UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Inelastic wave propagation in metal rods Santosham, Thomas V. 1969

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1969_A1 S26.pdf [ 13.15MB ]
Metadata
JSON: 831-1.0104174.json
JSON-LD: 831-1.0104174-ld.json
RDF/XML (Pretty): 831-1.0104174-rdf.xml
RDF/JSON: 831-1.0104174-rdf.json
Turtle: 831-1.0104174-turtle.txt
N-Triples: 831-1.0104174-rdf-ntriples.txt
Original Record: 831-1.0104174-source.json
Full Text
831-1.0104174-fulltext.txt
Citation
831-1.0104174.ris

Full Text

INELASTIC IN  WAVE METAL  PROPAGATION RODS  BY  THOMAS V . B.Sc.  SANTOSHAM  University of Strathclyde G l a s g o w , S c o t l a n d , 1963  M.A.Sc. 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 , V a n c o u v e r , B r i t i s h C o l u m b i a , 1966  A  THESIS THE  SUBMITTED IN PARTIAL REQUIREMENTS  FOR T H E D E G R E E  DOCTOR OF in  FULFILMENT OF  PHILOSOPHY  t h e Department of  Mechanical  We  accept  required  THE  this  Engineering  t h e s i s as c o n f o r m i n g  to the  standard  UNIVERSITY  OF  BRITISH  February,  1969  COLUMBIA  OF  In  presenting  this  an a d v a n c e d d e g r e e the I  Library  further  for  agree  in p a r t i a l  fulfilment  of  at  University  of  Columbia,  the  make  that  it  freely  this  representatives. thesis  for  It  financial  by  the  gain  Department Columbia  shall  not  the  requirements  reference copying of  Head o f  understood that  written permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  for  extensive  granted  is  British  available  permission for  s c h o l a r l y p u r p o s e s may be  by h i s of  shall  thesis  I agree and  be a l l o w e d  that  Study.  this  thesis  my D e p a r t m e n t  c o p y i n g or  for  or  publication  without  my  ABSTRACT  Experimental r e s u l t s of  s t r a i n waves i n long rods of m i l d s t e e l  (AA2024S) and copper order 500 The  are presented on the p r o p a g a t i o n  u in/in  average  second.  (soft e l e c t r o l y t i c ) .  Strain  p u l s e s of  amplitude were generated by mechanical  s t r a i n r a t e d u r i n g l o a d i n g was  The  (type-1020) aluminum  of o r d e r one  impact.  per  l e n g t h and diameter of the specimens and the domin-  ant f r e q u e n c i e s i n the s t r a i n p u l s e s were such t h a t one-dimensional  conditions prevailed.  Budd M e t a l f i l m r e s i s t a n c e  type  s t r a i n gauges were used f o r r e c o r d i n g the s t r a i n p u l s e s . magnetic  d i s t u r b a n c e s were e f f e c t i v e l y e l i m i n a t e d by  grounding  and  Electro-  proper  shielding.  E l a s t i c wave p r o p a g a t i o n i n m i l d s t e e l , aluminum, and copper was apparent  studied.  For the s t e e l specimen, there was  a t t e n u a t i o n or d i s p e r s i o n of e l a s t i c waves.  significant  no However,  a t t e n u a t i o n and d i s p e r s i o n were observed i n aluminum  and copper specimens,  a 30 p e r c e n t r e d u c t i o n i n amplitude  a c c u r r i n g i n aluminum over a d i s t a n c e of 8 f e e t .  Comparison of  the F o u r i e r transforms of the s t r a i n p u l s e s i n copper inum a t d i f f e r e n t  and alum-  p o s i t i o n s along each specimen r e v e a l e d t h a t  amplitude decreased e x p o n e n t i a l l y w i t h d i s t a n c e and t h a t phase angle v a r i e d l i n e a r l y with d i s t a n c e . a t t e n u a t i o n and phase v e l o c i t y results  conform  Furthermore,  the  observed  were frequency dependent.  These  to the behaviour of s t r a i n p u l s e s p r o p a g a t i n g  in linear visco-elastic  materials.  Complex compliances  for  aluminum AA2024S and s o f t e l e c t r o l y t i c copper were d e r i v e d over the frequency range  400-6000 c.p.s. from the v a r i a t i o n of  a t t e n u a t i o n and phase angle measured i n these t e s t s .  Approximate  three-parameter models s u i t a b l e f o r e s t i m a t i n g i n t e r n a l damping i n these two m a t e r i a l s were a l s o  determined.  P l a s t i c wave p r o p a g a t i o n i n s t a t i c a l l y p r e s t r e s s e d rods of  aluminum and copper was  investigated.  observed t h a t s t r a i n increments propagate  In copper, i t was a t constant v e l o c i t y  along the rod and t h a t the v e l o c i t y of p r o p a g a t i o n decreases with increasing s t r a i n .  S t r a i n - r a t e independent  theory i s thus  a p p l i c a b l e t o the d e s c r i p t i o n of p l a s t i c wave p r o p a g a t i o n i n copper, but the dynamic s t r e s s - s t r a i n curve f o r the m a t e r i a l l i e s w e l l above the q u a s i - s t a t i c one.  Furthermore,  a l l y observed l o a d i n g - u n l o a d i n g boundaries i n copper  experimentresemble  the shape p r e d i c t e d by Skobeev and c a l c u l a t i o n s based on these boundaries are compatible w i t h the s t r a i n - r a t e  independent  theory. I t was  found t h a t annealed aluminum  (AA2024S-0) does  not possess a smooth q u a s i - s t a t i c s t r e s s - s t r a i n r e l a t i o n e x h i b i t s u n s t a b l e behaviour under dynamic l o a d i n g .  and  CORRIGENDA  Page  Line  i i  8 up  vi  10 u p  3 4  2 up 8, 9, 13  5  2 up  8  6 up  Change occurring linear  not accurring  not linen  parentheses  not parenthesis  discrepancy  not discrepency  propagates_  not propagate  waves  at a  number....  10  6  quasi-static  14  4  reduce  21  8  monotonically  22  6 up  propagation  55  8  condition  62  7  . ...2024S-0 w e r e  65  3  negligible  not negligable  not quansi-static  not reduces not monotonicaly  not  propagagation  not conditions less....  70  14, 15, 16  U)/2TT  not  a)  75  7, 8, 12  OJ/2TT  not  co  77  13  transient  83  13  specimens^ n o t s p e c i m e n  not transcient  86  5 up  negligible  92  3  Taylor,  G.I_. n o t T .  92  10  Griffis  not Griffs  93  17  Bell,  93  6 up  W.  93  6 up  pp.  93  Last  94  17  not negligable  J . F . n o t T.  Prager  not  Pragar  101-117 n o t 166-182  (Ed. N.J. H u f f i n g t o n , J r . ) not Journal Vol.  88 not 8  TABLE  OF  CONTENTS  Chapter  1  Page  INTRODUCTION  1  1.1  P r e l i m i n a r y Remarks  1  1.2  Review  3  1.3  Critical of  2  . . .  THEORY  of Past  This  Work  Summary  of Past  Work  and Purpose  Investigation  10  .  13  . .  2.1  Introduction  13  2.2  Wave P r o p a g a t i o n  i n an E l a s t i c  2.3  Wave P r o p a g a t i o n  i n a  Material  . . .  Visco-elastic  Material 2.4  3  3.2  16  Wave P r o p a g a t i o n  i n a Plastic  Material  . . .  21  2.4.1  Loading  21  2.4.2  Unloading  23  2.4.3  Loading-Unloading  APPARATUS 3.1  15  AND  General  INSTRUMENTATION Outline  3.1.2  Instrumentation  3.2.2  .  . . . . . . . .  System  Static  Load  29 31 35  System  a)  Loading  b)  Top and bottom  Impact  29 29  Apparatus  Mechanical  25  .  3.1.1  3.2.1  Boundary  35  device  System  35 grips  35 37  Chapter  Page  3.3  4  5  a)  Hammer  and p l a t f o r m  b)  Hammer  height  c)  Anvil  Electronic  details  adjustment  . . . .  37  . . . . .  39 41  System  45  3.3.1  Static  Load  Measuring  3.3.2  Static  Strain  3.3.3  Dynamic  3.3.4  Electromagnet.Release  3.3.5  Cathoderay Circuit  Circuit  . . . .  Circuit  Strain  45  Circuit  48 Circuit  Oscilloscope  . . . .  Recording  3.5  Test  Triggering  System  . . .  TEST  PROCEDURE  4.1  Load  4.2  Specimen  4.3  Tests  on E l a s t i c  Waves  4.4  Tests  on P l a s t i c  Waves  4.5  Repeated  4.6  Motion  51  Specimen  R E S U L T S AND  49  49  3.4  Cell  45  51 53  Calibration  53  Preparation  53  Impact  54 . . . . .  Tests  of the Platform  59 After  Impact  . . . . . .  DISCUSSION  5.1  Pre-test  5.2  Wave P r o p a g a t i o n  59  60 61  Results  61 i n a Mild  Steel  5.2.1  Elastic  Loading  5.2.2  Elastic  Unloading  5.2.3  Comparison o f E l a s t i c U n l o a d i n g Waves v  Specimen  Waves  . .  62 62  Waves Loading  65 and  66 66  Chapter  Page 5.3  Wave P r o p a g a t i o n 5.3.1  5.3.2  5.3.3  5.3.4  5.4  5.6 6  E l a s t i c Loading Aluminum  Waves  E l a s t i c Loading Aluminum  Waves  E l a s t i c Loading Aluminum  Waves  P l a s t i c Loading Aluminum  Waves  Wave P r o p a g a t i o n  Specimens  . . .  i n 2024S-T4 66 i n  2024S-F 71  i n 2024S-0 71 i n 2024S-0 72  i n Copper  Specimens  . . . .  Loading  Waves  i n Copper  . . .  73  5.4.2  Plastic  Loading  Waves  i n Copper  . . .  76  5.4.3  P l a s t i c L o a d i n g Waves i n C o p p e r Specimen Under Repeated Impact Loading  84  Unloading  85  Waves  Comparison  of Elastic  Mild  Aluminum  Steel  Acceleration OF  i n Copper Loading  and Copper  Waves i n .  86  of the Platform  87  CONCLUSIONS  90  Bibliography  Appendix  A B  92 A Typical Representation of a Linen E l a s t i c Model To Determine  «(u>)  and  Visco-  f(u>)  99  Results  104  C  Pre-test  Appendix  D  Wave P r o p a g a t i o n i n a T h r e e - P a r a m e t e r V i s c o - E l a s t i c Model  E  96  the Nature of the Functions  Appendix  Appendix  73  Elastic  SUMMARY  Appendix  66  5.4.1  5.4.4 5.5  i n Aluminum  Determining Strain at a Point i n the U n l o a d i n g Domain U s i n g t h e L o a d i n g Unloading Boundary  vi  .  110  112  LIST OF FIGURES Figure  Page  1  Loading-Unloading boundary f o r c o n t i n u o u s l y d e c r e a s i n g l o a d a t the impact end  26  2  M e c h a n i c a l system  30  3  Schematic o f i n s t r u m e n t a t i o n  32  4  Loading beam arrangement  34  5  Grip d e t a i l  36  6  Hammer-platform d e t a i l s  38  7  Mounting system f o r the electro-magnet  8  I l l u s t r a t i o n o f the anvil-hammer-platform  42  9  Load c e l l  44  .  .  40  10  Dynamic and s t a t i c s t r a i n c i r c u i t f o r two gauges  11  Electro-magnet r e l e a s e and o s c i l l o s c o p e triggering  .  circuit  47  50  12  Test section  57  13 14  Instrumentation E l a s t i c l o a d i n g waves i n m i l d s t e e l with weight o f 2.69 l b s . . . .  58 a hammer 118  15  E l a s t i c l o a d i n g waves i n m i l d s t e e l with a hammer weight o f 3.32 l b s  118  16  E l a s t i c l o a d i n g waves i n m i l d s t e e l with to d e c e l e r a t e the p l a t f o r m  the a n v i l 119  Propagation  steel  17  of s t r a i n increments i n m i l d  under e l a s t i c l o a d i n g  120  18  E l a s t i c unloading  121  19  Comparison o f e l a s t i c l o a d i n g and unloading i n mild s t e e l E l a s t i c l o a d i n g waves i n 2024S-T4 aluminum  20  waves i n m i l d s t e e l waves . . . .  122 123  Figure  21  22  23  24  25  26  27  Page  E l a s t i c l o a d i n g waves i n 2024S-T4 aluminum w i t h the a n v i l to d e c e l e r a t e the p l a t f o r m  124  Loading-unloading boundary l o a d i n g f o r 2024S-T4 .  125  Fourier transform F i g u r e 20a  of  the  V a r i a t i o n of attenuation i n 2024S-T4 aluminum  Variation in  of  2024S-T4  first  elastic  wave  in 126  V a r i a t i o n s of amplitude d i s t a n c e of the F o u r i e r aluminum  V a r i a t i o n of phase 2024S-T4 aluminum  under  and p h a s e a n g l e w i t h components i n 2024S-T4 127 factor with  frequency 128  velocity  with  frequency  in 129  complex  compliance  with  frequency  aluminum  130  28  Elastic  loading  waves  in  2024S-F  aluminum  131  29  Elastic  loading  waves  in  2024S-0  aluminum  132  30  P l a s t i c l o a d i n g waves before work-hardening P l a s t i c l o a d i n g waves a f t e r work-hardening  in  2024S-0  aluminum,  31  32  Quasi-static  133 in  2024S-0  aluminum, 134  stress-strain  curve  for  2024S-0  aluminum  135  33  Elastic  34  E l a s t i c l o a d i n g waves i n c o p p e r w i t h the anvil to d e c e l e r a t e the p l a t f o r m ( S p e c i m e n No.2) Loading-unloading boundary under e l a s t i c loading for c o p p e r ( S p e c i m e n No.2)  35  36  37  loading  waves  Fourier transform F i g u r e 34a  of  in  the  copper  first  (Specimen  wave  No.l)  ^  .  136  137 138  in 139  V a r i a t i o n o f a m p l i t u d e and p h a s e a n g l e w i t h d i s t a n c e o f the F o u r i e r components i n copper  v i i i  .  .  .  140  Figure  38  39  40  41  42  Page  Variation i n copper  of attenuation  Variation copper  o f phase  Variation i n copper  o f complex  factor  velocity  with  46  47  48  compliance  Plastic  waves  13,400  lb/in . 2  i n copper (Specimen  with No.  P l a s t i c waves i n copper w i t h 17,400 l b / i n . ( S p e c i m e n No.  frequency  a prestress 2)  . . .  of  ••  a prestress 2)  144 of 145  increments i n the x - t a p r e s t r e s s o f 13,400 2)  146  Propagation of strain plane f o r copper with lb/in . ( S p e c i m e n No.  increments i n the x - t a p r e s t r e s s o f 17,400 2)  147  Dynamic s t r e s s - s t r a i n ( S p e c i m e n N o . 2)  curves  f o r copper 148  Comparison o f dynamic and q u a s i - s t a t i c s t r a i n curves f o r copper ( S p e c i m e n No. Loading-unloading ( S p e c i m e n N o . 2)  boundary . ,  i n  P l a s t i c waves i n copper w i t h 11,600 l b / i n . ( S p e c i m e n No. Propagation of strain plane f o r copper with lb/in . ( S p e c i m e n No. 2  50  with  Propagation of strain plane f o r copper with lb/in . ( S p e c i m e n No.  2  49  in  143  2  45  frequency  142  2  44  frequency 141  2  43  with  stress2)  149  copper 150  a prestress 3)  of 151  increments i n the x - t a p r e s t r e s s o f 11,600 3)  152  Propagation of s t r a i n increments i n the x - t plane f o r c o p p e r w i t h a p r e s t r e s s o f 18,800 l b / i n . ( S p e c i m e n N o . 3)  153  Dynamic s t r e s s - s t r a i n ( S p e c i m e n N o . 3)  154  2  51  52  Loading-unloading ( S p e c i m e n No. 3)  curves  boundaries  f o r copper  i n  copper 155  ix  Figure 53 54  Page A t t e n u a t i o n o f maximum p l a s t i c s t r a i n wave amplitude i n copper (Specimen No. 3)  156  P l a s t i c waves i n copper w i t h a p r e s t r e s s o f 13,400 l b / i n . (Specimen No. 3)  157  2  55 56 57 58 59 60  61 62  Comparison o f dynamic s t r e s s - s t r a i n curves f o r copper w i t h a p r e s t r e s s of 13,400 l b / i n Comparison o f dynamic s t r e s s - s t r a i n curves f o r copper w i t h a p r e s t r e s s of 15,200 l b / i n  2  . . .  158  2  . . .  159  P l a s t i c waves i n copper under repeated impact (2nd and 6th impacts i n Specimen No. 4)  160  P l a s t i c waves i n copper under repeated impact (8th and 17th impacts i n Specimen No. 4)  161  R i s e time o f s t r a i n wave i n upper a t f i r s t s t r a i n gauge p o s i t i o n (Specimen No. 4)  162  P r o p a g a t i o n v e l o c i t y o f s t r a i n increments i n copper under repeated impacts (Specimen No. 4)  163  E l a s t i c u n l o a d i n g waves i n copper (Specimen No. 5)  16 4  Unloading waves i n copper w i t h a p r e s t r e s s of 11,600 l b / i n . 2  (Specimen No. 5)  165  63  A t t e n u a t i o n o f e l a s t i c waves i n t h r e e metals  . . .  64  A c c e l e r a t i o n o f the p l a t f o r m a f t e r impact  65  M u l t i p l e impacts o f the hammer  168  66  E l a s t i c waves i n copper  169  67  Comparison o f maximum s t r a i n amplitude recorded by the a c c e l e r o m e t e r w i t h the s t r a i n gauge readings  . . . .  (Specimen No. 4)  x  166 167  170  LIST  OF  TABLES  Table  1  2  3  Page  Variation of Rockwell t e s t specimens . .  hardness  number  on 105  L o c a t i o n o f s t r a i n gauges on t e s t specimens a n d t h e hammer h e i g h t a n d w e i g h t u s e d d u r i n g test  117a  Comparison of s t r a i n  117b  of  calculated  and  observed  values  ACKNOWLEDGEMENT  I wish t o express my s i n c e r e thanks  and g r a t i t u d e  t o my s u p e r v i s o r Dr. H. Ramsey f o r h i s constant advice and encouragement. Mr.  The author wishes t o express a p p r e c i a t i o n t o  J . Hoar, Mr. P. Hurren  and t o the e n t i r e t e c h n i c a l  staff  of the M e c h a n i c a l E n g i n e e r i n g Department, whose e n t h u s i a s t i c a s s i s t a n c e g r e a t l y a c c e l e r a t e d the r e s e a r c h programme. a p p r e c i a t i o n i s a l s o expressed graduate  t o Mr. V . I . Johannes, a f e l l o w  student, f o r h i s h e l p f u l  i n i t i a l e x p e r i m e n t a l work.  Sincere  suggestions concerning the  Thanks must a l s o be expressed t o  the Department o f Mechanical E n g i n e e r i n g f o r the use o f t h e i r facilities. Much o f the experimental apparatus  used i n t h i s  i n v e s t i g a t i o n was designed by a p r e v i o u s graduate C.J. Anderson, and i s g r a t e f u l l y acknowledged.  student Mr.  LIST OF SYMBOLS E  Young's modulus Hammer weight  Wp  Weight o f p l a t f o r m Hammer v e l o c i t y  c  E l a s t i c bar v e l o c i t y  o  J  c(u)  Phase v e l o c i t y  c(e)  Velocity  e  Coefficient'of platform  f  Phase v e l o c i t y  m  Mass o f impacting body p e r u n i t area o f the r o d  r(u))  Modulus o f F o u r i e r  t  Time  u  Displacement i n x d i r e c t i o n  v =  Particle o  of propagation of p l a s t i c  r e s t i t u t i o n between hammer and factor  components  velocity  t  x  Distance  e - 4-^—  Strain  a  Stress  p  Mass d e n s i t y  *  Attenuation  u  Circular  <>j (w)  Argument o f F o u r i e r  m.sec. y in/in  strains  factor  frequency component  Milli-seconds M i c r o inches per i n c h  strain  C H A P T E R  I  I N T R O D U C T I O N  1. 1.1  INTRODUCTION  P r e l i m i n a r y Remarks S t r e s s wave p r o p a g a t i o n  described  as e l a s t i c  in solid  or i n e l a s t i c  m a t e r i a l s c a n be  d e p e n d i n g on w h e t h e r o r n o t  the m a t e r i a l obeys H o o k e s law. 1  The  theory  o f wave p r o p a g a t i o n  a l o n g and d i s t i n g u i s h e d h i s t o r y  in elastic  m a t e r i a l s has  and many o f t h e n i n e t e e n t h  century  physicists,  Kelvin,  Lamb and L o v e a r e a s s o c i a t e d w i t h much o f t h e f u n d a -  m e n t a l work. propagation  e . g . , Young, S t o k e s ,  Poisson,  I n c o n t r a s t , r e s e a r c h on t h e t h e o r y in inelastic  m a t e r i a l s which e x h i b i t  or h y s t e r e t i c  phenomenon i s much more r e c e n t .  this  i s t h a t a knowledge o f m e c h a n i c a l  interest  inelastic  both  engineering.  are  o f wave dissipative  The r e a s o n f o r behaviour of  m a t e r i a l s a t h i g h r a t e s o f l o a d i n g h a s become o f  importance in  Rayleigh,  comparible  i n military  application  and i n i m p a c t  problems  Under h i g h r a t e s o f l o a d i n g , i n e r t i a with  forces  t h e a p p l i e d f o r c e s and p r o p a g a t i o n  waves i s an i n s e p a r a b l e p a r t o f t h e d y n a m i c r e s p o n s e  of stress  of the  material. I n s t u d y i n g wave p r o p a g a t i o n materials,  i t i s s i m p l e s t t o c o n s i d e r t h e problem under one-  dimensional  conditions.  compared w i t h be  When t h e l e n g t h o f t h e r o d i s l o n g  the diameter,  wave  s t u d i e d under o n e - d i m e n s i o n a l  used  i n this The  phenomena i n d i f f e r e n t  propagation conditions.  i n the r o d can Such r o d s a r e  investigation. concept  of e l a s t i c  wave p r o p a g a t i o n  i n rods  i s , to  2  some e x t e n t , an i d e a l i z e d one and an e x a c t mathematical  treat-  ment s h o u l d i n c l u d e the d i s s i p a t i v e process i n the m a t e r i a l . In p r a c t i c e , however, i t has been assumed t h a t as long as the s t r e s s l e v e l i s kept below the y i e l d p o i n t , l o s s e s due t o i n t e r n a l f r i c t i o n i n most metals are s m a l l and can be n e g l e c t e d . In the study of p l a s t i c s t r e s s wave p r o p a g a t i o n i n metal r o d s , two c o n f l i c t i n g t h e o r i e s have a r i s e n i n the p a s t two decades.  These  t h e o r i e s correspond w i t h the two ways i n  which metals d e v i a t e from p e r f e c t l y e l a s t i c behaviour.  First,  the r e l a t i o n between s t r e s s and s t r a i n may be n o n l i n e a r . Secondly, the r e l a t i o n between s t r e s s and s t r a i n may be time dependent.  Behaviour of many metals under s t a t i c and dynamic  l o a d i n g can be approximated behaviour.  by n o n l i n e a r , but time  Such metals are c o n s i d e r e d t o e x h i b i t  independant strain-rate  independent e f f e c t s when the s t r e s s l e v e l i s above the y i e l d point.  Many i n v e s t i g a t o r s have q u e s t i o n e d the v a l i d i t y of the  s t r a i n - r a t e independent theory due t o some o f the apparent e r r o r s observed i n c o r r e l a t i n g e x p e r i m e n t a l and t h e o r e t i c a l results.  They i n t u r n have proposed simple models which  account o f the s t r a i n - r a t e e f f e c t o f the m a t e r i a l .  take  3 1.2  R e v i e w o f P a s t Work When t h e end  sufficient yield  impact  of  m a g n i t u d e t o p r o d u c e s t r e s s e s w h i c h a r e above  p o i n t of the m a t e r i a l , p l a s t i c  simple ing  o f a r o d i s s u b j e c t e d t o an  hypothesis  i n one  strains  f o r the p r o p a g a t i o n  direction  down an  are produced.  of p l a s t i c  initially  the  waves  undisturbed  A  travel-  rod given  by  * Donnell  [1]  relation  i s as  f o l l o w s : suppose the  i s assumed t o a p p l y  each increment  of s t r e s s  /p)^/^  stress  c o n s i d e r e d , and  form  of  a x i s , such increment formal based Von  travelling  solution on  by W h i t e  and  a priori  law,  Taylor  axis.  The  the  earliest  each  The  theory  form  t h a t i t s h o u l d be experimental  wave  but  of the  verification  A  propagation  and  by  Rakhmatulin  independently is a  does not r e q u i r e governing  concave toward the  Numbers i n p a r e n t h e s i s r e f e r s the B i b l i o g r a p h y .  strain  stress  assumes t h a t s t r e s s  a = a ( e ) ,  usual  p r e v i o u s ones.  [3] i n E n g l a n d  f u n c t i o n of s t r a i n ,  except  Thus f o r t h e  i d e a s were a l s o c o n s i d e r e d  [5].  the  obtained independently  knowledge of the e x p l i c i t  strain  modulus a t  i t travels,  more s l o w l y t h a n  Similar  with  i n t e n s i o n , concave t o the  h y p o t h e s i s was  Griffis  single-valued  p the d e n s i t y .  d i s p e r s e s as  [2] i n A m e r i c a ,  travels  tangent  f o r the problem of p l a s t i c  Donnell's  Karman  i s the  relation  a wave f r o n t  [4] i n R u s s i a .  in  where ^  stress-strain  stress-strain  under the dynamic c o n d i t i o n s ; then  i n t h e wave f r o n t  velocity level  static  of the  t o numbered  stressstrain  strain-  references  rate  independent  t h e o r y was c a r r i e d  who m e a s u r e d t h e d i s t r i b u t i o n  o u t by Duwez and C l a r k  o f permanent s t r a i n  in a  wire  w h i c h h a d b e e n i m p u l s i v e l y s t r e t c h e d by a f a l l i n g  They  found  that the d i s t r i b u t i o n  between t h e p a r t i c l e  velocity  the permanent s t r a i n  behind  good a g r e e m e n t w i t h examinations the  theory  attributed up  and e x p e r i m e n t .  t h e wave f r o n t  e n d o f t h e w i r e and t o be i n r e a s o n a b l y  predictions.  A more d e t a i l i  showed some d i s c r e p e n c i e s between T h e s e d i s c r e p e n c i e s i n p a r t were  to the complicated  Many i n v e s t i g a t o r s ,  nature  o f t h e wave p a t t e r n s e t  however, have a t t r i b u t e d  discrepency t o the n e g l e c t of s t r a i n  material. is  of the r e s u l t s  weight.  and t h e r e l a t i o n  a t the impact  the t h e o r e t i c a l  copper  d u r i n g u n l o a d i n g w h i c h was n o t c o n s i d e r e d i n t h e e a r l i e r  theory. the  of strain  [6] ,  To c o n c l u d e  inadequate  rate effects  that the s t r a i n - r a t e  i n e x p l a i n i n g the m a t e r i a l behaviour  assumptions  the q u a s i - s t a t i c  that the governing  from  Stuart  the  inadequacy  Sternglass  stressed  results  [8] were i n t e r p r e t e d  the propagation  beyond t h e i r y i e l d  point.  a n c e s was t o i n c r e a s e t h e s t r e s s  disturbances  law was  [7] and S t e r n g l a s s  theory.  of small  The e f f e c t  B e l l and  already  of the d i s t u r b -  According  wave p r o p a g a t i o n ,  t r a v e l with v e l o c i t y  t o show  amplitude  b a r s w h i c h were  further.  theory of p l a s t i c  should  by B e l l  independant  d i s t u r b a n c e s i n copper  rate-independent  this  made  stress-strain  b y many i n v e s t i g a t o r s  of the s t r a i n - r a t e  investigated  longitudinal  theory  curve.  Further experimental and  of the  independent  i n v e s t i g a t i o n was r a t h e r p r e m a t u r e s i n c e t h e a n a l y s i s additional  some o f  /^da^ de '  p  to the such  where da/de i s  5 the  t a n g e n t modulus o f t h e s t r e s s - s t r a i n c u r v e .  yield  point  of the material  considerably  has been exceeded  Since the  da/de w i l l  l e s s t h a n Young's modulus f o r t h e m a t e r i a l .  p r o p a g a t i o n v e l o c i t y o f t h e wave f r o n t s h o u l d t h e r e f o r e considerably  l e s s t h a n t h e v e l o c i t y o f an e l a s t i c  Experimental observation bances  always  showed, however, t h a t  propagated with the e l a s t i c  Analysis  ial  f o r which the p l a s t i c  the  dynamic o v e r - s t r e s s ,  the  stress  a t same s t r a i n  any  wave s h o u l d p r o p a g a t e  part  by r a t e  existence  Malvern's  propagamater-  function of over  Calculations the f i r s t  a t the e l a s t i c  theory.  of uniform s t r a i n  based  part of  speed.  a t a speed  This  f a s t e r than  Calculations  however,  near the impact  end i n a  t e s t , although experiments  o f such a p l a t e a u .  The s i g n i f i c a n t  indicated  features of  s t r a i n - r a t e s o l u t i o n a r e most a p p a r e n t i n t h e x - t  I t i s f o u n d t h a t when s t r a i n  constant s t r a i n  concave  indicated that  independent  constant v e l o c i t y impact  the  test.  o f t h e p l a s t i c wave p r o p a g a t e s  showed no p l a t e a u  plane.  Q  t o a g r e e w i t h some e x p e r i m e n t a l e v i d e n c e t h a t t h e  predicted  the  bar v e l o c i t y C .  r a t e was a l i n e a r  in a static  rate-dependence  early  these d i s t u r -  i s , the excess of s t r e s s  upon l i n e a r  appeared  be  [9] f o r a r a t e - d e p e n d e n t  strain that  The  wave.  o f o n e - d i m e n s i o n a l l o n g i t u d i n a l wave  t i o n was c a r r i e d o u t by M a l v e r n  plastic  be  upward.  lines  appear  Low m a g n i t u d e  rate  i n t h e x - t p l a n e as c u r v e s  strain  increments propagate a t  h i g h e r v e l o c i t y t h a n w o u l d be p r e d i c t e d independent  d e p e n d e n c y i s assumed,  from s t r a i n  t h e o r y , and t h e h i g h m a g n i t u d e s t r a i n  rate-  propagate a t  a lower v e l o c i t y than the r a t e - i n d e p e n d e n t t h e o r y would  predict.  6  S i m i l a r r e s u l t s u s i n g a r a t e - d e p e n d e n t model were a l s o i g a t e d by S o k o l o v s k y [ 1 0 ] . More r e c e n t l y E f r o n  invest-  [ 1 1 ] has shown  t h a t when t h e rate-dependence o f t h e s t r e s s - s t r a i n r e l a t i o n i s s m a l l , t h e p l a t e a u s appear as they would i n n o n - r a t e dependent material.  S i n c e t h e s t r a i n r a t e e f f e c t must be s m a l l t o observe  the c o n s t a n t s t r a i n p l a t e a u f o r a c o n s t a n t v e l o c i t y i m p a c t , i t i s l o g i c a l t o assume t h a t t h e s t r a i n - r a t e independent t h e o r y could e x p l a i n the m a t e r i a l behaviour adequately.  The r a t e  i n d e p e n d e n t t h e o r y , however, was s t i l l open t o q u e s t i o n due t o i t s i n a d e q u a c y i n p r e d i c t i n g t h e r e s u l t o b s e r v e d by B e l l i n w h i c h t h e wave f r o n t p r o p a g a t e d a l o n g t h e b a r a t t h e e l a s t i c b a r v e l o c i t y , even when t h e b a r was s t a t i c a l l y above t h e y i e l d  pre-stressed  point.  As mentioned b e f o r e , any e x p e r i m e n t a l checks on t h e v a l i d i t y o f t h e s t r a i n - r a t e independent t h e o r y were o n l y made w i t h t h e a d d i t i o n a l assumption t h a t t h e dynamic b e h a v i o u r o f a m a t e r i a l i s governed by t h e q u a s i - s t a t i c r e l a t i o n .  The f i r s t  e x p e r i m e n t a l i s t t o consider the p o s s i b i l i t y of a separate s t r e s s - s t r a i n c u r v e f o r r a t e independent f i n i t e a m p l i t u d e wave p r o p a g a t i o n was C a m p b e l l [ 1 2 ] .  He made an e f f o r t t o d e t e r m i n e  dynamic s t r e s s - s t r a i n c u r v e s from t h e s t u d y o f n o n l i n e a r wave f r o n t s , without i n t r o d u c i n g the q u a s i - s t a t i c hypothesis. His e x p e r i m e n t s succeeded i n q u a l i t a t i v e l y d e m o n s t r a t i n g t h a t i f the r a t e independent wave p r o p a g a t i o n t h e o r y a p p l i e d i n c o p p e r , the s t r e s s - s t r a i n law was n o t t h a t o f t h e q u a s i - s t a t i c one. Campbell i n d i c a t e d t h a t h i s d a t a was s u b j e c t experimental error.  to large  7 Bell  [13], [14] has c a r r i e d out e x t e n s i v e  experiments  on p l a s t i c wave p r o p a g a t i o n u s i n g h i s own extremely technique  [15].  elegant  The p r i n c i p l e o f t h i s technique i s t o r u l e  d i f f r a c t i o n g r a t i n g s on the specimen and o p t i c a l l y observe the continuous  change i n s p a c i n g o f the  d i f f r a c t i o n gratings.  Very s m a l l changes i n the g r a t i n g s p a c i n g can then be measured. These measurements g i v e changes i n s t r a i n d i r e c t l y .  