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UBC Theses and Dissertations

Dynamic buckling of plates under impact loading Loh, See-Kok 1970

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DYNAMIC BUCKLING OF PLATES UNDER IMPACT LOADING by SEE-KOK LOH B.Sc.(Eng.) National Taiwan Univers i ty T a i p e i , Taiwan, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept t h i s thes is as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1970 In presenting t h i s thes is in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at The Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make i t f r e e l y ava i lab le for reference and study. I fur ther agree that permission fo r extensive copying of t h i s thes is for scholar ly purposes may be granted by the Head of my department or by his representat ives, I t i s understood that copying or publ icat ion of t h i s thes is f o r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permission. See-Kok Loh Department of Mechanical Engineering The Univers i ty of B r i t i s h Columbia Vancouver 8, B r i t i s h Columbia i ABSTRACT A theory i s presented to examine the formation of wrinkles in plates when subjected to high rates of loading in the ax ia l d i r e c t i o n . The type of i n s t a b i l i t y examined occurs in metals when the s t ra ins are well beyond the e l a s t i c range. For t h i s reason the metals are assumed to be governed by the equations of a r i g i d p l a s t i c m a t e r i a l . In p a r t i c u l a r , the von-Mises y i e l d c r i t e r i o n i s used in conjunction with the Levy-Mises flow r u l e . A parameter i s introduced which measures the l a t e r a l r e s t r a i n t of the p la te . By g iv ing t h i s parameter d i f f e r e n t va lues , a l l p late widths can be examined. The theory predicts wavelengths of the buckled plates which are compared with some experimental resu l ts obtained in 1968 by Goodier. i i TABLE OF CONTENTS CHAPTER PAGE I INTRODUCTION . 1 1.1 P r e l i m i n a r y Remarks . . . . . . 1 1.2 Purpose and Scope o f I n v e s t i g a t i o n 2 1.3 Method o f I n v e s t i g a t i o n . . . . . . . . . . . 3 II THEORY . . . . . . . . . . . . . 4 2.1 The G e n e r a l T h e o r y . . . . . . . . . . . . . 4 2.2 U n p e r t u r b e d M o t i o n . . . . . . . . . . . . . 14 2.3 P e r t u r b e d M o t i o n . . . . . . 19 2.4 S o l u t i o n o f t h e E q u a t i o n o f M o t i o n . . . . . 23 2.4.1 S o l u t i o n f o r t h e Case f ( t ) = l 24 2.4.2 S o l u t i o n f o r t h e Case f ( t ) = l - t / t f . . 26 2.5 D e t e r m i n a t i o n o f t f 29 I I I RESULTS AND DISCUSSION . . . . . . . 30 3.1 G e n e r a l R e s u l t s O b t a i n e d f o r f ( t ) = l . . . . . 39 3.2 G e n e r a l R e s u l t s O b t a i n e d f o r f ( t ) = l - t / t f . . 45 3.3 Comparison o f T h e o r e t i c a l and E x p e r i m e n t a l R e s u l t s ...... 45 IV EXPERIMENTAL WORK REVIEW 51 V SUMMARY OF CONCLUSIONS . 54 BIBLIOGRAPHY . . . . . . . . . . . . 55 i i i L IST OF TABLES TABLE PAGE 1 Data o f Specimens Recorded f r o m [ 3 ] 48 2 Comparison o f T h e o r e t i c a l and E x p e r i m e n t a l R e s u l t s . 49 3 T h e o r e t i c a l and E x p e r i m e n t a l R e s u l t s Recorded from [ 3 ] 50 i v L IST OF FIGURES FIGURE PAGE 1 P l a t e - m a s s System A p p r o a c h i n g t h e T a r g e t a t a C o n s t a n t V e l o c i t y V ' 5 2 C o - o r d i n a t e System used f o r t h e P l a t e . . . . . . . . 6 3 T r u e S t r e s s v e r s u s L o g a r i t h m i c S t r a i n and t h e Assumed S t r e s s - S t r a i n Curve . . . . . . . . . . . . . . . 11 4 The Dynamic Model R e p r e s e n t a t i o n o f a R i g i d L i n e a r S t r a i n - H a r d e n i n g S t r e s s - S t r a i n R e l a t i o n . . . . . 11 5 The I n i t i a l and Subsequent Von-Mises Y i e l d E l l i p s e s . 12 6 The F o r c e E q u i l i b r i u m o f an Element Cut from t h e P l a t e 15 7 The Moment E q u i l i b r i u m o f an Element Cut from t h e P l a t e 16 8 A v e r s u s n f o r k = 0.20 31 n 9 A n v e r s u s n f o r k = 0.30 . . . . . 32 10 A n v e r s u s n f o r k = 0.40 33 11 A v e r s u s n f o r k = 0.50 . 34 n 12 B v e r s u s n f o r k = 0.20 35 n 13 B v e r s u s n f o r k = 0.30 . . . . . . . . . . . . . . . 36 n 14 B v e r s u s n f o r k = 0.40 37 n 15 B v e r s u s n f o r k = 0.50 38 n 16 The E f f e c t o f Impact V e l o c i t y on B u c k l i n g Mode Number 41 17 The E f f e c t o f S t r a i n - H a r d e n i n g Modulus on B u c k l i n g Mode Number . . . . . . . . . . 43 V FIGURE PAGE 18 The E f f e c t o f P l a t e T h i c k n e s s on B u c k l i n g Mode 19 A (0) v e r s u s n f o r P l a t e LAC-1 . . . . . . . . . . . . 46 n 20 B n ( ° ) v e r s u s n f o r P l a t e LAC-1 47 21 Ph o t o g r a p h o f Impact Machine . . . . . . . . 52 vi LIST OF SYMBOLS a n I n i t i a l displacement amplitude corresponding to nth mode. A n Ampl i f i ca t ion of i n i t i a l displacement imperfect ions, b Width of p l a t e . b n I n i t i a l v e l o c i t y amplitude corresponding to nth mode. B n Ampl i f i ca t ion of i n i t i a l v e l o c i t y imperfect ions. E^  Strain-hardening modulus. h Thickness of p l a t e . ^2 Second invar iant of dev ia tor ic s t r e s s . k A parameter introduced to measure the l a t e r a l r e s t r a i n t of the p l a t e . k-| Function of k, k^  = /k* - k + l kg Function of k, kg •= 2 - k„ k 3 Function of k, k 3 = 2k - l . L Length of p l a t e . M Mass attached to the p l a t e , m Mass of the p l a t e . M ,M Moments in the x , y d i rec t ions respect i ve l y , x y N ,N Membrane forces in the x , y d i rec t ions respect i ve l y . v i i n Number o f h a l f - w a v e s a l o n g t h e l e n g t h o f t h e p l a t e . p Mean s t r e s s , p = - ( ° x + a y + C T z)/3 q Sum o f components o f f o r c e s i n t h e z - d i r e c t i o n . r P o s i t i v e i n t e g e r used i n summation s i g n . t Time i n m i c r o s e c o n d s . t.p D u r a t i o n o f f l o w m o t i o n i n m i c r o s e c o n d s . V C o n s t a n t v e l o c i t y o f impact. W Weight o f t h e b l o c k i n t h e dynamic r e p r e s e n t a t i o n o f s t r e s s -s t r a i n r e l a t i o n . w D e f l e c t i o n o f p l a t e . x-y-z C a r t e s i a n c o o r d i n a t e s used t o d e s c r i b e t h e p l a t e . a F u n c t i o n o f k, a = ^ / k ^ B F u n c t i o n o f k, g = (4 - a )/k. Y F u n c t i o n o f k, y = ^ 3 / ^ n F u n c t i o n o f k, n = (2 + 2y)/k^ p D e n s i t y o f p l a t e m a t e r i a l 5 D i m e n s i o n l e s s t i m e v a r i a b l e , x, - 1 - t / t ^ v O r d e r o f m o d i f i e d B e s s e l ' s f u n c t i o n s , d e f i n e d as v = (l+P^ ) / 2 a G e n e r a l i z e d s t r e s s . Yie ld stress of the plate material in un iax ia l t e n s i l e Approximate value of o. Real s t ress component in x - d i r e c t i o n . Real st ress component in y - d i r e c t i o n . Real st ress component in z - d i r e c t i o n . Deviator ic st ress component in x - d i r e c t i o n . Deviator ic stress component in y - d i r e c t i o n . Deviator ic st ress component in z - d i r e c t i o n . Generalized s t r a i n ra te . S t ra in rate in x - d i r e c t i o n . S t ra in rate in y - d i r e c t i o n . S t ra in rate in z - d i r e c t i o n . Shear s t r a i n rate on xy plane. Shear s t r a i n rate on xz plane. Shear s t r a i n rate on yz plane. P ropor t iona l i t y f a c t o r , X = 3e/2a Roots of the equation X? + P x + Q? = 0 ^ n n n ^n ACKNOWLEDGMENT I wish to express my sincere thanks and appreciat ion to my supervisor Dr. H. Vaughan for the guidance and assistance given throughout the research program and during the preparation of the t h e s i s . His constant advice and encouragement in the past one and one-hal f years has been inva luable . Thanks are also due to the Department of Mechanical Engineering for the use of t h e i r f a c i l i t i e s . F inancial assistance was received from the National Research Council of Canada, Grant No. A5563. CHAPTER I INTRODUCTION 1.1 P r e l i m i n a r y Remarks