V-6*' T h e h o m o g e n e o u s s y s t e m (7.34) r e q u i r e s A t o s a t i s f y (1 — as2)[asx — c)Si s 2 C o r r e s p o n d i n g t o each r o o t of (7.35), we have t h e e i g e n f u n c t i o n 4>x = /O(A*)/A(*) = / 0 ( ^ ) / A W (7.35a) w i t h fx(z) = (1 — a s 2 ) c o s ( s i A ) c o s ( s 2 A z / / i ) — (1 — a s 2 ) c o s ( s 2 A ) c o s ( s l A z / / i ) . (7.36) F o r t h e s y m m e t r i c (w.r.t. t h e m i d p l a n e z = 0) stress a n d d i s p l a c e m e n t case, we have A = C = 0 a n d / 5 1 ( c - a s ^ ) c o s ( s 1 / i ) s 2 ( c - asl)cos(s2/i)\ (B\ _ (o\ , . \ ( l - a s 2 ) s i n ( S l ^ ) (1 - as 2,) s i n ( s 2 / i ) / \DJ~ \0J' T h e p a r a m e t e r /x m u s t t h e r e f o r e s a t i s f y t h e e q u a t i o n t a n ( s 2 / i ) = — t a n ( s ! / i ) . (7.38) s2 C o r r e s p o n d i n g t o e a c h r o o t of (7.38), we have t h e e i g e n f u n c t i o n 0 = /o ( / * * ) f y ( z ) = h(^r)g^z) OUTER AND INNER SOLUTIONS / 7.3 76 w i t h dpi2) = (1 - as2l)sm(slii)sm{s2fiz/h) - (1 - asl)sm{s2ri)sin(slriz/h). (7.39) I n c o n t r a s t t o t h e p l a n e s t r a i n d e f o r m a t i o n case, h e r e t h e e i g e n f u n c t i o n s c o r r e s p o n d i n g t o t h e z e r o e i g e n v a l u e does n o t p r o d u c e t h e c o m p l e t e o u t e r s o l u t i o n . T h e u n i f o r m a s y m p t o t i c e x p a n s i o n f o r t h e s t r e s s f u n c t i o n <f> as e —• 0 m a y now be w r i t t e n as f ~ fouler + dinner where 4>outer i s t h e s u m of (E.9) a n d (E.13) w h i l e <f>i„ner i s d e f i n e d b y * < — = E ^ / » w + E B / ^ S Y ^ - { 7 M ) x •'0I7) ft M J I n t h e e x p r e s s i o n (7.40), Ax, B^ are u n k n o w n c o n s t a n t s a n d t h e e i g e n f u n c t i o n s f\(z), g^z) are d e f i n e d i n eq.(7.36) a n d eq.(7.39) w i t h t h e e i g e n v a l u e s A , / i d e f i n e d b y (7.35) a n d (7.38). T h e e x p o n e n t i a l l y i n c r e a s i n g p r o p e r t y o f J 0 m a k e s i t s e l f s u i t a b l e t o r e p r e s e n t t h e i n n e r s o l u t i o n at t h e edge r = r 0 whereas t h e e x p o n e n t i a l l y d e c r e a s i n g p r o p e r t y of K0 i s u s e f u l f o r r e p r e s e n t i n g t h e i n n e r s o l u t i o n at t h e edge r = rx(< r 0 ) , t h e r a d i u s of a h o l e at t h e c e n t e r of t h e p l a t e . T h u s f o r a c i r c u l a r p l a t e w i t h a ho l e i n i t s c e n t e r , t h e p r o p e r i n n e r s o l u t i o n is ^ fanner = X ) -^ T^ T/AOO + £ TTfiW*) x 1o\7) p i f l w i (7 41) S i n c e t h e e q u a t i o n s g o v e r n i n g t h e eigenvalues, (7.35) a n d (7.38), are of t h e same f o r m as t h o s e f o r p l a n e s t r a i n d e f o r m a t i o n , t h e a l g o r i t h m d e v e l o p e d i n Sec.2.4 a l s o a p p l i e s f o r t h e a x i - s y m m e t r i c d e f o r m a t i o n . B y s e t t i n g s2 = 1 a n d sx —> 1, t h e l e a d i n g t e r m of t h e T a y l o r e x p a n s i o n s of (7.35) a n d (7.38) assume s i n 2A — 2A = 0 a n d s i n 2/x + 2/i = 0, r e s p e c t i v e l y . I n o t h e r w o r d s , f o r i s o t r o p i c m a t e r i a l s , t h e e i g e n v a l u e s f o r p l a n e s t r a i n a n d a x i - s y m m e t r i c d e f o r m a t i o n s are i d e n t i c a l . Some n u m e r i c a l r e s u l t s f o r s e v e r a l t r a n s v e r s e l y i s o t r o p i c m a t e r i a l s are g i v e n i n t h e f o l l o w i n g t a b l e : OUTER AND INNER SOLUTIONS / 7.3 77 m a t e r i a l o r t h o 1 o r t h o 2 o r t h o 3 o r t h o 4 i s o E 0.05E' 1.56 x 1 0 6 0 . 5 F 1.56 x 1 0 6 E' 1.56 x 1 0 6 0.05JE 1.56 x 1 0 6 0.5E V 0.287 0.287 0.287 0.287 u' 0.449 0.02245 0.449 0.2245 G' 0.078£' 0.078JS 0.078£' 0.078£ Si 0.7312 3.2702 2.5056 3.5434 1 s2 0.3176 1.4204 0.2794 0.3951 1 ^1 4.7379 + i l . 6 9 6 8 1.0594 + tO.3794 1.2703 0.8983 2.106 + 125 6.8408 1.5297 1.8010 1.2735 3.749 +11.384 I n t h e a b o v e t a b l e , t h e e l a s t i c p a r a m e t e r s E,E\ e t c . are d e f i n e d i n (7.2) a n d t h e u n i t f o r e l a s t i c i t y m o d u l u s is p o u n d / i n 2 . T h e v a l u e s f o r o r t h o t r o p i c 1 a r e t h o s e of D o u g l a s fir w i t h t h e a p p r o x i m a t i o n o f t r a n s v e r s e i s o t r o p y . T h e v a l u e s f o r o r t h o t r o p i c 2 are o b t a i n e d by i n t e r c h a n g i n g t h o s e f o r o r t h o t r o p i c 1 i n t h e a x i a l a n d t h e t r a n s v e r s e d i r e c t i o n s . W e th e n m o d i f y these d a t a t o have t h e same o r d e r o f m a g n i t u d e i n b o t h d i r e c t i o n s f o r o r t h o 3 a n d o r t h o 4. T h e q u a n t i t y nl is t h e e i g e n v a l u e w i t h t h e m i n i m u m p o s i t i v e r e a l p a r t f o r eq.(7.38) a n d A x is t h e c o r r e s p o n d i n g v a l u e f o r eq.(7.35). T h e d e c a y i n g r a t e is c h a r a c t e r i z e d by t h e r e a l p a r t s of / i j a n d A j . We see once a g a i n t h a t t h e d e c a y i n g r a t e f o r a n o r t h o t r o p i c m a t e r i a l m a y be l a r g e r o r s m a l l e r t h a n t h a t f o r i s o t r o p i c one. NECESSARY CONDITIONS FOR DECAYING STATES / 8.1 78 CHAPTER 8 NECESSARY CONDITIONS FOR DECAYING STATES T h i s c h a p t e r is t h e a n a l o g u e of c h a p t e r 3 f o r a x i - s y m m e t r i c d e f o r m a t i o n s of a t r a n s v e r s e l y i s o t r o p i c c i r c u l a r p l a t e . B y i n t e g r a t i n g t h e b a s i c e q u a t i o n s a n d u s i n g p r o p e r t i e s of t h e e x a c t s o l u t i o n s , we are a b l e t o d e r i v e five n e c e s sary c o n d i t i o n s f o r d e c a y i n g s t a t e s e x p r e s s e d i n t e r m s of e x p l i c i t i n t e g r a l s . T h e i r a s y m p t o t i c c o u n t e r p a r t s f o r t h e i s o t r o p i c case were first g i v e n i n [12]. T h e s e necessary c o n d i t i o n s f o r d e c a y i n g s t a t e s g i v e the p r e c i s e m e a n i n g of t h e w e l l k n o w n St.Venant's p r i n c i p l e f o r p l a t e p r o b l e m s [ 1 7 ] . S i x c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s f o r t h e t r a n s v e r s e l y i s o t r o p i c case are d e f i n e d here t o d e v e l o p a d d i t i o n a l a s y m p t o t i c necessary c o n d i t i o n s f o r b o t h m i x e d a n d d i s p l a c e m e n t b o u n d a r y c o n d i t i o n s w h e r e e x a c t , e x p l i c i t ones are n o t a v a i l a b l e . 8.1 SOME NECESSARY CONDITIONS FOR DECAYING STATES We b e g i n b y s e t t i n g (8.1) NECESSARY CONDITIONS FOR DECAYING STATES / 8.1 79 T h e n , we i n t e g r a t e t h e s e c o n d e q u a t i o n o f (7.5) w.r.t. z f r o m — h t o h, a n d use t h e b o u n d a r y c o n d i t i o n az = 0 at z = ±h t o get o r rQr — c o n s t . w h e r e t h e c o n s t a n t c a n be s h o w n t o be z e r o f o r d e c a y i n g s t a t e s . I n f a c t f o r t h e s y m m e t r i c ( e x t e n s i o n ) case i t is o b v i o u s l y t r u e . F o r t h e a n t i - s y m m e t r i c ( b e n d i n g ) case, d u e t o l i n e a r -i t y , o n l y a t y p i c a l t e r m o f t h e i n n e r s o l u t i o n needs t o b e checked. F o r t h i s t e r m we have f r o m (7.6) a n d (7.35a) Qr = fk arzdz = r 0 A /* { < ( * ) + £fx(z)}dz (8.3) J-h c r 0 J-h n' whose s e c o n d f a c t o r , b y (7.36) a n d (7.35), is e q u a l t o A 2 fh ^ ( l — a s 2 ) ( l — a s 2 ) J {cos(sl\) c o s ( s 2 A z / A t ) — c o s ( s 2 A ) c o s ( s 1 A z / / i ) } d z = — 4 - ( l - a s i ) ( l - c o s f s j A ) c o s ( s 2 A ) { t a n ( s 2 A ) — — t a n ( s x A ) } = 0. S 2 / l Sj R e c a l l i n g t h e a s y m p t o t i c i n n e r s o l u t i o n is a l s o t h e e x a c t e x p r e s s i o n of t h e d e c a y i n g s o l u t i o n , we have f o r d e c a y i n g s t a t e s n e c e s s a r y c o n d i t i o n 1 )r = f arzdz = 0 (8.4) J-h •ft -h w h i c h is j u s t t h e o v e r a l l e q u i l i b r i u m e q u a t i o n i n t h e z d i r e c t i o n . N o w we m u l t i p l y t h e first e q u a t i o n o f (7.5) b y z a n d i n t e g r a t e t o get M9 = ^ ( r M r ) (8.5) where we have m a d e use of t h e b o u n d a r y c o n d i t i o n arz = 0 at z = ±h a n d (8.4) t o get >h « fh arzdz = 0. NECESSARY CONDITIONS FOR DECAYING STATES f 8.1 80 T h e first e q u a t i o n o f (7.4), w h e n c o m b i n e d w i t h (7.1), gives Q anar + al2a<> + alzaz - — {r(al2(rr + ancre + aizaz)} = 0 (8.6) w h i c h , a f t e r m u l t i p l i c a t i o n b y z a n d i n t e g r a t i o n w.r.t. z, becomes anMr + al2M0 + al3Mz - ^ { r ( a 1 2 M r + anMd + al3Mz)} = 0. E l i m i n a t i o n o f M$ f r o m t h e above e q u a t i o n b y (8.5) y i e l d s «n W - l [ r A ( r M r ) ] } + al3{Mz - ±(rMz)} = 0 (8.7) where u p o n i n t e g r a t i o n b y p a r t s a n d a p p l i c a t i o n s o f b o u n d a r y c o n d i t i o n s , Mz = J_hezzdz = - J _ - — d z . T h i s e x p r e s s i o n f o r Mz c a n be t r a n s f o r m e d b y t h e e q u i l i b r i u m e q u a t i o n i n t o -h S u b s t i t u t i n g (8.8) i n t o (8.7) a n d i n t e g r a t i n g o nce w.r.t. r g i v e 3 dMT o dMTZ - . aur — 1- a 1 3 ( r — r M f 2 ) = cons t . dr dr I n t e g r a t i n g once m o r e l e a d s t o (8.8) Mr + ^ M r z = ^ + C2 (8.9) w h e r e t h e c o n s t a n t s Cx a n d C2 c a n b e p r o v e d t o b e ze r o f o r d e c a y i n g s t a t e s . I n d e e d , o n l y j , t y p i c a l . t e r m of t h e i n n e r s o l u t i o n i n t h e a n t i - s y m m e t r i c case needs t o be checked. F o r t h i s t e r m , we have -h + ( l - 6 ) r0[Xr/h) r I0{\r0/h) J_h f Zf'x(z)d2 J-h NECESSARY CONDITIONS FOR DECAYING STATES / 8.1 81 w h e r e t h e first t e r m of t h e r i g h t h a n d s i d e t u r n s o u t t o b e z e r o u p o n i n t e g r a t i o n b y p a r t s a n d a p p l i c a t i o n s of eq.(8.3) a n d t h e b o u n d a r y c o n d i t i o n arz = 0 at z — ±.h w h i c h c o r r e s p o n d s t o jpfxi±h) + afi{±h) = 0. (8.10) T h u s we h ave ^ , a i 3 J X > _ ip(Ar//>) 1 fh i u f / 1 3 2 V 2 / x f l f ) U , w h i c h c a n be p r o v e d t o b e z e r o b y (7.36), (7.35) a n d t h e i d e n t i t y *13 = ( l -6 )o an d — ac T h u s f o r d e c a y i n g s t a t e s we a r r i v e at n e c e s s a r y c o n d i t i o n 2 (8.11) M r + ^ M r z = 0 o r aTzdz + — - / -0rzz2dz = 0 (S.12) •h a \ \ r J-h 2 whi c h , i n t h e i s o t r o p i c case, is i d e n t i c a l t o t h e one i n [12] o b t a i n e d b y a n a p p l i c a t i o n of the r e c i p r o c a l t h e o r e m . a n d n e g l e c t i n g a n e x p o n e n t i a l l y s m a l l e r r o r as e —• 0. H e r e we were abl e to p r o v e t h a t (8.12) is e x a c t l y t r u e . T o o b t a i n a n e c e s s a r y c o n d i t i o n f o r t h e s y m m e t r i c case we d e f i n e ./» rh NR — I ardz} NQ = j c$dz N z = azdz, Mrz = / zarzdz J-h J-h a n d i n t e g r a t e t h e first e q u a t i o n of (7.5) t o get No = ~ ( r t f r ) (8.14) NECESSARY CONDITIONS FOR DECAYING STATES / 8.1 82 w h e r e t h e b o u n d a r y c o n d i t i o n art = 0 at z = ±h has been used. I n t e g r a t i o n o f eq.(8.6) gives o n ^ r + «i2-N* + a i 3 ^ - ^{r{ai2Nr + anN9 + a13Nz)} = 0 w h i c h a f t e r t h e s u b s t i t u t i o n o f (8.14) becomes « n W - Y r [ r T r { r N r ) ] } + a M ~ 3^rJV*» = ° (8'15) w h e r e u p o n i n t e g r a t i o n b y p a r t s a n d a p p l i c a t i o n s o f b o u n d a r y c o n d i t i o n s at z = ±h, Nz = I azdz = - I z?