He has  shown t h a t f o r very pure aluminum which has been c a r e f u l l y annealed, no o b s e r v a b l e r a t e e f f e c t o c c u r s . t h a t the r a t e independent  He a l s o s t a t e s [14]  theory a p p l i e s t o copper,  aluminum, 4  l e a d and magnesium f o r s t r a i n r a t e s between 10/sec and 10 / s e c without  any a p r i o r i assumptions w i t h r e s p e c t t o the s t r e s s -  s t r a i n law, and without assuming t h a t the wave p r o p a g a t i o n was n e c e s s a r i l y one-dimensional. many experiments  Finally, Bell  [14] s t a t e s t h a t  which c l a i m t o show s t r a i n r a t e e f f e c t s have  been e r r o n e o u s l y i n t e r p r e t e d by q u a s i - s t a t i c  theory.  Using dynamic p r e s t r e s s s i m i l a r t o t h a t o f A l t e r and Curtis  [16], B e l l and S t e i n  [17] from d i r e c t measurements i n  a m a t e r i a l known t o be s t r a i n r a t e independent,  have shown  t h a t the reason t h a t i n c r e m e n t a l waves t r a v e l w i t h the e l a s t i c 1/2 wave v e l o c i t y r a t h e r than w i t h the v e l o c i t y that y i e l d i n g i s a discontinuous process.  [ .(do/de)/p ] Furthermore,  , is  when  the s t r e s s - s t r a i n r e l a t i o n i s measured c a r e f u l l y i t i s found t o be d i s c o n t i n u o u s and t o c o n s i s t of e l a s t i c p o r t i o n s i n t e r spread w i t h r e g i o n s of almost p e r f e c t l y p l a s t i c  yielding.  To s i m p l i f y a n a l y s i s o f s t r e s s wave propagation i n o b t a i n i n g the dynamic behaviour  of m a t e r i a l s , a number of  8 "extended  quasi-static"  developed.  [14] e x p e r i m e n t a l techniques have been  These extended q u a s i - s t a t i c experiments  always  i n v o l v e the use of s h o r t specimens, w i t h the hope t h a t somehow i n t h i s s i t u a t i o n , c o m p l i c a t i o n s of wave i n t e r a c t i o n s by r e f l e c t i o n s from the boundaries of the specimen  produced  are s m a l l .  Hence, s t r e s s and s t r a i n throughout the whole complicated dynamic process w i l l remain uniform.  Such techniques were  f i r s t d e v i s e d by T a y l o r [18], V o l t e r r a  [19] and Kolsky [20].  These methods have been used more r e c e n t l y by Kolsky and Douch  [21] , B i a n c h i [22] and Malvern  [23].  A l l their  studies  show t h a t dynamic s t r e s s - s t r a i n curves f o r most metals  differ  from q u a s i - s t a t i c t e s t s observed i n o r d i n a r y mechanical machines.  testing  D i f f e r e n c e s , however, f o r annealed aluminum and  copper are found t o be q u i t e s m a l l . one aluminum a l l o y t h e r e was  Kolsky and Douch found f o r  no measurable  the s t a t i c and dynamic c u r v e s .  d i f f e r e n c e between  K o l s k y , Malvern and B i a n c h i  have a l s o shown t h a t use of rate-independent theory with such a dynamic s t r e s s - s t r a i n r e l a t i o n g i v e s r e l i a b l e p r e d i c t i o n s of the observed p l a s t i c wave p r o p a g a t i o n . Anderson  [24] i n v e s t i g a t e d dynamic behaviour of metals  by s i m u l t a n e o u s l y o b s e r v i n g f i n i t e t r a n s i e n t p l a s t i c waves a t number o f p o s i t i o n s on r e l a t i v e l y l o n g r o d s .  Although h i s  e x p e r i m e n t a l apparatus proved to be extremely u s e f u l f o r i n v e s t i g a t i o n s of s t r e s s wave propagations i n long rods, the s t r a i n waves observed i n the o s c i l l o s c o p e had unwanted n o i s e superimposed  on them.  Hence, no c o n c l u s i v e evidence  was  o b t a i n e d t o prove the a p p l i c a b i l i t y o f s t r a i n r a t e or r a t e dependent theory f o r m a t e r i a l he C l i f t o n and Bodner of one-dimensional  independent  used.  [25] p r e s e n t e d a t h e o r e t i c a l  rate-independent  analysi  theory f o r e l a s t i c - p l a s t i c  wave p r o p a g a t i o n w i t h a smooth s t r e s s - s t r a i n curve concave toward the s t r a i n a x i s .  T h i s was  a p p l i e d t o a problem of a  long uniform bar loaded a t one end by a p r e s s u r e p u l s e of s h o r t duration. infinite is  They o b t a i n e d the s o l u t i o n s f o r the case of a  semi-  bar and f o r the case of a f i n i t e bar whose other  end  s t r e s s - f r e e , by u s i n g the methods of c h a r a c t e r i s t i c s i n the  x-t plane. boundaries  The  a n a l y s i s a l s o i n c l u d e d the g e n e r a l shape of  i n the x - t plane which separate r e g i o n s governed by  the dynamic e l a s t i c equations p l a s t i c equations.  To compare the theory w i t h  r e s u l t s Bodner and C l i f t o n  temperature  experimental  [26] i n v e s t i g a t e d wave propagation  of e l a s t i c - p l a s t i c p u l s e s due of l o n g , annealed,  from r e g i o n governed by dynamic  t o e x p l o s i v e l o a d i n g a t one  commercially  end  pure, aluminum rods a t room  and at e l e v a t e d temperature  up t o 750°F.  Stress  waves were d e t e c t e d by a condenser microphone at the f a r end of the rod and,  i n some cases, by s t r a i n gauges a t a c r o s s s e c t i o n  d i s t a n t from the impact  end.  Experimental r e s u l t s  indicated  t h a t e s s e n t i a l f e a t u r e s of the recorded v e l o c i t y - t i m e p r o f i l e s and s t r a i n - t i m e p r o f i l e s are i n agreement with the p r e d i c t i o n of r a t e independent  elastic-plastic  theory.  R e c e n t l y , S c h u l t z [27] i n v e s t i g a t e d m a t e r i a l behaviour of 1100  aluminum s u b j e c t to impact  loading.  He observed  the  10 dynamic strain  stress-strain and deformation  created His  theory,  relation  departed  percent  pure  aluminum  relation  strain  waves i n such  levels  propagate  presented Hopkins  that  does  such  olies  slowly  should  propagation depends material avoid  rates  wave  Work  a smooth  stress-  Furthermore,  discrete  stress  have  been  [31], Craggs [32],  and Purpose  dependence plays  i n recent  taking  measured,  many  and the" r o l e  i n e x p l a i n i n g t h e anomhave  years.  Under  account  materials.  of  o f metals  propagation  o f the dynamic  i s being  difficulty  99  waves [ 2 9 ] .  Lee  of loading.  be a n a l y s e d  on a knowledge  this  curve.  annealed  behaviour.  i n v o l v e d i n measuring  i n inelastic  which  stress-strain  wave p r o p a g a t i o n  dependence  i n plastic  to the d i f f i c u l t y  results  rate-indepen-  [34] a n d Symonds e t . a l . [ 3 5 ] .  of strain-rate  at high  front  wire.  stress-strain  not possess  et.al.[30],  considerable controversy  relations  on a  the  shown t h a t  as shock  of plastic  strain-rate  observed  using  the dynamic  unstable  C r i t i c a l Summary o f P a s t this Investigation  that  due  analysed  impact  a material at certain  by Abramson  Studies  of  (1100)  very  [33], Kolsky  transverse  [28] h a v e  and e x h i b i t s  surveys  t h e t r a n s v e r s e wave  the quansi-static  and D i l l o n  strain  Many  were  concluded  from  o f t h e m a t e r i a l by r e c o r d i n g  behind  velocity  results  and he  Kenig  1.3  angle  by a c o n s t a n t  experimental  dent  behaviour  been  This  conditions,  of stress this,  experimental  wave  i n turn,  stress-strain  one moves  i s partly  stress-strain these  Since  subjects  law o f t h e  i n a circle.  investigators  have  To  11 introduced nonlinear  "extended wave  neglected. open  [14] a s s u m p t i o n s  reflection  obtained  under  strain-rate  does  are  conditions are  stress-strain  strain  axis.  relations:  law o t h e r  constant;  b)  constant  wave  ity.  check  To  wave  Solutions using  a) w a v e  speed  that  propagation  these  this  relation  between  of Taylor  s t i l l  than  that be  theory  strain  concave  positions  must  toward  two  of strain i n terms  should  of  these  and p a r t i c l e  conditions are satisfied,  at different  be  form  stress i s  predict  level  exists,  dynamic  and Von  of the e x p l i c i t  and s h o u l d  c(e) f o r each  an i n t e g r a l  speeds,  theory  knowledge  single-valued function of strain  be  and i n t e r a c t i o n  such  independent  not require a priori  the governing  the  i n which  to question.  Karman  a  propagation,  Results  The  of  quasi-static"  veloc-  transient  observed  simultaneously. Bell, wave  propagation  Campbell  results,  rods.  propagation  r e s i s t a n c e type  wire  strain  has used  have  test  b u t the form  i s n o t known.  waves,  i t i s difficult  shown  Both that  system  at only  Although  i s an  and B e l l ,  the behaviour  Bell's  extremely  up t o 4 f e e t  o f t h e l o a d i n g wave  two  His  t o use t h e method  specimens  Bodner  transient  gauges.  to large error.  grating  the  simultaneously.  wave  technique,  investigation,  observed  the transient  of a diffraction  Bodner  have  positions  therefore, are subject  technique elegant  using  and Bodner  at different  observed  positions  end  Campbell  on  long  a t the  by o b s e r v i n g  of the material  long i nh i s impact  transient they  t e s t e d may be e x p l a i n e d by u s i n g the s t r a i n - r a t e theory.  Anderson,  independent  [24] by o b s e r v i n g the t r a n s i e n t wave a t  f o u r p o s i t i o n s s i m u l t a n e o u s l y , demonstrated t h a t h i s apparatus may be used  f o r s t u d y i n g s t r e s s wave p r o p a g a t i o n i n rods.  T h i s i n v e s t i g a t i o n was undertaken ing  as p a r t of a c o n t i n u -  programme i n s t u d y i n g p l a s t i c s t r e s s wave p r o p a g a t i o n i n  s e m i - i n f i n i t e rods.  I n i t i a l l y , equipment o r i g i n a l l y  designed  by Anderson was m o d i f i e d t o g i v e a s t r a i n wave amenable t o t h e o r e t i c a l a n a l y s i s and t o r e c o r d the wave w i t h no i n t e r f e r ence from e l e c t r o - m a g n e t i c d i s t u r b a n c e s .  F i n a l l y , change i n  shape o f the p r o p a g a t i n g e l a s t i c and p l a s t i c s t r a i n waves i n m i l d s t e e l , aluminum and copper were i n v e s t i g a t e d ally.  Furthermore,  observed ally.  experiment-  using a s u i t a b l e c o n s t i t u t i v e equation,  change i n shape o f the waves was analysed  theoretic-  C H A P T E R  T H E O R Y  I I  2.  2.1  THEORY  Introduction When t h e wave l e n g t h o f l o n g i t u d i n a l waves i n a r o d i s  large  compared  process  t o the diameter  o f t h e r o d , t h e wave  c a n be t r e a t e d a d e q u a t e l y  which n e g l e c t s l a t e r a l  inertia.  propagation  by a o n e - d i m e n s i o n a l The  equation  theory  of motion i s  then  The  3a  av  3x  at  particle velocity  continuity  v  i s related  t o the s t r a i n  =  f o l l o w s from  (2.1.2)  at  the r e l a t i o n s  e  = ~ o  particle velocity  (2.1.1) ly  by t h e  -ll  3x  The  e  equation,  fl  which  (2.1.1)  v  and ( 2 . 1 . 2 ) .  and  =  — d  c a n be e l i m i n a t e d by  The e q u a t i o n  v  X  relating  . u  combining a  and  e  direct-  is  2  The formulation  2  a a  3  ax  3t  additional  e  e q u a t i o n which completes  of the problem  i s the c o n s t i t u t i v e  (2.1.3)  the  mathematical  equation  of the  material. and  This  d e p e n d s on  the r e l a t i o n pendent, There  between s t r e s s  a r e , h o w e v e r , two elastic  the r e l a t i o n d e p e n d on strain  and  ways i n w h i c h  and  be  time dependent  c r e e p and  stress  show b o t h  types of d e v i a t i o n  strain  nonlinear plasticity There  relaxation  i s n o n l i n e a r and time  on  1  behaviour  have s t r e s s - s t r a i n differential termed  linear  and  b e h a v i o u r and  the assumption  with  solids  Hooke's law. be  r e l a t i o n s which  equation involving visco-elastic  First,  n o n l i n e a r and between  Many  stress as  materials  i . e . , the  Many m e t a l s  stress-  exhibit  the simple t h e o r y of  approximate  independence.  to l i n e a r ,  time  small deformations  c a n be d e s c r i b e d by  time.  inde-  law.  s u c h phenomenon  law;  If  d e v i a t e from  of such time  for sufficiently  Such m a t e r i a l s  a  linear  are  solids.  In o r d e r t o t r e a t semi-infinite  time.  e ,  time 1  the r e l a t i o n  from Hooke s  which  and  to Hooke s  become i m p o r t a n t .  a r e a l s o many p l a s t i c s  dependent  by  so t h a t  d e p e n d s oh  independent  i s based  i s linear  s t r a i n may  Second,  strain  of the m a t e r i a l .  real  behaviour described  loading path.  may  strain  and  equations reduces  between s t r e s s  the  a  stress  the mechanical p r o p e r t i e s  the c o n s t i t u t i v e  perfectly  and  equation relates  the problem  r o d s , the c o n s t i t u t i v e  the equations of motion  A suitable  boundary  condition  introduced  to solve  specific  (2.1.1)  o f wave p r o p a g a t i o n i n  equation and  must be  continuity  a t t h e l o a d i n g end problems.  combined  (2.1.2).  must t h e n  be  2.2  Wave P r o p a g a t i o n The  i n an E l a s t i c  constitutive  equation  Material  i s given  by t h e Hooke's  law  relation.  a  where  Equation to  E  =  (2.2.1)  the familiar  =  Young's  E  e  modulus  c a n be combined one-dimensional  with  The  solution  of  where  and  g, G  ditions. the  direction  ponds the of  velocity frequency,  dispersion.  3 u z  =  E  (2.2.2)  r  3x  may  be w r i t t e n as  u  =  g ( c t-x) o  c  =  / E/p  g  travelling  of propagation t h e wave  + G ( c t+x) o  (2.2.3)  f u n c t i o n s d e p e n d i n g on t h e i n i t i a l corresponds  of increasing  t o a wave  leads  equation,  (2.2.2)  function  and t h i s  2  3 u 5. 3t  are arbitary The  (2.1.3)  wave  2  P  (2.2.1)  x, w h i l s t  t o a wave  C  q  travel's  travelling in  the function  i n the opposite  G corres-  direction.  (2.2.3)  con-  given  by  i s  along  the rod without  Since  independent any  The  stress  a,  hence d i s p l a c e m e n t , along  2.3  attenuation  Wave P r o p a g a t i o n  material  E and is  e  a  =  i s the F o u r i e r  5  =  /a(x,t)  E*, which i s f u n c t i o n convenient  to write  E j and E  2  be w r i t t e n  as  (2.3.1)  transform of s t r e s s  f"  /ir  exp  a(x  (iut) dt,  f  t)  (2.3.2)  transform of s t r a i n  e  (x , t )  c o , i s t h e complex m o d u l u s .  of  E* i n t e r m s o f i t s r e a l and  It  imaginary  respectively,  E*  The  dispersion.  E* e  i s the corresponding Fourier  parts,  (2.2.2);  constitutive relation for a linear visco-elastic  1  --  satisfy  i n a Visco-elastic Material  a  i  also  and w i t h o u t  u n d e r u n i a x i a l s t r e s s may  where  e  s t r e s s , and s t r a i n waves a l l p r o p a g a t e  a rod without  The  and s t r a i n  complex compliance  =  j *  Ei + i E  i s defined  2  .  (2.3.3)  by  (2.3.4) E* The  r e a l and i m a g i n a r y  part  of  J*  are designated  by  J  and  j  , giving  J*  =  J  + i j 1  j  and 1  j  are both  positive  (2.3.5)  2  u>  when  i s r e a l and p o s i t i v e .  2  (See A p p e n d i x A.) An can  ordinary d i f f e r e n t i a l  equation which  be o b t a i n e d b y a p p l y i n g t h e F o u r i e r  (2.3.2)  t o (2.1.3),  and t h e n  (2.3.1) a n d ( 2 . 3 . 4 ) .  