The b u c k l i n g o f s i m p l e s t r u c t u r a l forms such as p l a t e s , s h e l l s and columns p r o v i d e s the f o u n d a t i o n f o r the more g e n e r a l t h e o r y o f s t a b i l i t y o f c o m p o s i t e s t r u c t u r e s . When a s t r u c t u r e i s l o a d e d beyond i t s l o a d c a r r y i n g c a p a c i t y , e i t h e r d y n a m i c a l l y o r s t a t i c a l l y , i t v e r y o f t e n happens t h a t the f a i l u r e o f the s t r u c t u r e o c c u r s , n o t b e c a u s e o f t h e h i g h s t r e s s l e v e l s p r e s e n t b u t because o f t h e low s t a b i l i t y o f the s t r u c t u r e . Study o f s t r u c t u r a l s t a b i l i t y under s t a t i c l o a d s has d e v e l o p e d r a p i d l y i n the l a s t f i f t y y e a r s t o meet th e t e c h n o l o g i c a l advancement i n b u i l d i n g s h i p s , b r i d g e s , a i r c r a f t and m a c h i n e r y . The modern t r e n d i n i n d u s t r y f u r t h e r r e q u i r e s know-l e d g e o f dynamic s t a b i l i t y i n o r d e r t o d e v e l o p h i g h speed machines and v e h i c l e s . T h e r e f o r e , the p r o b l e m o f dynamic s t a b i l i t y o f s t r u c -t u r e s becomes more and more i m p o r t a n t and i n v o l v e d i n modern d e s i g n . The e l a s t i c s t a b i l i t y o f s t r u c t u r e s has been e x t e n s i v e l y d i s c u s s e d and many r a t h e r c o m p l e t e l i t e r a t u r e s i n t h e o r y and p r a c t i c e [1] [ 2 ] a r e a v a i l a b l e . In c o n t r a s t , the t h e o r y o f dynamic p l a s t i c s t a b i l i t y o f s t r u c t u r e s i s s t i l l i n i t s e a r l y s t a g e o f d e velopment. T h i s i s m a i n l y b e c a u s e o f t h e l a t e r d e velopment o f t h e t h e o r y o f p l a s t i c i t y i n r e l a t i o n t o the t h e o r y o f e l a s t i c i t y . 2 1.2 Purpose and Scope o f I n v e s t i g a t i o n In t h i s t h e s i s , a t h e o r y i s p r e s e n t e d t o examine the dynamic p l a s t i c b u c k l i n g o f p l a t e s under i n - p l a n e i m p a c t l o a d i n g . T h i s t h e o r y p r o v i d e s a method o f o b t a i n i n g the w a v e l e n g t h o f t h e b u c k l e d f o r m . In 1968, J . N. G o o d i e r p r o p o s e d a t h e o r y [ 3 ] t o s e r v e t h i s p u r p o s e f o r the c a s e o f p l a n e s t r e s s i n the l a t e r a l d i r e c t i o n . The e x p e r i m e n t a l r e s u l t s r e c o r d e d were i n f a i r l y good agreement w i t h h i s t h e o r y . In our work, a p a r a m e t e r k i s i n t r o d u c e d t o measure the l a t e r a l r e s t r a i n t o f the p l a t e . By g i v i n g t h i s p a r a m e t e r d i f f e r e n t v a l u e s which c o r r e s -pond t o d i f f e r e n t s i z e s o f p l a t e , a l l p l a t e w i d t h s can be examined. The p a r a m e t e r k depends on t h e p l a t e d i m e n s i o n s . I t w i l l be shown t h a t when k = 1/2, we r e c o v e r t h e c a s e t h a t was c o n s i d e r e d i n [ 3 ] . T h u s, our work ext e n d s t h a t d e v e l o p e d by G o o d i e r . By v a r y i n g the p a r a m e t e r k and o t h e r i n f l u e n t i a l f a c t o r s such as i m p a c t v e l o c i t y , s t r a i n h a r d e n i n g modulus, e t c . , we examine many ca s e s t h e o r e t i c a l l y . The time dependence i s d e s c r i b e d by a f u n c t i o n f ( t ) . We s t u d y two d i f f e r e n t f u n c t i o n s o f time t , namely f ( t ) = 1 and f ( t ) = 1 - t / t ^ which d e s c r i b e the f l o w m o t i o n . The f o r m e r was used i n [ 3 ] and [ 4 ] w h i l e the l a t t e r was c o n s i d e r e d i n [ 5 ] f o r c y l i n d r i c a l s h e l l s . The r e s u l t s o b t a i n e d f r o m the p r e s e n t t h e o r y a r e compared w i t h t h e e x p e r i m e n t a l r e s u l t s r e c o r d e d i n [ 3 ] . 1.3 Method o f I n v e s t i g a t i o n The a n a l y s i s i s based on the c o u p l i n g o f c l a s s i c a l p l a t e t h e o r y and the t h e o r y o f p l a s t i c i t y . S i n c e emphasis i s p l a c e d on o b t a i n i n g an a p p r o x i m a t e t h e o r y f o r a p p l i c a t i o n , a number o f a s s u m p t i o n s and a p p r o x i m a t i o n s a r e made t h r o u g h o u t t h e a n a l y s i s . When t h e p l a t e i s p r o j e c t e d towards a r i g i d t a r g e t a t a c o n s t a n t v e l o c i t y , t h e r e s u l t i d i s t o r t i o n i s c o n s i d e r e d as the c o m p o s i t i o n o f a u n i f o r m s h o r t e n i n g and t h i c k e n i n g o f the p l a t e t o g e t h e r w i t h a s m a l l s u p e r i m p o s e d non-u n i f o r m b e n d i n g o r p e r t u r b e d m o t i o n . In t h i s a n a l y s i s , t h e u n i f o r m m o t i o n i s examined f i r s t and th e n t h e b e n d i n g m o t i o n d e s c r i b i n g t h e b u c k l i n g i s t r e a t e d as a p e r t u r b a t i o n . CHAPTER I I THEORY The a n a l y s i s o f the phenomenon o f dynamic p l a s t i c b u c k l i n g o f p l a t e s under i m p u l s i v e l o a d i n g i n v o l v e s both t h e t h e o r y o f p l a s t i c i t y and the w e l l - d e v e l o p e d t h e o r y o f p l a t e s . P r e v i o u s i n v e s t i g a t i o n s have been made i n v o l v i n g many ass u m p t i o n s and s i m p l i f i c a t i o n s . U s u a l l y t h e s e a s s u m p t i o n s a r e the outcome o f c a r e f u l e x p e r i m e n t s and the s i m -p l i f i c a t i o n s a r e f o r the sake o f m a t h e m a t i c a l s i m p l i c i t y . However, i n some c a s e s t h e s e s i m p l i f i c a t i o n s impose u n d e s i r a b l e l i m i t a t i o n s on t h e a p p l i c a b i l i t y o f t h e t h e o r y . On the o t h e r hand the a s s u m p t i o n s do e n a b l e a c o m p l i c a t e d p h y s i c a l p r o c e s s t o be d e s c r i b e d by a f a i r l y s i m p l e t h e o r e t i c a l model which g i v e s s a t i s f a c t o r y r e s u l t s . In o u r a n a l y s i s , a number o f a s s u m p t i o n s a r e made, some o f which have been p r o p o s e d by e a r l i e r workers and some o f which a r e o r i g i n a l and n e c e s s a r y t o d e s c r i b e the p a r t i c u l a r problem b e i n g c o n s i d e r e d . 2.1 The G e n e r a l T h e o r y C o n s i d e r a p l a t e w i t h a mass M a t t a c h e d a t one end t r a v e l l i n g a t a c o n s t a n t v e l o c i t y towards a r i g i d t a r g e t ( F i g u r e 1 ) . The s t r a i n r a t e s i n t r o d u c e d by the i m p a c t o f the p l a t e - m a s s s y s t e m on the r i g i d t a r g e t a r e d e n o t e d by e , e and t where the s u b s c r i p t s x, y , and z x y z r e f e r t o the c o o r d i n a t e axes used t o d e s c r i b e the e v e n t ( F i g u r e .2). The d i s t o r t i o n a l s t r a i n r a t e s e , e and a r e s m a l l f o r s m a l l z 1 / / / / R i g i d T a r g e t P l a t e E n d M a s s Figure 1 Plate-mass system approaching the target at a constant velocity V F i g u r e 2 C o - o r d i n a t e system used f o r t h e p l a t e 7 a m p l i t u d e f l e x u r a l motions and a r e t a k e n as z e r o i n the t h e o r y . The p l a t e m a t e r i a l i s assumed t o be i n c o m p r e s s i b l e , and t h u s , t h e f o l l o w i n g e q u a t i o n o f i n c o m p r e s s i b i l i t y , r e l a t i n g t h e p r i n c i p a l s t r a i n r a t e s h o l d s £x+ £y+ ez = o ( 2 . i . i ) The g e n e r a l i z e d s t r a i n r a t e i s d e f i n e d as The v o n - M i s e s y i e l d c r i t e r i o n i s used and may be e x p r e s s e d i n the form <r 2=3j 2=4K 2+ <2+ a-;2) (2.i.3> where a t h e s t r e s s a t which t he m a t e r i a l f i r s t y i e l d s i n a u n i a x i a l t e n s i o n o r c o m p r e s s i o n t e s t , has been r e p l a c e d by a, t h e g e n e r a l i z e d s t r e s s which d e f i n e s t h e s u b s e q u e n t y i e l d s t r e s s d u r i n g l o a d i n g . A l s o , i i i J ? i s t h e s e c o n d i n v a r i a n t o f d e v i a t o r i c s t r e s s and a , a and a a r e c. x y z the d e v i a t o r i c s t r e s s components i n the x, y , and z d i r e c t i o n s r e s p e c t i v e l y . The d e v i a t o r i c s t r e s s e s a r e r e l a t e d t o the r e a l s t r e s s components a^, and by the f o l l o w i n g e q u a t i o n s < = ° ; + p , v i * 0 ? * ? , v~z= °~z* P ( 2 , 1 , 4 ) where p i s t h e mean s t r e s s which i s d e f i n e d as p = -l/3 ( a +a +a ). x y z In t h i n p l a t e t h e o r y ( t h i c k n e s s o f p l a t e s m a l l compared t o o t h e r d i m e n s i o n s ) , t h e s t r e s s component normal t o the m i d d l e p l a n e i s n e g l i g i b l e when compared t o o t h e r s t r e s s e s and may be t a k e n as z e r o , P u t t i n g o z = 0 i n t o e q u a t i o n (2.1.4) and s o l v i n g f o r t h e s t r e s s e s , t h e 8 f o l l o w i n g e x p r e s s i o n s a r e o b t a i n e d . cr* = 2 c r / + ay cry = 2 cTy + <rx' ( 2 . 