~dz. J-h J-h dz T h i s e x p r e s s i o n f o r Nz c a n be t r a n s f o r m e d b y t h e e q u i l i b r i u m c o n d i t i o n i n t o -h _ F o l l o w i n g a s i m i l a r d e d u c t i o n l e a d i n g t o (8.9), we get f r o m (8.15) a n d (8.16) Nz = j_ ~(rarz)dz = l^(rMrz)- (8.16) J V r + ^ i j G r „ = § + C 4 . (8.17) the c o n s t a n t s C 3 a n d C 4 c a n also be p r o v e d t o b e z e r o f o r d e c a y i n g s t a t e s b y a n a r g u m e n t s i m i l a r t o t h e one g i v e n above. T h u s we have f o r d e c a y i n g s t a t e s n e c e s s a r y c o n d i t i o n 3 a n r o r /h a 1 [ h ardz +—- zarzdz = 0. (8.18) -h o n r J-h T o get n e c e s s a r y c o n d i t i o n s i n v o l v i n g d i s p l a c e m e n t s , we o b t a i n f r o m eq.(7.3) a n d eq.(7.1) u = r ( a 1 2 a r + anae + a13cr.) NECESSARY CONDITIONS FOR DECAYING STATES / 8.2 83 a n d t h e r e w i t h f udz = r{a12Nr + anN0 + a 1 3JV*J J-h '-h = r{a12Nr + an—{rNr) + a13~{rMrt)} = r{al2Nr + au±{rNr) - a ^ ^ r 2 * , ) } w h e r e eqs.(8.14), (8.16), a n d (8.18) have b e e n used. T h u s we have •h udz - (al2 - an)rNr (8.19) J - h w h i c h t u r n s (8.18) i n t o . n e c e s s a r y c o n d i t i o n 4 / 1 f^1 a C^1 / udz+ — zarzdz = 0. (8.20) - a n J-h a n J-h 'h „ rh— I °12 S i m i l a r l y , we c o n s i d e r t h e i n t e g r a l I uzdz = r{ai2Mr + anM() + atfMz) J-h = r{a12Mr + o n ^ ( r M r ) + o 1 3 ^ ( r M r J } = r{al2Mr + an^-(rMr) - a n ^ ( r 2 M r ) } where eqs.(8.5), (8.8) a n d (8.12) have b e e n used. T h u s , we have / uzdz = r ( a 1 2 - a n ) M r (8.21) J-h w h i c h changes (8.12) i n t o n e c e s s a r y c o n d i t i o n 5 1 t^* a /*'* 1 / uzdz + — / -<rr,z2dz = 0. (8.22) 3.2 ASYMPTOTIC NECESSARY CONDITIONS FOR DECAYING STATES We h a v e a l r e a d y seen t h a t i t is n o t a l w a y s p o s s i b l e t o get e x a c t , e x p l i c i t n e c e s s a r y c o n d i t i o n s s i m i l a r t o t h o s e i n s e c t i o n 8.1. F o r e x a m p l e so f a r we have n o t NECESSARY CONDITIONS FOR DECAYING STATES / 8.2 84 o b t a i n e d s u c h n e c e s s a r y c o n d i t i o n s f o r e i t h e r m i x e d edge b o u n d a r y c o n d i t i o n s u>, ar o r d i s p l a c e m e n t edge b o u n d a r y c o n d i t i o n s u>, fi. O n t h e o t h e r h a n d , we k n o w f r o m [11,12] t h a t b y a n a p p l i c a t i o n o f t h e r e c i p r o c a l t h e o r e m , we m a y d e v e l o p n e c e s s a r y c o n d i t i o n s f o r b o t h m i x e d a n d d i s p l a c e m e n t edge d a t a w i t h a n e r r o r e x p o n e n t i a l l y s m a l l as 6 —• 0, i.e. a s y m p t o t i c n e c e s s a r y c o n d i t i o n s . T o d e v e l o p n e c e s s a r y c o n d i t i o n s f o r d e c a y i n g s t a t e s w i t h t h e m i x e d edge b o u n d a r y c o n d i t i o n s w = to, ar = aT at r = r 0, t h r e e c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s w i l l b e d e f i n e d . E a c h has b o u n d a r y c o n d i t i o n s w = 0, ar = 0 at r = r 0 a n d d i f f e r e n t s i n g u l a r i t i e s at t h e c e n t e r c o r r e s p o n d i n g t o d i f f e r e n t o u t e r s o l u t i o n s , i.e. C B V P l : A = 0, D = 0, C = 1; C B V P 2 : A = 1, D = 0, C = 0; (8.23) C B V P 3 : A = 0, D = 1, C = 0. S u p p o s e ar, ar:, u a n d w d e n o t e s t r e s s a n d d i s p l a c e m e n t edge d a t a f o r a d e c a y i n g s t a t e , a n d <7R,-, crr.,-, u; a n d ti), f o r t h e c a n o n i c a l b o u n d a r y v a l u e p r o b l e m i ( i = l , 2, 3). S i m i l a r t o c h a p t e r 3, a n a p p l i c a t i o n o f t h e r e c i p r o c a l t h e o r e m t o t h e d e c a y i n g s t a t e a n d each c a n o n i c a l b o u n d a r y v a l u e p r o b l e m d e f i n e d above ov e r a n a n n u l a r r e g i o n , (0 <)rt- < r < r 0, g i v e s [ arUidz= f aTziwdz + 0(e-x°le) ( i = 1,2,3) (8.24) J-h J-h where t h e s e c o n d t e r m o n t h e r i g h t r e p r e s e n t s s o l u t i o n c o n t r i b u t i o n s f r o m t h e i n t e r i o r r e g i o n of t h e p l a t e , say at r,- = r 0/2, i n t h e r e c i p r o c a l t h e o r e m . T h e c o n s t a n t A 0 i s t h e e i g e n v a l u e w i t h m i n i m u m p o s i t i v e r e a l p a r t of e i t h e r t a n ( s 1 A 0 ) = — t a n ( s 2 A 0 ) (8.25) s2 o r t a n ( s 2 A 0 ) = — t a n ( s 1 A 0 ) . (8.26) s 2 NECESSARY CONDITIONS FOR DECAYING STATES / 8.2 85 T h u s , f o r d e c a y i n g s t a t e s w i t h edge d a t a to, aT a t r = r 0 , we h a v e asymptotic necessary conditions h (arUi - arziw)dz = 0 (» = 1,2,3) (8.27) '-h u p t o a n e x p o n e n t i a l l y s m a l l e r r o r as e —* 0. I n a s i m i l a r w a y we m a y d e f i n e t h r e e c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s f o r d e c a y i n g s t a t e s w i t h t h e d i s p l a c e m e n t edge d a t a to = to, u = fi at r — TQ. E a c h c a n o n i c a l p r o b l e m has t h e b o u n d a r y c o d i t i o n s at t h e p l a t e edge: to = 0, u = 0 at r = r 0 a n d t h e f o l l o w i n g s i n g u l a r i t i e s at t h e ce n t e r : C B V P 4 : A = 0, D = 0, 6=1; C B V P 5 : A = l, D = 0, 6 = 0; (8.28) C B V P 6 : i = 0, D=l, 6 = 0. F o r d e c a y i n g s t a t e s i n d u c e d b y t h e edge d a t a w, u at r = T Q , t h e r e c i p r o c a l t h e o r e m gi v e s us t h e f o l l o w i n g asymptotic necessary conditions h (<7r,u + drziw)dz = 0 (i = 4,5,6) (8.29) '-h u p t o an e r r o r o f o r d e r 0 ( e ~ A ° / e ) . T h e q u a n t i t i e s <rri a n d a r i i are d i r e c t a n d t r a n s v e r s e s h e a r st r e s s at r = r 0 i n t h e c a n o n i c a l b o u n d a r y v a l u e p r o b l e m i (i=4,5,6). CORNER STRESS SINGULARITIES / 9.1 86 CHAPTER 9 CORNER STRESS SINGULARITIES S i m i l a r t o t h e p l a n e s t r a i n case, c o r n e r ( o r m o r e c o r r e c t l y , r i m ) s i n g u l a r i -t i e s , i f t h e y e x i s t , m u s t be d e t e r m i n e d a n d e x t r a c t e d f o r a n a c c u r a t e n u m e r i c a l s o l u t i o n of c e r t a i n c a n o n i c a l p r o b l e m s u s e f u l i n t h e c o r r e c t f o r m u l a t i o n a n d s o l u t i o n o f p l a t e p r o b -lems. B y u s i n g t w o d i f f e r e n t l o c a l p o l a r c o o r d i n a t e s a n d c o n s i d e r i n g t h e l e a d i n g t e r m of a s y m p t o t i c s o l u t i o n b e h a v i o r i n t h e n e i g h b o u r h o o d o f t h e r i m c o r n e r , we are l e d t o t w o t r a n s c e n d e n t a l e q u a t i o n s g o v e r n i n g t h e s i n g u l a r i t y e x p o n e n t s i n t h e stresses at a r i g h t c o r n e r w i t h c l a m p e d a n d m i x e d edge c o n d i t i o n s r e s p e c t i v e l y . W h e n r e d u c e d t o t h e c o r r e -s p o n d i n g i s o t r o p i c cases, we find t h a t a s i n g u l a r s o l u t i o n e x i s t s o n l y f o r t h e c l a m p e d edge case b u t not f o r t h e case o f m i x e d edge c o n d i t i o n s , a r e s u l t n o t f o u n d i n t h e l i t e r a t u r e . F o r t h e o r t h o t r o p i c case, t h e same c o n c l u s i o n is c o n f i r m e d n u m e r i c a l l y . I n a d d i t i o n , we obse r v e d f r o m o u r n u m e r i c a l r e s u l t s t h a t t h e s t r e s s s i n g u l a r i t y e x p o n e n t r e m a i n s t h e s ame i f t h e d a t a of m a t e r i a l p r o p e r t y i n t h e a x i a l a n d t h e t r a n s v e r s e d i r e c t i o n s a r e i n t e r c h a n g e d . T h e c o n t e n t s o f t h i s c h a p t e r a r e t o a p p e a r i n [21]. 9.1 MIXED BOUNDARY CONDITION CASE We c o n s i d e r a r i g h t r i m c o r n e r f o r c a n o n i c a l p r o b l e m s 1-3 f o r b o u n d a r y c o n d i t i o n s az = arz = 0 at z = h a n d w = 0, cr = 0 at r = r 0 . L e t t h e l o c a l c a r t e s i a n CORNER STRESS SINGULARITIES / 9.1 87 c o o r d i n a t e s y s t e m ( i , y) have i t s o r i g i n at t h e r i m c o r n e r a n d x = r0-r, y = h- z. (9.1) T h r o u g h o u t t h i s c h a p t e r , <f>{xty) d e n o t e s t h e l e a d i n g t e r m of t h e a s y m p t o t i c e x p a n s i o n of t h e str e s s f u n c t i o n i n t h e n e i g h b o u r h o o d of t h e r i m c o r n e r , ( x , y) = ( 0 , 0 ) . T h u s , <f> i s g o v e r n e d b y d2 1 B 2 d2 1 d2 T h e c o r r e s p o n d i n g l e a d i n g t e r m e x p r e s s i o n s f o r t h e s t r e s s c o m p o n e n t s g i v e n b y (7.6) be-come d .d2<f> d2<}>. d lud2<f> d2<f>. ° ° = o - y { b M + a d ? ) > d , d2* J 2 [ 9 ' 3 ) S i m i l a r t o the m e t h o d d e v e l o p e d i n s e c t i o n 4.1, we i n t r o d u c e t w o l o c a l p o l a r c o o r d i n a t e s x = pc o s y ? , s1y = ps\n<p (9.4) and x / s 2 = r c o s 0 , y = r s i n 0 (9.5) and w r i t e t h e s o l u t i o n o f (9.2) as 4>{r, 0) = r A + 2 { C i ( A ) cos(A + 2)9 + C 2 ( A ) s i n ( A + 2)9 + C3(\)f(\,9) + CA(\)g(\,9)} (9.6) =rx+2Gx{9) / ( A , 9) = / (p4,)x c o s ( A W 0 ) s i n ( A + 2){9 - V)<ty, Jo f9 g{\, 9) = J {p^)x s i n t A c ^ ) s i n ( A + 2){0 - 1>)drP, ( Q ? ) pj, = (s2 c o s 2 ip + s2 s i n 2 VO1/2, UJ^ = A r c t a n ( s 1 / s 2 t a n ^ ) . w h e r e CORNER STRESS SINGULARITIES / 9.1 88 A detailed calculation shows that at 9 = 0 <\>xxx = <f>rrr/s2> and at 9 = n/2 <t>xyy = {-2 W 3 + <t>0Or/r2 - 4>rlr2 + <f>rrM/s2, <t>xxy = {4>rre - + 2^/ r 2 }/ ( rs? , ) , <t>yyy = <t>oeo/r* + Wrefr2 - 2 ^ / r 3 <t>xXX = -{Hrelr - 2<f>0/r2 + <t>000/r2}/(rs\), <t>xyy = ~{^rrO - Wro/r + 2fo/r2}/(r32), <f>xxy = {<f>rr/r - 4>Jr2 + <t>reo/r2 - 2<f>00/rz}I's\, Furthermore, we have from the definition of f and g in (9.7) / (A,0) = 0, <7(A,0) = 0, / ' (A,0) = 0, o'(A,0) = 0, / , ( A , * / 2 ) = (A + 2)/(A,ir /2) , a'(A,7r/2) = (A + 2)ff(A,7r/2), / " (A,0) = (A + 2) S 2 A, g"(\,0) = 0, / " ( A , n/2) = (A + 2)sxcos(A7r/2) - (A + 2) 2 / (A, *r/2), g"{\, TT/2) = (A + 2)sx sm{Xir/2) - (A + 2) 2 f f(A, TT/2), /" ' (A,0) = 0, / ' ( A , 0 ) = (A + 2 ) A S l s J - 1 , / ' " (A , TT/2) = - ( A + 2) 3 / (A, TT/2) - (A + 2 )AsJ - 1 s 2 sin(A7r/2), g"'(X, TT/2) = - ( A + 2)3g{\,n/2) + (A + 2 )As A " 1 s 2 cos(A*r/2) where a prime denotes j ^ , and / (A , 5) = f ( ^ ) A cos(Au^,) cos(A + 2)(9 - tp)dtp, Jo fe g{\,9)= / (p v,)Asin(Aw v,)cos(A + 2 ) ( 0 - ^)drp. Jo (9.8) (9.9) (9.10) (9.11) CORNER STRESS SINGULARITIES / 9.1 89 F r o m (9.3), (9.6), (9.8) a n d t h e f o r m u l a e l i s t e d above, t h e b o u n d a r y c o n d i t i o n orz = 0 at 6 = 0 g i v e s Z?i(A + l ) C j + sxCz = 0 (9.12) w i t h Dx = \ - 1. a$l S e t t i n g az = 0 at 6 = 0 y i e l d s a n o t h e r e q u a t i o n D2{\ + 1 ) C 2 + SiS^Ct = 0 (9.13) w i t h O n t h e o t h e r h a n d , ar = 0 at 6 = TT/2 gives t h e e q u a t i o n (A + l ) C 1 c o s ( A + 2)£ + (A + l ) C 2 s i n ( A + 2 ) -2 2 + {(A + l ) / ( A , ^ ) + - ^ c o s ( A | ) } C 3 ( 9 > 1 4 ) + {(A + l M A , | ) + ^ s i n ( A ^ ) } C 4 = 0 w i t h = q s 2 — 1. F i n a l l y , w = 0 at 6 = ir/2 y i e l d s |^ = e 2 = 0 w h i c h is (A + l)Ci cos(A + 2) I + (A + 1 ) C 2 s i n ( A + 2) £ + ( ( A + 1) / ( A , |) + ^  c o s ( A | ) } C 3 ( 9. 1 5 ) + {(A + l M A , | ) + ^ 8 i n ( A | ) } C 4 = s 0 w h e r e 2 a a 1 3 - d a 3 3 2 = — — SO — 1. (6 + l ) q 1 3 - c q 3 3 T h e e x i s t e n c e of a n o n t r i v i a l s o l u t i o n f o r {C,-} r e q u i r e s F(X) = 0 (9.16) CORNER STRESS SINGULARITIES / 9.2 90 w h e r e (9.17) A c c o r d i n g t o eqs.(9.3) a n d (9.6), t h e e x i s t e n c e of s t r e s s s i n g u l a r i t i e s c o r r e s p o n d s t o t h e s o l u t i o n s o f eq.