e  satisfies  t r a n s f o r m a t i o n as i n  substituting  e/ * J  fora  from  Hence  dl 2  dx  =  and, f o r a wave p r o p a g a t i n g the  solution  x = o,  e  Q  (u>)  and  2  and  2«f  f  x,  as  e (to)  =  exp(-*+if)x  0  i s the Fourier <*  (2.3.6)  i n the d i r e c t i o n of i n c r e a s i n g  o f (2.3.6) i s t a k e n  e  where  - po) J*e  2  (2.3.7)  t r a n s f o r m o f t h e s t r a i n wave a t  a r e g i v e n by t h e r e l a t i o n s  =  pco J 2  2  (2.3.8)  18  f  Finally, obtain  e  (  x  ,  t  2  —  =  2  by means o f t h e  from  inverse Fourier transformation,  I e U)  jjzj-  )  that  must a l w a y s be f  must be  function  co .  of  and  positive  for  strain  alternative  Fourier  We  x ,  and  and  must be  f  are  i n Appendix  an  co ; odd  establish-  B.  o f the F o u r i e r t r a n s f o r m a t i o n of  for treating  later.  for  the experimental  s t a r t with  data to  the u s u a l statement  of  be the  transformation,  e(x,o))  =  / /2T  Since  form  result  an e v e n f u n c t i o n o f  «  expansions  (2.3.10)  of i n c r e a s i n g  positive,  These p r o p e r t i e s of  i s useful  presented  must be co  ] dco  (fx-t)  a meaningful  i n the d i r e c t i o n  positive  we  strain,  [-«x+iu  (2.3.10) y i e l d  e d by means o f power s e r i e s An  exp  0  a s t r a i n wave p r o p a g a t i n g  also,  (2.3.9)  2  (2.3.7) t h e e x p r e s s i o n f o r  In o r d e r  «  pco J  e'(x,t) = o  velocity  e(x,t)  exp  (icot)  dt  .  (2.3.11)  -  for  of propagation  t < x/c  where  c  o f t h e wave f r o n t ,  i s the  observed  i t i s convenient  to  19 introduce  a new  variable  t  t'  and  to describe  related  to  strain  e(x,t)  e  1  1  ,  =  t - |  by a new  (2.3.12)  function  e'(x,t')  which i s  by  (x,f)  =  e(x,t'+-) .  (2.3.13)  c  Now  e  1  (x,t')  e(x,u>)  =  = o  exp (icox/c) /  Equations  I  0  for  [-°cx+i  /E  1  ,  and  (x,t')  (2.3.11) c a n be  exp(iu)t')  dt'  rewritten  (2.3.14)  2TT  (2.3.7) and  (to) exp  t'<o  (2.3.14) c a n be combined t o  (f--)x]  =  C  — /~27  e'Uft')  give  exp  (icot ) d f 1  oi  (2.3.15)  N e x t we w r i t e polar  e  0  (w)  and t h e r i g h t  hand s i d e  of  (2.3.15) i n  form,  e  0  (OJ)  =  r  Q  (oi)  exp  [i<f> (u))J 0  (2.3.16)  20 and  'e'(x,t')  1  exp  (lot )  dt  1  1  =  r(x,to)  exp  [i<j>(x,oj)] (2.3.17)  We  note  that  4>(x,oo) and  r  (oo)  o  and  respectively  (2.3.17)  into  at  (2.3.15)  d> (OJ)  are the values  0  x = o  .  lends  to  r(x,(D)  =  r  -(a))  + (X,OJ)  =  • (u)+  Substitution  exp  o f r(x,cj) from  and  (2.3.16)  (-<*x)  (2.3.18)  and  when m o d u l i a n d a r g u m e n t s that  i n a linear  Fourier  dependent;  angle  expression  equations  the amplitude  are also  frequency  dependent vary  with  velocity x  i s  i s known  oj/f . f  indicate of  and decay i s  angles  The phase  phase  These  the phase  with of  (2.3.19)  also  which  .  material,  x  o f waves decay e x p o n e n t i a l l y  components x  (f-3')  are equated.  visco-elastic  components  frequency  0  of the Fourier linearly  I f the rate  c a n be c a l c u l a t e d  of  change  from the  21  f  =  !± + fr (x,.ai) dx  c  2.4  Wave P r o p a g a t i o n  2.4.1  •  d(  (2.3.20)  "  in a Plastic  Material  3o ( -r-r- > o)  Loading:  ot  The c o n s t i t u t i v e  e q u a t i o n 'of t h e m a t e r i a l may  be w r i t t e n  as,  a  =  where F i s a m o n o t o n i c a l y assumed t h a t  for a l l  F(e)  increasing  e ,  ^  (2.4.1)  function  of e .  is a monotonicaly  It is  decreasing  function. 1  a  Figure  After  (2.4.1) i s i n t r o d u c e d i n t o  2.1  (2.1.1), the equation of motion  becomes 8 u 2  1 da  3 u  p de  3x  2  = 3t  2  (2.4.2) 2  22 This  i s a quasi-linear  given  wave  equation  with  the  characteristics  by  dx <— dt  =  +  c(e)  (2.4.3)  where  c (e)  The  differential  1 -  =  2  p  relations  da — de  (2.4.4)  s a t i s f i e d along  the  characteristics  are  dv  The  upper  each  =  and lower  +  signs  i n (2.4.3)  and  (2.4.5)  (2.4.5)  correspond  to  other. From  gation  (2.4.4)  of a strain  i t follows  increment  slope  of the stress-strain  which  i s concave c(e)  decreases  strain  a t the impact  the strain  that  curve.  when  the velocity  i s constant  to the s t r a i n  tion  then  c ( e ) de  and i s governed  For a stress-strain  axis,  the velocity  the s t r a i n  increases.  end o f t h e r o d i n c r e a s e s  increments  of propaga-  generated  by t h e  relation  of propagaThus, i f  continuously,  successively  a t the end  23 w i l l propagate w i t h c o n t i n u o u s l y d e c r e a s i n g v e l o c i t y .  Further-  more, c o r r e s p o n d i n g wave f r o n t s w i l l be r e p r e s e n t e d i n the x-t  plane by a d i v e r g e n t f a m i l y of s t r a i g h t l i n e s whose slope  will 2.4.2  increase with  strain.  Unloading:  ('-TIT  o  -  O)  t  For most metals, the u n l o a d i n g process i s e s s e n t i a l l y p e r f e c t l y e l a s t i c as i l l u s t r a t e d below. 1  CT  o  B  (x)  e  e  m Figure  2.2  Unloading which begins a t p o i n t A takes p l a c e along the s t r a i g h t l i n e AB w i t h a s l o p e g i v e n by the Young's modulus of the m a t e r i a l . The c o n s t i t u t i v e equation f o r d e s c r i b i n g unloading i n the rod i s g i v e n by  24  o (x) m  At each s e c t i o n  x ,  a m  and  [e-E  + E  (x) a n d m .  M  (x)]  e (x)  (2.4.6)  a r e t h e maximum  stress  s t r a i n r e l a t i n g t o p o i n t A. When  of.motion  (2.4.6) i s i n t r o d u c e d i n t o  the equation  becomes  3 u 2  3t  (2.1.1),  3 u 2  2  3x  2  The c h a r a c t e r i s t i c s  2  1 + p .  d o  m  - c  ( x )  2 d e 0  _ dx  m  (2.4.7)  ( x )  dx  o f (2.4.7) a r e  dx +  c  (2.4.8)  o  dt  where  c  The r e l a t i o n s  d  v  o  =  /_,  /  (2.4.9)  E / P  satisfied  along the c h a r a c t e r i s t i c s are  =  de + - i pc  + c  G  da  m  - c  0  de  m  (2.4.10)  0  The u p p e r and l o w e r s i g n s o f (2.4.8) and (2.4.10) to  each o t h e r .  correspond  2.4.3  Loading-Unloading By  definition,  geometrical giving bar. in  locus  Boundary the loading-unloading  of points i n the c h a r a c t e r i s t i c  t h e t i m e o f maximum s t r a i n Skobeev  the  occurs  x-t  boundary i s the  [36] h a s p r o v e d plane  above t h i s  x-t  a t each s e c t i o n  the existence  f o r continuously  x  of this  decreasing  plane  of the boundary  load.  Unloading  b o u n d a r y and t h e l o a d i n g b e l o w i t , i n a  semi-infinite bar. The related  strain  history  at a section  to the loading-unloading  x  o f t h e b a r c a n be  boundary.  Let  4- (x)  t  represent (Figure  the loading-unloading  1).  a time  t = x_  From t h i s until  moment  t = *Mx) .  section as  x  stress  ,  elastic  has e l a p s e d  wave r e a c h e s  At this  x-t  time,  increases  the f i r s t  plane  the s e c t i o n  since the beginning  onwards, t h e s t r a i n  and t h e e l a s t i c  the exact  known, s t r a i n  after  o f impact.  continuously  unloading  component o f s t r a i n  wave  then  shape o f t h e l o a d i n g - u n l o a d i n g  a t any p o i n t i n t h e u n l o a d i n g  calculated  analytically.  the  at a given  strain  observed  boundary i n t h e  reaches  decreases  decreases.  If is  The f i r s t  (2.4.11)  This  section  loading-unloading  domain may be  a p p r o a c h was u s e d x  boundary  i npredicting  from the e x p e r i m e n t a l l y  boundary.  The s t e p s  involved i n  this  calculation  are described  Two a r b i t a r y  points  P^x^tj)  t a k e n on t h e l o a d i n g - u n l o a d i n g characteristic domain.  of negative  Similarly  drawn  through  point  P(x,t)  P  2  =  x-x  x  =  -  from the unloading  of positive  slope i s  intersect  (2.4.8) i t f o l l o w s  at a  that  c (t-t )  (2.4.13)  2  c  (2.4.13).  Along  From  P i , the  (2.4.12)  0  and  1  (2.4.10) i s s a t i s f i e d a l o n g  the p o s i t i v e  o f . (2.4.10) a p p l y .  characteristics  Through  i s drawn  are  2  c (t-t )  differential relation  in  slope  1).  The  upper s i g n s  boundary.  P2(x2/t )  T h e s e two c h a r a c t e r i s t i c s  (Figure  x-x  and  the c h a r a c t e r i s t i c .  2  below.  characteristic  Similarly  (2.4.13), t h e lower  signs  along  (2.4.12)  (2.4.12) t h e t h e lower  apply.  The  differences  p a r t i c l e v e l o c i t i e s may be w r i t t e n as  v-v  2  =  c (e-e ) 0  2  +  [a (x)-a (x )] -c m  m  2  0  [e (x)-e (x )] m  m  2  (2. 4,. 14)  v-vi  =  -c (e0  £ l  )  - ~ -  I%( )-^ ( l) J + ° x  m  x  c  [  e  m  ( >x  c  m  ( l>] x  (2.4.15)  Eliminating  v  from  •'  (2.4.14) and ( 2 . 4 . 1 5 ) , a n d t a k i n g  into  . account and  e  that, -  2  =  E ( x ) , we may w r i t e m  v  If  side can  2  2  _  1  — 2E  Q  from  using  be  E J = e(xj), e  as,  (2.4.16)  2  A  b o u n d a r y and t h e v a r i a t i o n o f  known e x p e r i m e n t a l l y ,  (2.4.4).  evaluated.  for  8  [2o ( x ) - a ( x ! ) - o ( x ) •] + e ( x ) m m m ^ m  (2.4.5) and  o f (2.4.16)  boundary,  an e x p r e s s i o n  the loading-unloading  e are both  calculated obtained  v  r  2c  c(e)with  on t h e l o a d i n g - u n l o a d i n g  2  a (x) , m  Then e v e r y  Vj  a (xi) m  and and  x  v  2  may be  o ( x ) may be m ^ 2  J  q u a n t i t y on t h e r i g h t  i s known, a n d t h e s t r a i n h i s t o r y  at  hand  P(x,t)  C H A P T E R  A P P A R A T U S  AND  I I I  I N S T R U M E N T A T I O N  3.  3.1  General  3.1.1  APPARATUS  Apparatus: apparatus  built  studying  incremental stress  stressed  rods  [8].  was  similar  apparatus  mechanical  part  stress to  grips,  the specimen,  1/4"  i n diameter Strain  the  hammer  amplitude the  anvil. which  used  rods  which  which  i s situated  feet  this  The  nearly  below the  i n the specimen  were  by  attached t o the specimen.  by v a r y i n g  the drop  i t i s possible  dropping  The wave  h e i g h t a n d mass o f i s limited  t o produce  i s a t t a c h e d t o the specimen  from  the lower  ing  waves c a n be o b s e r v e d  and  u n l o a d i n g waves i n t h e lower a t the lower  i s attached  by t h e  a wave  form  rectangular.  platform  o f t h e way  pre-  long.  waves a r e p r o d u c e d  system  con-  self-  i n the present investigation  20  to  2, a n d  a dead-load  The d i s p l a c e m e n t o f t h e p l a t f o r m  i s very  platform  a platform  made The  self-aligning, applying  and  were  i s shown i n F i g u r e  beam.for  and an a n v i l  i s controlled  With  one-third  by S t e r n g l a s s  and improvements  a hammer,  on t h e p l a t f o r m  hammer.  used  and lower  a loading  [24] f o r  propagation i n statically pre-  to that  o f upper  Specimens  by Anderson  f o r the present investigation.  t o the specimen,  platform.  wave  of the apparatus  essentially  tightening  previously  Some m o d i f i c a t i o n s  Anderson's  sists  INSTRUMENTATION  Outline  The  Stuart  AND  end o f t h e specimen.  i n the upper  one-third  at a  portion  portion. point  of the  The l o c a t i o n  i s optimum  point Load-  specimen, of the  f o r studying  n J3  30  Coarse  Adjustment  Fine Adjustment Lower  Grip  P i n Connections  Loading Beam  Figure  2  M e c h a n i c a l system  Lateral Guide  loading grips tion  waves b e c a u s e r e f l e c t e d waves f r o m t h e  a r r i v e simultaneously i n the  upper p o r t i o n  i n g waves c a n the  be  of  rod.  means o f  of  resistance  static  strain  curing  test  p r o p a g a t i o n of of  8 feet  read  one  the The  epoxy.  secload-  without  by  a m u l t i - p o s i t i o n two-pole  Two-arm b r i d g e s  SR-4  with are  located  in  the  strain  indicator is switch  by  accomplished  located  in  the  box. Either  switched  of  the  two  sets  i n t o dynamic c i r c u i t s  switch  bridges  specimen  a B a l d w i n SR-4  Switching  control  taken  t w e l v e gauges u s e d on  indicator. means o f  are  dummy g a u g e s b e i n g  a t a t i m e on i n t o the  records  bonded t o t h e  Budd GA-2  central control unit.  specimen are  strain  gauges  u s e d i n a l l gauge c i r c u i t s ,  are  amplifier  located  and  s i x BAM-1  cope t r a c e s  meter  units using  four  gauges u s i n g  s i x a c t i v e gauges a multiposition  c o n t r o l box.  to s t a t i c a l l y  The  calibrated E l l i s  are  each impact.  recorded m.m.  565  gauges s i m u l t a n e o u s l y .  six  The  bridge-  static outputs from  used  and  one  for recording  Single-sweep traces  can  be  used  Tektronix the are  strain from  oscillos-  s i n g l e - l e n s r e f l e x camera.  o s c i l l o s c o p e which  are  the  photographically  a chopping device,  together  are  s i x dynamic h a l f -  (BAM-1) u n i t s w h i c h p e r m i t  a 35  trace Tektronix  of  through  balanced before  dual  oscilloscope  i n the  connected  component t o be the  the  r e f l e c t e d waves.  d y n a m i c and  room-temperature  pole  The  of  lower  Instrumentation: The  the  the  extremities  observed over a distance  interference  3.1.2  a t the  u p p e r and  One for 502-A  s i x dynamic displayed  on  Strain Gauge To Electro  Magnet BAM-1 Meter  BAM-1 Meter  BAM.l Meter • —  1  '  BAM-1 Meter  BAM-1 Meter  BAM.l Meter  Striking Platform  5 R-4 Tektronix - 5 6 5 Oscilloscope  Strain Indicator  External  Figure  3  Trigger  Schematic  of Instrumentation  Tektronix 502 A Oscilloscope  the  oscilloscope  the  instant  screen  of impact.  by t r i g g e r i n g t h e h o r i z o n t a l With t h i s  instrumentation  sweep a t  the response  f r o m any number o f gauges up t o s i x c a n be r e c o r d e d  simultan-  eously. A in  Figure  schematic  diagram of the instrumentation  3.  i  i s shown  1. T e s t  specimen  5. Frame t o f a s t e n t h e s c r e w  2. Lower g r i p 3.  jack  Screw-jack  6. L o a d i n g beam 7.  4. L a t e r a l g u i d e  Figure  4  t o the concrete  L o a d i n g beam  Weights  arrangement  floor  35 3.2  M e c h a n i c a l System  3.2.1  S t a t i c Load System: a) Loading The  [24]  device l o a d i n g system used was designed by Anderson  and i n c o r p o r a t e s  a simple-beam l o a d i n g mechanism.  I t was  designed t o p r e s t r e s s a 1/4" diameter r o d s t a t i c a l l y up t o a s t r e s s o f 100,000 p s i .  The l o a d i n g beam has a 10 t o 1 mechanical  advantage and r e q u i r e s a l o a d of 500 pounds t o achieve the above stress l e v e l .  The fulcrum  by means o f a connecting  o f the beam i s connected t o a bracket  pin.  The bracket  r i d e s i n a bearing  i n the lower end o f the male p o r t i o n o f a screw j a c k . female p o r t i o n o f the jack i s f a s t e n e d i s anchored t o the concrete  floor.  The  r i g i d l y t o a frame which  The l o a d i n g beam overhangs  the p i v o t p o i n t and i s guided by means of r o l l e r s t o p r o v i d e lateral stability An Figure  f o r the beam.  i l l u s t r a t i o n o f the above assembly i s shown i n  4.  