1 . 5 ) H a v i n g i n t r o d u c e d the i n c o m p r e s s i b i l i t y c o n d i t i o n ( 2 . 1 . 1 ) and t h e y i e l d c r i t e r i o n ( 2 . 1 . 3 ) we now i n t r o d u c e the f l o w law. Here we use t h e L e v y - M i s e s f l o w law o f i n c r e m e n t a l p l a s t i c i t y which may be e x p r e s s e d i n t h e f o r m ^ ! x = _ d ^ =de_z d A tfy ay CT-; where de , de and de a r e the i n c r e m e n t a l s t r a i n s and dx i s the p r o -x' y z p o r t i o n a l i t y f a c t o r d e f i n e d by dx = 3 d e / 2 a . Upon d i v i d i n g by d t , e q u a t i o n ( 2 . 1 . 6 ) becomes = JL*. = JU. = A ( 2 . 1 . 7 ) °~* < °~z where X i s g i v e n by x = 3 e / 2 a . E l i m i n a t i n g and from e q u a t i o n s ( 2 . 1 . 5 ) by u s i n g ( 2 . 1 . 7 ) , and u s i n g the above e x p r e s s i o n f o r X i t f o l l o w s t h a t t h e s t r e s s e s a r e r e l a t e d t o the s t r a i n r a t e s t h r o u g h t h e f o l l o w i n g r e l a t i o n s : CT = 2 ( 2 6 + £y) cr/3k ( 2 . 1 . 8 ) cr = 2(26 y-r Gx)cr/3e 9 The g e n e r a l p r o c e d u r e t o be f o l l o w e d i s now o u t l i n e d . A r e l a -t i o n must be o b t a i n e d between the g e n e r a l i z e d s t r e s s a and the g e n e r -a l i z e d s t r a i n e. C o n c u r r e n t l y the s t r a i n r a t e components must be found f r o m k i n e m a t i c a l c o n s i d e r a t i o n s and the g e n e r a l i z e d s t r a i n r a t e e f o u n d f r o m ( 2 . 1 . 2 ) . The s t r e s s e s can t h e n be c a l c u l a t e d f r o m (2 . 1.8) which i n t u r n y i e l d t h e membrane f o r c e s and b e n d i n g moments by i n t e -g r a t i o n . The e q u a t i o n s o f m o t i o n o f the p l a t e t h e n y i e l d d i f f e r e n t i a l e q u a t i o n s i n t h e d i s p l a c e m e n t s . B u c k l i n g i s then c o n s i d e r e d t o be p r e s e n t i f n o n - u n i f o r m s o l u t i o n s can be f o u n d . S i n c e t h e b u c k l i n g o f t h e p l a t e i s c o n s i d e r e d as a p e r t u r b a t i o n on a u n i f o r m m o t i o n t h e a n a l y s i s i s composed o f two p a r t s . F i r s t t h e u n i f o r m c o m p r e s s i o n o f t h e p l a t e i s examined f o l l o w i n g t he above p r o c e d u r e . Then a s m a l l d i s p l a c e m e n t p e r t u r b a t i o n o r v e l o c i t y p e r t u r b a t i o n i s s u p e r i m p o s e d on t h i s known u n i f o r m f l o w . B u c k l i n g i s t h e n d e t e c t e d i f the imper-f e c t i o n s grow r a p i d l y w i t h i n c r e a s i n g t i m e . We now p r o c e e d a c c o r d i n g l y . F i g u r e 3 shows a t y p i c a l one-d i m e n s i o n a l t r u e s t r e s s - s t r a i n c u r v e i n an u n i a x i a l t e n s i o n o r com-p r e s s i o n t e s t f o r aluminum. The s t r a i g h t l i n e i s the c o r r e s p o n d i n g s t r e s s - s t r a i n c u r v e used i n t h i s a n a l y s i s . In p a r t i c u l a r we i g n o r e c o m p l e t e l y t h e e l a s t i c r e g i o n o f the c u r v e and a p p l y t he a p p r o x i m a t e s t r a i g h t l i n e r e l a t i o n t o a r i g i d p l a s t i c t h e o r y . T h i s i s r e a s o n a b l e s i n c e we s h a l l examine impacts s u f f i c i e n t l y l a r g e t o p r o d u c e t o t a l s t r a i n s much g r e a t e r t h a n t he e l a s t i c s t r a i n . F o r many m a t e r i a l s , such as aluminum and magnesium a l l o y s , t he s t r a i g h t l i n e a p p r o x i m a t i o n o f t he s t r a i n h a r d e n i n g p o r t i o n o f the a-e c u r v e i s s a t i s f a c t o r y [ 6 ] . 10 The main a d v a n t a g e o f t h i s a p p r o x i m a t i o n i s i n the r e s u l t i n g mathe-m a t i c a l s i m p l i f i c a t i o n s , s i m p l i f i c a t i o n s which do n o t a f f e c t the v a l i d i t y o f the a n a l y s i s t o any g r e a t e x t e n t . More s p e c i f i c a l l y we s h a l l show t h a t b u c k l i n g i s f a i r l y i n s e n s i t i v e t o t h e magnitude o f t h e s l o p e o f the above a-e c u r v e . T h i s i s n o t so f o r q u a s i - s t a t i c b u c k l i n g . With the a s s u m p t i o n o f l i n e a r h a r d e n i n g , the g e n e r a l i z e d s t r e s s and s t r a i n i n c r e m e n t a r e r e l a t e d by da = E^de. I n t e g r a t i o n o f t h i s e q u a t i o n l e a d s t o t h e f o l l o w i n g s t r e s s - s t r a i n r e l a t i o n (T = CT + B. 6 (2.1.9) o n where aQ i s the y i e l d s t r e s s i n an u n i a x i a l t e n s i o n o r c o m p r e s s i o n t e s t and E^ i s the s l o p e o f the l i n e which d e s c r i b e s s t r a i n h a r d e n i n g . A s p r i n g - m a s s dynamic model r e p r e s e n t a t i o n o f (2.1.9) i s a l s o shown i n F i g u r e 4 i n which the s t r e s s i s r e p l a c e d by the f o r c e P and s t r a i n i s r e p l a c e d by the d i s p l a c e m e n t o f w e i g h t W. We can see f r o m F i g u r e 4 t h a t the b l o c k W can o n l y be moved a l o n g t h e p l a n e when the f o r c e P r e a c h e s a c e r t a i n v a l u e , say P . Then s t a r t i n g f r o m P Q t h e d i s p l a c e m e n t o f W v a r i e s l i n e a r l y w i t h the a p p l i e d f o r c e P because o f the r e s t r a i n t o f the s p r i n g . The f i r s t p a r t c o r r e s p o n d s t o the v e r t i c a l l i n e p o r t i o n up t o e l a s t i c l i m i t i n t h e a p p r o x i m a t e a-e d i a g r a m ( F i g u r e 3) whereas t h e l a t t e r p a r t c o r r e s p o n d s t o the s t r a i g h t l i n e beyond the e l a s t i c l i m i t shown i n F i g u r e 3. I t i s i m p o r t a n t t o note t h a t aQ i s a l s o t h e i n i t i a l v a l u e o f the g e n e r a l i z e d s t r e s s which f i x e s the i n i t i a l s i z e o f the von-Mises y i e l d e l l i p s e i n the o^-a p l a n e as shown i n F i g u r e 5. F i g u r e 3 T r u e s t r e s s v e r s u s l o g a r i t h m i c s t r a i n and t h e assumed s t r e s s - s t r a i n curve 777TT77TT A/VWvV F i g u r e 4 The dynamic model r e p r e s e n t a t i o n o f a r i g i d 1 i n e a r s t r a i n - h a r d e n i n g s t r e s s - s t r a i n r e l a t i o n 12 F i g u r e 5 The i n i t i a l and s u b s e q u e n t von-Mises y i e l d e l l i p s e s 13 The s u b s e q u e n t y i e l d l o c i ( F i g u r e 5) a r e d e t e r m i n e d by the s u b s e q u e n t v a l u e s o f a, which i n g e n e r a l , i s a f u n c t i o n o f the h i s t o r y o f l o a d i n g and o f p o s i t i o n . I t i s assumed t h a t , f o r i s o t r o p i c h a r d e n i n g , t h e y i e l d s u r f a c e w i l l expand w i t h s t r e s s and s t r a i n h i s t o r y b u t w i l l r e t a i n t h e o r i g i n a l shape as i t f i r s t y i e l d s . However t h i s a s s u m p t i o n does n o t t a k e i n t o a c c o u n t the B a u c h i n g e r e f f e c t which would t e n d t o d i s t o r t the shape o f the y i e l d s u r f a c e as the y i e l d i n g p r o g r e s s e s [ 7 ] . I t has been m e n t i o n e d t h a t a i s a f u n c t i o n o f time and p o s i t i o n . I t s v a r i