(9.17) f o r 0 < R e { A } < 1. F o r a n i s o t r o p i c m a t e r i a l (9.16) r e d u c e s t o w h i c h has n o s o l u t i o n f o r 0 < R e { A } < 1. Hence, we c l a i m t h a t for an isotropic plate in axi-symmetric deformation, corner stress singularities do not exist for the case of mixed edge conditions. A s p o i n t e d o u t i n t h e p l a n e s t r a i n case, a n i m a g i n a r y p a r t of A w o u l d r e s u l t i n a s o l u t i o n o s c i l l a t i n g w i t h a n i n f i n i t e l y l a r g e f r e q u e n c y as r —• 0, a n d t h e r e f o r e seems t o b e u n r e a l i s t i c . We s e a r c h o n l y r e a l s o l u t i o n s f o r s i n g u l a r i t y e x p o n e n t s f o r t h e o r t h o t r o p i c case. Eq.(9.17) has b e e n p r o g r a m m e d a n d c o m p u t e d i n t h e U B C C o m p u t i n g C e n t e r . N o stress s i n g u l a r i t y ( w i t h a r e a l e x p o n e n t A) has b e e n f o u n d n u m e r i c a l l y f o r the o r t h o t r o p i c case. 9.2 DISPLACEMENT BOUNDARY CONDITION CASE s i n ( A i r ) = 0 I n t h e d i s p l a c e m e n t edge b o u n d a r y c o n d i t i o n case, eqs.(9.12), (9.13) a n d (9.15) r e m a i n . T o r e p l a c e (9.14), we n o t e t h a t d~1rz d2u d ,dw d2u dez dz dz2 dr dz dz2 dr w h e n e v a l u a t e d at 9 = TT/2 ( w h e r e u = 0), t h i s r e l a t i o n b ecomes T h i s c a n be f u r t h e r w r i t t e n as fottx*' + a<f>xyy)e=n/2 = ~ g g ( + xxy CORNER STRESS SINGULARITIES / 9.2 91 T h e s u b s t i t u t i o n of t h e e x p r e s s i o n f o r <f>, (9.6), y i e l d s t h e r e q u i r e d f o u r t h c o n d i t i o n f o r {<?,}: Cx s i n ( A + 2 ) | - C2 cos(A + 2 ) | - C 3 / ( A , | ) - C 4 ? ( A , | ) = 0. (9.18) T h e e x i s t e n c e of a n o n t r i v i a l s o l u t i o n f o r (9.12), (9.13), (9.15) a n d (9.18) r e q u i r e s t h e d e t e r m i n a n t of t h e c o e f f i c i e n t m a t r i x of these e q u a t i o n s t o v a n i s h so t h a t we have a g a i n (9.16) b u t now w i t h F(X) = ( A ~ | " 1 ) 2 / " / 2 ( ^ ) A ( ^ ) A s i n A ( ^ - u+)sin(A + 2){<f> - rp)d<f>d1> * Jo Jo sx r / 2 + (A + l ) - f / (p v,) Asin(Aw v,)cos(A-r2 ) V r f V u \ Jo " ( A + 1 ) f l H — T W ) A c o s ( ^ ) s i n ( A + 2 ) ^ V u2 Jo + (A + 1 ) ^ ^ ( ^ ) A s i n A ( | - ^ ) c o s ( A + 2 ) ( | - ^ V ( 9- 1 9) A+l X-l _ _ + - ^ D r c o s ( A 2 ) c o s ( A + 2 ) 2 - -2A-1 ^ 1 ^ 2 ' F o r a n i s o t r o p i c m a t e r i a l (9.19) r e d u c e s t o F(X) = c o s ' ( A - ) - - J - ^ - i + — . (9.20) F r o m t h i s e q u a t i o n we have F{0) = 4(1 - i / ) 2 / ( 3 - 4i/) > 0, F ( l ) = - 4 i / ( l - z/)/(3 - 4i/) < 0 w h i c h shows t h e e x i s t e n c e of a s i n g u l a r i t y , i.e. a r e a l s o l u t i o n 0 < A < 1. T o p r o v e t h e u n i q u e n e s s of t h e s i n g u l a r s o l u t i o n i n 0 < R e { A } < 1, we w r i t e A = a + ib (0 < a < 1), a n d c o n s i d e r a r e c t a n g u l a r c o n t o u r i n t h e c o m p l e x A p l a n e c o n s i s t i n g o f t h e s egments of s t r a i g h t l i n e s : R e { A } = 0, R e { A } = 1, I m { A } = ±H CORNER STRESS SINGULARITIES / 9.2 92 w h e r e H is a l a r g e p o s i t i v e n u m b e r . We t h e n s t u d y t h e i m a g e of t h i s c o n t o u r m a p p e d b y t h e a n a l y t i c f u n c t i o n „ 1 3-Au b2-a? 1 Z = F[X) =-y- — r H h 1- — cos air c o s h bir v ' A(3-Av) A 3-Au 2 — — + x s i n an s i n h bn] l3-Au 2 J w h i c h has t h e p r o p e r t y of lm{Z} < 0 f o r 6 > 0 a n d lm{Z) > 0 f o r 6 < 0. A s a p o i n t A moves a l o n g t h e c o n t o u r i n t h e u p p e r h a l f A p l a n e (6 > 0) f r o m A = 1 t o A = 0, t h e c o r r e s p o n d i n g p o i n t Z moves a l o n g a c u r v e i n t h e l o w e r h a l f Z p l a n e f r o m F ( l ) < 0 to F ( 0 ) > 0 (Z is o n t h e p o s i t i v e r e a l a x i s as A i s o n t h e l i n e Re{A} = 0); T h e n as A c o n t i n u e s t o move a l o n g t h e c o n t o u r i n t h e l o w e r h a l f A p l a n e (6 < 0) b a c k t o A = 1, Z c o n t i n u e s t o m ove a l o n g a c u r v e i n t h e u p p e r h a l f Z p l a n e b a c k t o F(l). B y t h e A r g u m e n t P r i n c i p l e , we c o n c l u d e t h a t t h e r e is o n l y one z e r o f o r t h e a n a l y t i c f u n c t i o n Z = F(A) i n t h e r e g i o n 0 < Re{A} < 1. T h a t i s , we have o n l y t o s e a r c h f o r a r e a l s o l u t i o n b e t w e e n 0 a n d 1. Hence, for an isotropic plate in axisymmetric deformation, there exists an unique solution of the equation governing corner stress singularities in the case of a clamped edge. F o r t h e o r t h o t r o p i c case, t h e e x i s t e n c e o f a s t r e s s s i n g u l a r i t y is c o n f i r m e d n u m e r i c a l l y . Eq.(9.19) has been p r o g r a m m e d a n d c o m p u t e d i n t h e U B C C o m p u t i n g C e n t e r . F o r a n i s o t r o p i c m a t e r i a l i t l e a d s t o a r o o t w h i c h s a t i s f i e s eq.(9.20) CORNER STRESS SINGULARITIES / 9.2 93 a c c u r a t e l y . S o me a d d i t i o n a l n u m e r i c a l r e s u l t s are l i s t e d b e l o w f o r f u t u r e references. m a t e r i a l E E1 G' V 7 = 1 - A i s o t r o p i c 0.0 0.0 i s o t r o p i c 0.1 0.1330 i s o t r o p i c 0.2 0.2189 i s o t r o p i c 1 1.56 x 1 0 6 0.3 0.2888 i s o t r o p i c 0.4 0.3501 i s o t r o p i c 2 1.56 x 1 0 6 0.5 0.4054 o r t h o t r o p i c 1 0.05£' 1.56 x 1 0 6 0.078JJ' 0.449 0.287 0.1544 o r t h o t r o p i c 2 1.56 x 1 0 6 0.05£ 0.078£ 0.02245 0.287 0.1544 o r t h o t r o p i c 0.05E' 1.56 x 1 0 6 0 . 0 0 7 8 ^ 0.449 0.287 0.1637 o r t h o t r o p i c 1.56 x 1 0 6 0.05E 0.0078E 0.02245 0.287 0.1637 o r t h o t r o p i c 3 0.5E' 1.56 x 1 0 6 0 . 0 7 S F 0.449 0.287 0.2813 o r t h o t r o p i c 4 1.56 x 1 0 6 0.5£ 0.078J2 0.2245 0.287 0.2813 o r t h o t r o p i c 0.5E' 1.56 x 1 0 6 0 . 0 0 7 8 ^ 0.449 0.287 0.2434 o r t h o t r o p i c 1.56 x 1 0 6 0.5E 0.0078E 0.2245 0.287 0.2434 o r t h o t r o p i c 0.005 E' 1.56 x 1 0 6 0.0078£' 0.449 0.287 0.0851 o r t h o t r o p i c 1.56 x 1 0 6 0.005E 0.0078E 0.002245 0.287 0.0851 o r t h o t r o p i c 0.005i?' 1.56 x 1 0 6 0.00078£' 0.449 0.287 0.098 o r t h o t r o p i c 1.56 x 1 0 6 0.005E 0.00078E 0.002245 0.287 0.098 In t h e above t a b l e , 7 d e n o t e s t h e stre s s s i n g u l a r i t y e x p o n e n t . F r o m t h e a b o v e n u m e r i c a l r e s u l t s , we o b s e r v e t h a t t h e s t r e s s s i n g u l a r i t y e x p o n e n t 7 r e m a i n s t h e s a m e i f t h e d a t a i n t h e a x i a l a n d t h e t r a n s v e r s e d i r e c t i o n s are i n t e r c h a n g e d . CANONICAL PROBLEMS / 10.1 94 CHAPTER 10 CANONICAL PROBLEMS I n t h i s c h a p t e r , we solv e s i x c a n o n i c a l p r o b l e m s d e f i n e d b y eqs.(8.23) a n d (8.28) b y a p p l y i n g t h e finite H a n k e l t r a n s f o r m t o t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n a n d d e r i v i n g a n i n t e g r a l e q u a t i o n of t h e s e c o n d k i n d . T h i s i n t e g r a l e q u a t i o n i s t h e n s o l v e d n u m e r i c a l l y b y t a k i n g t h e r i m c o r n e r s i n g u l a r i t y i n t o c o n s i d e r a t i o n . T h e s i n g u l a r i t y at t h e c e n t e r of t h e p l a t e is r e m o v e d b y s u b t r a c t i n g off t h e c o r r e s p o n d i n g o u t e r s o l u t i o n b e f o r e t h e finite H a n k e l t r a n s f o r m s . A s o l u t i o n b y t h e m e t h o d of e i g e n f u n c t i o n e x p a n s i o n s is also d i s c u s s e d b r i e f l y . O u r F O R T R A N p r o g r a m s d e v e l o p e d f o r t h e o r t h o t r o p i c case a l s o a p p l i e s to t h e i s o t r o p i c case. N u m e r i c a l r e s u l t s are d i s p l a y e d g r a p h i c a l l y f o r d i f f e r e n t sets of e l a s t i c p a r a m e t e r v a l u e s l i s t e d at t h e e n d of c h a p t e r 9. 10.1 FINITE HANKEL TRANSFORM METHOD O b s e r v e t h a t az s a t i s f i e s t h e b a s i c e q u a t i o n f o r <f> , i.e. A r A r ^ + (a + c ) A r ^ j < r , + d - ^ f = 0 (10.1) w i t h h o m o g e n e o u s b o u n d a r y c o n d i t i o n s a2 = 0, -4L = 0atz = ±h (10.2) dz CANONICAL PROBLEMS / 10.1 95 where t h e s e c o n d one comes f r o m t h e b o u n d a r y c o n d i t i o n arz = 0 a n d eq.(7.5). T h e f i n i t e H a n k e l t r a n s f o r m of az ( r , z) i s g i v e n b y E,(p,-, z) = H(az) = / raz{r, z)J0(rPi)dr (10.3) Jo w h e r e J 0 is t h e z e r o t h o r d e r B e s s e l f u n c t i o n o f t h e first k i n d a n d pt- i s d e f i n e d b y M'oPi) = 0. (10.4) It has t h e f o l l o w i n g i n v e r s e t r a n s f o r m 2 /_ Mw) ( r , „ - j ; B . t e , . ) p « w i ( 1 0. 5 ) w h e r e Jx is t h e first o r d e r B e s s e l f u n c t i o n of t h e first k i n d . B y t h e r e l a t i o n s H{Ar<rz) = roP.-ff^ro.aJJifroP,-) - p 2 E 2 ( p , - , z ) , H{ArArcz) = pjE,(pf-,«) - r o p J a ^ r o . a J J ^ r o p , - ) + r o P . - A ^ f o i ^ f o P t ) . the b o u n d a r y v a l u e p r o b l e m is t r a n s f o r m e d i n t o dE'2"' - ( a + e ) p ? E j + P ? E * = f(Pi, z) (10.6) w i t h E z(p,,±/i)=0, E',(p,-,±A)=0 a n d /(P.. z ) = '•oP,«/i(r0p,){p2o-2(ro,z) - (a + c ) c r " ( r 0 , z ) - Araz{r0,z)} (10.7) where ( )' = T h e s o l u t i o n of Y>z is g i v e n i n t e r m s of a Green's f u n c t i o n ( d e r i v e d i n a p p e n d i x F ) as f o l l o w s : Vt{pitz)=[ G(pity,z)f{Pily)dy. (10.8) J-h S u b s t i t u t i n g (10.7) i n t o (10.8) a n d u s i n g t h e r e l a t i o n s (F.13), (F.15) (see a p p e n d i x F ) , (10.8) b e comes /h dw ti {G*D-a7r + + G0<TrO}dy (10.9) CANONICAL PROBLEMS / 10.2 96 where t h e s u b s c r i p t 0 den o t e s e v a l u a t i o n at r = r 0 a n d GW = 9iP2G(pi,y}z) kiDY2G(Pi,y, * ) , G« = ff2P,?G,(p.->y>*) k2DY2G(Pi,y, »). G„ = QzPiGipi^z) k3GY2G(p{,y, w i t h gi,k{{i = 1,2,3) d e f i n e d b y 01 = -°ll/( 013 ~ all°33)i 02 = a 1 3 / ( a 1 3 ~ a l l a 3 3 ) . 93 ~ ( a l l - a l 2 ) a 1 3 / ( a 1 3 ~ all°33)» ki = ( O + c ) & + hi a n d w i t h / i , g i v e n i n a p p e n d i x F. S u b s t i t u t i o n o f (10.9) i n t o (10.5) y i e l d s 2 PiMrPi) where <rgi{z)= f { G w ^ + G ^ + Gaar0}dy. J-h °y r0 Eq.(lO.lO) c a n al s o be w r i t t e n as <r»(r,z)=[ { K ^ ^ + K ^ + K^a^dy (10.11) J-h °y H) where 2 \-*PiMrPi). etc.. 10.2 INTEGRAL EQUATION OF THE SECOND KIND H a v i n g eq.( 10.10) a n d n o s i n g u l a r i t y at t h e c e n t e r o f t h e p l a t e , we are a b l e t o e x p r e s s orz b y i n t e g r a t i n g eq.(7.5) so t h a t r J0 dz CANONICAL PROBLEMS / 10.2 97 S u b s t i t u t i n g (10.10) i n t o (10.12) a n d u s i n g t h e i d e n t i t y [ 3 0 ] f rJQ{TPi)dT = fj^rp.) (10.13) Jo Pi we get o r °rz{r,z) = / {Krsw?P- + KrzH — + K^a^dy (10.15) J-h °y ro where _ 2 ^  ^i(rpt) 5 etc.. N o w we are r e a d y t o d e v e l o p a n i n t e g r a l e q u a t i o n f o r s o l v i n g e i t h e r OTQ o r u 0 . F i r s t , eq.(7.5) gives N e x t , t h e f i r s t t w o e q u a t i o n s of (7.1) c o m b i n e d w i t h (10.16) y i e l d du d . . d(Trz = anar + a12—(r(Tr) + a12r-^-+ ai3(Tz, u d . . d<jr, - = a12ar + an—(rar) + anr—— + aizcrz. r or oz T h e s u m of t h e a b o v e t w o e q u a t i o n s c a n be i n t e g r a t e d t o g i v e ru = (an + ai2)r2(Tr + (an+a12) J T2^^-dr + 2aiS J razdr w h i c h , b y (10.13) a n d / r2Jl{rPi)dT=r-J2{rpi), (10.17) JO Pi becomes -2 ru = ( « n + a 1 2 )rV r - (a„ + « „ ) ( £ ) £ f j ^ ' M P, ; P.