b) Top and bottom g r i p s The  top and bottom o f the 20 f o o t t e s t specimen are  connected t o the l a b o r a t o r y r o o f and l o a d i n g beam r e s p e c t i v e l y by means o f s e l f - a l i g n i n g g r i p s . tapered  Each g r i p c o n s i s t s of a  c i r c u l a r s e c t i o n made of Keewatin t o o l s t e e l hardened  to Rockwell C50.  As the l o a d i s a p p l i e d t h i s s e c t i o n i s drawn  i n t o a mating tapered  section.  In i t s f i n a l  g r i p , which i s threaded i n t e r n a l l y t o p r o v i d e surface, i s s p l i t  i n t o three  segments.  form the tapered a good g r i p p i n g  The mating tapers are  36  Pin Connect ion  Load C e l l S e c t i o n  «  G r i p Body  Inner Taper  Outer Taper  «  Grip  cap  Specimen  Figure  5  Grip  detail  37 clamped t o g e t h e r The in  turn  i n t h e main body as shown i n F i g u r e  upper g r i p i s pin-connected  i s pin-connected  to a piece angles  to a bracket  of circular  tubing.  Since  two p i n s  the  s p e c i m e n due t o m i s a l i g n m e n t  of the grips i s eliminated.  The  circular  four  tube has h o l e s  every  the tube-bracket-grip  rigidly  bolted  3.2.2  roof.  incorporates  and p l a t f o r m  details  hammer c o n s i s t s o f a aluminum h e a d a n d a  a l u m i n u m arm w h i c h  i s pivoted  i s equipped with  on a r o l l e r  two h a r d e n e d h i g h  impact  t r a n s m i t t i n g p a d s , l o c a t e d 36 i n c h e s  point,  and i s d e s i g n e d  of weights.  The hammer's p i v o t p o i n t  adjustment  a fine  screw adjustment a l l o w i n g  lateral the  and l e v e l l e d  The impact  through  carbon  seven  before  The  steel  from the p i v o t  i s equipped with  feet of v e r t i c a l  pounds a  movement a n d  t h e hammer t o be a c c u r a t e l y  each  impact.  movement o f t h e hammer, s t a y w i r e s  hammer h e a d  steel  allowing  light  bearing.  f o r t h e a d d i t i o n o f up t o t h r e e  coarse  positioned  two p i n s a t  System:  Hammer  hammer h e a d  which a r e used t o  arrangement t o a p l a t e which i s  lower g r i p s i m i l a r l y  The tubular  any b e n d i n g o f  t o e l i m i n a t e bending o f the specimen.  Impact a)  inches  t o beams on t h e l a b o r a t o r y  The right-angles  t o each other  which  the  fasten  are at right  5.  To r e d u c e t h e are attached  between  and p i v o t i n g s h a f t . platform  i s f a b r i c a t e d f r o m aluminum w i t h  r e c e i v i n g s c r e w s c o n n e c t e d t o two s t e e l  two d i f f e r e n t i a l  s c r e w s as shown i n F i g u r e  aluminum p l a t f o r m has a t a p e r e d  hole  at i t s centre  6.  two  inserts, The  and mates  38  Additional Weights  Hammer Head  Impact Transmitting Pad  Threaded Steel Insert Piece  Lock Nut  Impact  Receiving Screw  Platform Body  Differential Screw for Transmitting Impact to fhe A n v i l  Lock N u t  F i g u r e  6  H a m m e r - p l a t f o r m  d e t a i l  with  a hardened  sections. was  and tempered  Keewatin  The s e l f - t i g h t e n i n g  found  t o be a d e q u a t e  steel  effect  of the threaded taper  t o t r a n s m i t the impulse  i m p a c t w i t h o u t any a p p r e c i a b l e s l i p p a g e . receiving screws  screws  which  differential each of  screw  are connected  i n turn  to the s t e e l  could  result.  The d i f f e r e n t i a l  i n d e p e n d e n t l y from t h e impact  permitting  t h e gap b e t w e e n t h e a n v i l  t o be a d j u s t e d . differential  Once t h e i m p a c t  screws  striking b)  platform  held  i n position  laboratory pivoting  point.  a specific  impact,  receiving  screws,  and t h e d i f f e r e n t i a l  receiving  thus screw  s c r e w s and position,  o f t h e hammer h e a d  t h e hammer i s h e l d  During i n i t i a l  tests,  by b r a c k e t s w h i c h  The m o u n t i n g allowing  height.  the lock  and t h e  i n position  were c o n n e c t e d  problem,  i n two  t h e hammer t o be p o s i t i o n e d that,  i n the t e s t  a c o n s t a n t hammer-height  t o the  o f t h e hammer  s y s t e m was a d j u s t a b l e  However, i t was f o u n d  by a  the electro-magnet  were i n d e p e n d e n t  due t o t h e p e r m a n e n t s t r a i n  overcome t h i s  bending  c a n be l o w e r e d  t h e h e i g h t o f t h e hammer above t h e s t r i k i n g  increased To  thus  Otherwise  adjustment  frame s t r u c t u r e which  directions, at  The d e t a i l s  impact  D.C. e l e c t r o - m a g n e t . was  The  a r e shown i n F i g u r e 6 .  Hammer h e i g h t Before  inserts.  screws  are s e t i n the required  are tightened.  differential  i s n e c e s s a r y t o g i v e e q u a l i m p a c t on  or r a i s e d  nuts  The two i m p a c t  b o t h b y t h e hammer and t h e a n v i l .  specimen  o f t h e hammer  t o l a r g e r diameter  are connected  arrangement  taper cut i n three  accurately  after  each  platform specimen. mechanical  1.  Electro-magnet  5.  2.  Bracket t o hold e l e c t r o  3.  A d j u s t a b l e frame s t r u c t u r e  magnet for  mounting t h e electro-magnet 4. Hammer  7. Cord 8.  10.  7  Mounting  system  wire  6. Hammer head to raise  hammer  Striking platform  9. T e s t  pivot  Figure  Stay  specimen  Anvil  f o r the electro-magnet  s y s t e m was d e s i g n e d .  The m e c h a n i c a l s y s t e m c o n s i s t s o f a frame  s t r u c t u r e which i s a d j u s t a b l e locked to The  i n a specific  a bracket  which  assembly  levelled.  c)  With  t h e hammer when height  the platform  above t h e s t r i k i n g  s t e e l which  after  impact.  of the platform, The a n v i l  rigidly  i s connected  and t h u s  c o n t r o l s t h e shape o f  c o n s i s t s o f two l a r g e b l o c k s o f  The s m a l l e r  Two m a t i n g t a p e r e d  a horizontal direction  When t h e t o p wedge  the smaller  The l a r g e s t o f  block  pieces  r i d e s on t o p o f  of steel  arrangement.  i s moved h o r i z o n t a l l y , i t s l i d e s  block.  a n d , due t o i t s t a p e r , The t a p e r  o f the wheel allows direction  by l e s s  o f t h e wedge  the smaller than  a r e wedged  The t o p wedge c a n be moved  by a w h e e l and s p i n d l e  o f t h e b o t t o m wedge  a vertical  the clearance  t o a s t e e l p l a t e which i n t u r n i s  to the f l o o r .  l a r g e r one.  turn  Varying  absorb the impulse of the p l a t f o r m .  two b l o c k s  surface  by A n d e r s o n was u s e d t o  changes t h e d e c e l e r a t i o n  b e t w e e n t h e t o p and b o t t o m b l o c k s .  one  platform i s  s y s t e m i s shown i n F i g u r e 7 .  and t h e p l a t f o r m  l o a d i n g wave.  lowers  i t i s p o s i t i o n e d and  a n v i l w h i c h was d e s i g n e d  characteristics  in  mount-  Anvi1  between t h e a n v i l  the  arrangement the electro-magnet  A picture of this  decelerate  bolted  this  rides with  The  the  The t o p o f t h e frame i s c o n n e c t e d  i n turn i s connected t o the electro-magnet.  Thus c o n s t a n t  assured.  the  position.  d i r e c t i o n s and c a n be  b o t t o m o f t h e f r a m e s t r u c t u r e i s mounted on t o p o f t h e  hammer p i v o t . ing  i n three  .001".  top block Two  on t h e u p p e r raises or  i s such  that  t o move i n  connecting  rods,  1. T o p  block  2. T a p e r e d 3. B o t t o m  9. H a r d e n e d  wedges  10.  block  Striking  11. Hammer  steel  pad  platform  head  4. C o n n e c t i n g r o d  12. A d d i t i o n a l w e i g h t s  5. G u i d e  13. G r o u n d i n g w i r e  rings  14. T e s t  6 . Steel plate 7. Wooden  specimen  floor  8. Wheel and s p i n d l e f o r r a i s ing  or lowering  Figure  8  the a n v i l  I l l u s t r a t i o n o f the anvil-hammer  platform  fastened are  to the  attached  block  top  block,  slide  to the  block.  The  inside guide  from moving i n a t r a n s v e r s e  vertical  motion.  A  special  steel  pad  i s attached  ceive  the  impact  the  platform.  the  anvil  An  of  to the  the  the  guide r i n g s which  rings restrain  direction,  hardened  and  but  tempered  upper s u r f a c e of  the  the  permit Kewatin block  d i f f e r e n t i a l screws i n c o r p o r a t e d  i l l u s t r a t i o n o f the  i s shown i n F i g u r e  8.  top  to  re-  in  hammer h e a d , p l a t f o r m  an  Figure  9  Load  cell  3.3  Electronic  3.3.1  S t a t i c Load Measuring C i r c u i t : To  strain the  test  the  the  measure the  gauge l o a d  Two  i s incorporated  the  Due  to  the  reading  9.  The  cell is  the  tension  to  Static  The  opposed are  o t h e r two  into  arrangement  the  load  strain  cell  long-  of  the  load The  oppos-  arms.  a simple  strain indicator. f o r the  two  the  sensitivity  as  arm  other  bridge,  reading of  in  direction  gauges i n t h e  arm  shank  mounted  a four  gauges i n t h e  the  strain  diametrically  circumferential wired  arm of  a r e a i n the  s t r a i n r e a d i n g on  a B a l d w i n S.R.4  load  cell  four  arm  i s shown i n  Strain Circuit: static  resistance  length the  times the  upper g r i p  C6-121 s e r i e s  reduced  four  four  9.  The  the  the  a simple  i n the  longitudinal  s t r a i n g a u g e , and  wheatstone b r i d g e  twelve  2.6  increased.  m e a s u r e d by  3.3.2  the  c h a r a c t e r i s t i c s of  i s thus  Figure  gauges a r e  circumferential  i s magnified  itudinal  of  the  mounted i n t h e  Wheatstone b r i d g e w i t h arms, and  section  d i r e c t i o n , and  gauges a r e  ing  prestress,  gauges d i a m e t r i c a l l y  longitudinal  shown i n F i g u r e  static  F o u r Budd M e t a f i l m  mounted on  grip.  opposed  cell  specimen.  gauges are of  System  of  specimen.  specimen, are  main c o n t r o l  the  leads  are  are  r e c o r d e d by  t y p e s t r a i n gauges w h i c h a r e  test  the  s t r a i n readings  unit  used  as  connected by the  long  The to  twelve  mounted  leads.  common g r o u n d and  are  The  of  along  s t r a i n gauges,  a multiposition  shielded  means  switch  cemented in  shields  grounded to  on the  c a s i n g o f t h e main c o n t r o l in and  the c o n t r o l read.  specimen  bridges.  any s p e c i f i c s t r a i n gauge  The c o m p e n s a t i n g  gauges mounted  gauge, w h i c h  selected  on t h e t e s t  i s located  inside  box and i s d i r e c t l y c o n n e c t e d t o a s t r a i n i n d i c a t o r ,  t h e o t h e r arm f o r e a c h o f t h e t w e l v e s t r a i n gauge The  strain indicated  by e a c h o f t h e t w e l v e s t r a i n  gauges i s r e a d i n d i v i d u a l l y u s i n g s w i t c h on a B a l d w i n S.R.4 for  c a n be  switch  p r o v i d e s one o f t h e two arms n e c e s s a r y f o r t h e s t r a i n  control  provides  With the m u l t i p o s i t i o n  Each o f the twelve a c t i v e  gauge b r i d g e . the  unit,  box.  the m u l t i p o s i t i o n  strain indicator.  a s e t o f two g a u g e s i s shown i n F i g u r e  selector  A schematic 10.  diagram  47  Active Gauge 1  11 o  N  Ao  r  Off  O  Compensating-** Gauge o  1 2  n O10  Active Gauge 2  4 O  5« 9  8 O  7 O  6 O  Any  Static Strain Output to SR Strain Indicator  Gauge Between 1 to 12  Common  MM  Compensating Gauge  Figure  1 0  Dynamic and s t a t i c two gauges  strain  circuit for  48  3.3.3  Dynamic S t r a i n  Circuit:  The twelve s t r a i n gauges mounted on the specimen, which measure s t a t i c s t r a i n , are a l s o used f o r r e c o r d i n g the dynamic  strain.  T h i s i s accomplished by means o f a three  p o s i t i o n s w i t c h i n the c o n t r o l u n i t .  The two p o s i t i o n s  indicated  by A and B i n F i g u r e 10, denote two s e t s of s i x gauges 1A, 2A, 6A, and IB, 2B, 'N"  6B r e s p e c t i v e l y .  The n e u t r a l  position  i s used when the twelve s t r a i n gauges are used f o r r e c o r d -  i n g the s t a t i c s t r a i n .  When the three-way  p o s i t i o n A, the s i x gauges 1A, 2A  switch i s s e t on  6A p r o v i d e s one o f the  two arms n e c e s s a r y f o r each o f the s i x dynamic bridges.  s t r a i n gauge  S i m i l a r l y , the p o s i t i o n B p r o v i d e s one of the two arms  f o r each o f the o t h e r s i x dynamic  s t r a i n gauge b r i d g e s .  Six  compensating s t r a i n gauges p r o v i d e the o t h e r arm f o r both sets of  s i x dynamic  control unit.  s t r a i n gauge b r i d g e s and are l o c a t e d i n s i d e the A 2.5 ohm r e s i s t a n c e i s connected i n s e r i e s w i t h  each of the s i x dynamic  compensating gauges t o extend the s t r a i n  range over which the dynamic anced.  r e c o r d i n g instrument can be b a l -  An a d d i t i o n a l 2.5 ohm r e s i s t a n c e can be e i t h e r added t o  or removed from the compensating arm o f the b r i d g e by a s w i t c h . The two arms formed by the a c t i v e and compensating gauges are connected t o h a l f b r i d g e s o f the s i x b r i d g e s i n the a m p l i f i e r meters  (BAM.l)  The outputs o f the s i x BAM-1 meters are connected  to the o s c i l l o s c o p e s by means o f standard phone j a c k s . shows the dynamic  F i g u r e 10  s t r a i n c i r c u i t f o r e i t h e r 1A o r IB and the  static strain circuit  f o r two gauges.  3.3.4  Electromagnet Release  Circuit:  Before dropping, the hammer i s h e l d i n p o s i t i o n by a D.C. magnet, l o c a t e d mobile b a t t e r y  as shown i n F i g u r e  2.  A 12 v o l t auto-  i s used t o e n e r g i z e the magnet.  A 12 v o l t bulb  connected i n a p a r a l l e l c i r c u i t serves as an i n d i c a t o r  light  when the magnet i s e n e r g i z e d . 3.3.5  Cathode Ray O s c i l l o s c o p e The  Triggering  Circuit:  cathode r a y o s c i l l o s c o p e t r i g g e r i n g c i r c u i t i s  designed t o p r o v i d e a s i n g l e h o r i z o n t a l sweep on the two beams of each o s c i l l o s c o p e s , w i t h the sweep b e g i n n i n g a t the i n s t a n t of hammer p l a t f o r m  impact.  d u a l beam T e k t r o n i c  The two o s c i l l o s c o p e s used are the  565 and T e k t r o n i x  502A which are capable  of a s i n g l e sweep o f the h o r i z o n t a l beam when a c t i v a t e d by an external  source.  By u s i n g  the chopping d e v i c e i n the T e k t r o n i c  565 scope, and the d u a l beam 502A, the outputs from the BAM-1 meters o f the s i x s t r a i n gauges are recorded The  external  ted by i n t r o d u c i n g  simultaneously.  t r i g g e r i n g o f the o s c i l l o s c o p e i s e f f e c -  a D.C. v o l t a g e  between the p l a t f o r m  and the  hammer which i s brought t o zero p o t e n t i a l when the hammer s t r i k e s the p l a t f o r m . t r i g g e r the p r e - s e t The  This  drop i n D.C. v o l t a g e  i s used to  h o r i z o n t a l sweep o f the two o s c i l l o s c o p e s .  12 v o l t D.C. b a t t e r y  used f o r e n e r g i z i n g  i s a l s o used f o r the t r i g g e r i n g c i r c u i t .  the electro-magnet  Two r e s i s t a n c e s are  shunted a c r o s s the power supply t o reduce the D.C. t r i g g e r to below  10 v o l t s .  