a t i o n t h r o u g h o u t the m o t i o n i s found t h r o u g h i t s dependence on e. The membrane f o r c e s can be f o u n d by e v a l u a t i n g t h e i n t e g r a l s o f t h e s t r e s s e s o v e r t he p l a t e t h i c k n e s s . In p a r t i c u l a r we have J-Vz (2.1.10) S i m i l a r l y , t h e b e n d i n g moments a r e rhlz Mx=-\ crx zdz (2.1.11) rh/z My*-I cridz J-h/z The g e n e r a l e q u a t i o n o f mo t i o n o f a p l a t e under t h e i n - p l a n e f o r c e s N x and g i v e n by (2.1.10) and b e n d i n g moments M x and g i v e n by (2.1.11) i s g i v e n i n [ 8 ] , as f o l l o w s ^ - ^ 2 - 2 T V ^ ~ ( i (2.1-12) 3>X2 2>y2 £X2>y 14 where q i s t h e sum o f t h e z-components o f the membrane f o r c e s ( F i g u r e s 6a and 6b) and the i n e r t i a f o r c e due t o d e f l e c t i o n o f t h e p l a t e ( F i g u r e 7) and i s g i v e n by S i n c e the d i s t o r t i o n a l s t r a i n r a t e s a r e t a k e n as z e r o , i t f o l l o w s f r o m t h e L e v y - M i s e s f l o w law t h a t the s h e a r s t r e s s e s a r e a l s o z e r o . Com-b i n i n g e q u a t i o n s (2.1.12) and (2.1.13) and p u t t i n g M = N = 0, xy xy we g e t 7 T + -ZZT = f h w - N * - ^ - N y ^ - (2.1.14) T h i s i s the f u n d a m e n t a l e q u a t i o n o f motion which must be examined i n terms o f d i s p l a c e m e n t s f o r the d e t e c t i o n o f f l e x u r a l m o t i o n s . We e l i m i n a t e N N and M . M i m m e d i a t e l y from (2.1.14) by u s i n g (2.1.10) x y x y and (2.1.11) r e s p e c t i v e l y . 2.2 U n p e r t u r b e d M o t i o n F i g u r e 2 shows a p l a t e w i t h a mass M a t t a c h e d a t one end a p p r o a c h i n g a r i g i d t a r g e t a t a c o n s t a n t v e l o c i t y V. The c o m b i n a t i o n o f V and M i s such t h a t p l a s t i c f l o w w i l l o c c u r . As mentioned p r e v i o u s l y , t h e e n s u i n g m o t i o n i s c o n s i d e r e d as a c o m b i n a t i o n o f an u n p e r t u r b e d and p e r t u r b e d m o t i o n . In t h i s s e c t i o n we examine the r e g u l a r u n p e r t u r b e d m o t i o n . A f t e r s t r i k i n g t he r i g i d t a r g e t , the s t r a i n r a t e e i n t h e p l a t e 4 x = - f f f t ) (2.2.1) 15 F i g u r e 6 The f o r c e e q u i l i b r i u m o f an e l e m e n t c u t f r o m t h e p l a t e z I Y F i g u r e 7 The moment e q u i l i b r i u m o f an e l e m e n t c u t f r o m t h e p l a t e •cn 17 The minus s i g n i n t r o d u c e d i n (2.2.1) i n d i c a t e s t h a t t h e s t r a i n r a t e i s c o m p r e s s i v e . The f u n c t i o n f ( t ) d e s c r i b e s the f l o w f r o m t h e i n s t a n t o f impact u n t i l the c o m p l e t i o n o f t h e m o t i o n . T h us, t i s d e f i n e d i n t h e r e g i o n o <_ t <_ t ^ where t ^ i s the d u r a t i o n o f t h e mot i o n and f ( o ) = 1.. Due t o the c o m p r e s s i o n i n the x - d i r e c t i on we a l s o e x p e c t f l o w t o o c c u r i n t h e y and z d i r e c t i o n s . We assume t h a t , a t the m i d s u r f a c e c a n be r e p r e s e n t e d by where k i s a c o n s t a n t which v a r i e s between 0 and 0.5. When k = 0. we have E = 0 f o r a l l time c o r r e s p o n d i n g t o a s t a t e o f p l a n e s t r a i n . T h i s s t a t e o f p l a n e s t r a i n may be due t o c o n s t r a i n t s a c t i n g a t the edges y = ± b / 2 o r i t may c o r r e s p o n d t o a v e r y wide p l a t e . When k = 1/2, we see from e q u a t i o n (2.1.8) t h a t = 0. Hence k = 1/2 c o r r e s p o n d s t o t h e c a s e o f p l a n e s t r e s s , o r a v e r y narrow p l a t e . We e x p e c t p l a t e s o f a l l w i d t h s t o have a c o r r e s p o n d i n g v a l u e o f k between 0 and 0.5. The v a l u e o f k f o r a p a r t i c u l a r p l a t e may be d e t e r m i n e d e x p e r i m e n t a l l y from s u r f a c e measurements. S i n c e we a r e c o n s i d e r i n g a u n i f o r m com-p r e s s i o n , e q u a t i o n (2.2.2) can be ex t e n d e d t o g i v e good a p p r o x i m a t i o n o v e r the t h i c k n e s s . E q u a t i o n (2.1.1) now y i e l d s kV fft) (2.2.2) L VfCt) L U-k) (2.2.3) The g e n e r a l i z e d s t r a i n g i v e n by (2.1.2) becomes (2.2.4) 18 D e n o t i n g / ^ f ( t ) d t by F ( t ) and n o t i n g t h a t e = 0 a t t = 0, e q u a t i o n (2.2.4) can be i n t e g r a t e d w i t h r e s p e c t t o t t o g i v e e = i m n j i ^ T j (2.2.5) With t h i s r e s u l t , t h e g e n e r a l i z e d s t r e s s can be found from (2.1.9) as f o l l o w s o-= a- + lMtXEVj/<*-K+I = a-0 (2.2.6) where a 0 i s c u r r e n t v a l u e o f t h e g e n e r a l i z e d s t r e s s and d e t e r m i n e s t h e s i z e o f the c u r r e n t y i e l d s u r f a c e . The t r u e s t r e s s e s a x and a^, t h e membrane f o r c e s N and N , the b e n d i n g moments M and M a r e c a l c u l a t e d x y x y from (2.1.8), (2.1.10) and (2.1.11) r e s p e c t i v e l y and a r e found t o be 2 (2.2.7) / V = - Hzhcr0/MK. (2.2.8) Ny= K2hcr0/J3K, The c o n s t a n t s k-j, k^ and k^ used i n the above e q u a t i o n s a r e d e f i n e d by Kj^fic-K-n , K2= 2-K , K2=2H-I (2.2.10) As we e x p e c t f o r u n i f o r m c o m p r e s s i o n , t h e b e n d i n g moments M x and a r e both z e r o . 19 2.3 P e r t u r b e d M o t i o n As m e n t i o n e d e a r l i e r e x p e r i m e n t a l r e s u l t s r e p o r t e d i n [ 3 ] showed t h a t no c u r v a t u r e d e v e l o p s i n the y - d i r e c t i o n f o r a r e c t a n g u l a r s t r i p under imp a c t i n the x - d i r e c t i o n . T h i s r e s u l t i n d i c a t e s t h a t t h e p e r t u r b a t i o n a l d e f l e c t i o n w i s i n d e p e n d e n t o f y and i s o n l y a f u n c t i o n o f x and t . Due t o t h i s c y l i n d r i c a l b e n d i n g t he s t r a i n r a t e i i s A g i v e n by £ = -^f-fCt) + zw" (2.3.1) The a d d i t i o n a l s t r a i n r a t e zw" i n (2.3.1) i s the p e r t u r b a t i o n a l c u r v a t u r e r a t e i n t r o d u c e d by the b e n d i n g d i s p l a c e m e n t w ( x , t ) and i s s m a l l compared w i t h t h e u n p e r t u r b e d s t r a i n r a t e . Thus we may be s u r e t h a t no s t r a i n r e v e r s a l o c c u r s w h i l s t w ( x , t ) i s s m a l l . I t i s p o s s i b l e t h a t t h e c o m b i n a t i o n o f V and M i s such t h a t v e r y l a r g e a m p l i t u d e b u c k l i n g o c c u r s . The t h e o r y c a n n o t d e s c r i b e such a s i t u a t i o n s i n c e t h e n u n l o a d i n g i n the p l a s t i c s e n s e must have t a k e n p l a c e . However, i t i s p o s s i b l e t h a t t h e t h e o r y w i l l s t i l l p r e d i c t c o r r e c t w a v e l e n g t h s s i n c e w a v e l e n g t h i s d e t e r m i n e d e a r l y i n the mo t i o n when a m p l i t u d e s a r e s t i l l s m a l l , t h a t i s , when w(x , t ) i s s m a l l . The prime i n (2.3.1) d e n o t e s p a r t i a l 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 x. For c y l i n d r i c a l b e n d i n g , £^ i s u n a f f e c t e d by the p e r t u r b a t i o n and hence i s a g a i n g i v e n by k y ^ f l t ) (2.3.2, The s t r a i n r a t e i n t h e z d i r e c t i o n can be found from t he i n c o m p r e s s i b i l i t y 20 r e q u i r e m e n t (2.1.1), t h a t i s iz = -{i^+i ). S u b s t i t u t i n g t h e s e new v a l u e s o f and i n t o (2.1.2) and u s i n g (2.2.10) t h e g e n e r a l i z e d s t r a i n r a t e i s found t o be - L 73 (2.3.3) T a k i n g t h e f i r s t term o f (2.3.3) o u t o f t h e r o o t s i g n and assuming / / w r w ^ 1 ^ 1 ( 2 - 3 - 4 ) t h e l a s t term o f (2.3.3) becomes n e g l i g i b l y s m a l l when compared w i t h u n i t y . A p p l y i n g t h e B i n o m i a l E x p a n s i o n t o e q u a t i o n (2.3.3) and n e g l e c t i n g h i g h e r o r d e r t e r m s , the f o l l o w i n g s i m p l e form o f (2.3.3) i s o b t a i n e d : w" . (2.3.5) I t i s a p p a r e n t t h a t i f f ( t ) i s such t h a t the i n t e g r a t i o n o f (2.3.5) i s p o s s i b l e , e can be found w i t h o u t d i f f i c u l t y . R e c a l l i n g f r o m the l a s t s e c t i o n t h a t t h e i n d e f i n i t e i n t e g r a l o f f ( t ) i s d e n o t e d by F ( t ) , we see t h a t i n t e g r a t i o n o f (2.3.5) l e a d s t o where w = w(x,o) i s the i n i t i a l d e f l e c t i o n o f the p l a t e b e f o r e i m p a c t . The g e n e r a l i z e d s t r e s s i s thus g i v e n by (2.1.9) as c r = cr°- -±—r z (w-ti)'' (2.3.7) 21 where a 0 i s i n t e r p r e t e d as i n ( 2 . 