->i(roP.) + 4 ^ V ^ , ( 2 ) . ro < J i ( r o P . ) F i n a l l y , e v a l u a t i n g (10.18) at r = r 0 a n d u s i n g MroPi) 2 (10.18) J i ( r o P i ) roP,' (10.19) CANONICAL PROBLEMS / 10.3 98 g i v e ^ (TW (Z) r 0 u 0 = (a n + al2)rZar0 - 4 ( a n + a12) £ ( 1 0 . 2 0 ) o r w h e r e r 0 u 0 =(a n + a12)rjfcr0 ( 1 0 - 2 1 ) J-h °y *o = ] [ > 4 ( f l " t ° 1 2 ) y,«) + 4a 1 8 g,fo,y, z ) } etc.. Eq.( 1 0 . 2 1 ) i s t h e r e q u i r e d i n t e g r a l e q u a t i o n o f t h e s e c o n d k i n d f o r finding u 0 i n c a n o n i c a l p r o b l e m s 1 - 3 , w h e r e a r 0 a n d WQ are k n o w n , a n d f o r finding <r r 0 i n c a n o n i c a l p r o b l e m s 4 - 6 , w h e r e u 0 a n d wQ are k n o w n . E q . ( 1 0 . 1 5 ) t h e n d e t e r m i n e s crrz0. 10.3 M E T H O D O F E I G E N F U N C T I O N E X P A N S I O N S T h e c a n o n i c a l p r o b l e m s c a n also be s o l v e d b y t h e m e t h o d o f e i g e n f u n c t i o n e x p a n s i o n s . F o r o u r p u r p o s e , we need o n l y c o n s i d e r a n e x p a n s i o n f o r t h e d e c a y i n g p a r t of (j> of t h e f o r m w h e r e f\{z) i s t h e e i g e n f u n c t i o n d e f i n e d b y e i t h e r fx i n ( 7 . 3 6 ) o r i n ( 7 . 3 9 ) . fx(z) s a t i s f i e s t he b o u n d a r y c o n d i t i o n s at z = ±h hx(z) = cX2f'x + dh?f'x" = 0 , ( 1 0 . 2 3 ) 9x(z) = A 2 / A + ah2f'l = 0 ( 1 0 . 2 4 ) a n d t h e d i f f e r e n t i a l e q u a t i o n dhAfi" + (a + c)X2h2f'x' + A 4 / A = 0 . ( 1 0 . 2 5 ) ( 1 0 . 2 5 ) c a n now be w r i t t e n as h2h'x(z) + X2gx(z) = 0 . CANONICAL PROBLEMS / 10.3 99 T h u s we h a v e h'x{z) = 0 (10.26) at z = ±h. O b v i o u s l y hx(z) s a t i s f i e s dh*ti'x" + ( a + c)X2h2h"x + X4hx = 0. (10.27) A g e n e r a l o r t h o g o n a l i t y c o n d i t i o n m a y be d e r i v e d b y c o n s i d e r i n g / {dh*h"x" + ( a + c)X2k2h'{ + X*hx)hwdz = 0 J-h w h i c h becomes, u p o n i n t e g r a t i o n b y p a r t s a n d a p p l i c a t i o n o f b o u n d a r y c o n d i t i o n s , -h B y s u b t r a c t i n g t h e f o l l o w i n g e q u a t i o n f r o m eq.(10.28) r h u2 [dh*kX-(a + c)X2h2hxhl + X*hxhlj:]dz = 0. (10.28) J-h fh X2 / [dhAhK - (a + c)u2h2k'xhl + uAhxh„)dz = 0, (10.29) J-h we o b t a i n t h e g e n e r a l o r t h o g o n a l i t y c o n d i t i o n / [dhthlh-l - w2X2hxhu)]dz = 0 (10.30) J-h •h f o r u / A . A s i n t h e p l a n e s t r a i n case, a c o n s i d e r a t i o n of t h e s u m •h J2 ^ j {dhAh'X - u2X2hxhJdz = 0 l e a d s t o Ax = - §- j\{Y,A^2fl){dh2hl + cX2hx) { ! fh 'A J-h + dh2C£Aija(ah2h'x, + X2hx)}dz (10.31) where C x = f {dh*(h"x)2 - X*h2x}dz. (10 .32) J-h B y u s i n g t h e s u b s c r i p t D t o d e n o t e t h e s o l u t i o n f o r stresses a n d d i s p l a c e m e n t s c o r r e s p o n d -i n g t o <f>D a n d t h e s u b s c r i p t 0 f o r t h e e v a l u a t i o n o f a f u n c t i o n at r = r 0, we h a v e T , ( ^ « 2 0 = h'{BA{an - a12)arD0 + - B2^f}/D, ° a (10.33) = - { B 8 ( a „ - *I*KDO + B Z ^ - B ^ } / D CANONICAL PROBLEMS / 10.4 100 w h e r e Bx =-an(b + 1) + cal3, B2 = — 2aan + a 1 3 d , #3 = ~ais(b + 1) + ca33> BA = - 2 a a 1 3 + da^, D = Bj J 5 4 — B2Bs. Eqs.(10.31), (10.33) a n d t h e e x p r e s s i o n f o r <r rp 0 l e a d t o DcrrD0 =errD0(an - an)(aBz - B4) + — ( a J ? 3 - B4) + ^ ( ^ 2 - ^ 1 ) (10.34) ro —A <*y ro where Ka = ( ° n - a ^ ) ^ -Eq.(10.34) gives t h e r e q u i r e d i n t e g r a l e q u a t i o n of t h e s e c o n d k i n d f o r finding e i t h e r <7rpo o r t/po i n a l l c a n o n i c a l p r o b l e m s . W i t h <?TD0iuD0 an<^ WD0> -^ A 1 S o b t a i n e d f r o m (10.31) a n d (10.33). T h e n crrzr,0 a n d a l l o t h e r p h y s i c a l q u a n t i t i e s m a y be e v a l u a t e d . N o t e t h a t two u n k n o w n c o n s t a n t s c h a r a c t e r i z i n g o u t e r s o l u t i o n s m a y b e i n c l u d e d i n t h e d a t a f o r d e c a y i n g s t a t e s , a n d t w o necessary c o n d i t i o n s s h o u l d be c o u p l e d w i t h (10.34) t o s o l v e t h e p r o b l e m . N u m e r i c a l r e s u l t s have s h o w n t h e same k i n d of c o n v e r g e n c e d i f f i c u l t i e s as i n t h e p l a n e s t r a i n case. 10.4 NUMERICAL ANALYSIS F O R T R A N p r o g r a m s f o r t h e n u m e r i c a l s o l u t i o n s of t h e s i x c a n o n i c a l p r o b -l e m s by t h e finite H a n k e l t r a n s f o r m m e t h o d have been d e v e l o p e d a n d i m p l e m e n t e d . Sym-CANONICAL PROBLEMS / 10.4 101 m e t r y r e d u c e s eq. (10.21) t o t h e f o l l o w i n g f o r m r0u0(z) = («„• + a 1 2 ) r 0 V r 0 ( z ) + /'\[K 1 B{y,z) ± Kw(-y)Z)ft^ h d y (10.35) + \Ktt(y,z) ± K u ( - y , z ) } ^ + \K0(y)Z) ± K.(-y,*)\*M))dy w h e r e t h e " + " s i g n is f o r t h e s y m m e t r i c case a n d t h e " — * s i g n i s f o r t h e a n t i - s y m m e t r i c case. T h e p r o b l e m s o l v e d b y (10.35) i s i n f a c t t h e d e f i n e d b o u n d a r y v a l u e p r o b l e m w i t h t h e c o r r e s p o n d i n g s i n g u l a r o u t e r s o l u t i o n s u b t r a c t e d . N u m e r i c a l m e t h o d s f o r s o l v i n g eq.(10.35) are s i m i l a r t o t h o s e f o r t h e p l a n e s t r a i n case; t h e d e t a i l s a r e g i v e n i n Sec.5.5. T h e first 20 p,- d e f i n e d i n (10.4) are t a k e n f r o m [37] a n d o t h e r s come f r o m t h e a s y m p t o t i c e x p a n s i o n 1 31 3779 , roPi ~ T i + W r 3 M f + TtitiT? ( 1 ° - 3 6 ) w h e r e Tt = (•• - \)*. S i m i l a r t o t h e p l a n e s t r a i n case, o u r p r o g r a m c a n also t r e a t t h e i s o t r o p i c case b y t a k i n g E ^ E , v' = v, R a p i d , s t a b l e c o n v e r g e n c e was a c h i e v e d i n p r a c t i c e . F o r d i f f e r e n t sets of p a r a m e t e r v a l u e s l i s t e d at t h e e n d of c h a p t e r 9, n u m e r i c a l r e s u l t s are d i s p l a y e d g r a p h i c a l l y b e l o w . Figs.10.1-3 e x h i b i t u 0 , t h e r a d i a l d i s p l a c e m e n t at r = r 0, i n c a n o n i c a l p r o b l e m s 1-3. A l l t h e c u r v e s t h e r e l o o k l i k e s t r a i g h t l i n e s . T h i s i n d i c a t e s t h a t t h e St.Venant's s o l u t i o n s a r e v e r y g o o d a p p r o x i m a t i o n s i n these cases. A s a m a t t e r of f a c t , t h e c u r v e s i n Figs.10.1-2 are c l o s e r t o s t r a i g h t l i n e s t h a n t h o s e i n Fig.10.3. T h e o r t h o t r o p i c 1 c u r v e h a s t h e l a r g e s t d i s p l a c e m e n t i n these cases. T h e c u r v e s i n Figs.10.4-6 r e p r e s e n t ar0i n o r m a l stresses at r = r 0, i n t h e c a n o n i c a l p r o b l e m s 4-6. T h e y are a l s o c l o s e t o s t r a i g h t l i n e s e x c e p t i n t h e r e g i o n n e a r the c o r n e r w h e r e a s t r e s s s i n g u l a r i t y e x i s t s . T h e i s o t r o p i c 2 c u r v e has t h e l a r g e s t n o r m a l stress. T h e c u r v e s i n Figs.10.7-12 represent crzQ) t h e s h e a r s t r e s s at r = r 0 i n t h e c a n o n i c a l p r o b l e m s 1-6. T h e i s o t r o p i c 2 c u r v e has t h e l a r g e s t s h e a r stress. CANONICAL PROBLEMS / 10.4 r FIGURE 10.1 uQ in canonical BVP 1. CANONICAL PROBLEMS / 10.4 103 r FIGURE 10.2 u 0 in canonical BVP 2. CANONICAL PROBLEMS / 10.4 i O * r= o.o -1.0--2.0-"0 -3.0 -4.0 Legend orthotropic 1  isotropic 1 isotropic 2 orthotropic 2 orthotropic 3 or tho tr opic _4 -5.0--6.0-0.2 —1 1— 0.4 0.6 r 0.8 FIGURE 10.3 u 0 in canonical BVP 3. CANONICAL PROBLEMS / 10.4 0 0.2 0.4 0.6 0.8 1 r FIGURE 10.4 cvo in canonical BVP 4. CANONICAL PROBLEMS / 10.4 106 0.025 0.020 0.015 A o.ow A 0.005 0.000 Legend orthotropic 1  isotropic 1 isotropic 2 orthotropic 2 orthotropic 3 orthotropic _4 FIGURE 10.5 a r 0 in canonical BVP 5. CANONICAL PROBLEMS / 10.4 107 0.035 0.030 0.025-Legend orthotropic 1  isotropic 1 isotropic 2 orthotropic 2 orthotropic 3 orthotropic _4 0.020 0.015- ==* 0.010 —r— 0.2 — i — : r— 0.4 0.6 r 0.8 FIGURE 10.6 fffO in canonical BVP 6. CANONICAL PROBLEMS / 10.4 108 0.6 0.4 A 0.2 A 0.0 -0.2 A -0.4 A -0.6 -0.8 Legend orthotropic 1  isotropic 1  isotropic 2 orthotropic 2 orthotropic 3 orthotropic _4 — i — 0.2 —l r— 0.4 0.6 r 0.8 FIGURE 10.7 arz0 in canonical BVP I. CANONICAL PROBLEMS / 10.4 r FIGURE 10.8 arz0 in canonical BVP 2. CANONICAL PROBLEMS / 10.4 110 0.000 -0.002 -0.004 TrzO -0.006 -0.008--0.010 Legend orthotropic 1 isotropic 1 _ isotropic 2 orthotropic 2 orthotropic 3 c^tJhotropic j4 \ \ \ \ \ \ \ \ \ •i1 V \ I \ ! 0.2 0.4 0.6 0.8 FIGURE 10.9 crz(i in canonical BVP 3. CANONICAL PROBLEMS / 10.4 0.8-\ 0.6 0.4 A 0.2 -0.2 -0.4-\ 0 Legend orlhoiropic 1 isotropic 1 isotropic 2  orthotropic 2 orthotropic 3 ort^> trop ic _4 / A I 0.2 0.4 0.6 r 0.8 FIGURE 10.10 arz0 in canonical BVP 4. CANONICAL PROBLEMS / 10.4 ' o ? s.o-i 6.0 A 4.0A 2.0 A 0.0 -2.0 A -4.0 A Legend orthotropic 1  isotropic 1  isotropic 2 orthotropic 2 orthotropic 3 orthotropic _4 / \ 0.2 0.4 0.6 0.8 FIGURE 10.11 c r „ 0 in canonical BVP 5. CANONICAL PROBLEMS / 10.4 113 0.012 o.owA 0.008 A <7rzO 0.006 A 0.004 A 0.002 A 0.000 Legend orthotropic 1 isotropic 1 _ isotropic 2 orthotropic 2 orthotropic 3 c^tJhot£opic 4^ \ FIGURE 10.12 c r „ 0 in canonical BVP 6. APPLICATIONS / 11.1 114 CHAPTER 11 APPLICATIONS T h e n e c essary c o n d i t i o n s f o r d e c a y i n g s t a t e s d e v e l o p e d i n c h a p t e r 8 c a n be u s e d t o d e t e r m i n e t h e o u t e r s o l u t i o n f o r a p l a t e p r o b l e m w i t h o u t ref e r e n c e t o t h e c o r r e s p o n d i n g i n n e r s o l u t i o n i n a way s i m i l a r t o t h a t s h o w n i n c h a p t e r 6. I n t h i s c h a p t e r we s o l v e t h e p r o b l e m of a n o r t h o t r o p i c c i r c u l a r p l a t e w i t h a c l a m p e d edge a n d a v e r t i c a l p o i n t l o a d a p p l i e d at t h e c e n t e r of i t s u p p e r surface. We c o m p a r e o u r r e s u l t s w i t h t h e c l a s s i c a l r e s u l t s i n [22] t o i l l u s t r a t e t h e u t i l i t y a n d e f f i c i e n c y of o u r theory. 11.1 SINGULARITY CAUSED BY CONCENTRATED FORCE AT CENTER i n Fig.7.1. T h e p l a t e is c l a m p e d at t h e edge r = r 0 a n d i s s u b j e c t t o a v e r t i c a l p o i n t l o a d P d o w n w a r d at i t s c e n t e r of t h e u p p e r surface. A s i t has b e e n p o i n t e d o u t i n c h a p t e r 7, the o u t e r s o l u t i o n of t h i s p r o b l e m is c h a r a c t e r i z e d b y s i x c o n s t a n t s , i n w h i c h A, C a n d D are a s s o c i a t e d w i t h p a r t i c u l a r s o l u t i o n s w h i c h are s i n g u l a r at t h e center. Eq.