level  Tek. 565  - 0 X 0 o o oo  Indicalto Bulb  0  co 4  CN  0  o  Scope  h  Hammer  o o oo o  Figure  1 1  Electro-magnet  release  and o s c i l l o s c o p e  triggering  circuit  o  A schematic diagram of the o s c i l l o s c o p e  triggering  c i r c u i t and the electro-magnet r e l e a s e c i r c u i t i s shown i n Figure 3.4  11.  Recording Due  metal,  System t o h i g h p r o p a g a t i o n v e l o c i t y of a s t r a i n wave i n  (10,000 f . p . s . t o 20,000 f.p.s.) and the u n d e s i r a b i l i t y  of r e p r o d u c i n g the r e f l e c t e d waves on the o s c i l l o s c o p e s c r e e n s , a f u l l sweep time of the o s c i l l o s c o p e i n the o r d e r of m i l l i - s e c o n d was  used.  Because  1-2  o f t h i s r a p i d d i s p l a y of the  s t r a i n v e r s u s time p l o t s on the phosphor-P2 o s c i l l o s c o p e s c r e e n s , it  i s n e c e s s a r y t o photograph the t r a n s i e n t s t r a i n waves f o r  analysis.  Photographs were taken u s i n g two Pentax Model V  cameras w i t h an o s c i l l o s c o p e adapter and Pentax c l o s e - u p l e n s . S a t i s f a c t o r y r e p r o d u c t i o n o f the o s c i l l o s c o p e d i s p l a y s  was  o b t a i n e d by u s i n g Kodak T r i - X  film  with lens s e t t i n g of f 4 .  (ASA-400) b l a c k and white  Time exposures were used, o b t a i n e d by  manually opening and c l o s i n g the s h u t t e r s b e f o r e and a f t e r g i v i n g an e f f e c t i v e exposure time o f approximately one 3.5  second.  T e s t Specimens All  20 f e e t l o n g .  t e s t specimens  used were 1/4  i n c h diameter and  T e s t were done on m i l d s t e e l , copper and aluminum.  For the t e s t s on m i l d rod'  impact,  (type 1020  steel)  d i f f e r e n t tempers  s t e e l , commercially specimens  o f A l c a n 24S  available  were used.  'nail  Three  aluminum rods which r e p r e s e n t  d i f f e r e n t l e v e l s of work-hardening were used, 2024S-0, 2024S-F, and 2024S-T4.  The  '0' r e p r e s e n t s annealed m a t e r i a l and hence  i s the l e a s t work-hardened d e s i g n a t e "fabricated"  of the t h r e e ;  and "heat  treated"  1  F  and  1  'T4'  respectively.  The  heat t r e a t e d 2024S-T4 a l l o y i s one of the h i g h e s t S t r e n g t h A l c a n 2 4S aluminum a l l o y s  which i s a v a i l a b l e commercially.  The 2024S-F aluminum i s the o r i g i n a l form of the rod a f t e r i t i s drawn through the d i e and i s work-hardened facturing process.  due to the manu-  Compositions o f 2024s aluminum a l l o y i n  terms o f the percentage o f weights are g i v e n i n the t a b l e below.  Composition  Aluminum  Percentage  Weight  92.3  Copper  4.5  Magnesium  1.5  Manganese  0.6  Iron  0.5  Silicon  0.5  Other C o n s t i t u e n t s  0.1  For the copper specimens, s o f t e l e c t r o l y t i c rods manufactured  copper  by Noranda Copper M i l l s L i m i t e d were used.  T h i s m a t e r i a l i s 99.9 p e r c e n t by weight pure copper.  C H A P T E R  T E S T  IV  P R O C E D U R E  4.  4.1  Load  Cell  a  load  testing  load  was  load  cell  load  was  4.2  calibrate  static  Instron  read with  rod  lengths, positions which  gave  pulled  in  small  strain  S.R.4  sensitivity  increments  corresponding  strain  the  upper  using to  indicator.  f o r measuring  described  in  an  the  A  applied  second  smaller  static  above.  range  of  load  cell  no.l  is  0-5000  pounds.  The  range  of  load  cell  no.2  is  0-1000  pounds.  Preparation check  the  longitudinal uniformity  t e s t s were  arbitrarily cut  into  diameter  along  each  the  lowest  an  in  incorporated  The  was  the  cell  The  Baldwin  higher  various  selected  sample  a  a  To  was  load  applied  c a l i b r a t e d as  Specimen  specimen,  was  the  machine.  on  also  PROCEDURE  Calibration  To grip,  TEST  Instron  carried  from twenty  and of  each  and  testing  inch  highest machine  a  of  pieces  number  twelve  the  on  group  equal  hardness  the  out  the  sample  test  rod  which  specimens. of  twelve  were  pieces.  The  two  number  record  the  The  inch  recorded  hardness to  of  at  three  pieces  were  stress-strain  curve. To  prevent  when  the  initial  men,  the  two  nut cess  before was  ends  the  found  static of  the  specimen to  the  be  top  and  prestress rod was  were  bottom was  satisfactory  applied  threaded  mounted  in  for  grips  the  to  from  to  sliding  the  f i t a  test 1/4  apparatus.  applying  static  speci-  inch This  pro-  prestress  54 loads which were l a r g e enough t o s t r e t c h the specimen above i t s y i e l d point.  A f t e r the ends of the t e s t rods had been  threaded,  s t r a i n gauges were mounted l o n g i t u d i n a l l y a t d i f f e r e n t  positions  along the l e n g t h of the r o d .  In a l l t e s t s , Budd M e t a - f i l m CX-121  s e r i e s s t r a i n gauges were used. was  Budd G.A.2  room-curing  epoxy  used t o bond the gauges t o the specimen. The  s u r f a c e s on t o which the gauges were bonded were  c l e a n e d w i t h Budd S o l v e n t to degrease abraded  the s u r f a c e and  w i t h f i n e emery paper w h i l e the s u r f a c e was  G.A.-1B N e u t r a l i z e r .  The  abraded  s u r f a c e was  then  soaked with  then wiped with  G.A.-^lB n e u t r a l i z e r u n t i l the products of a b r a s i o n disappeared. A f t e r repeated a p p l i c a t i o n of the n e u t r a l i z e r , the s t r a i n gauge was  bonded t o the t e s t specimen w i t h G.A.-2 epoxy and a p i e c e of  Mylar  tape f a s t e n e d around the s t r a i n gauge and the r o d .  s p e c i a l one  i n c h long clamp designed  12 p s i  then mounted on top of the s t r a i n gauge and  was  A  t o apply a p r e s s u r e of  10-  Mylar  tape. A f t e r the epoxy adhesive had cured, the clamp and Mylar tape were removed, s m a l l l e a d wires were s o l d e r e d t o the gauges, and the p l a t f o r m was  mounted on the t e s t specimen.  t e s t specimen w i t h the p l a t f o r m a t t a c h e d was top and bottom g r i p s f o r f u r t h e r experimental 4.3  connected  The  to the  tests.  T e s t s on E l a s t i c Waves The  t e s t specimens of m i l d s t e e l  e l e c t r o l y t i c copper  and three d i f f e r e n t tempers of A l c a n  (2024) aluminum a l l o y were used propagation.  (nail rod), soft 24S  f o r s t u d y i n g e l a s t i c wave  E l a s t i c s t r a i n waves were o b t a i n e d by a p p l y i n g a  small  s t a t i c prestress  t h a t the  and  t o t a l s t r e s s i n the  s t r e s s of the m a t e r i a l . the weight of the  The  loading  imen i n alignment. i n the  due  to the weight of the  the  t o t a l loading  loading  m.v.  a p p l i e d t o the  which was  the  number up  BAM-1  dynamic l o a d i n g . by p u l l i n g a c o r d ,  spec-  prestress p  in/in, the  S.R.4 The  statically  A weight to produce  loading  arm  and  the  strain indicator c o n t r o l u n i t was  to s i x gauges i n t o the  then  dynamic  meters were a d j u s t e d to  recorded on the o s c i l l o s c o p e .  0-30,000 c.p.s. the  the p l a t f o r m  the  outputs of the BAM-1  response of the  static  i s l e s s than 200  t e s t specimen.  static circuit.  used to connect any  100  arm  the  specimen were c a l i b r a t e d by  s t r a i n gauges were read u s i n g  The  by  s t r a i n waves were r e c o r d e d , the s t r a i n  i n / i n of s t r a i n was  circuit.  employed to h o l d  y i n / i n and  a known l o a d to the  connected t o the  was  produced  t e s t specimen.  gauges mounted on the  p  and  was  yield  under dynamic c o d i t i o n s d i d not exceed  Before the  200  below the  s t a t i c prestress  arm  range of 400-800  s t r e s s of the  applying  specimen was  such  Since the dynamic s t r a i n wave amplitude  was  yield  then superposing dynamic s t r e s s  Since  give  the  meter i s f l a t over a frequency range of above s t a t i c c a l i b r a t i o n i s a l s o v a l i d under  The and  measured.  hammer was  then r a i s e d up  to the  solenoid  the h e i g h t of the hammer head above On  the  two  o s c i l l o s c o p e s , which together  r e c o r d e d the dynamic s t r a i n f o r s i x s t r a i n gauges s i m u l t a n e o u s l y , the  sweep c o n t r o l was  the beam.  The  s e t to give  a s i n g l e h o r i z o n t a l sweep of  hammer-platform t r i g g e r i n g c i r c u i t was  connected  to  the external  time.  By o p e r a t i n g  noid, the  t h e hammer  single  lens  were  opened  cope after  height  anvil  adjusted shape .015 top  inches. block  gauge which  adjusted  i s accurate  were  field  superimposed  disturbances,  this  disturbance,  i t was  platform  to the anvil  a s shown  The ing ed  loading  i n mild  materials steel  found  waves.  t h e gap a t  by u s i n g  the a  inches.  created  strain  by t h e c o l l a p s e  waves.  sufficient  To e l i m i n -  t o ground the  8.  procedure  was u s e d  The l o a d i n g  but the unloading  and copper.  satisfactory  or lowering  ±0.001  was  w h e n t h e hammer was r e -  i n Figure  experimental  and u n l o a d i n g  i n a l l three  studied  above  A  by s e t t i n g  to within  on r e c o r d e d  on t h e bottom  p a d on t o p o f a n v i l  by r a i s i n g  i n the solenoid  the  o f t h e wave,  screws  a n d t h e g a p was m e a s u r e d  electro-magnetic  the magnetic  leased, ate  T h e g a p was  amplitude  below the  o f s u i t a b l e shape.  was o b t a i n e d  the  o r by v a r y i n g t h e  was m o u n t e d  absorbing  manually  o r lowering  the d i f f e r e n t i a l  a pulse  anvil  by r a i s i n g  The  of  oscillos-  To analyse  and p r i n t e d .  the platform  pulse  closed  To c o n t r o l t h e shape  and impulse  of the  Initially, of  head.  on each  were  platform.  the platform  The gap between  of the strain  feeler  developed  the sole-  the shutters  mounted  The s h u t t e r s  above  t o generate  f o renergizing  cameras  changed  to decelerate  the platform  switch  h i t the striking  wave was  of the oscilloscope at this  Simultaneously  Pentax  manually.  o n t h e hammer  platform. of  reflex  o f t h e hammer  weights the  the on-off  t h e f i l m s were  the strain  system  was d r o p p e d .  t h e hammer  results, of  triggering  f o r record-  waves were  waves were  record-  only  57  1.  Loading frame  2. Lower g r i p s 3. T e s t  specimen  4. Anvi1 5.  Platform  6. Hammer 7.  Solenoid  8. S o l e n o i d mounting system 9. Cord and p u l l e y t o r a i s e the hammer  Figure 1 2  Test  Section  Baldum S.R.4  s t r a i n i n d i c a t o r f o r load strain indicator for static  Main c o n t r o l Solenoid  unit  cell  5.  Tektronic  565  strain  6.  Tektornic  5 0 2 A  7.  6 B A M - 1 meters  and t r i g g e r c o n t r o l  Figure  13  Instrumentation  oscilloscope oscilloscope  59  An o v e r a l l view of the t e s t s e c t i o n and an  illustra-  t i o n of the i n s t r u m e n t a t i o n used are shown i n F i g u r e 12 and i n F i g u r e 13 4.4  respectively.  T e s t s on P l a s t i c Waves P l a s t i c waves were generated  specimen  loaded  i n a prestressed test  s t a t i c a l l y t o a l e v e l s l i g h t l y below the y i e l d  p o i n t of the m a t e r i a l .  Only the s o f t e l e c t r o l y t i c copper and  annealed  (20245-0) rods were used f o r these  tests.  aluminum a l l o y  To p r e s t r e s s the t e s t specimen, weights were added t o  the l o a d i n g arm and the corresponding  s t r e s s c a l c u l a t e d by  r e c o r d i n g the l o a d c e l l r e a d i n g on the upper g r i p .  The t e s t  procedure f o l l o w i n g the s t a t i c p r e s t r e s s was i d e n t i c a l t o the procedure d e s c r i b e d f o r the e l a s t i c wave s t u d i e s i n S e c t i o n 4.3. to  Because of the permanent s t r a i n produced i t was  l e v e l the hammer a f t e r each impact.  necessary  The BAM^-1 meters were  zeroed a f t e r each impact a l s o . Loading  and unloading waves'were i n v e s t i g a t e d i n the  e l e c t r o l y t i c copper rods, but only l o a d i n g waves were gated i n annealed 4.5  aluminum  investi-  (20245-0) rods.  Repeated Impact T e s t To c o n s i d e r the workhardening of a m a t e r i a l under  repeated  dynamic l o a d i n g , a copper specimen was s t a t i c a l l y  pre-  s t r e s s e d t o a s t r e s s l e v e l which was a l i t t l e below the y i e l d p o i n t of the m a t e r i a l and the s t r a i n waves were recorded f o r twenty-five  consecutive  impacts.  Since i t was necessary t o  have a f i x e d hammer h e i g h t f o r each impact, the constant hammer  60 h e i g h t mechanical system f o r mounting the s o l e n o i d was employed i n the apparatus.  The t e s t procedure f o r each impact was  i d e n t i c a l t o the procedure d e s c r i b e d i n S e c t i o n 4.4.  Although  both the o s c i l l o s c o p e s were used f o r r e c o r d i n g the outputs o f the s t r a i n gauges, the s t r a i n waves were recorded p o s i t i o n s simultaneously 4.6  a t only  four  along the l e n g t h o f the r o d .  Motion o f the P l a t f o r m A f t e r Impact To determine the motion of the p l a t f o r m a f t e r  a commercially attached  a v a i l a b l e p i e z o e l e c t r i c accelerometer  t o the bottom b r a c k e t o f the p l a t f o r m .  meter used was a B r u e l and K j a e r  (B & K) type  impact,  was  The a c c e l e r o -  4336, having a  weight o f 2 grams and a s e n s i t i v i t y o f 4.08 m.v./g.  Since the  accelerometer  i s a h i g h impedance d e v i c e , i t was connected t o  an impedance  matching B & K type  output  2616 p r e - a m p l i f i e r and the  from the p r e - a m p l i f i e r was connected t o the T e k t r o n i c 565  oscilloscope.  To observe the time d u r i n g which the hammer i s  i n c o n t a c t w i t h the p l a t f o r m a f t e r impact, the hammer-platform t r i g g e r i n g s i g n a l was a l s o connected t o the v e r t i c a l a m p l i f i e r of the T e k t r o n i c 565 o s c i l l o s c o p e .  By t r i g g e r i n g the h o r i z o n t a l  sweep o f the o s c i l l o s c o p e as d e s c r i b e d i n S e c t i o n 4.3  the  a c c e l e r a t i o n o f the p l a t f o r m and the time of c o n t a c t between the platform-hammer were recorded a n a l y s i s , these S e c t i o n 4.3.  simultaneously.  For further  t r a c e s were photographed as d e s c r i b e d p r e v i o u s l y ,  C H A P T E R  R E S U L T S  AND  V  D I S C U S S I O N  5. 5.1  Pre-test  calibration  are  shown i n F i g u r e  the  applied load  strain  reading  plying  by  diameter  along mild  C4,  different  two  yield  are  the  2.28;  pieces  point  by  load c e l l s  the  load c e l l  multiplying  f o r load  cell  specimens the the  used no.l  the  no.2,  by  multi-  v a r i a t i o n s of  r o d were l e s s  C5  sample t e s t  i n Table show t h e  For  lowest  and  thanO.0005  had  of the  yield  was  curves  specimen the  pieces  occur  curves  than  The  f o r the  check  there  for  the  stressthe  In a l l  7 percent  curves  for  below  the  when  2024S-0 aluminum  the  specimen,  and  a large variation  two  pieces  exceeded.  