2 . 6 ) . S u b s t i t u t i o n o f (2.3.1) and (2.3.2) i n t o (2.1.8) l e a d s t o x 3 6 1 3 6 y 3 e z w L ,-1 To e v a l u a t e O/E i t i s n e c e s s a r y t o a p p r o x i m a t e ( e ) ~ by u s i n g B i n o m i a l e x p a n s i o n and n e g l e c t i n g p r o d u c t s o f w" and (w-w)". way we f i n d t h a t (2.3.8) the In t h i s a- E 6 6 J3K,8 J3K, where £ = ^ h ^ i V f ^ l m S u b s t i t u t i n g t h i s r e s u l t i n t o (2, / V 1— t h e t r u e s t r e s s components a r e found t o be 3 . 8 ) , 3 J J3 2JS Vflt) - ,//7 (2.3.9) where a, 3 , y and n a r e d e f i n e d as Z -A/3 r=k3/K, > 7j = (*+2r)/Kf The membrane f o r c e s and b e n d i n g moments a r e fo u n d by u s i n g e q u a t i o n s (2.1.10) and ( 2 . 1 . 1 1 ) . They a r e cr°hck (2.3.10) X A/3 / y J3 M = 12 I 2J3 Vf(V w 22 (2.3.11) 3 K ZjSVfCU E q u a t i o n s (2.3.11) may be p a r t i t i o n e d i n t o two groups o f ter m s . The f i r s t g roup i n each e q u a t i o n depends l i n e a r l y on E ^ - - t h e s t r a i n h a r d e n i n g modulus. These c o n t r i b u t i o n s t o the b e n d i n g moments a r e termed t he " s t r a i n - h a r d e n i n g moments". When E h = 0 the b e n d i n g moments do n o t v a n i s h s i n c e t he second group o f terms a r e i n d e p e n d e n t o f E^. These p a r t s o f the b e n d i n g moments have been termed t h e " d i r e c t i o n a l moments" by G o o d i e r i n h i s work on p l a s t i c p l a t e s [ 3 ] . They a r i s e t h r o u g h t he f a c t t h a t t he y i e l d s u r f a c e i s l o c a l l y c u r v e d and because d i f f e r e n t p o i n t s t h r o u g h t he p l a t e t h i c k n e s s c o r r e s p o n d t o d i f f e r e n t l o c a l p o i n t s on t h i s y i e l d s u r f a c e ( F i g u r e 5 ) . The n o r m a l i t y c o n d i t i o n g o v e r n i n g t h e s t r a i n r a t e i n c r e m e n t v e c t o r t h e n e n s u r e s t h a t t he s t r a i n r a t e i s not u n i f o r m t h r o u g h t he p l a t e t h i c k n e s s t h e r e b y c a u s i n g t he " d i r e c t i o n a l " b e n d i n g moments. I t a l s o f o l l o w s t h a t i f we use the T r e s c a y i e l d c r i t e r i o n which i s p i e c e w i s e l i n e a r t h e n no d i r e c t i o n a l moment a r i s e s . For c y l i n d r i c a l b e n d i n g w i s i n d e p e n d e n t o f y . Hence 2 2 2 2 9 My/9y and 9 w/9y a r e both z e r o . The e q u a t i o n o f m o t i o n (2.1.14) r e d u c e s t o the form + A/ x w" = f h W (2.3.12) D i f f e r e n t i a t i n g t w i c e the f i r s t o f (2.3.11) and s u b s t i t u t i n g t h i s r e -s u l t t o g e t h e r w i t h the v a l u e o f the f i r s t o f (2.3.10) i n t o ( 2 . 3 . 1 2 ) , 23 we g e t J2 V f(t) ^-^Jgr^-") f**0 (2.3.13) For k = 0.5 and f ( t ) = 1, e q u a t i o n (2.3.13) r e d u c e s t o 3GfV~  + -^Tf-C"'") + + l V = ° (2-3.14) E q u a t i o n (2.3.14) i s e x a c t l y t he same as t h e one o b t a i n e d i n [ 3 ] and [ 4 ] . 2.4 S o l u t i o n o f The E q u a t i o n o f M o t i o n The d i f f e r e n t i a l e q u a t i o n g o v e r n i n g the f l e x u r a l m o t i o n o f t h e p l a t e i s g i v e n by ( 2 . 3 . 1 3 ) . We seek a s o l u t i o n o f t h e form WC*, t ) = * £ (2.4.1) The i n i t i a l d e f l e c t i o n s and v e l o c i t i e s a r e t a k e n t o be 'I \ / mi L mix (2.4.2) where a and b a r e c o n s t a n t s . The s u b s t i t u t i o n o f (2.4.1) and the n n f i r s t o f (2.4.2) i n t o (2.3.13) l e a d s t o t h e f o l l o w i n g d i f f e r e n t i a l e q u a t i o n f o r each n, K + + fi^= K°-n (2-4-3' 2 where P , Q and R a r e g i v e n by n n n 3 J 24 n2 - C*L ckEhh*?tn? _ V l g - 0 ^ (2.4.4) The s o l u t i o n o f (2.4.3) l e a d s t o a c o m p l e t e s o l u t i o n o f ( 2 . 3 . 1 3 ) . I t can be e x p e c t e d t h a t t h e s o l u t i o n o f e q u a t i o n (2.4.3) depends on t h e c h o i c e o f the f u n c t i o n f ( t ) . The c h o i c e o f f ( t ) s h o u l d be such t h a t i t can a d e q u a t e l y d e s c r i b e the r e a l b u c k l i n g m o t i o n . Some i d e a l i z e d a s s u m p t i o n s a r e employed h e r e t o d e t e r m i n e f ( t ) . F i r s t the p l a s t i c f l o w m o t i o n i s c o n s i d e r e d as a u n i f o r m m o t i o n i n which the p l a t e deforms u n i f o r m l y w i t h c o n s t a n t v e l o c i t y f r o m t h e i n s t a n t o f i m p a c t and t h i s m o t i o n i s a b r u p t l y c u r t a i l e d when t = t ^ , where t ^ i s t h e d u r a t i o n o f t h i s m o t i o n and i s t o be d e t e r m i n e d s e p a r a t e l y . Secondly,, we c o n s i d e r a c o n s t a n t d e c e l e r a t i o n o f the f r e e end o f t h e p l a t e so t h a t the v e l o c i t y d e c r e a s e s l i n e a r l y w i t h time from V = V Q a t t = 0 t o V = 0 a t t = t p . The f o r m e r c a s e was c o n s i d e r e d i n [3] and [ 4 ] whereas the l a t t e r c a s e was c o n s i d e r e d i n [ 5 ] . In the f o l l o w i n g a n a l y s i s , both o f t h e c a s e s w i l l be c o n s i d e r e d s e p a r a t e l y . 2.4.1 S o l u t i o n For The Case f ( t ) = 1 . As d e s c r i b e d above t h i s c a s e c o r r e s p o n d s t o the p l a t e b e i n g compressed a t a c o n s t a n t v e l o c i t y . F or t h i s c a s e e q u a t i o n (2.4.3) becomes K + + fi* w„=. X^a* (2.4.1,1) 25 and the i n i t i a l c o n d i t i o n s become W»(t>) = (ZK and W„(o)= bK The s o l u t i o n o f (2.4.1.1) i s W„(t) = An(t)CLn + 3 n ( t ) l > n (2.4.1.2) where 7 1 P l * - Pin -tit aVnt (2.4.1.3) Pn ~ Plw + - 2 2 where x^, x^ a r e t h e r o o t s o f the e q u a t i o n A n + P n x n + Q n = 0, t h a t i s , a;»-i(p»-y^ *af) (2.4.1.4) Z \ T h e r e f o r e , t he c o m p l e t e s o l u t i o n o f e q u a t i o n (2.3.13) f o r f ( t ) = 1 i s g i v e n by W(tft) ^^(^(t)^ 3n(t)bn) S W ^ H (2.4.1.5) E q u a t i o n (2.4.1.5) g i v e s t he p e r t u r b a t i o n a l d e f l e c t i o n o f t h e p l a t e i n terms o f t h e i n i t i a l i m p e r f e c t i o n s . I t a l s o shows t h a t A n ( t ) and B ( t ) a r e t h e a m p l i f i c a t i o n s o f t h e i n i t i a l d e f l e c t i o n a m p l i t u d e s and v e l o c i t y a m p l i t u d e s r e s p e c t i v e l y . We see t h a t t he i n i t i a l d i s p l a c e m e n t imper-f e c t i o n s a r e e x p r e s s e d by u s i n g F o u r i e r s i n e s e r i e s w i t h c o e f f i c i e n t s a n - The v a l u e s o f a , i n g e n e r a l , a r e d i s t i n c t , and c o u l d be d e t e r m i n e d 26 i n t he u s u a l way i f the i n i t i a l forms o f d i s p l a c e m e n t s were known. The d i s p l a c e m e n t a t any l a t e r time c o u l d t h e n be f o u n d by the a p p r o p r i a t e F o u r i e r t y p e summation T h e r e f o r e , we c o u l d examine p l a t e b u c k l i n g when known i n i t i a l d i s p l a c e -ments a r e g i v e n t h e i r a p p r o p r i a t e F o u r i e r s e r i e s r e p r e s e n t a t i o n . T h i s r e s u l t i s e q u a l l y t r u e f o r i n i t i a l v e l o c i t y i m p e r f e c t i o n s b n . In our c a s e , we l e t a = 1 so t h a t A d e s c r i b e s the m otion o f the p l a t e n n r d e f l e c t i o n and the mode number d e t e r m i n e d by A n i s t a k e n as the number o f h a l f - w a v e s i n t o w hich the p l a t e w i l l b u c k l e . S i m i l a r l y , we l e t b n = 1 f o r the i n i t i a l v e l o c i t y i m p e r f e c t i o n s and the m a g n i f i c a t i o n f a c t o r B n d e t e r m i n e s the magnitude o f d e f l e c t i o n and the number o f h a l f - w a v e s . 