(8.4) gi v e s We c o n s i d e r t h e p r o b l e m of a c y l i n d r i c a l l y o r t h o t r o p i c c i r c u l a r p l a t e s h o w n A,B,C, D, E a n d w0 115 o r ^ = 3ro_ 4hs d — etc j_h i y'1  / *rzod* (11-1) w h e r e <r r 2 0 is aTZ at r = r 0. O v e r a l l e q u i l i b r i u m r e q u i r e s •h P = 27rr0 / <xrz0dz J-h so t h a t 3 P C = ~7T"o' ~ ; ~ r . (11.2) 4 A 3 2?r(d - ac) v ' If t h e r e were a h o l e at t h e c e n t e r of t h e p l a t e , t h e n t h e a p p l i c a t i o n of n e c e s s a r y c o n d i t i o n s f o r a d e c a y i n g s t a t e a l o n g b o t h t h e o u t e r a n d i n n e r edge w o u l d y i e l d e n o u g h e q u a t i o n s t o d e t e r m i n e t h e o u t e r s o l u t i o n . F o r a d i s k w i t h o u t a h o l e at t h e c e n t e r , t h e c o n s t a n t s A a n d D w h i c h give r i s e t o t h e s t r e s s s i n g u l a r i t y at t h e c e n t e r c a n b e d e t e r m i n e d b y a n a p p l i c a t i o n of t h e r e c i p r o c a l t h e o r e m as i n [13] f o r t h e i s o t r o p i c case. G e n e r a l l y t h e s t r e s s s i n g u l a r i t y a s s o c i a t e d w i t h A a n d D m a y b e c a u s e d by c o n c e n t r a t e d heat sources o r o t h e r i n i t i a l s t r a i n s a n d c a n be d e t e r m i n e d b y t h e p r e s c r i b e d l o a d i n g c o n d i t i o n at t h e c e n t e r . I n t h e present case, A a n d D are d e t e r m i n e d b y c o n c e n t r a t e d f o r c e P as n o i n i t i a l s t r a i n s are i n v o l v e d . T o see t h i s , a ssume t h a t a d o w n w a r d f o r c e P i s a p p l i e d at t h e p o i n t r = 0 o n t h e t o p s u r f a c e . T h e n , t h e b e n d i n g ( a n t i - s y m m e t r i c ) p a r t of t h e o u t e r s o l u t i o n r e p r e s e n t e d b y A, B a n d C is c a u s e d by a n a n t i - s y m m e t r i c l o a d , i.e. t h e d o w n w a r d forces of m a g n i t u d e P/2 a p p l i e d at r = 0 a n d z = ±h r e s p e c t i v e l y . We a p p l y t h e r e c i p r o c i a l t h e o r e m t o t w o d e f o r m a t i o n s t a t e s of t h e c i r c u l a r p l a t e b o u n d e d b y r = r 0/2. T h e s t a t e 1 i s t a k e n t o be t h e w h o l e a n t i - s y m m e t r i c d e f o r m a t i o n d e f i n e d a bove a n d t h e s t a t e 2 is t o be t h e o u t e r s o l u t i o n a s s o c i a t e d w i t h B, w h i c h does not have a s i n g u l a r i t y at t h e c e n t e r . T h u s , we h a v e h P P arB(AuA + BuB + CuG)2irrdz = — —wB(0, h) — —wB(0, —h) < n - 3 > + / {(Aar\ + BarB + CarC)uB + CarzCwB}2nrdz J-h w h e r e q u a n t i t i e s i n t h e i n t e g r a l s are e v a l u a t e d at r = TQ/2 a n d a n e x p o n e n t i a l l y s m a l l e r r o r as e —* 0 has b e e n n e g l e c t e d . S u b s t i t u t i o n of (11.2) i n t o (11.3) gi v e s 4 0 * a n a h APPLICATIONS / 11.2 116 w h i c h i s i n d e p e n d e n t of t h e c h o i c e o f r = TQ/2. S i m i l a r l y , t h e e x t e n s i o n a l ( o r s y m m e t r i c ) p a r t o f t h e o u t e r s o l u t i o n r e p r e -s e n t e d b y D a n d E i s c a u s e d b y a s y m m e t r i c l o a d , i.e. a d o w n w a r d f o r c e o f m a g n i t u d e P/2 a p p l i e d at r = 0 , z = k a n d a n u p w a r d f o r c e of m a g n i t u d e P/2 at r = 0, z = — h. We ap p l y t h e r e c i p r o c a l t h e o r e m t o t w o d e f o r m a t i o n s t a t e s o f t h e c i r c u l a r p l a t e b o u n d e d b y r = r 0 / 2 , i.e. t h e s t a t e o f t h e w h o l e s y m m e t r i c d e f o r m a t i o n d e f i n e d a bove a n d t h e p a r t of it, t h e o u t e r s o l u t i o n E, w h i c h does n o t have a c e n t e r s i n g u l a r i t y . T h u s , we h a v e ' h P P arE(DuD + EuE)2nrdz = ~—wE(Q,h) + —wE(0,-h) + / (DarD + E(TrE)uE2nrdz J-h w h i c h i m p l i e s 4JT a n i n d e p e n d e n t o f t h e c h o i c e of r = r 0 / 2 . F o r a n i s o t r o p i c m a t e r i a l , e q s . ( 1 1 . 4 ) a n d ( 1 1 . 6 ) r e d u c e t o A — ^ + s>T' (I1-7» Pu D = - - . (.1.8) ( 1 1 . 8 ) is e x a c t l y t h e same as ( 6 . 3 ) of [13] whereas ( 1 1 . 7 ) w o u l d agree w i t h t h o s e i n [13] i f ( 5 . 2 , 3 ) of [13] are c o r r e c t e d t o r e a d as f o l l o w s : A = P_ 5 = 2 ( 8 - 3 ^ ) P 64TTI>' 5 ( 1 - u) 64nD' N o t e t h a t a t y p e of s t r e s s t e r m i s c o n t r i b u t e d b y b o t h A a n d B t o ar i n ( 5 . 7 ) of [13] . 11.2 OUTER SOLUTION FOR CLAMPED CIRCULAR PLATE Now, a n a p p l i c a t i o n of n e c e s s a r y c o n d i t i o n s f o r a d e c a y i n g s t a t e , i.e. ( 8 . 2 9 ) , t o t h e p a r t i c u l a r d e c a y i n g s t a t e c o r r e s p o n d i n g t o t h e d i f f e r e n c e b e t w e e n t h e e x a c t s o l u t i o n a n d t h e o u t e r s o l u t i o n , u — AuA — Cuc — Duj) — BuB — EuE APPLICATIONS I 11.2 117 a n d xv — AwA — CWQ — Dxvj) — BxoB — EwE — w0 e v a l u a t e d at t h e c l a m p e d edge, i.e. u = 0, w = 0 at r = r 0 , gives >h {ori[AuA + CuG + DuD + BuB + EuE] h (11.9) + arzi\AwA + CwG + DwD + BwB + EwE + w0]}dz = 0 f o r i=4,5,6. A d e t a i l e d c a l c u l a t i o n r e d u c e s (11.9) t o t h e f o l l o w i n g t h r e e e q u a t i o n s f o r t h e r e m a i n i n g unknowns,E , B a n d wQ: {{a12 + o n ) r 0 J V 6 0 + 2al3M61}E = -N60D{al2 - o n ) / r 0 , { ( a 1 2 + an)rQNbl + axzMb2)B = -NblA(al2 - a u ) / r 0 -C{cN3Nb3 + c N l N 5 l + cM2Mb2} (11.10) A^40^o + {(<*i2 + an)rQNu + al3Mi2}B = -N^A{al2 - a n ) / r 0 w h e r e A*40 = / Vrzidz, J-h .h ^41 = / Pr4zdz> ^43 = / aRIZ3dz, J-h J-h / arbzdz, Nb3 - / arbz3dz, J-h J-h J d^dz, M 6 1 = J ^rzezdz, (11-11) fh ~ 2 ~ fh - 2 A^ 42 = / oTziz dz, Mb2 = / arzbz dz, J-h J-h CJVI = {a12 + an)[(b + l)d - 2ac]r0lnr0 + an(b - l ) d r 0 , c M 2 = a 1 3 { [ ( 6 + l ) d - 2oc] In r 0 + [b - l ) d / 2 } , cm = ( ai2 - «u)(l - b)c/{3r0). F o r a c l a m p e d c i r c u l a r p l a t e w i t h r 0 = 10 a n d h = 1, we h a v e M40 = a n d t h e f o l l o w i n g APPLICATIONS / 11.2 n u m e r i c a l r e s u l t s f o r t h e m a t e r i a l s l i s t e d i n c h a p t e r 9: 118 m a t e r i a l o r t h o 1 iso 1 iso 2 o r t h o 2 o r t h o 3 o r t h o 4 ^41 .1683+1 .1693+1 .1774+1 .1700+1 .1692+1 .1702+1 #43 .1010+1 .1019+1 .1070+1 .1025+1 .1019+1 .1027+1 M 4 2 .6750-1 .1234 .2601 .4338-1 .9357-1 .8250-1 .9368-2 .9544-2 .1340-1 .9370-2 .9365-2 .9369-2 #53 .5623-2 .5741-2 .8069-2 .5631-2 .5627-2 .5632-2 .2452-3 .5680-3 .1850-2 .9613-4 .3876-3 .3206-3 #co .2812-1 .2872-1 .4074-1 .2814-1 .2816-1 .2819-1 .1105-2 .2151-2 .7037-2 .3291-3 .1677-2 .1280-2 In t h e above t a b l e , .1683+ 1 denotes .1683 x 1 0 + 1 , etc.. S i m i l a r l y , t h e f o l l o w i n g c o n s t a n t s are n e e d e d f o r finding o u t e r s o l u t i o n s of c i r c u l a r p l a t e s w i t h t h e k n o w n b o u n d a r y c o n d i t i o n s <7R, to at r = r0: M o = f °rzidz, J-h Un = I u1zdz, J 7 1 3 = / uxzzdz, J-h J-h ^12 = / cr~~\Z2dzy U2i = I u2zdz, (11.12) J-h J-h U~23 = / u2z3dz, M 2 2 = / ar.2z2dz, J-h J-h ^30 = / tt3<kf ^31 = / Vrs^zdz. J-h J-h F o r t h e m a t e r i a l s l i s t e d i n c h a p t e r 9, we have A f 1 0 = a n d t h e f o l l o w i n g n u m e r i c a l APPLICATIONS / 11.2 119 r e s u l t s : m a t e r i a l o r t h o 1 is o 1 iso 2 o r t h o 2 o r t h o 3 o r t h o 4 Un -.6834-3 -.3249-4 -.1671-4 -.3449-4 -.6857-4 -.3443-4 Uiz -.4099-3 -.1946-4 -.1002-4 -.2069-4 -.4112-4 -.2065-4 Ml2 -.4723-1 -.1653 -.1758 .1539-1 -.1089 -.8033-1 U2l -.3809-5 -.1836-6 -.1270-6 -.1903-6 -.3803-6 -.1900-6 -.2284-5 -.1100-6 -.7611-7 -.1140-6 -.2280-6 -.1138-6 M22 -.5596-3 -.1235-2 -.1743-2 -.2054-3 -.8967-3 -.7332-3 -.1142-4 -.5485-6 -.3774-6 -.5704-6 -.1138-5 -.5675-6 M31 -.2582-2 -.4641-2 -.6543-2 -.7169-3 -.3930-2 -.2969-2 B y u s i n g t h e a b o v e r e s u l t s , we have s o l v e d t h e p r o b l e m of a c i r c u l a r p l a t e m a d e of o r t h o t r o p i c 1 m a t e r i a l , w i t h i t s edge c l a m p e d a n d a n u n i t v e r t i c a l l y c o n c e n t r a t e d f o r c e a p p l i e d d o w n w a r d at t h e c e n t e r of i t s u p p e r surface. I n Fig.11.1 t h e s o l i d c u r v e r e p r e s e n t i n g u>(r,0) of o u r r e s u l t s i s c o m p a r e d w i t h t h e d a s h e d one o f t h e c l a s s i c a l r e s u l t s i n [22]. T h e agreement is r e m a r k a b l e . O u r s o l u t i o n , as a n o u t e r s o l u t i o n , i s n o t a c c u r a t e i n t h e n e i g h b o u r h o o d s of t h e edge a n d t h e c e n t e r of t h e p l a t e . N e v e r t h e l e s s , t h e s i n g u l a r i t y at t h e c e n t e r a n d t h e c l a m p e d b o u n d a r y c o n d i t i o n at t h e edge are i n d i c a t e d b y o u r s o l u t i o n . APPLICATIONS / 11.2 120 FIGURE 11.1 Comparison with the classical result. CONCLUSIONS / 12.0 121 CHAPTER 12 CONCLUSIONS I n t h i s t h e s i s , we have p r e s e n t e d a n a s y m p t o t i c a n a l y s i s of t h e t h r e e d i m e n -s i o n a l t h e o r y of e l a s t i c o r t h o t r o p i c p l a t e s i n p l a n e s t r a i n a n d a x i - s y m m e t r i c d e f o r m a t i o n s , whose c o u n t e r p a r t f o r i s o t r o p i c p l a t e s was i n i t i a t e d i n [6-8,29,11-14]. We s u m m a r i z e t h e c o n t r i b u t i o n s m a d e i n t h e t h e s i s below. F o l l o w i n g t h e g e n e r a l m e t h o d f o r s o l v i n g s i n g u l a r l y p e r t u r b e d e q u a t i o n s , we o b t a i n e d u n i f o r m a s y m p t o t i c e x p a n s i o n s of t h e e x a c t s o l u t i o n s f o r t h e g e n e r a l i z e d b i -h a r m o n i c e q u a t i o n i n b o t h C a r t e s i a n a n d c y l i n d r i c a l c o o r d i n a t e s , w h i c h g o v e r n p l a n e s t r a i n a n d a x i - s y m m e t r i c d e f o r m a t i o n s of o r t h o t r o p i c p l a t e s . F o r p l a n e s t r a i n d e f o r m a t i o n , we fo u n d t h a t t h e o u t e r s o l u t i o n c o n s i s t s of t h e e i g e n f u n c t i o n s c o r r e s p o n d i n g t o z e r o e i g e n v a l u e a n d t h e i n n e r s o l u t i o n c o n s i s t s of t h e e i g e n f u n c t i o n s w i t h d e c a y i n g p r o p e r t y . I n f a c t , t h e u n i f o r m a s y m p t o t i c e x p a n s i o n of t h e s o l u t i o n i s i d e n t i c a l i n f o r m t o i t s e i g e n f u n c t i o n e x p a n s i o n i n t h i s case. T h e o u t e r s o l u t i o n f o r a x i - s y m m e t r i c d e f o r m a t i o n s is here g i v e n f o r t h e first t i m e a n d is c o n s i s t e n t w i t h t h o s e f o r t h e i s o t r o p i c case. A l t h o u g h t h e i n n e r s o l u t i o n f o r a x i - s y m m e t r i c d e f o r m a t i o n s c o n s i s t s of t h e e i g e n f u n c t i o n s w i t h d e c a y i n g p r o p e r t y , t h e e i g e n f u n c t i o n s c o r r e s p o n d i n g t o z e r o e i g e n v a l u e do n o t g i v e t h e c o m p l e t e o u t e r s o l u t i o n i n t h i s case. A n e f f i c i e n t a l g o r i t h m f o r finding e igenvalues has b e e n d e v e l o p e d a n d n u m e r i c a l r e s u l t s are o b t a i n e d . CONCLUSIONS / 12.0 122 N e c e s s a r y c o n d i t i o n s f o r edge d a t a t o i n d u c e o n l y a d e c a y i n g s t a t e i n a n o r t h o t r o p i c p l a t e have been e s t a b l i s h e d . T h e s e c o n d i t i o n s p l a y a n i m p o r t a n t r o l e i n de-t e r m i n i n g o u t e r s o l u t i o n s w i t h o u t reference t o i n n e r s o l u t i o n s . T h e y a l s o d e l i n e a t e t h e a p p l i c a b i l i t y o f t h e w e l l k n o w n St.Venant's p r i n c i p l e t o p l a t e p r o b l e m s . F o r th o s e c o n d i -t i o n s i n t h e i s o t r o p i c case w h i c h a p p e a r i n e l e m e n t a r y a n d e x p l i c i t f o r m , we f o u n d t h e i r c o u n t e r p a r t s i n t h e o r t h o t r o p i c case. I n t h e i s o t r o p i c case, t h e y were d e r i v e d b y a n ap-p l i c a t i o n of t h e r e c i p r o c a l t h e o r e m a n d , t h e r e f o r e a r e c o r r e c t i n a n a s y m p t o t i c sense as e —» 0. F o r t h e o r t h o t r o p i c case, t h e y a r e t h e e x a c t consequences of t h e b o u n d a r y v a l u e p r o b l e m s . T h e a s y m p t o t i c i t y h a s been removed. We gave a c o m p l e t e d i s c u s s i o n of c o r n e r s t r e s s s i n g u l a r i t i e s i n b o t h p l a n e s t r a i n a n d a x i - s y m m e t r i c d e f o r m a t i o n s . B y u