and  C3,  which recorded  stress-strain  uniformity  m a t e r i a l was  To  less  point.  found  pieces.  stress-strain  Figure  f o r a s t r e s s w h i c h was  a variation  d i d not  except  R o c k w e l l h a r d n e s s numbers.  have t h i s  curve  C2,  stress-strain  f o r two  recorded  specimens, Figure  2024S-0 aluminum t h e  above t h e  two  1.  e a c h sample  were i d e n t i c a l  variation  between t h e  a l l the  rods.  except  point  length of  were o b t a i n e d  the  stress-strain  the  by  sample t e s t  Figure  however, d i d not  this  in/in  tabulated  sample  s t r e s s was  yield  For  the  v a r i a t i o n s o f R o c k w e l l h a r d n e s s number  and  and  materials the  u  a l l the  curves  highest  C l i n A p p e n d i x C.  l e n g t h on  steel,  strain  f o r both  i n pounds i s o b t a i n e d  along  The  the  Figure  curves  0.438. On  inches.  DISCUSSION  Results  The  the  RESULTS AND  was  after  Below t h e good  f u r t h e r on  o b s e r v e d between t h e  in  the  yield  point  agreement  the two  variation pieces  of  the  t e s t s were r e p e a t e d and in  i n each case stress  strain  on  four other pieces  i t was  found  curve  from  t h a t t h e r e was  f o r each p i e c e .  the  same  specimen,  a large variation  In a l l c a s e s ,  a  strain  _3  rate  of  6 x 10 The  / s e c . was  used.  pre-test results  showed t h a t v a r i a t i o n s  m a t e r i a l p r o p e r t i e s along the  l e n g t h of the  mens e x c e p t  than  5.2  2024S-0  was  Wave P r o p a g a t i o n  5.2.1  Elastic To  i n the M i l d  Loading find  mentation,  initial  specimen.  The  Table  the  strain  of  stress  due  Specimen  of the  apparatus  a mild steel  f o u r gauges s i t u a t e d  simultaneously.  The  and  F i g u r e 15  show t h e of  2.69  instru-  (nail  above  rod)  the of  the  are given i n response  l b s . and  T h e s e f i g u r e s were o b t a i n e d w i t h  t o the weight of the  and  locations  t h e h e i g h t o f hammer u s e d  gauges f o r hammer w e i g h t s  respectively.  Steel  limitations  responses  F i g u r e 14  for a l l speci-  percent.  t e s t s were made on  g a u g e s and 2.  7  the  Waves:  p l a t f o r m were r e c o r d e d strain  less  rod  of  l o a d i n g arm.  3.32  lbs.  a small  For the  pre-  elastic  theory,  that  p o i n t above t h e p l a t f o r m a l o n g  the  dynamic s t r a i n  l e n g t h o f the  E  V. = -  x  o  °  where c the  Q  rod  W, —  i s given  (1+e)  exp  the  result  by  [(x-c t)/  (5.2.1)  0  W, +W  m  h p  r e p r e s e n t s the  impacting  a t any  obtained  four  case, Anderson, using a simple the  [24]  o f the  body  elastic  per u n i t  bar  velocity,  cross-sectional  m  i s t h e mass o f  area of the  rod  and  e  i s the coefficient  platform. impact  In obtaining  specimen.  the  weight  constant,  = o.  the  should  increase  treatment which  the strain  in  Figure  in  determining  14  wave  no  the various  with  wave a n d  Also,  (5.2.1)  a n d a = Ee  at .  time Hence,  velocity  at  acceleration  of  and t h e  initial  i s proportional to the  predicts  an e x p o n e n t i a l  i s observed  qualitatively  to the d i f f i c u l t i e s  parameters analysis  decay  involved  i n the expression was  the theoretical  made  t o compare t h e  results  obtained  by  (5.2.1). The  second  wave, w h i c h  m.  sec. after  the i n i t i a l  by  the second  impact  Further propagation  spectral of  Due  quantitative  results  x = o  increasing  magnitude  physically  and t h i s  15.  15.  t o an i n f i n i t e  also  amplitude  and F i g u r e  experimental  at  0  hammer  remaining  an i n s t a n t a n e o u s  i s impossible  The treatment  of  accurately,  predicts  that  of the strain  of finite a/pc  the  and  the platform to  factors  i n Figure  v =  t h e hammer  that  i t i s seen  a l l other  strain  case  o f t h e wave  acceleration.  on  assumed  t h e magnitude  corresponds  This  9e/3t  with  qualitatively  For the e l a s t i c  platform.  using  i t was  expression  an i n s t a n t a n e o u s  t = o  slope  this  o f t h e hammer,  idealized  time the  From  i s confirmed  predicts t  (5.2.1)  between  i s transmitted instantaneously through  the  this  of restitution  definite  analyses  approximately  i n Figure  15, i s  0.37 produced  on t h e p l a t f o r m .  of the experimental  waves were  a n d i t was  duration  front  o f t h e hammer  of elastic  analysis,  wave  arrives  performed  advantageous  results by u s i n g  t o produce  to carry out the analysis  obtained Fourier  a wave  numerically.  The  s t r a i n wave o f f i n i t e  decelerating time. time its  duration  the s t r i k i n g  When t h e s t r i k i n g  ( F i g u r e 16) was o b t a i n e d by  platform after  p l a t f o r m i s stopped  in a  finite  by t h e a n v i l a t  t = t , t h e p l a t f o r m d e c e l e r a t e s i n a s i m i l a r manner t o acceleration  level  propagates  duration  and t h e c o r r e s p o n d i n g  o f t h e wave  i s controlled  increments  propagation  o f t h e wave  t h e gap b e t w e e n  A gap o f 0.015 i n c h e s  strain  levels  and a  i n o b t a i n i n g t h e wave o f  x-t  plane.  16,  the constant  strain  Since each increment  determines  at different  level  shown a r e s t r a i g h t  The s t r a i g h t  plots  propagates lines  of propagation  the  variations  velocity  the  region  the  limit  i n propagation  Results  along the  in/in  are less  f o r each s t r a i n  The r e s u l t s of strain  than  of the o s c i l l o s c o p e  f o r propagation  at a constant  velocity  incre-  to the experimental  squares.  of accuracy  positions  the con-  and t h e s l o p e o f t h e l i n e  l i n e s were f i t t e d  u  by p l o t t i n g  are' g i v e n i n F i g u r e 17.  p o i n t s by t h e method o f l e a s t  0-600  strain  F o r t h e s t r a i n waves shown i n F i g u r e  of strain  the v e l o c i t y  of the d i f f e r e n t  i s calculated  observed  i n the  curves  velocity  front  rod  ment.  Thus t h e  Q  0.4 m . s e c s . shown i n F i g u r e 16. The  stant  (x-c t).  by c h a n g i n g  hammer w e i g h t o f 2.6 9 l b s . were u s e d duration  reduction i n strain  along the c h a r a c t e r i s t i c s  t h e p l a t f o r m and t h e a n v i l .  the  t h e impact  show t h a t  increments i n  2 percent which i s used  for recording.  o f s t r a i n waves i n t h e s t e e l  s p e c i m e n show t h a t t h e r e i s no d i s p e r s i o n o r a t t e n u a t i o n o f t h e waves as t h e y one  thing,  travel  along  the rod.  t h a t t h e r e i s no b e n d i n g  This indicates, f o r  of the rod during  impact  since  b e n d i n g waves a r e  wave showed t h a t  the  dispersive.  amplitude  10,000 c . p . s . were n e g l i g a b l e component. which of  are  the  of F o u r i e r  significant This  i n the  are  the  a n a l y s i s of  components  when compared  Under t h e s e c o n d i t i o n s ,  rod.  effect  A Fourier  to the  result justifies  e q u a t i o n of motion  lateral  The  diameter inertia  experimental  of  steel  d e t e r m i n e d h e r e i s 16,600 f t / s e c w h i c h a g r e e s c l o s e l y the  value  for  the  law  under dynamic  the  yield  5.2.2  nail  of  rod  17,000 f t / s e c  given  loading the  when t h e  steel the  Unloading  in  the  of  the  the  transient  the  the  18.  show any  rod.  This  small arm  bar  unloading  static as  waves was  platform  prestress  before  waves, and This  was  Hooke's  exceed  investi-  on  the  applied  t o a l i g n the  responses of  f i g u r e shows t h a t  attenuation  v e l o c i t y was  the  that  there  hammer i m p a c t .  found  t o be  16700  as  by  platform gauges  unloading i t travels  i s no In  mild  specimen  these the  or d i s p e r s i o n  further confirms  t e s t specimen d u r i n g  elastic  Results  obeys  F o u r gauges were mounted b e l o w t h e  shown i n F i g u r e  along  steel  l o a d does not  gauges below the  loading  apparatus.  wave d o e s n o t  of  the  A  total  [37].  mild  Waves:  strain  t e s t specimen.  to record are  mounting  weight  Kolsky  n  material.  P r o p a g a t i o n of e l a s t i c g a t e d by  by  wave f r o n t i  specimen i n d i c a t e t h a t m i l d  s t r e s s of  Elastic  elastic  lengths  value  with  an  dominant  times the  neglecting  (2.1.1).  propagation v e l o c i t y of  above  s h o r t e s t wave  more t h a n a h u n d r e d  the  bending  compression,  f.p.s.  66 5.2.3  Comparison  of the E l a s t i c  Figure approximately the  19 s h o w s  3.25  l o a d i n g wave  the platform.  waves  observed  of  velocity value  recorded  observed The  demonstrate waves a  Elastic  problem  platform.  test  without  gauges  were r e c o r d e d  gauges  propagation was  obtained  responses is  using  o f gauges  s e t to a value  there  i s a strong  elastic  wave  the steel used  1  percent  the  strain  c a n be t r e a t e d as  the responses  Responses  above t h e  and  a wave  fifth  and t h e o t h e r  To i n v e s t i g a t e  gap between inches.  propagation  of thes i x  o f the second  Figure  along  wave  at s i x positions  o f 0.3  t h e wave  m.sec. d u r a t i o n  21a a n d 21b  show t h e  the anvil  and p l a t f o r m  These  figures  attenuation of the strain  as i t t r a v e l s  Elastic  specimen  t o observe  the transient  material further  o f 0.015  feet  unloading  i s within  o n t h e 502A o s c i l l o s c o p e  when  on  Specimens  observed  the anvil.  4  are identical.  o n t h e 565 o s c i l l o s c o p e . i n this  imposed  i n 2024S-T4 Aluminum:  specimen  the anvil.  super  propagation.  2 0 a a n d 20b show  gauges  four  Waves  w a v e s was  Figure  i n wave  position  l o a d i n g and  the problem  i n Aluminum  Loading this  the e l a s t i c  from  and t h a t  Waves:  wave.  the instrumentation  Wave P r o p a g a t i o n  In of  obtained  i s satisfactory,  5.3.1  two p o s i t i o n s  f o r the unloading  one-dimensional  5.3  show t h a t  at a  approximately  f o r t h e l o a d i n g wave  results  that  wave  the platform,  at a position  Results  a t these  and Unloading  the unloading  below  recorded  above  bar  feet  Loading  the rod.  indicate  amplitude Since  there  that  i n the i s a  67 large  a t t e n u a t i o n of the  successive c a n n o t be  strain  strain  increments  determined  amplitude,  at d i f f e r e n t  accurately.  Due  method d e s c r i b e d i n S e c t i o n 5.2.1 characteristics mining  i n the  the p r o p a g a t i o n  elastic  bar  time  t h e wave f r o n t  of  Figure that  22  the  velocity  elastic  t o be  the  elastic  calculated  plot  velocity by  the  by  the  gauge p o s i t i o n s  slope of  used here  bar-velocity.  arrival  i n the  x-t  line, The  plane.  indicating  elastic  straight  bar  line  is  closely with  the  [37]. of a t t e n u a t i o n of  at a v e l o c i t y two  for deter-  the  were  in  the  different  velocities  the  plotted  done t o d e t e r m i n e w h e t h e r  I f the  the  plotting  l o a d i n g - u n l o a d i n g b o u n d a r y was  propagating  the  The  which agrees Kolsky  of  increments.  i s constant.  g i v e n by  T h i s was  wave was  the  i s a straight  slope of t h i s  times  difficulty,  e = +0)  i n v e s t i g a t e t h e mechanism  plane.  unloading  of s t r a i n  16,800 f t / s e c ,  amplitude,  x-t  be  to t h i s  c a n n o t be  (i.e. Strain  16,700 f t / s e c To  strain  bar  determined  calculated of  may  strain  b a s e d on  plane  velocity  shows t h a t t h i s  velocity,  value  x-t  arrival  from  the  different,  a complicated  l o a d i n g - u n l o a d i n g phenomenon w o u l d be i n d i c a t e d .  Such w o u l d be  the  rate-independent at  m e d i a , and  observed.  The  than  It  may  seen t h a t the  by  a straight  (e = +o)  shown i n t h e  x-t  they  and  do  to the  propagate in  elastic-  boundary the  plane,  loading front,  would  loading-  Figure  l o a d i n g - u n l o a d i n g boundary  parallel  exhibiting  waves w o u l d  loading-unloading  loading front  boundary are  line  Unloading  l o a d i n g waves as  a similar  unloading be  f o r example, o f a m a t e r i a l  hysteresis.  a higher v e l o c i t y  plastic be  case,  is  22.  represented  indicating  68 that  unloading  This  result  mechanism  waves p r o p a g a t e a t t h e e l a s t i c  r e j e c t s the o r i g i n a l  of attenuation,  hypothesis  bar  velocity.  concerning  the  and r a t e - d e p e n d e n t b e h a v i o u r i s  suggested. Next, t o analyse the in  strain Figure  ents  observed  by a p p l y i n g  measured  resolved  2.3.  For this  from the f i r s t  strain  gauge  i n Figure  Table  2.  s t r a i n wave a t This  result  amplitude the  i n d i c a t e s t h a t above a f r e q u e n c y  observed  The v a r i a t i o n  components  as t h e y  i n amplitude  an e x p o n e n t i a l  t h e F o u r i e r components  wave p r o p a g a t i o n  velocity  with  variation  I t was  than 5 percent  w h i c h show t h e  x, a r e i n i n amplitude  shown i n S e c t i o n  be d e s c r i b e d  m o d e l , and t h a t t h e a t t e n u a t i o n  o f t h e F o u r i e r components  of  of the  reasonable and  2.3  are given  by a  linear  that  had t h e above c h a r a c t e r i s t i c s ,  phenomenon c o u l d  23.  t h e r o d a r e shown i n  The r e s u l t s ,  and p h a s e a n g l e  i n phase angle.  visco-elastic  of the  o f 7000. c .p. s. , t h e  and p h a s e a n g l e  propagate along  change  if  are given i n  c . p . s . , and h e n c e c a n be  of amplitude  24a and 24b r e s p e c t i v e l y .  variation  i s less  at o  Figure  agreement w i t h  The  the F o u r i e r transformation  o f t h e F o u r i e r components  neglected.  22a.  x  x = 0, i n i t s p o l a r f o r m , i s shown i n F i g u r e  maximum a m p l i t u d e  Fourier  result,  compon-  a n a l y s i s the distance  l o c a t i o n s o f t h e gauges above t h e p l a t f o r m a typical  t h e r o d , shown  t o t h e s t r a i n wave as  exact  As  amplitudes,  into their Fourier  Fourier transformation  i n Section  of s t r a i n  at s i x , p o s i t i o n s along  21a and 21b, were  discussed was  pulses  the attenuation  the  linear  f a c t o r and p h a s e by  (2.3.18)  and  (2.3.19) .  These e x p r e s s i o n s a l s o  curves  fitted  are  ents  observed  line  curves  t o the  amplitude  experimentally  are  fitted  show t h a t i f e x p o n e n t i a l  (Figure 24a),  experimentally  attenuation  °=  calculated. (2.3.20) .  