2.4.2 S o l u t i o n F o r The Case f ( t ) = 1 - t / t f We now c o n s i d e r t h e c a s e i n which f ( t ) = 1 - t / t ^ . The b u c k l i n g m o t i o n so d e s c r i b e d i m p l i e s t h a t the r a t e o f d e f o r m a t i o n o f the p l a t e d e c r e a s e s as time i n c r e a s e s and t h i s r a t e i s i n v e r s e l y p r o p o r t i o n a l t o the d u r a t i o n o f the m o t i o n . Now e q u a t i o n (2.4.3) l e a d s t o •f (2.4.2.1) I n t r o d u c i n g the new v a r i a b l e ? where ?= t-t/t f (2.4.2.2) e q u a t i o n (2.4.2.1) becomes 27 n. (2.4.2.3) w i t h t he i n i t i a l c o n d i t i o n s i i 2 and P , Q and R a r e d e f i n e d as f o l l o w s n' ^n n (2.4.2.4) (2.4.2.5) Note t h a t (') used i n (2.4.2.3) denotes 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 £. T h i s n o t a t i o n w i l l be used onward from ( 2 . 4 . 2 . 3 ) . The s o l u t i o n i2 o f (2.4.2.3) depends on the s i g n o f Q n . However, the s o l u t i o n t a k e s i 2 the f o r m below r e g a r d l e s s o f the s i g n o f Q K i d ^ - ^ r ^ + ^ f ? ; ^ . (2.4.2.6) .2 F o r Q n >o, A n and B n a r e g i v e n by -^-/O/yCQ**)] + fcfQin (2.4.2.7) i i where J (Q ?) and Y (Q t) a r e the B e s s e l f u n c t i o n s o f the f i r s t and second k i n d o f o r d e r v. • 2 F o r Q <o, A and B a r e a l t e r n a t i v e l y g i v e n by x n n n J • J 28 + i^ca'jKyCa^i) + ti/a'n (2.4.2.8) where I (Q d and K (Q <;) a r e t h e m o d i f i e d B e s s e l f u n c t i o n s o f the v n v n f i r s t and second k i n d o f o r d e r v and the v a l u e o f v i s g i v e n by (2.4.2.9) The c o m p l e t e s o l u t i o n o f (2.3.1.13) f o r f ( t ) = 1 - t / t p i s nitx L. (2.4.2.10) where A n and B n a r e g i v e n by e i t h e r (2.4.2.7) o r (2.4.2.8) d e p e n d i n g '2 on whether Q n i n (2.4.2.3) i s p o s i t i v e o r n e g a t i v e . We a l s o l e t a = b = 1 as f o r t h e c a s e f ( t ) = 1. n n To d e t e c t t h e e x i s t e n c e o f b u c k l i n g , i t i s o n l y n e c e s s a r y t o examine t h e a m p l i t u d e s A n ( ? ) and B n ( c ) a t t h e end o f t h e f l o w m o t i o n . As t->t^, t h a t i s as c-*o, t h e f u n c t i o n s and Y y a p p r o a c h z e r o , "2 hence f o r Q n >o, t h e r e i s no b u c k l i n g . C o n s e q u e n t l y o n l y t h e c a s e ' 2 f o r w hich Q n <o i s c o n s i d e r e d and the c o r r e s p o n d i n g a s y m p t o t i c form i s now o b t a i n e d . As s+o I v ( Q n c ) ^ o and ? v K v ( Q n c h 2 v _ 1 r ( v ) / Q n v where r(v) i s the Gamma f u n c t i o n . E x p r e s s i n g I ( Q n ) and I ] ( Q n ) i n power s e r i e s , we o b t a i n 29 * fc> r-'tr+v)! ~^ r f 0 r.'(r+»)! (2 .4 .2 .11) S u b s t i t u t i n g (2 .4 .2 .11) i n t o (2 .4 .2 .8) and n o t i n g t h a t r(v) = ( v - 1 ) ! , 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 o f A n ( o ) and B n ( o ) a r e o b t a i n e d (2 .4 .2 .12) 3n(o) when S i s d e f i n e d as n (2 .4 .2 .13) 2.5 D e t e r m i n a t i o n o f t ^ For t h e c a s e shown i n F i g u r e 1 i t i s r e a s o n a b l e t o d e s c r i b e the time dependence by the f u n c t i o n f ( t ) = 1 - t / t ^ . T h i s i m p l i e s t h a t t he f o r c e a c t i n g on the mass M i s c o n s t a n t d u r i n g d e f o r m a t i o n . The a c t u a l f o r c e i s o f c o u r s e /a xd2dy A r e a o f p l a t e S i n c e a x does n o t change s i g n i f i c a n t l y ( E ^ not s t e e p i n F i g u r e 3) i t f o l l o w s t h a t t he f o r c e a c t i n g on M i s i n d e e d s e n s i b l y c o n s t a n t . A s i m p l e c o n s e r v a t i o n o f e n e r g y c a l c u l a t i o n then g i v e s f z (2.5.1) CHAPTER I I I RESULTS AND DISCUSSION The s o l u t i o n o f (2.3.13) f o r both c a s e s o f f ( t ) t a k e s t h e g e n e r a l f o r m g i v e n by ( 2 . 4 . 1 5 ) . However, t h e a m p l i f i c a t i o n f a c t o r s A and B a r e d i f f e r e n t f o r t h e d i f f e r e n t f u n c t i o n s f ( t ) . E q u a t i o n s n n (2.4.1.3) g i v e t h e v a l u e o f t h e s e f a c t o r s f o r f ( t ) = l and (2.4.2.12) y i e l d t h e c o r r e s p o n d i n g f a c t o r s f o r f ( t ) = l - t / t f . I f b u c k l i n g o f t h e p l a t e under t h e g i v e n impact l o a d i n g c o n d i t i o n does e x i s t , t h e g r a p h i c a l p l o t s o f A n v e r s u s n and B n v e r s u s n a t any t f o r 0<t<t^. w i l l e x h i b i t s h a r p peaks. These peaks i n d i c a t e t h e w a v e l e n g t h i n t o which t h e p l a t e w i l l d eform. F i g u r e s 8 t o 11 a r e p l o t s f o r A n ( t ) v e r s u s n and F i g u r e s 12 t o 15 a r e f o r B n ( t ) v e r s u s n a t d i f f e r e n t v a l u e s o f t h e p a r a m e t e r k f o r f ( t ) = l . The v a l u e s o f k, t h e d u r a t i o n s o f b u c k l i n g m o t i o n t ^ and t h e a x i a l s t r a i n s e a r e g i v e n on each g r a p h . The s t a t i c v a l u e s used i n t h e c a l c u l a t i o n t o o b t a i n t h e r e s u l t s f o r F i g u r e s 8 t o 15 a r e l i s t e d below: Y i e l d s t r e s s OQ= 45,000 p s i . Impact v e l o c i t y V=l,500 i n / s e c . S t r a i n h a r d e n i n g modulus E^=l50,000 p s i . P l a t e l e n g t h L=3 i n c h e s . 9 l b -sec* 1 D e n s i t y o f t h e m a t e r i a l f = 0.00025 j i n P l a t e t h i c k n e s s h = 1/8 i n c h . 31 0 2 4 6 8 10 12 14 16 18 20 22 24 M O D E N U M B E R M F i g u r e 8 A n v e r s u s n f o r k = 0.20 K = 0-30 t f = 69 00 usee. Strain = 3 45% 0 8 10 12 14 16 MODE NUMBER SI F i g u r e 9 A p v e r s u s n f o r k = 0.30 40 30 MODE NUMBER II F i g u r e 10 A v e r s u s n f o r k = 0.40 4 0 3 0 2 0 i — i — [ — r K = 0-50 t f = 70-82 usee. Strain = 3-54% n io 9 8 7 6 T = t 2H 6 8 10 12 14 MODE NUMBER D 2 0 2 2 24 Figure 11 A n versus n for k = 0.50 37 MODE NUMBER U Figure 14 B versus n for k = 0.40 38 39 3.1 G e n e r a l R e s u l t s O b t a i n e d f o r f ( t ) = 1 F i g u r e s 8 t o 11 show A n ( t ) v e r s u s n f o r v a r i o u s k r a n g i n g f r o m 0.20 t o 0.50 as i n d i c a t e d . The f a m i l y o f c u r v e s drawn i n t h e f i r s t f i g u r e i s r e p r e s e n t a t i v e o f a l l t h e o t h e r c u r v e s . The o n l y d i f f e r e n c e b e i n g i n t h e p o s i t i o n o f t h e peaks ( t h e d e g r e e o f a m p l i f i c a t i o n and t h e a s s o c i a t e d v a l u e s o f t h e mode number). T h e r e f o r e , any i n t e r p r e t a t i o n o f one such f a m i l y o f c u r v e s w i l l be e q u a l l y a p p l i e d t o o t h e r s w i t h o u t l o s s o f g e n e r a l i t y . F o r example i n F i g u r e 8 t h e f i v e c u r v e s show t h e c o n t i n u o u s change o f t h e b u c k l i n g a m p l i t u d e and t h e mode number f r o m t / t f = 0.20 t o t/t.p = 1.0. However, t h e mode number c o r r e s p o n d i n g t o t h e peak a t v a r i o u s t i m e s does n o t change by more t h a n one i n t h e above m e n t i o n e d t i m e i n t e r v a l . T h a t i s , t h e pr e d o m i n a n t b u c k l i n g mode, as fo u n d t h e o r e t i c a l l y , i s s e l e c t e d e a r l y i n t h e m o t i o n . T h i s has al s o , been o b s e r v e d e x p e r i m e n t a l l y by e x a m i n i n g f i l m s equences t a k e n o f some o f t h e e x p e r i m e n t s c o n d u c t e d i n t h e l a b o r a t o r y u s i n g t he impact machine shown i n F i g u r e 21. Comparing F i g u r e s 8, 9, 10 and 11, we o b s e r v e t h e e f f e c t o f t h e changes o f k on t h e number o f h a l f waves and t h e magnitudes o f a m p l i -f i c a t i o n s . The i n c r e a s e i n t h e v a l u e o f k c o n t r o l s t h e d e c r e a s e s o f mode number as w e l l as t h e magnitude o f a m p l i f i c a t i o n , The v a r i a t i o n o f s t r a i n i s s m a l l . I f we a r r a n g e t h e mass M such t h a t t h e s t r a i n s i n t h e s e f i g u r e s a r e t h e same, t h e e f f e c t o f k on t h e mode number and mag n i t u d e s o f a m p l i f i c a t i o n s would be s i m i l a r t o t h o s e shown i n F i g u r e s 8 t o 11. 