s i n g t w o d i f f e r e n t l o c a l p o l a r c o o r d i n a t e s a n d c o n s i d e r i n g t h e l e a d i n g t e r m of t h e a s y m p t o t i c e x p a n s i o n o f t h e e x a c t s o l u t i o n i n th e n e i g h b o u r h o o d o f t h e c o r n e r , we were a b l e t o d e v e l o p t h r e e t r a n s c e n d e n t a l e q u a t i o n s g o v e r n i n g t h e e x p o n e n t s of t h e c o r n e r stress s i n g u l a r i t y . I n p l a n e s t r a i n d e f o r m a t i o n we pr o v e d t h e e x i s t e n c e of t h e s o l u t i o n of t h e r e l e v a n t e i g e n v a l u e p r o b l e m f o r st r e s s s i n g u l a r -i t i e s i n o r t h o t r o p i c p l a t e s . I n a x i - s y m m e t r i c d e f o r m a t i o n t h e n o n e x i s t e n c e of t h e s o l u t i o n of t h e r e l e v e n t e i g e n v a l u e p r o b l e m f o r stress s i n g u l a r i t i e s f o r m i x e d c o n d i t i o n s o n t h e p l a t e edge a n d t h e e x i s t e n c e of t h e same f o r c l a m p e d c o n d i t i o n s were p r o v e d f o r i s o t r o p i c p l a t e s a n d s h o w n n u m e r i c a l l y f o r o r t h o t r o p i c p l a t e s . T h e u n i q u e n e s s o f t h e s o l u t i o n f o r s t r e s s s i n g u l a r i t i e s f o r i s o t r o p i c p l a t e s i n p l a n e s t r a i n a n d a x i s y m m e t r i c d e f o r m a t i o n s was p r o v e d b y t h e A r g u m e n t P r i n c i p l e . F O R T R A N p r o g r a m s were d e v e l o p e d a n d a c c u r a t e n u m e r i c a l r e s u l t s o b t a i n e d . We f o u n d t h a t t h e stress s i n g u l a r i t y e x p o n e n t r e m a i n s t h e same i f t h e o r t h o g o n a l m a t e r i a l p r i n c i p a l axes are s u b j e c t t o a r o t a t i o n b y 90 degrees. T h i s p a r t of o u r w o r k b e l o n g s t o t h e t o p i c o f s o l v i n g e l l i p t i c b o u n d a r y v a l u e p r o b l e m s i n n o n - s m o o t h d o m a i n s . We h a v e s o l v e d t h e c a n o n i c a l b o u n d a r y v a l u e p r o b l e m s f o r t h e g e n e r a l i z e d b i - h a r m o n i c e q u a t i o n i n C a r t e s i a n c o o r d i n a t e s f o r a s e m i - i n f i n i t e s t r i p i n p l a n e s t r a i n de-f o r m a t i o n a n d i n p o l a r c o o r d i n a t e s f o r a d i s k i n a x i - s y m m e t r i c d e f o r m a t i o n . I n p l a n e CONCLUSIONS I 12.0 123 s t r a i n deformation,, we u s e d a F o u r i e r s i n e (cosine) t r a n s f o r m c o m b i n e d w i t h a s y m p t o t i c e x p a n s i o n s f o r a l a r g e p a r a m e t e r t o d e r i v e a s i n g l e g e n e r a l i z e d C a u c h y - t y p e s i n g u l a r i n -t e g r a l e q u a t i o n of t h e f i r s t k i n d g o v e r n i n g t h e d e f o r m a t i o n s of a s e m i - i n f i n i t e o r t h o t r o p i c s t r i p u n d e r e x t e n s i o n , b e n d i n g a n d flexure, r e s p e c t i v e l y . T h e e q u a t i o n o b t a i n e d i s s l i g h t l y m o r e g e n e r a l t h a n t h o s e d i s c u s s e d i n [4,5] a n d i t s s i n g u l a r i t y was a n a l y s e d i n a s i m i l a r w a y y i e l d i n g r e m a r k a b l y a c c u r a t e n u m e r i c a l r e s u l t s a n d i n agreement w i t h t h o s e o b t a i n e d b y s o l v i n g t h e t r a n s c e n d e n t a l e q u a t i o n m e n t i o n e d above. T h e m e t h o d d e v e l o p e d here r e p r e s e n t s a s u b s t a n t i a l i m p r o v e m e n t ov e r t h o s e i n [2,16] f o r t h e i s o t r o p i c case. I n a x i -s y m m e t r i c d e f o r m a t i o n , b y first c h o o s i n g az as a b a s i c v a r i a b l e a n d t h e n a p p l y i n g t h e finite H a n k e l t r a n s f o r m , we were l e d t o a n i n t e g r a l e q u a t i o n of t h e s e c o n d k i n d f o r t h e b o u n d a r y v a l u e p r o b l e m . T h e F O R T R A N p r o g r a m s d e v e l o p e d here c a n d e a l w i t h t h e o r t h o t r o p i c case as w e l l as t h e i s o t r o p i c case whose s o l u t i o n f o r a x i - s y m m e t r i c d e f o r m a t i o n has n o t been a v a i l a b l e so f a r . N u m e r i c a l r e s u l t s were d i s p l a y e d g r a p h i c a l l y f o r t h e first t i m e . I n a d d i t i o n , we have e x p l o r e d t h e p o s s i b i l i t y f o r a p p l i c a t i o n s of t h e m e t h o d of e i g e n f u n c t i o n e x p a n s i o n s . T h e a c c u r a c y of n u m e r i c a l s o l u t i o n s f o r t h e c a n o n i c a l b o u n d a r y v a l u e p r o b -l e m s a n d the e f f i c i e n c y of t h e p l a t e t h e o r y were c o n f i r m e d t h r o u g h t w o s p e c i f i c a p p l i c a t i o n s : (1) t h e f l e x i b i l i t y c o e f f i c i e n t of a s h e a r e d o r t h o t r o p i c b l o c k , a n d (2) a c l a m p e d c y l i n d r i c a l l y o r t h o t r o p i c c i r c u l a r p l a t e l o a d e d b y a c o n c e n t r a t e d f o r c e at t h e center. F u r t h e r w o r k is needed t o d e t e r m i n e t h e s i g n i f i c a n c e , i f any, of t h e case p < 0 i n p a r t 1 of t h i s t h e s i s . We have i m p o s e d t h e r e s t r i c t i o n o f p > 0 b a s e d o n o u r o b s e r v a t i o n s o n t h e a v a i l a b l e t e c h n i c a l d a t a . I n p a r t I I we i n f a c t d e a l w i t h t h e m a t e r i a l i s o t r o p i c i n t h e 0 a n d r d i r e c t i o n s f o r t h e sake of a v a i l a b i l i t y o f t h e s t r e s s f u n c t i o n . T h e d e v e l o p m e n t of t h e t h e o r y f o r m a t e r i a l s w i t h g e n e r a l o r t h o t r o p y i s e x p e c t e d . T h e s i t u a t i o n s o t h e r t h a n p l a n e s t r a i n a n d a x i - s y m m e t r i c d e f o r m a t i o n s f o r o r t h o t r o p i c o r a n i s o t r o p i c p l a t e s a n d t h e g e n e r a l i z a t i o n of t h e t h e o r y t o s h e l l s t r u c t u r e s o f f e r o p p o r t u n i t i e s f o r f u r t h e r s i g n i f i c a n t work. 124 REFERENCES C. T.H.Baker, " T h e N u m e r i c a l T r e a t m e n t of I n t e g r a l E q u a t i o n s " , C l a r e n d o n P r e s s , O x f o r d , 1977. D. B.Bogy, " S o l u t i o n o f t h e P l a n e E n d P r o b l e m f o r a S e m i - i n f i n i t e S t r i p " , Z.A.M.P., V.26, 1975, 749-769. R . H . B r y a n t , " A n E x a m i n a t i o n of t h e E d g e E f f e c t i n a C a n t i l e v e r B e a m " , J . Engng. Math., V.7, 1973, 351-360. F . E r d o g a n , G . D . G u p t a & T.S.Cook, " N u m e r i c a l S o l u t i o n o f S i n g u l a r I n-t e g r a l E q u a t i o n s " , Chap.7 of Methods of Analysis and Solutions to Crack Problems, ed. G.C.Sih, N o o r d h o f f , 1972, 368-425. F . E r d o g a n , " C o m p l e x F u n c t i o n T e c h n i q u e " , C l i a p . 3 of Treatise on Contin-uum Physics, ed. A . C . E r i n g e n , A c a d e m i c P r e s s , N e w Y o r k , 1972, 523-603. K . O . F r i e d r i c h s & R.F . D r e s s i e r , "A B o u n d a r y L a y e r T h e o r y f o r E l a s t i c B e n d -i n g o f P l a t e " , Comm. Pure. Appl. Math., V.14, 1961, 1-33. A . L . G o P d e n v e i z e r , " D e r i v a t i o n of a n A p p r o x i m a t e T h e o r y of B e n d i n g o f a P l a t e b y t h e M e t h o d of A s y m p t o t i c I n t e g r a t i o n of t h e E q u a t i o n of t h e T h e o r y of E l a s t i c i t y " , P.M.M., V.26(4), 1962, 668-686. A . L . G o l ' d e n v e i z e r & A . V . K o l o s , "On D e r i v a t i o n of T w o D i m e n s i o n a l E q u a -t i o n s i n t h e T h e o r y of T h i n E l a s t i c P l a t e s " , P.M.M., V . 2 9 ( l ) , 1965, 141-155. R.D.Gregory, " T h e T r a c t i o n B o u n d a r y V a l u e P r o b l e m f o r t h e E l a s t o s t a t i c S e m i - i n f i n i t e S t r i p , E x i s t e n c e of S o l u t i o n , a n d C o m p l e t e n e s s of t h e P a p k o v i c h F a d l e E i g e n f u n c t i o n s " , J . of Elasticity, V.10, 1980, 295-327. R . D . G r e g o r y & I . G l a d w e l l , " T h e c a n t i l e v e r B e a m U n d e r T e n s i o n , B e n d i n g o r F l e x u r e at I n f i n i t y " , J . of Elasticity, V.12(4), 1982, 317-343. R . D . G r e g o r y & F.Y.M.VVan, " D e c a y i n g S t a t e s of P l a n e S t r a i n i n a Semi-i n f i n i t e S t r i p a n d B o u n d a r y C o n d i t i o n s f o r P l a t e T h e o r y " , J. of Elasticity, V.14, 1984, 27-64. R . D . G r e g o r y & F.Y.M.Wan, "On P l a t e T h e o r i e s a n d S a i n t - V e n a n t ' s P r i n c i -p l e " , Int. J. Solids Structures, V.21(10), 1985, 1005-1024. R . D . G r e g o r y k F.Y.M.Wan, "E d g e E f f e c t i n t h e S t r e t c h i n g o f P l a t e s " , Lo-cal Effects in the Analysis of Structures, ed. P. L a d e v e z e , E l s e v i e r S c i e n c e P u b l i s h e r s , B . V . A m s t e r d a m , 1985, 35-54. REFERENCES 125 [14] R . D . G r e g o r y & F.Y.M.Wan, "On t h e I n t e r i o r S o l u t i o n f o r L i n e a r l y E l a s t i c P l a t e s " , Tech. Report, No. 86-7, D e p t . o f A p p l . M a t h . , U n i v . o f W a s h i n g -t o n , J u l y 1, 1986. [15] C . W . G r o e t s c h , " C o m e n t s o n Mo r o z o v ' s D i s c r e p a n c y P r i n c i p l e s " , Improperly Posed Problems and their Numerical Treatment, ed. G . H a m m e r l i n & K.-H.Ho f f m a n n , 1983, 97-104. [16] G .D.Gupta, " A n I n t e g r a l E q u a t i o n A p p r o a c h t o t h e S e m i - i n f i n i t e S t r i p P r o b -l e m " , J . Appl. Mech., V . 4 0 ( 4 ) , 1973, 948-954. [17] C.O.Horgan & J. K . K n o w l e s , "Recent D e v e l o p m e n t s C o n c e r n i n g St.Venant's P r i n c i p l e " , A d v a n c e s in Appl. Mech., V.23, 1983, 179-269. [18] V . A . K o n d r a t ' e v & O . A . O l e i n i k , " B o u n d a r y V a l u e P r o b l e m s f o r P a r t i a l D i f -f e r e n t i a l E q u a t i o n s i n N o n - s m o o t h D o m a i n s " , Russian Mathematical Sur-veys, V.38, No.2, 1983, 1-86. [19] S . G . L e k h n i t s k i i , " T h e o r y of E l a s t i c i t y of a n A n i s o t r o p i c B o d y " , H o l d e n - D a y , Inc. S a n F r a n c i s c o , 1963. [20] Y . H . L i n & F.Wan, " T h e c o n s e r v a t i o n L a w s a n d t h e R e s i d u a l S o l u t i o n s f o r O r t h o t r o p i c P l a t e s i n P l a n e S t r a i n a n d A x i - s y m m e t r i c D e f o n n a t i o n s " , T e c h . R e p t . N o . 8 7 - 1 , D e p t . A p p . M a t h . , U n i v . of W a s h i n g t o n , S e a t t l e , J a n . 