40 F i g u r e s 12 t o 15 show B n ( t ) v e r s u s n. The e x p l a n a t i o n and r e s u l t s g i v e n above a r e e q u a l l y a p p l i c a b l e t o t h e s e c u r v e s . F i g u r e s 8 and 12 show t h e d i s p l a c e m e n t and v e l o c i t y m a g n i f i c a -t i o n s f o r k = 0.20. I t i s e v i d e n t t h a t t h e number o f h a l f - w a v e s p r e d i c t e d by d i s p l a c e m e n t p e r t u r b a t i o n i s a p p r o x i m a t e l y one h i g h e r t h a n t h a t g i v e n by v e l o c i t y p e r t u r b a t i o n . The c o m p a r i s o n o f any o t h e r group f o r a p a r t i c u l a r v a l u e o f k shows t h e s i m i l a r r e s u l t . F i g u r e 16 shows t h e e f f e c t o f t h e impact v e l o c i t y on t h e mode number f o r 2 v a l u e s o f k namely k = 0.25 and k = 0.5. I t i s c l e a r t h a t t h e s l o p e ( t h e change o f n w i t h r e s p e c t t o v e l o c i t y ) o f t h e k = 0.25 c u r v e b e i n g l a r g e r , i m p l i e s t h a t t h e e f f e c t o f impact v e l o c i t y on mode number i s more s i g n i f i c a n t when k i s s m a l l , t h a t i s , when t h e p l a t e i n q u e s t i o n has a l a r g e r b/L r a t i o where b i s t h e w i d t h o f t h e p l a t e . The mode numbers shown i n t h i s f i g u r e a r e o b t a i n e d f r o m t h e p l o t s o f A p ( t ) v e r s u s n a t t h e v a r i o u s v e l o c i t i e s and t h e y a r e t h e v a l u e s ( n o t n e a r e s t i n t e g e r s ) c o r r e s p o n d i n g t o t h e f i n a l s t a g e o f t h e f l o w m o t i o n ( t = t ^ ) . Any o t h e r c u r v e f o r k between 0.25 and 0.5 w i l l be between t h e two c u r v e s shown i n F i g u r e 16 w i t h s l o p e d e c r e a s e s as k i n c r e a s e s . I t i s a l s o c l e a r t h a t t h e mode number i n c r e a s e s w i t h i n c r e a s i n g impact v e l o c i t y . However, when impact v e l o c i t y i s v e r y h i g h , f r a c t u r e o f t h e m a t e r i a l may o c c u r and a l s o wave p r o p a g a t i o n becomes i m p o r t a n t . M o r e o v e r , F i g u r e 16 i s a l s o i n agreement w i t h t h e p r e v i o u s r e s u l t , t h a t t h e mode number d e c r e a s e s as k i n c r e a s e s , o b t a i n e d i n c o n s i d e r i n g t h e a m p l i f i c a t i o n c u r v e s . I t i s e x p e c t e d t h a t a s e t o f r e s u l t s s i m i l a r t o t h o s e shown i n F i g u r e 16 would be o b t a i n e d i f v e l o c i t y p e r t u r b a t i o n mode numbers a r e t a k e n . 41 VELOCITY ( in/sec.) F i g u r e 16 The e f f e c t o f impact v e l o c i t y on b u c k l i n g mode number F i g u r e 17 shows t h e e f f e c t o f -- t h e s t r a i n h a r d e n i n g modulus -- on t h e d i s p l a c e m e n t p e r t u r b a t i o n mode number f o r v a r i o u s v a l u e s o f k. The impact v e l o c i t y , t h e t h i c k n e s s and t h e l e n g t h o f t h e p l a t e a r e a l l k e p t c o n s t a n t a t 1500 i n / s e c , 1/8 and 3 i n c h e s r e s p e c t i v e l y . From t h i s f i g u r e , we see t h a t t h e number o f h a l f - w a v e s i s q u i t e i n s e n s i t i v e t o t h e change o f t h e s t r a i n - h a r d e n i n g modulus, e s p e c i a l l y f o r l a r g e v a l u e o f k. Even i n t h e c a s e f o r k = 0.20, t h e r e d u c t i o n i n n i s o n l y 2 when E^ changes from 0 p s i t o 300,000 p s i . The r e s u l t l e a d s t o t h e c o n c l u s i o n t h a t t h e s t r a i n - h a r d e n i n g e f f e c t ( b e n d i n g moment c o n t a i n i n g E^ term) on mode number i s s e c o n d a r y when compared t o t h e d i r e c t i o n a l e f f e c t . A s i m i l a r r e s u l t was c o n c l u d e d i n [ 3 ] and [ 4 ] . A g a i n , t h e p r e v i o u s c o n c l u s i o n t h a t t h e mode number d e c r e a s e s as k i n c r e a s e f o r c o n s t a n t E^ w i t h a l l o t h e r p a r a m e t e r s f i x e d i s e v i d e n t from F i g u r e 17. I t s h o u l d be no t e d t h a t t h e i n s e n s i -t i v i t y o f mode number t o s t r a i n - h a r d e n i n g modulus o n l y h o l d s f o r r e a s o n a b l y s m a l l v a l u e s o f E^/a, s a y E^/a<5. F i g u r e 18 shows t h e e f f e c t o f p l a t e t h i c k n e s s on t h e mode number. As t h e p l a t e t h i c k n e s s h i n c r e a s e s , t h e mode number d e c r e a s e s f o r a l l v a l u e s o f k. The c u r v e s shown i n F i g u r e 18 a r e f o r k = 0.1, 0.25 and 0.50 w i t h v e l o c i t y o f i m p a c t , p l a t e l e n g t h , e t c . , b e i n g h e l d f i x e d . F o r t h e c u r v e shown f o r k = 0.1, mode number d e c r e a s e s r a p i d l y f r o m 11 f o r h = 0.1 i n c h t o 4.6 f o r h = 0.25 i n c h . The o t h e r two c u r v e s i n d i c a t e a s i m i l a r r e s u l t . O 1 5 0 0 0 0 3 0 0 0 0 0 E h (psi) F i g u r e 17 The e f f e c t o f s t r a i n - h a r d e n i n g modulus on b u c k l i n g mode number 45 3.2 G e n e r a l R e s u l t s O b t a i n e d f o r f ( t ) = 1 - t / t f In g e n e r a l , r e s u l t s s i m i l a r t o t h o s e o b t a i n e d f o r f ( t ) = 1 a r e o b t a i n e d f o r f ( t ) = 1 - t / t ^ . F i g u r e s 19 and 20 show t h e r e s u l t s o f A (0) and B n ( 0 ) o b t a i n e d by u s i n g ( 2 . 4 . 2 . 1 2 ) . A g a i n , t h e c u r v e s i n d i c a t e s h a r p peaks f o r some mode numbers i n d i c a t i n g t h e p h y s i c a l wave-l e n g t h . The g e n e r a l r e s u l t s p r e s e n t e d i n t h e l a s t s e c t i o n a r e g e n e r a l l y a p p l i c a b l e t o t h i s c a s e . The e f f e c t o f t h e mass r a t i o on t h e mode number has n o t y e t been shown. In g e n e r a l , t h e mode number d e c r e a s e s as t h e mass r a t i o i n c r e a s e s , o t h e r p a r a m e t e r s b e i n g k e p t c o n s t a n t . M o r e o v e r , h i g h e r mass r a t i o c o n t r i b u t e s t o l a r g e r m a g n i f i c a t i o n s . 3.3 Comparison o f T h e o r e t i c a l and E x p e r i m e n t a l R e s u l t s T a b l e 1 shows t h e d a t a c o r r e s p o n d i n g t o 6 p l a t e s r e p o r t e d i n [ 3 ] . In T a b l e 2, t h e t h e o r e t i c a l r e s u l t s f o r f ( t ) = 1 and f ( t ) = 1 - t / t f a r e g i v e n i n t h e l a s t two columns. I t s h o u l d be no t e d t h a t t h e a x i a l s t r a i n s a r e t h e same f o r both c a s e s . The c o m p a r i s o n o f t h e l a s t two columns shows t h a t t h e a v e r a g e h a l f - w a v e l e n g t h s o b t a i n e d f o r f ( t ) = 1 - t / t f a r e somewhat l a r g e r t h a n t h a t p r e d i c t e d f o r f ( t ) = 1. A c o m p a r i s o n o f t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s i s a l s o shown i n T a b l e 2. A g e n e r a l good agreement i s o b t a i n e d . I t i s e x p e c t e d t h a t t h e p r e d i c t e d h a l f - w a v e l e n g t h s u s i n g t h e p r e s e n t t h e o r y ( f ( t ) = 1 ) a r e l e s s than t h o s e p r e d i c t e d i n [ 3 ] because s m a l l e r v a l u e s o f k a r e used i n t h e p r e s e n t t h e o r y . B e a r i n g t h i s i n mind, we see t h a t i n most c a s e s i n T a b l e 3 which i s r e p r o d u c e d f r o m [ 3 ] , t h e p r e s e n t t h e o r y would p r e d i c t more s a t i s f a c t o r y r e s u l t s . 46 18 F i g u r e 19 A (0) v e r s u s n f o r p l a t e LAC-1 1 0 0 0 8 0 0 47 1 I I I I I I I 2 4 6 8 10 12 14 16 18 n F i g u r e 20 B (0) v e r s u s n f o r p l a t e LAC -1 TABLE 1 DATA OF SPECIMENS RECORDED FROM [3] Specimen Designation Impact Velocity ft/sec. Plate Size in x in x in 10 3 psi 10 3 psi 1 +M/m k SAC-1 4 0 0 5x1/2x1/16 3 0 4 8 3 5 2 5 0 - 4 2 7 SAC-2 3 0 0 5x1/2x1/16 3 0 4 8 - 3 5 - 2 5 0 4 2 7 SAC-3 199 5x1/2x1/16 3 0 4 8 - 3 5 - 2 5 0 4 2 7 LAC-1 184 5x3/4x1/16 2 8 - 7 5 8 - 6 3 - 5 0 0 - 3 9 5 LAC-2 310 5x3/4x1/16 3 0 4 8 - 3 3 - 5 0 0 -395 LAC-3 344 5x3/4x1/16 3 0 4 8 - 3 6 0 0 0-395 TABLE 2 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS Experiment Results Obtained From (3) Present Theories Specimen Designation Impact Velocity ft/sec Axial Strain % Experimental half-wavelength in Axial Strain % f(t)=l f(t) = l - t/t f Average half-wavelength in Average half-wavelength in S A C - 1 400 36 0-31 38 0-23 0-29 S A C - 2 300 23 0-26 24 0-29 0-37 S A C - 3 199 10 0-26 1 1 0-39 0-48 LAC-1 184 7 0-38 7 0-38 0 47 LAC -2 310 16 0-41 18 0-30 0-39 L A C - 3 344 30 0-36 33 0-25 0-31 TABLE 3 THEORETICAL AND EXPERIMENTAL RESULTS RECORDED FROM [3] | Specimen Designation Impact Velocity Ft/sec Axial Strain % E h 10 3 psi 10 3 psi Calculated Average Half-wavelength in Experimental Average Half-wavelength in SAC -3 199 10 4 8 - 3 3 0 0 - 4 3 0 - 2 6 SAC-2 3 0 0 2 3 4 8 - 3 3 0 0 - 3 4 0 - 2 6 SAC-1 4 0 0 3 6 4 8 - 3 3 0 0 - 2 5 0 - 3 1 LAC-1 184 7 5 8 - 6 2 8 - 7 0 - 4 9 0 3 8 LAC-2 310 16 4 8 - 3 3 0 0 - 3 7 0 - 4 1 LAC-3 3 4 4 3 0 4 8 3 3 0 0 - 2 9 0 - 3 6 4 C S C - 3 5 9 1 2 6 0 0 2 3 - 8 0 - 8 1 0 - 6 1 4 C S C - 3 100 3 1 2 1 - 7 2 5 - 7 0 - 5 8 0 - 5 1 4 CSC-3 1 15 3 1 2 1 - 7 2 5 - 7 0 - 5 8 0 - 4 8 SAC-3 5 7 8 2 9 4 8 - 3 3 0 0 - 3 9 0 - 3 8 LAC-4 3 7 7 15 4 8 - 3 3 0 0 - 5 7 0 - 3 7 LAC-5 4 6 0 2 0 4 8 - 3 3 0 0 - 4 7 0 4 9 CHAPTER IV EXPERIMENTAL WORK REVIEW In o r d e r t o s t u d y t h e b u c k l i n g phenomenon o f p l a t e s under impact l o a d i n g , an impact machine has been p u r c h a s e d by t h e d e p a r t -ment. The impact machine i s a " V a r i p u l s e 1500 shock machine" so c a l l e d because o f i t s a b i l i t y t o g e n e r a t e s i n u s o i d a l and s q u a r e p u l s e s which a r e u s e f u l i n v i b r a t i o n a n a l y s i s . The machine has a •heavy t a b l e which i s f r e e t o move up and down a l o n g two v e r t i c a l g u i d e s . The t a b l e can f a l l f r e e l y under g r a v i t y and i n t h i s way i m p a r t s an impact t o t h e specimen which i s p l a c e d on t h e l o w e r t a b l e o f t h e machine. F i g u r e 21 shows a p h o t o g r a p h o f t h e machine. Some t e s t s were c a r r i e d o u t u s i n g t h i s machine as f o l l o w s : R e c t a n g u l a r s h e l l s , c o n s i d e r e d as f o u r j o i n e d p l a t e s , were g r i d e d i n b o t h x and y d i r e c t i o n s . G r i d measurements were made b e f o r e and a f t e r i m p a c t and r e s u l t i n g s t r a i n o b t a i n e d . A t r a v e l l i n g m i c r o s c o p e was used f o r t h i s p u r p o s e . Some q u a l i t a t i v e t e s t s were o b t a i n e d but a r e n o t i n c l u d e d h e r e . In g e n e r a l , i t was f e l t t h a t t h e impact v e l o c i t y was t o o low and i n o r d e r t o p r o d u c e l a r g e s t r a i n s t h e d u r a t i o n was r e l a t i v e l y h i g h . C o n s e q u e n t l y l a t e r a l i n e r t i a e f f e c t s were a l s o q u i t e s m a l l . One f e a t u r e o f dynamic b u c k l i n g i s t h e i m p o r t a n t r o l e t h a t l a t e r a l i n e r t i a p l a y s i n c o n t r o l l i n g t h e a m p l i t u d e s o f t h e f l e x u r a l m o t i o n s . The e x p e r i m e n t a l r e s u l t s o b t a i n e d by G o o d i e r f o r h i g h impact v e l o c i t i e s 52 F i g u r e 21 P h o t o g r a p h o f impact machine r e s u l t e d i n narrow p l a t e s ( s t r u t s ) b e i n g compressed by a x i a l l o a d s f o u r o r f i v e t i m e s t h e c o r r e s p o n d i n g E u l e r b u c k l i n g l o a d . With t h e impact machine a v a i l a b l e h e r e t h e v e l o c i t i e s were so low t h a t o n l y E u l e r b u c k l i n g c o u l d be o b t a i n e d and f o r t h i s r e a s o n t h e r e s u l t s a r e not i n c l u d e d . One o t h e r d i f f i c u l t y e n c o u n t e r e d was t h a t o f p l a t e a l i g n m e n t . The i n i t i a l w e l l - a d j u s t e d a l i g n m e n t c o u l d be d i s t u r b e d by s m a l l v i b r a t i o n s o c c u r r i n g d u r i n g f r e e f a l l o f t h e t a b l e and e c c e n t r i c i m p a c t l o a d i n g t h e n r e s u l t e d . The e f f e c t o f t h i s e c c e n t r i c i t y t h e n c a u s e s f u r t h e r b e n d i n g r a t h e r t h a n b u c k l i n g o f t h e p l a t e . A n o t h e r p o s s i b l e f a c t o r was t h e heavy mass o f t h e impact t a b l e L a r g e amounts b f p l a s t i c s t r a i n s c o u l d be o b t a i n e d even a t a low v e l o c i t y o f i m p a c t . T h e r e f o r e , t h e combined e f f e c t o f low v e l o c i t y and l a r g e M/m r a t i o c o u l d p o s s i b l y p roduce l a r g e t o t a l s t r a i n b u t a s m a l l mode number. I t i s r a t h e r d i f f i c u l t , a t t h i s p o i n t , t o j u d g e which o f t h e above c a u s e s i s most i m p o r t a n t . I t i s b e l i e v e d t h a t a l l t h e f a c t o r s have c o n t r i b u t e d t o make t h e e x p e r i m e n t a l work d i s a p p o i n t i n g . CHAPTER V SUMMARY OF CONCLUSIONS The f o l l o w i n g c o n c l u s i o n s a r e drawn from t h e above r e s u l t s g i v e n i n C h a p t e r I I I . 1. The b u c k l i n g o f p l a t e s a t s u f f i c i e n t l y h i g h v e l o c i t y e x i s t s and t h e most r e s p o n s i v e mode number i s d e t e c t a b l e . 2. The l a t e r a l r e s t r a i n t p a r a m e t e r k a f f e c t s t h e b u c k l i n g mode number. The mode number d e c r e a s e s as k i n c r e a s e s . 3. The e f f e c t o f t h e s t r a i n - h a r d e n i n g modulus E^ on t h e mode number i s s m a l l . 4. In g e n e r a l , t h e mode number i n c r e a s e s as impact v e l o c i t y i n c r e a s e s . However, f o r v e r y h i g h v e l o c i t y f r a c t u r e o f t h e m a t e r i a l may o c c u r . 5. The mode number d e c r e a s e s as t h e p l a t e t h i c k n e s s i n c r e a s e s . T h i s e f f e c t i s more s i g n i f i c a n t f o r s m a l l e r v a l u e o f k. 6. The c o m p a r i s o n o f t h e p r e s e n t t h e o r e t i c a l r e s u l t s t o t h e e x p e r i m e n t a l ones o b t a i n e d i n [3] i s i n g e n e r a l agreement. 55 B i b ! i o g r a p h y 1. Timoshenko and G e r e , T h e o r y o f E l a s t i c S t a b i l i t y , M c G r a w - H i l l , 1961. 2. H. L. Cox, The B u c k l i n g o f P l a t e s and S h e l l s , Pergamon P r e s s , 1963. 3; J . N. G o o d i e r , Dynamic B u c k l i n g o f R e c t a n g u l a r P l a t e s i n Sub- s t a i n e d P l a s t i c C o m p r e s s i v e Flow, E n g i n e e r i n g P l a s t i c i t y , Cambridge U n i v e r s i t y P r e s s , 1968. P r o c e e d i n g s o f an i n t e r -n a t i o n a l c o n f e r e n c e on p l a s t i c i t y h e l d i n Cambridge, England., March, 1968. 4. A. L. F l o r e n c e and J . N. G o o d i e r , Dynamic P l a s t i c B u c k l i n g o f C y l i n d r i c a l S h e l l s i n S u b s t a i n e d A x i a l C o m p r e s s i v e Flow, J . App. Mech. 35, T r a n s . ASME, S e r i e s E, 80-86, 1968. 5. H. Vaughan, The Response o f a P l a s t i c C y l i n d r i c a l S h e l l t o A x i a l Impact, J . App. Math, and P h y s i c s , V o l . 20, pp. 321-328, 1969. 6. H. S c h w a r t z b a r t and W. F. Brown, J r . , N o t c h - B a r T e n s i l e P r o p e r t i e s o f V a r i o u s M a t e r i a l s and T h e i r R e l a t i o n t o t h e Unnotch Flow  Curve and Notch S h a r p n e s s , T r a n s . ASM, 46, 998, 1954. 7. P. M. N a g h d i , F. E s s e n b e r g and W. K o f f , An E x p e r i m e n t a l S t u d y o f I n i t i a l and Subsequent Y i e l d S u r f a c e s i n P l a s t i c i t y , J . A p p l . Mech. 28, 1958. 8. S. P. Timoshenko, W o i n o s w s k y - K r i e g e r , T h e o r y o f P l a t e s and S h e l l s , M c G r a w - H i l l Book Co., 1959. G e n e r a l R e f e r e n c e s 1. R. H i l l , M a t h e m a t i c a l T h e o r y o f P l a s t i c i t y , O x f o r d P r e s s , 1950. 2. Mendel s o n , P l a s t i c i t y , T h e o r y and A p p l i c a t i o n , M a c M i l l a n , 1968. 

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