1987, s u b m i t t e d t o S t u d i e s i n A p p l i e d M a t h e m a t i c s . [21] Y . H . L i n & F.Wan, " S t r e s s S i n g u l a r i t i e s at a R i m of C i r c u l a r C y l i n d e r s " , t o a p p e a r i n t h e p r o c e e d i n g s of C . C . L i n S y m p o s i u m , J u n e 1987. [22] A.E.H.Love, T h e M a t h e m a t i c a l T h e o r y of E l a s t i c i t y , ( 4 t h ed.) D o v e r P u b l i -c a t i o n s , N e w Y o r k , 1944. [23] V . A . M o r o z o v , "On t h e S o l u t i o n of F u n c t i o n a l E q u a t i o n s b y t h e M e t h o d of R e g u l a r i z a t i o n " , Soviet math. Dokl., V.7, 1966, 414-417. [24] N . I . M u s k h e l i s h v i l i , " S i n g u l a r I n t e g r a l E q u a t i o n s " , N o o r d h o f f , G r o n i n g e n , T h e N e t h e r l a n d s , 1953. [25] S . N a i r & E. R e i s s n e r , " I m p r o v e d U p p e r a n d L o w e r B o u n d s f o r D e f l e c t i o n s of O r t h o t r o p i c C a n t i l e v e r Beams", I n t . J. Solids Structures, V . l l , 1975, 961-971. [26] S . N a i r & E. R e i s s n e r , "On t h e D e t e r m i n a t i o n o f S t r e s s a n d D e f l e c t i o n s f o r A n i s o t r o p i c H o m o g e n e o u s C a n t i l e v e r B e a m " , J. of Appl. Mech., V.43, 1976, 75-80. REFERENCES 126 [27] S.Nair, " E f f e c t o f Stress-free E d g e s i n P l a n e S h e a r of a R e c t a n g u l a r Or-t h o t r o p i c R e g i o n " , J. of Appl. Mech., V.45, 1978, 307-312. [28] D . L . P h i l l i p s , "A T e c h n i q u e f o r t h e N u m e r i c a l S o l u t i o n o f C e r t a i n I n t e g r a l E q u a t i o n s of t h e F i r s t K i n d " , J . Assoc. Comput. Mach., V.9, 1962, 84-96. [29] E . L . R e i s s , " S y m m e t r i c B e n d i n g o f T h i c k C i r c u l a r P l a t e s " , J . of S.I.A.M., V.10, 1962, 596-609. [30] I.N.Sneddon, F o u r i e r T r a n s f o r m s , M c g r a w - H i l l B o o k C o m p a n y , Inc., 1951. [31] A . H . S t r o u d & D.Secrest, " G a u s s i a n Q u a d r a t u r e F o r m u l a s " , P r e n t i c e - H a l l , N e w Y o r k , 1906. [32] A . T i k h o n o v , " S o l u t i o n o f I n c o r r e c t l y F o r m u l a t e d P r o b l e m s a n d t h e R e g u l a r -i z a t i o n M e t h o d " , Soviet Math. Dokl., V.4, 1963, 1035-1038. [33] A . T i k h o n o v , " R e g u l a r i z a t i o n o f I n c o r r e c t l y P o s e d P r o b l e m s " , Soviet Math. Dokl., v.4, 1963, 1624-1627. [34] A . N . T i k h o n o v & V . Y . A r s e n i n , " S o l u t i o n s of I l l - P o s e d P r o b l e m s " , W i n s t o n & Sons, 1977. [35] P . J . T o r v i k , " T h e E l a s t i c S t r i p w i t h P r e s c r i b e d E n d D i s p l a c e m e n t s " , J . Appl. Mech., V.38, 1971. 929-936. [36] I.I.Vorovich & V. V . K o p a s e n k o , "Some P r o b l e m s i n t h e T h e o r y of E l a s t i c i t y f o r a S e m i - i n f i n i t e S t r i p " , P.M.M., V . 3 0 ( l ) , 1966, 109-115. [37] G.N.Watson, A t r e a t i s e o n t h e T h e o r y of B e s s e l F u n c t i o n s , C a m b r i d g e , 1922. [38] M . L . W i l l i a m s , " S t r e s s S i n g u l a r i t i e s R e s u l t i n g f r o m V a r i o u s B o u n d a r y C o n -d i t i o n s i n A n g u l a r C o r n e r s of P l a t e s i n E x t e n s i o n " , J. of Appl. Mech., T r a n s . ASME, V.74, 1952, 526-528. DERIVATION OF KERNEL IN CHAPTER 5 / A.O 127 APPENDIX A DERIVATION OF KERNEL IN CHAPTER 5 T h e c o m p l e m e n t a r y s o l u t i o n o f t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n * « « - s 2 ( 2 + p ) * „ + = g{a, z) (A.1) "where c a n be w r i t t e n as 4 = £ A , ( S ) $ , ( z ) (A.2) t=i $ i ( z ) = sinh(sA;z) $2(2) = cosh(sA:0) $ 3 ( z ) = s i n h ( - z ) $4(2) = cosh(-2r). A p a r t i c u l a r s o l u t i o n o f e q . ( A . l ) i s , b y t h e m e t h o d of v a r i a t i o n o f p a r a m e t e r s , i n t h e f o r m *(•,*) = ^B t-(«)*.-(«) (A.S) i=l w h e r e t h e J3,(z) are g o v e r n e d b y e q u a t i o n / $ j $ 2 $3 #4 $'j $'2 $ 3 $ 4 $» $» $» $« V*'/' $ 2 " $ 3 " $'4" ( o \ 0 0 J v 0 ( 3 , 2 ) / (A.4) DERIVATION OF KERNEL IN CHAPTER 5 / A.O 128 w i t h a p r i m e d e n o t i n g d i f f e r e n t i a t i o n w i t h r e s p e c t t o z. L e t t h e d e t e r m i n a n t of t h e c o e f f i -c i e n t m a t r i x of (A.4) be A a n d A , = $ t + 1 $, + 2 $,-+3 <b'. *' vi+2 vi+3 &" <f>" S>" wt + l vt+2 *t+3 w h e r e | • | d e n o t e s d e t e r m i n a n t a n d = $ , + 4 f o r i = l , 2, 3 a n d 4, t h e s o l u t i o n o f (A.4) b ecomes B>2 A 2 -£9{s,z)} 9{s>z), B3 = -^g(s,z), B\ = ^g{s,z). D e t a i l e d c a l c u l a t i o n s g i v e 8 A j = — A;4) cosh(sA;z), s 3 A 2 = ^ ( l - A: 4)sinh(sA : 2 ) , s3 s A 3 = — ( f c 4 - l ) c o s h ( - z ) , A 4 = ^ ( * 4 - l ) s i n h ( | z ) , A = £ ( 1 - * 4 ) 2 , w here ^~ = 0 a n d A = cons t . = A (0) have been used. T h e g e n e r a l s o l u t i o n of ( A . l ) f o r t h e s y m m e t r i c case ( C a n o n i c a l B o u n d a r y V a l u e P r o b l e m 1) is 4 $ ( s , z) = A 2 $ 2 + A 4 $ 4 + ]T t=i w i t h Bi d e f i n e d b y i f i=l, 3 a n d Bi = ^j *i(y)g(s,y)dy i f i = 2 , 4 a n d w i t h A 2 , A 4 g i v e n b y (**{h) * M \ ( A 2 \ _ (-B^h)*^) - B3(h)*3(h) \V2(k) *'4(h)) { A J - [-Bl(h)$[(h)-B3(k)*3(h) (A . 5 ) (A .G) (A.7) (A.8) DERIVATION OF KERNEL IN CHAPTER 5 / A.O 129 (A.8) are d i r e c t c o nsequences of t h e b o u n d a r y c o n d i t i o n s $ ( s , ±h) = ±h) = 0. T h e e x p r e s s i o n s i n (5.13) were o b t a i n e d b y s u b s t i t u t i n g ( A . 6 ) , (A.7) a n d t h e s o l u t i o n o f (A.8) i n t o ( A . 5 ) . ASYMPTOTIC EXPANSIONS IN CHAPTER 5 / B.l 130 APPENDIX B ASYMPTOTIC EXPANSIONS IN CHAPTER 5 B.l SYMMETRIC CASE A s m e n t i o n e d i n Sec.5.2, t h e i m p r o p e r i n t e g r a l s o f Pi(k) a n d Qi(k) ( i = l , 2) can be s p l i t i n t o t w o p a r t s , s u c h as f c o s ( i s ) „,,. , fM cos(£s) „,, v. where AT is a l a r g e p o s i t i v e n u m b e r t o be chosen l a t e r i n n u m e r i c a l c o m p u t a t i o n s . T h e n the a s y m p t o t i c e x p a n s i o n s f o r l a r g e s i n t h e s y m m e t r i c case, A , = ( L l A ! ) e ( * + 1 A ) ^ { i + 0(e-2h°'h)} 4 a n d P^k) = -e(k+llk)h*{e-k'(z~y) - e~k»(z+y) _ e~k*(2h-y-z) + o{e-ehlk}} 8 etc., p r o v i d e t h e f o l l o w i n g a s y m p t o t i c e x p a n s i o n s f o r P{(k) a n d Qi(k) ( i = l , 2) a n d Q i { k ) = C ^ A V Q ^ d 3 + 2 0 ^ ) ^ f l { ( - l ) ^ 5 + 90-91 + 29&}ds + o ( l ) ASYMPTOTIC EXPANSIONS IN CHAPTER 5 / B.l 131 where o ( l ) = 0(e~Nhlk) a n d (B.l) 9 l = e-kt\*s\ g2 = e-k»(*+y) gA = e-*(h-z)/k-k,(h-y) gb = e-lr-«l/* gt = e-(*+»)/* g7 = e-Vh-„-,)/k g8 = e-ki{h-z)-i(h-9)/kt B y a p p l y i n g the o p e r a t o r s Dx = a n d Dz = £7 t o P t ( f c ) a n d Q,(fc), e v a l u a t i n g at x = 0 an d s e t t i n g e-fctf|y-*| /1 y - * e-kN(y+z) y + z e-kN(2h-y-z) ^ = 2h-y-z e-N(h-z)/k-kN(h-y) ^ = h - z + k2(h - y) e-N\y-z\/k h = (B.l) / 6 = / 7 = h = y - z e-N(z+y)/k y + z e-N{2h-y-z)/k 2h-y-z e-kN(h-z)-N(h-y)/k k2(h -z) + h-y * we get t h e f o l l o w i n g a s y m p t o t i c e x p a n s i o n s u s e f u l i n d e r i v i n g t h e s i n g u l a r i n t e g r a l equa-ASYMPTOTIC EXPANSIONS IN CHAPTER 5 / B.2 t i o n s : \DMk)h = - j" ^  + s^zy {/. + A + /.} + 0(.-*V*) [B,Q,(*)1O = - j" ^ - d s - jprny ( A + A - A + 2/s> + o(«-"v*) [*>,4(*)lo = *2 - jj^nj (A + A + A) + of.-"*/*) P.ftWI. = p jf ^ * + jjj^rij {/. + A - A + if/.} + 0(.-*V», w h e r e Qi(A;) = s i n h ( - y ) { c o s h ( s f c / i ) c o s h ( - ( z - h)) - A: 4cosh(sA:z)} Q2{k) = c o s h ( ^ z ) c o s h ( s A ; / i ) s i n h ( ^ ( y - h)) - k4 cosh(sfcz) s i n h ^ y ) K k k a n d Pi(k), Q;(k) a n d A , are t h o s e d e f i n e d i n t h e s y m m e t r i c case. B.2 ANTI-SYMMETRIC CASE T h e c o u n t e r p a r t s o f t h e r e s u l t s i n s e c t i o n B . l f o r t h e a n t i s y m m e t r i c case a r e l i s t e d b e l ow f o r reference U>.A(*)1. - - / " ^ + s j ^ i j W . - /, + A) + [B,<?,(*)Io = - f" + - J L — ) { - h + /, + /,_ 2A) + 0(e-"*/*) P.ftWl. = p f + ^ i - f , {A - A - A + 2*4A} + 0(.-*V*, w h e r e = rosh(£y){smh(«*A)cosh(|(fc - z)) - k4smh(skz)} s s A S (B.G) Q2{k) = s i n h ( s A ; / i ) s i n h ( - z ) s i n h ( - ( y - h)) - A ; 4 s i n h ( s A ; z ) c o s h ( - y ) K K k a n d Pi(k), <?,(&)> a n d A , are t h o s e f o r t h e a n t i - s y m m e t r i c ca3e. 132 (5.3) (5:4) KERNELS OF SINGULAR INTEGRAL EQUATIONS / C.l 133 APPENDIX C KERNELS OF SINGULAR INTEGRAL EQUATIONS C.l SYMMETRIC CASE {DI)g a n d {DJ)„, t h e s i n g u l a r p a r t s o f (DI)0 a n d (DJ)Q i n eq.(5.24), h a v e t h e f o l l o w i n g u n i f i e d e x p r e s s i o n s f o r 0 < y < z a n d z < y < h: where 1 4 8 f'=l 1 + uk2 (k2 + 1)(1 + uk2) °z ~ 2k{k2 - \)E _ k{u + k2)  C* ~ (k2 - 1)E k2 + v C b = C^~lkE (k2 + l)(v + k2) C7 « 8 2k(k2 - 1)E k[l + uk2) (k2 - 1)E (C.l) (C.2) KERNELS OF SINGULAR INTEGRAL EQUATIONS / C.J 134 a n d 1 f u - f u - y _ z 2h — y — z 1 4* A-z+Jfc 2(/i-y) f 1 A d i r e c t s u b s t i t u t i o n of these e x p r e s s i o n s i n t o t h e i n t e g r a l e q u a t i o n g i v e s us c o e f f i c i e n t s f o r t h e s i n g u l a r k e r n e l i n (5.25): A 2 = A x A ^ 2 ^ - l ) ^ { ( 1 + 1 / t 2 ) 2 + * V + ^ > (C.4) _ afc(l + uk2)(u + k2)  4 ~ (A:2 — l)(fc4 — ak3{u + k2)(\ + vk2) Ar 5 (A; 2 - 1 ) ( A 4 - l)E ' We now set fiR = /,• — /,-, w i t h /,• d e f i n e d i n a p p e n d i x B. T h e n t h e r e g u l a r k e r n e l i n (5.25) is g i v e n b y uk2 8 KN(z,y) = ^{u{DI)N + (DJ)N + -£—J2cif<R i=l (C.5) t'=l »'=5 w h e r e (DP)^ a n d (DJ)jf are d e f i n e d b y r n n -1(DI)NU f o r O < y < z ; 1 l(Z>/)jV2, for z<y<fe. m n _{{DJ)NU f o r O < y < z ; KERNELS OF SINGULAR INTEGRAL EQUATIONS / G.2 135 w i t h = lkfz{jE{C[Piik) + Q i i k ) + + k 2 ^ \ ^ A ' d s { D J ) N i = ( k * - \ ) E { I { P i { k ) + + k * Q i { k ) + k ' P i ( l ) V A . d s + vj"[k2Pi(k) + k2Qi(k) + + k'Qify/A.ds} f o r i = l , 2. N o t e t h a t Pi{k),Qi(k)tQi(k) a n d A , are t h o s e f o r t h e s y m m e t r i c case. C.2 ANTI-SYMMETRIC CASE T h e f o r m u l a s g i v e n i n Sec. C . l c a n also be u s e d f o r t h e a n t i - s y m m e t r i c case w i t h t h e f o l l o w i n g changes: 1. c 2 = -Ci a n d c 6 = - c 5 i n e q . ( C 2 ) . 2. A 2 = - A i i n eq.(C.4). 3. Pi{k)fQi(k),Qi(k) a n d A, i n eq.(C.6) are now those f o r t h e a n t i - s y m m e t r i c case. THEORY FOR GENERALIZED CAUCHY TYPE SIE / D.O 136 APPENDIX D THEORY FOR GENERALIZED CAUCHY TYPE SIE T h e m e t h o d f o r d e t e r m i n i n g s i n g u l a r i t i e s o f C a u c h y t y p e s i n g u l a r i n t e g r a l e q u a t i o n s , w e l l d e v e l o p e d i n [24],has b e e n a p p l i e d t o t h e g e n e r a l i z e d C a u c h y t y p e s i n g u l a r i n t e g r a l e q u a t i o n s i n [5]. I n t h i s a p p e n d i x , we giv e a co n c i s e i n t r o d u c t i o n o f t h e t h e o r y w i t h some m o d i f i c a t i o n s t o a c c o m m o d a t e o u r s l i g h t l y m o r e g e n e r a l e q u a t i o n s . Eq.(5.25) is i n t h e g e n e r a l f o r m of where c 0 , cy, dg are k n o w n c o n s t a n t s a n d ZXj, Z2l are k n o w n p o i n t s n o t o n t h e l i n e ab. T h e p r o p e r t i e s of Z\j —• a, as x —• a a n d Z2t —* bf as x —• b c a n e a s i l y b e seen f r o m t h e e x p r e s s i o n s ' Zlj = a + {x-a)k,-ei0i (D.2) Z2t = b + (b- x)h,e™> where 0 < Oj < 2n, —n < ue < n, a < x < b a n d fcy, ht are p o s i t i v e c o n s t a n t s . T h e s p e c i a l case kj = ht = 1 has b e e n c o n s i d e r e d i n [5]. T h e r e g u l a r p a r t o f t h e i n t e g r a l e q u a t i o n h a s THEORY FOR GENERALIZED CAUCHY TYPE SIE / D.O 137 been i n c l u d e d i n t h e r i g h t h a n d s i d e w h i c h is s u p p o s e d t o s a t i s f y a H o l d e r c o n d i t i o n . T h e s o l u t i o n f(t) w i t h s i n g u l a r i t i e s at t h e ends a a n d b m a y b e e x p r e s s e d as W-(«-.W-0». l x u ) w h e r e g(t) s a t i s f i e s a H o l d e r c o n d i t i o n i n a < t < b a n d 0 < Re{a,/?} < 1. T h e n t h e s e c t i o n a l l y h o l o m o r p h i c f u n c t i o n '6 fit) _ i fb gjt) *Ja {t-a)°{b-t)P(t-Z) dt t - Z b m DT (DA) is a n a l y t i c e v e r y w h e r e e x c e p t at x o n t h e l i n e ab w h e r e t h e t w o one-sided l i m i t s $ + ( x ) a n d $~{x) as Z —> x are g e n e r a l l y n o t e q u a l . It has b e e n p r o v e d i n [24] t h a t $(Z) has a n a s y m p t o t i c e x p a n s i o n as Z —• a w i t h l e a d i n g t e r m g{a)e*ia (6 - af s i n *a{Z - a)a ^D'^ a n d a n a s y m p t o t i c e x p a n s i o n as Z —• b w i t h l e a d i n g t e r m -9(b) . . {b-a)Qsm,n(3(Z -b)P' ( } B y t h e P l e m e l j f o r m u l a , t h e C a u c h y i n t e g r a l i n e q u a t i o n ( D . l ) c a n be w r i t t e n as I [ = c 0 * ( x ) = | ( * + ( . ) + •"(«)) w h i c h , b y (D . 5 ) a n d (D.6), has a n a s y m p t o t i c e x p a n s i o n as x —• a w i t h l e a d i n g t e r m c o g ( q ) e " ° 1 1 , 2(6 - a)» s i n ira{(Z - a)° (Z - a) ° ' - 1 ( l + e - » " ) [D.t) 2(6 — a)P s i n Tra ( x — a) c0<7(a) co t not (6 - a)*(s - o)° a n d a n a s y m p t o t i c e x p a n s i o n as x —• 6 w i t h l e a d i n g t e r m c o g ( 6 ) c o t *P (b-aY(b-x)f*' 1 J THEORY FOR GENERALIZED CAUCHY TYPE SIE / D.O 138 T h e s e c o n d t y p e of i n t e g r a l i n t h e i n t e g r a l e q u a t i o n ( D . l ) hj *Ja ( t - a n b - t n t - z ^ ) is o f t h e f o r m (D.4) a n d , s i n c e ZXj i s n o t o n t h e l i n e ab, has a n a s y m p t o t i c e x p a n s i o n as x —* a w i t h l e a d i n g t e r m cjg{a)e*ia _ cjg(a)e*iakjae-ia$> (b - a)!3 s i n na(Zlj - a)a ~ (b - a)" s i n * r a ( x - a)a S i m i l a r l y , f o r i n t e g r a l s o f t h e t h i r d t y p e i n t h e e q u a t i o n d8g{t)dt (D.9) I f d J { t ) dt=1 f ^ nja t-Z2, irja (t-a)a(b-(D.6) gives a n a s y m p t o t i c e x p a n s i o n as x —• 6 w i t h l e a d i n g t e r m -9(b)d, d M b ^ e - ^ (D.10) (b - a)a s i n irp(Z2$ - bf (b - a)a s i n ir(3{b - " B y p u t t i n g a l l of t h e a b o v e a s y m p t o t i c e x p a n s i o n s of t h e i n t e g r a l s i n t o t h e i n t e g r a l e q u a t i o n a n d c o l l e c t i n g l e a d i n g t e r m s as x —* a a n d z —+ 6 r e s p e c t i v e l y , o ne o b t a i n s t h e f o l l o w i n g e q u a t i o n s g o v e r n i n g a a n d f3 J c 0 cos Tra + ^ Cjkjaeial*-*A = 0 (£.11) l a n d S c 0 cos nP + Y^ d,h-0e~ipu' = 0. (D.U) OUTER SOLUTIONS IN AXISYMMETRIC CASE / E.J 139 APPENDIX E OUTER SOLUTIONS IN AXI-SYMMETRIC CASE I n t h i s a p p e n d i x we giv e d e t a i l s f o r t h e d e r i v a t i o n of t h e o u t e r s o l u t i o n i n Ch.7. E.l ANTI-SYMMETRIC CASE T h e g e n e r a l s o l u t i o n of t h e first e q u a t i o n of (7.17) = #03(*)'3/6 + #o2(s)*2/2 + Roi(x)t + RQO(Z) reduces t o = #o2(s)*2/2 + R0o(x) f o r a n t i - s y m m e t r i c a l stresses a n d d i s p l a c e m e n t s . T h e s e c o n d e q u a t i o n of (7.17) is s a t i s f i e d a u t o m a t i c a l l y a n d i t s t h i r d e q u a t i o n gives RQ2(X) = con s t . w here t h e c o n s t a n t c a n be set t o zero s i n c e t h e f u n c t i o n t2 g i v e s n o c o n t r i b u t i o n s t o stresses. T h u s we have 4>0 = #00(x) (E.l) OUTER SOLUTIONS IN AXI-SYMMETRIC CASE / El 140 w h e r e t h e f u n c t i o n RQQ(X) is t o b e d e t e r m i n e d . E q u a t i o n s (7.18) a n d ( E . l ) g i v e 4>i = Ri2<x)t2/2 + R10{x) aRl2(x) + AxR00(x) - Ax w h e r e Ax i s a c o n s t a n t a n d c a n be set t o zero. T h u s we have 4>x = - A x J * o 0 ( x ) t 2 / ( 2 a ) + R10{x) (E.l) where Rl0(x), RQ0(z) are t o be d e t e r m i n e d . S u b s t i t u t i n g ( E . l ) a n d (E.2) i n t o (7.19), we o b t a i n dfamt = c / a A x A x J 2 o 0 ( x ) dfattt = c / a A x A I i E 0 0 ( x ) (at t = 1) a<f>2xll = AxAxRQ0(x)/(la) - AxR'l0(x) (at t = 1). T h e s o l u t i o n of t h i s s y s t e m i s 4>2 = ^ A , A x J R o 0 ( x ) < 4 / 2 4 + R22(x)t2/1 + # 2 0 ( x ) R22(x) = ^Tj-A*AxR0(>(x) ~\AxRio(*) + M where A2 c a n be d r o p p e d . F i n a l l y we have, a f t e r e l i m i n a t i n g R22 f r o m <f>2, 4>2 = ^AxAxR00(x)t*/24 + {d^^AxAxR00(x) - ±AxR10(x)}t2/2 + ^ ( x ) . (E.3) I n o r d e r t o get a n e q u a t i o n f o r RQ0(x), we s o l v e t h e first e q u a t i o n o f (7.20) t o get + ^A IA,B1 0(i)}(4/24 + *31(i)f2/2 + a n d t h e s a t i s f a c t i o n of t h e s e c o n d a n d t h i r d e q u a t i o n s of (7.20) g i v e s A x A x A x i ? 0 0 ( x ) = 0 #3i(*) = - ^ A x i 2 2 0 ( x ) - A x A x i ? 1 0 ( x ) (EAa,b) w h i c h i n t u r n s i m p l i f y <f>3 t o get ^3 = ^AxAxRl0(x)t*/14 + { i - ^ A x A x J R 1 0 ( x ) -^AxR20(x)}t2/2 + R20(x). (E.S) OUTER SOLUTIONS IN AXISYMMETRIC CASE / El 141 w h e r e Now, t h e s i m i l a r i t y b e t w e e n <f>2 a n d <f>$ makes i t p o s s i b l e t o w r i t e o u t t h e w h o l e s o l u t i o n f o r <f>: <f> = <f>o + £2<l>i + €4^2 + fi6^3 H = i2oo(^) + e2R10{x) + eAR20{x) + e 6 £ 3 0 ( x ) + • • • 2 2 + ^ { - - ( A ^ O O + €2AxR10{x) + £ 4 A xi22o (x) + • • •) + ^ p ( A * A *£oo(*) + e 2 A x A x i E 1 0 ( x ) + •••)} 4 4 AxAxAxRoQ{x) = 0 AxAxAxR10(x) = 0 AxAxAxR2(i{x) = 0 RQ(X) = R00(x) + e2Rl0{x) + e*R20{x) + ••• (E.6) AxAxAxR0(x) = 0, etc.. L e t w h i c h s a t i s f i e s t h e s t r e s s f u n c t i o n <f> c a n t h e n be w r i t t e n as <t> = W + €^{-lAxRQ(x) + d^e2AxAxR0(x)} 4 4 o r z2 . „ , „ d — ac 4 where A - * 1 - — r ~ dr2 + rdr' A r A r A r / 2 o ( r ) = 0. (E.7) OUTER SOLUTIONS IN AXI-SYMMETRIC CASE / E.2 142 T h e g e n e r a l s o l u t i o n f o r Ro(r) is R0(r) = C1+C2lnr + C 3 r 2 + AT2 In r + Br* + Cr* In r (£.8) wher e C1,C2 a n d C 3 m a y be set t o z e r o s i n c e t h e y g i v e n o c o n t r i b u t i o n s t o stresses. F o l l o w i n g t h i s , t h e o u t e r s o l u t i o n f o r t h e a n t i - s y m m e t r i c case i s <f> = i { - - z 2 ( l n r + 1) + r 2 l n r } a + C{ ^ ^ 4 ( 2 In r + 3) - V r 2 ( 2 In r + 1) 3 a a a + 8 { d ~ a c ) z 2 h 2 ( 2 In r + 3) + r 4 In r } a2d where A, B, C are c o n s t a n t s t o b e d e t e r m i n e d . T h i s s o l u t i o n r e d u c e s t o t h e k n o w n r e s u l t f o r t he i s o t r o p i c case u p o n s p e c i a l i z a t i o n . E.2 SYMMETRIC CASE S i m i l a r l y we m a y d e r i v e t h e o u t e r s o l u t i o n f o r t h e s y m m e t r i c (w.r.t. t h e m i d p l a n e of t h e p l a t e ) s t r e s s a n d d i s p l a c e m e n t cases. N o w eq.(7.17) gives 4>0 = tRQl(x) (£.10) where #01 (x) i s t o be d e t e r m i n e d . Eq.(7.18) g i v e s t h e e x p r e s s i o n tz ^ = - i 2 1 3 ( x ) + * i ? u ( x ) a n d t h e b o u n d a r y c o n d i t i o n s dRlz(x) _+cAxR0l{x) = 0 aR'13(x) + A X i ( i ) = 0 f r o m w h i c h we get OUTER SOLUTIONS IN AXISYMMETRIC CASE / E.2 143 a n d AxR0i(x) = tx w h e r e e± is a c o n s t a n t . Eq.(7.19) gi v e s t h e e x p r e s s i o n ^2 = ^ 3 ( ^ + ^ 1 3 ( 2 : ) a n d t h e b o u n d a r y c o n d i t i o n s dR23{x) + cAxRn{x) = 0 f r o m w h i c h we get a n d aR!23{x) + AxR'n(x) = 0 t3c where e 2 is a c o n s t a n t . Now, t h e s o l u t i o n f o r <f> c a n b e w r i t t e n as 4> = 4>0 + e24>i + e*fa + ••• = ~ C r 2 { e 1 + e 2 e 2 + ...} + t{Rm(x) + e2Rn{x) + e 4 i ? 1 3 ( x ) + • • •} where RQ^X),i?u(x), etc. c a n be e a s i l y d e t e r m i n e d : Raiix) = e 1 x 2 / 4 + d 1 l n x + C i Rn(x) = e2z2/4 + d2 l n x + C2 etc., w h e r e t h e c o n s t a n t s C,- w i l l be set t o zero as t h e y d o n o t c o n t r i b u t e t o t h e stresses. T h e r e f o r e t h e s u b s t i t u t i o n of .RQ^X), Rn(x)t e t c . t h e n l e a d t o t h e f o l l o w i n g o u t e r s o l u t i o n , . / > { £ , , , > + * { - J L , » + £ } (,.„) where t w o c o n s t a n t s D a n d E are t o be d e t e r m i n e d . T h i s s o l u t i o n r e d u c e s t o t h e k n o w n r e s u l t f o r t h e i s o t r o p i c case u p o n s p e c i a l i z a t i o n . DERIVATION OF GREEN'S FUNCTION / F.O 144 APPENDIX F DERIVATION OF GREEN'S FUNCTION T h e Green's f u n c t i o n G(pi} y, z) of eq.(10.6) i s d e f i n e d b y dG"" - (a + c)p2G" + p\G = 6{y - z) (F.l) G = 0, G' = 0 {at y = ±h) where ( )' = F o r —h<y<z, t h e c o m p l e m e n t a r y s o l u t i o n G — Cx s i n h ( s 1 p l y ) + C2 s i n h ( s 2 p , y ) + Dl cosh(s!p tj/) + D2 c o s h ( s 2 p , y ) w h i c h s a t i s f i e s t h e b o u n d a r y c o n d i t i o n s at y = —h is G = ihDi + f2D2}/A {F.2) where A = s2 s i n h ( s x p , 7 i ) cosh(s 2p,7i) — sx s i n h ( s 2 p , / i ) cosh(sxp,7i), fi = /i(«ii»2.p»y) = c o s h ( s 1 p , y ) A + s i n h ^ p . - y ) A ' - sx s i n h ( s 2 p , y ) , h = /2(*i,*2»Pi»y) = /i(«2i*iiPtiy)i A ' = s 2 c o s h ( s 2 p , / i ) cosh(sip t-A) — sx s i n h ( s 2 p , A ) s i n h ( s 1 p , / i ) . (F.S) DERIVATION OF GREEN'S FUNCTION / F.O 145 O n t h e o t h e r h a n d , f o r z < y < h, t h e c o m p l e m e n t a r y s o l u t i o n w h i c h s a t i s f i e s t h e b o u n d a r y c o n d i t i o n s at z = h i s G={f1D1 + f2D2}/A (FA) w h e r e fx = fxi3x>s2,Pi,y) = / i ( s 1 , s 2 , p , , - y ) , h = / 2(*i » * 2 i P«y) = fi(s2,SuPi,-y). T h e c o n t i n u i t y a n d j u m p c o n d i t i o n s at y = z G = G, G' — G\ G" = G", G'" - G'" = 1/d r e q u i r e w here D2 Dx D2 A i = A 2 = A 3 = -A . = -Al A d A 2 _ A AD d _ A a _ A A D d A 4 A A n d h fx h f'2 fx h fS f" % fx h h fi fx h fx' fx' fS fx h h fx r2 h fx' is is fx h fx fx f'2 fx fll fll fll JX J2 JX fx h fx h fx f2 fx f'2 fx' f'2' fx' fi fill fill fill fill JX 12 JX J2 (F.5) DERIVATION OF GREEN'S FUNCTION / F.O 146 T h e p r o p e r t y A # = A p ( 0 ) g i v e s AD = 4 s l 5 2 p f («} - s ^ A 2 * ^ (F.6) where A j 3 = -3xs 2 — A ' A " a n d A " = s 2 s i n h ( s 2 p , / i ) s i n h ( s 1 p , / i ) — sx cosh(s 2p,7i) c o s h ( s i p t 7 i ) . A d e t a i l e d c a l c u l a t i o n shows t h a t A i = A 1(a 1 , * 2,Pi,«) A 2 = -A l(s 2,*i,P,-,«) A 3 = Al(s1,s2,pi,-z) A 4 = - A 1 ( s 2 , s 1 , p , - , - z ) (F.7a,6,c,d) w h e r e t h e r e l a t i o n s ^ l ( * l i * 2 i P . - | * ) = 2pf(*2 -*l ) a 2^(*l » * 2 i P«) A( ali* 2 i P i i*)i A ( s j , 5 2,p,,z) = S j 5 2 s i n h ( s 1 p , z ) + s 1 A c o s h ( s 2 p , z ) + A A " c o s h ( s 1 p , - z ) + A , A " s i n h ( s 1 p , - 2 ) , {F.Ba, b) a n d A(a 2!«nP.-) = - A(*i » * 2»P.-) have been used. W i t h t h e c o n s t a n t s .D, a n d g i v e n b y (F. 5 ) , t h e e x p r e s s i o n s (F.2) a n d (F.4) m a y be c o m b i n e d t o g i v e r(n *\ fG(Pi'y>z)> torz<y<h; DERIVATION OF GREEN'S FUNCTION / F.O 147 w h e r e G{Pi,y,z) = G c { s 2/ 1(s 1 >s 2,p,-,-y)A ( 3 1,s 2,P , - , 2 ) + s i / i ( s 2 , supit - y ) A ( s 2 , 8 U p i t z ) } , G{Pi,y,z) = G(pi,-y,-z), Gc = l/{2dsls2pi{s2l - s22)AAD}. (F.10) (F.ll) (F.12) S o m e i n t e g r a l s r e l a t i n g t o t h i s Green's f u n c t i o n u s e d i n c h a p t e r 10 are d e r i v e d b e l o w . F i r s t , i n t e g r a t i o n b y p a r t s a n d t h e use of t h e b o u n d a r y c o n d i t i o n s at y = ±h g i v e N e x t we have a n d w h e r e It f o l l o w s t h a t /ft /"ft QlQ ^ G(Pi, y, z)<Tz\rQ, y)dy = j -Q^^z^O, y)dy-&r°z = A r — (cAr<f> + d<f>zz) g = — ( c A r A r ^ + d A r ^ ) Q = g^{c(-(c + a)*r<t>zz ~ d<f>Z2ZZ) + dAr<fizz} g2 = T~2 -ae- c2)Ar<f>. - cd<f>zzz) (\*,\ = ( B n B i 2 B I S \ I ** \ 4>zzz J \B2\ B22 B 2 Z ) I ^ Bn = dais — 2aan B\2 = - d a 3 3 + 2 o a 1 3 B\z = ( ai2 - °n)(^a33 _ 2 a o 1 3 ) B21 = - c a 1 3 + (6 + l ) a n B22 = c a 3 3 - (6 + l)o 1 3 B23 = (a12 - a n ) ( - c o 3 3 + (6 + l)a 1 3). (F.13) (F.14) DERIVATION OF GREEN'S FUNCTION / F.O 148 w h e r e hi = ( d — ac- c2)Bu — cdB2i f o r 1=1,2,3 a n d t h e s u b s c r i p t 0 de n o t e s e v a l u a t i o n at r = r 0. T h e n , i n t e g r a t i o n b y p a r t s a n d o b s e r v i n g t h e p r o p e r t i e s o f t h e Green's f u n c t i o n y i e l d / G{pi,y,z)ATaz(r0,y)dy= f ^ { h ^ p - + h 2 ^ + hzar0}dy. ( F . 1 5 ) J-h J-h °y °y r0 

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