The  by  t h e method o f  as  full  and  fitting  least  that i f straight of the F o u r i e r  c  then  factor  i s calculated  of experimental  squares  F o u r i e r compon-  ( F i g u r e 24b),  phase v e l o c i t y  phase v e l o c i t y  A l l curve  and  t o the phase v a r i a t i o n  components o b s e r v e d factor  decay of the  and  the  f  can  by  using  curves  i n F i g u r e 24a  and  above p r o c e s s  for different  f r e q u e n c i e s , the v a r i a t i o n  mined. In for  factor  and  These r e s u l t s  S e c t i o n 2.3, relating  factor  <=  two  phase v e l o c i t y w i t h are p l o t t e d  expressions  with  the  factor  From t h e s e  e x p r e s s i o n s , the v a r i a t i o n  q u e n c y was  calculated.  functions of  of  w  .  Voigt  element  Bland  meters  and  from  proposed  was  linear  in series Lee  the  J  was  deter-  Figure  26.  the a t t e n u a t i o n  J  2  2  ) .  with  fre-  shown i n F i g u r e 2 7 as  description  of the  problem  complete. made t o f i t e x p e r i m e n t a l  visco-elastic with  [38] was  of  (J* = J i +i J  o f J j and  are  2  mathematical  i s now  attempt  a three parameter  by  The  wave p r o p a g a t i o n An  and  J*  and  the  (2.3.9.) were d e r i v e d  f , and  complex c o m p l i a n c e  shown  repeating  frequency  i n F i g u r e 25  (2.3.8) and  the phase v e l o c i t y  By  done  are  lines  attenuation  be  r e s u l t s was  fitted  F i g u r e 24b.  the  a spring.  used  experimental  t h r e e parameter model,  model c o n s i s t i n g The  an  for  Ji  of  with a  method d e s c r i b e d  for calculating  values  results  and  the J  2  three .  For  parathe  expression for attenuation  70 factor  <=  and p h a s e v e l o c i t y  the  t h r e e model p a r a m e t e r s  the  t h r e e parameters used  of  <*  and  c , obtained  full  lines  that  there i s reasonably i n the frequency  smaller  frequency  with  c a n be d e r i v e d i n t e r m s o f  (See A p p e n d i x D ) .  range  calculated  values  f o r v a r i o u s f r e q u e n c i e s a r e shown as These r e s u l t s  good a g r e e m e n t w i t h  By c h o o s i n g  t h e t h r e e p a r a m e t e r model  results  show  the experimental  r a n g e o f 0-4000 c . p . s .  good agreement w i t h  experimental  The v a l u e s f o r  and t h e c o r r e s p o n d i n g  i n F i g u r e 25 and F i g u r e 26.  results  extremely  c  the experimental  could  a  yield  results.  Agreement  c o u l d be i m p r o v e d by u s i n g a  five  p a r a m e t e r model-, b u t t h i s was n o t c o n s i d e r e d t o be t o o u s e f u l . Experimentally in  F i g u r e 25, i n d i c a t e  Typical  values of  «  observed  attenuation factors,  t h e s t r o n g dependence o f  co = 6000  Variation  from  a t a frequency  frequency ponents  i s closer  components (co+o).  frequency ones.  frequency  frequency  0 0  than  increases  o f 17,300  I t i s interesting to calculated  t o the lower  t o be  of the higher frequency  [39] has shown t h a t f o r a t h r e e  applied load w i l l  a b a r v e l o c i t y which  plotted  Phase v e l o c i t y  to a velocity  bar v e l o c i t y  (co-* )  and  components o f t h e  t o t h e phase v e l o c i t y  Morrison  meter model, a suddenly  with  o f 6000 c . p . s .  that the experimental  16,800 f t / s e c  with  the lower  a t zero  c.p.s.,  co = 4300 c . p . s . ,  o f phase v e l o c i t y  than  15,900 f t / s e c  ft/sec note  faster  frequency.  c.p.s.  i n F i g u r e 26 shows t h a t t h e h i g h e r wave t r a v e l  on  co = o  a r e 0.0007 p e r i n c h a t  a maximum v a l u e o f 0.00 87 p e r i n c h a t 0.0044 p e r i n c h a t  <*  shown  travel  i s g i v e n by / E j / p .  along  This  compara-  the rod  result  "71 corresponds o)->°° .  The  t o t h e e x p r e s s i o n g i v e n by e x p e r i m e n t a l l y observed  ably w e l l with t h i s The behaviour elastic  5.3.2  foregoing results  illustrate  velocity this  the s t r a i n  o f t h e g a u g e s , i t was  and  specimen  specimen  agree  dynamic  linear  visco-  found  that  No  used  t e s t s were done on similar  565  The  results  the e l a s t i c  third  and  and  oscilloscope. show t h a t  specimen two  observed  obtained for this  to the  specimen  2024S-T4  5.2.1.  simultaneously.  502-A o s c i l l o s c o p e the  bar  L o a d i n g Waves i n 2024S-0 Aluminum:  to produce of the  amplitude  further  R e s p o n s e s o f s i x gauges s i t u a t e d were o b s e r v e d  the e l a s t i c  w i t h the r e s u l t s  so c l o s e l y  four  In a n a l y s i n g  closely  described i n section  Elastic  other  the  gauges mounted a t  the a t t e n u a t i o n of the s t r a i n  s i n c e wave p r o p a g a t i o n was  this  reason-  L o a d i n g Waves i n 2024S-F Aluminum:  2024S-T4 s p e c i m e n .  sponses  that  a l o n g t h e r o d a r e shown i n F i g u r e 28.  response  was  agrees  o f 2024S-T4 aluminum c a n be d e s c r i b e d by  Elastic  5.3.3  bar v e l o c i t y  theory.  positions  in  i n A p p e n d i x D when  result.  R e s p o n s e s on  the  (6)  A hammer w e i g h t waves i n t h i s  o f 1.4  specimen.  Rethe  gauges were r e c o r d e d  T h e s e a r e shown i n F i g u r e 29a the a t t e n u a t i o n of s t r a i n  and  amplitude  to the a t t e n u a t i o n observed  specimens.  pounds  gauges were r e c o r d e d on  the remaining  i s similar  aluminum  fifth  above t h e p l a t f o r m  on 29b. in  i n the  72 5.3.4  Plastic  Loading  Plastic along 7200 waves  When  tests  elastic  were  waves  observe  waves  specimen  were  These  waves  "staircase" phenomenon  has  idealized shown  propagating  i n Section  would  5.3.3.  have on  t e n times  to this  with  stress  31a and 31b.  i n Section  This  waves  characteristics  by s t u d y i n g 32, w h i c h  was o b t a i n e d  discrete  jumps, thus  unusual  response  material  f o r this  stress-strain  above t h e y i e l d producing  [40] o b s e r v e d  aluminum a l l o y  stress-strain  shown  Dillon  rate point, a  similar  (AA1100).  relation,  o f t h e form  .  indicates  at a strain  i n small  Dillon  2  are similar  i n this  the quasi-static  that,  pure  lbs/in  Some e x p l a n a t i o n  indicate  99 p e r c e n t  transient  of the material  Results  curve.  level of  Results  5.3.3.  of the plastic  to unstable  level,  a p r e s t r e s s o f 7200  i n Figure shown  to a stress  plastic  l o a d i n g and u n l o a d i n g t h e  "staircase"  that  tested.  repeatedly  stress-strain with  random b e -  After  i n Figure  increases  Such  shapes  loaded  i s obtained  3  a n d wave  statically  waves  6 x 10' /sec.  stress  an  described  that work-hardening  a r e shown  shown  plastic  prestress levels  a r e removed by work-hardening.  behaviour curve  impacts.  the prestress,  t h e waves  recorded  be a t t r i b u t e d  waves,  chaotic behaviour,  without  the chaotic behaviour  which  of  again  the elastic  that can  .  A prestress of  was  statically  waves  to  resembled  2  at s i x positions  30a and 30b.  f o rthe various  the effect  lb/in  recorded  consecutive  repeated  the specimen  11,600  exhibit  with  was o b s e r v e d  waves  Aluminum:  In contrast to e l a s t i c  specimen  not reproduced  i n 2024S-0  i n Figure  was u s e d .  2  i n this  haviour  To  transient  t h e r o d a r e shown lb/in  were  Waves  Using [29]  i n F i g u r e 30a  73 and  30b  may  be  obtained.  gauge  i n Figure  there  is a  30a  Over  cent  strain  the  further u  from  in/in,  reached Kenig does  the  at  and  any  occur  not  affected,  at  is  an  5.4  and  Wave  Elastic  waves  responses of and  the  attenuates  Loading  the  by  the  30b,  35  while  per-  of  is  is 1060  not  substantiates  large  other  two  which  for a material  relation,  that  between  gauge  This  second  show  amplitude  rod.  that  the  increase of  i n Figure  specimen  which  strains  points.are  inhomogeneities.  study  carried  of  plastic  out  indicated  in  the  since by  the  length have  of  shown  wave  the  propagation  specimens  were  variation  in  the  Test  rod.  in not  Rockwell results,  that  2024S-0  aluminum  used  in analyzing  Copper  Waves  different  in  Copper:  specimens  were  In  both  specimens  s i x gauges  were  recorded  Results as  gauge  30b,  strain  strain  stress-strain in a  an  of  material.  s i x gauges  3 3b.  foot,  along  however,  i n copper. of  maximum  observation  along  Propagation  Two ing  1),  as  in Figure  Maximum  slight  not  static,  unstable  5.4.1  to  1  response  i s recorded  section  [28]  uniformly, (Table  of  first  theoretical  m a t e r i a l was  hardness  the  point  due  i n the  end.  smooth  one  A  dynamic  by  Dillon's  may  gauge  amplitude  other  possess  annealed  distance  impact  recorded  not  this  a  transient  first  large difference  positions. in  and  The  for show  specimen that  i t propagates  the  No.l  (No.l  are  the  No.2)  simultaneously. shown  amplitude  along  and  rod.  of  i n Figure the  strain  Propagation  load-  the Responses 33a wave velocity  74 of  the  in  the  wave f r o n t , c a l c u l a t e d f r o m s l o p e x-t  ft/sec given  plane  which by  To  For by  investigate  decelerating  plicated  the  to the  by  i n the  no  significant  it  i s unloaded.  wave r e c o r d e d Figure  36.  12,100 f t / s e c  x-t  plane  Fourier  the  first  simultaneously  result.  finite  duration the  fifth  34a  and  not  was  anvil.  Copper  Responses  34b.  the  To  the  35).  The  boundary  r e s u l t s show  the  parallel  loading-unload-  elastic  bar  This  clearly  indicates that  v a r i a t i o n s of  velocity  e f f e c t i n the in polar  (Figure  amplitude  34a) and  that  material  the  gauge  one  a com-  is a straight line  transform  in  independent  loading-unloading  (Figure  of  confirm  r e s u l t of  to a rate  de-  produced  gauges r e c o r d e d  indicates that  The  stress  the  boundary  This  of  six positions  investigation.  dynamic h y s t e r e s i s  by  attenuation  for analysing  5.3.1, t h e  wave f r o n t .  The  of  s t r e s s - s t r a i n curve of  boundary propagates w i t h loading  value  means o f  phenomenon due  front.  the  5.2.1, i s 12,200  method  s e c o n d and  loading-unloading  by  line  The  a m p l i t u d e was  i n Section  loading  straight  components.  wave o f  shown i n F i g u r e  p l o t t e d i n the  that  used  for this  the  strain  loop  discussed  was  are  of  was  platform  used  unloading  hysteresis  ing  the  s i x gauges, w i t h  attenuation  as  5.3.1  required  was  oscilloscope,  at  into their Fourier  i n Section  s p e c i m e n No.2  the  f u r t h e r the  s t r a i n waves r e c o r d e d  a n a l y s i s , the  the  i n Section  the  [37].  were decomposed scribed  described  agrees c l o s e l y with  Kolsky  amplitude,  as  of  given there  material  when  form f o r  the  i s shown i n  phase angle o f  the  is  Fourier 37b. the  components a l o n g  By  attenuation  various in  f o l l o w i n g the  the  38  °=  Figure  and  39.  attenuation  dent.  It increases  c.p.s.  to a value  from a value  of  0.0082 p e r  of phase v e l o c i t y  frequency  of  than  the  decreases  lower frequency.  a small  of  lowest  attenuation  solid  curve  curves  obtained  A p p e n d i x D. is and To  39  Although  reasonably  the  the  results  results depenu> =  0  c.p.s. to  a  components  travel  d i f f e r e n c e between  highest  than  and  velocity  5000 c . p . s . , p h a s e  i n c r e a s i n g frequency This  by  the  38  and  and  i s due  t h a n aluminum,  be  39  to  for  calculating  greater  show t h e  i n copper. theoretical in  q u a n t i t a t i v e agreement i s poor, a g r e e m e n t between t h e frequency  problem, experimental  range  values  data.  the  t h r e e - p a r a m e t e r model d e s c r i b e d  i n the  vari-  experimental  method u s e d  Figure  200  change i n t h e  phase v e l o c i t y  attenuation  good q u a l i t a t i v e  experimental complete  the  shown  shows t h a t up  phase v e l o c i t y w i l l  i n Figure  using  the  inch at  w = 6000  frequency  The  f a c t o r and  percentage e r r o r introduced f a c t o r and  for  experimental  0.0009 p e r  sharply.  c o p p e r shows much l e s s  attenuation  5.3.1,  These are  e r r o r i n v o l v e d i n o b t a i n i n g them f r o m t h e  Since  The  c  and  (to = 0 c . p . s . ) i s l e s s  amount w i t h  factor increases  the  38,  inch at  higher  Between 4000 c . p . s .  attenuation  of  i n Figure  4000 c . p . s . t h e  6000 c . p . s . ) and  ft/sec.  the  i n Section  phase v e l o c i t y  In F i g u r e  37a  f a c t o r i s s t r o n g l y frequency  Variation  ation  shown i n F i g u r e  procedure described  factor  and  show t h a t t h e  (co =  are  F o u r i e r components were c a l c u l a t e d .  Figure  faster  rod  there  theoretical  0-4000  o f J i and  c.p.s. J  2  obtained This  using  figure  loading.  (2.3.8) and  describes the behaviour  Strain  (2.3.8),  a t any p o i n t  (2.3.9) and Results  attenuation  of strain  The b e h a v i o u r  total  load  of copper  specimen  amplitude  o f copper  have shown t h a t  as a wave p r o p a g a t e s  under impact  i s below t h e s t a t i c  Plastic  yield  point  L o a d i n g Waves  and N o . 3 ) .  different lb/in  2  levels  In the  tests  of static  were u s e d .  prestress  Throughout  l b s . were u s e d .  Typical  different  of s t a t i c  and  42.  o f 13,400 l b / i n  wave  42a and 42b.  f r o n t , which  initial  wave  arrives  front,  i n t h e range  the r o d .  To i n v e s t i g a t e  11,500-20,000 on t h i s  spec-  o f 2.69  u n d e r two  s t r a i n waves when a  Similarly,  o f 17,400 l b / i n  figures  the observed  2  the observed  approximately  0.7 m.  by t h e s e c o n d  a r e shown i n secondary  sec a f t e r the impact  R e s u l t s shown i n F i g u r e 41a  there i s a broadening  five  a r e shown i n F i g u r e s 41  i s used.  i s produced  specimens  (No.2),  and a hammer w e i g h t  prestress  In these  hammer on t h e p l a t f o r m . that  i n two c o p p e r  experimental tests  t r a n s i e n t waves f o r a p r e s t r e s s Figure  o f the m a t e r i a l , can  l o a d i n g waves o b s e r v e d  2  along the  model.  F i g u r e 41a and 41b show o b s e r v e d  prestress  there i s  l o a d i n g , when t h e  on one s p e c i m e n  i m e n , a hammer h e i g h t o f 2.0 f e e t  levels  usin  i n Copper:.  L o a d i n g waves were o b s e r v e d (No.2  dynamic  on t h e r o d c a n be c a l c u l a t e d  r e p r e s e n t e d by a l i n e a r v i s c o - e l a s t i c  5.4.2  under  (2.3.10).  i n copper  rod.  be  (2.3.9) a r e shown i n F i g u r e 40.  o f t h e wave  of the indicate  as i t p r o p a g a t e s  the p o s s i b i l i t y  that broadening  along of the  w a v e may  be  constant  strain  is  explained  accomplished  by  levels  were  o f impact  strain  at a distance  to  the s t r a i n - r a t e  each  strain  any  a  of  the s t r a i n  In  Figure  in  a  levels  43  full  plane  are used  drawn the  lb/in  44,  for fitting Curves  i n Figure  43  them  waves  11,600  lb/in  strain  levels  2  ,  waves  The  observed 15,500  plotted  lb/in  i n the  level  plots  i n the  any  . x-t  levels  observed  The  2  straight  using  prestress  error  plane  should  the line  both  from  repeated f o r  In a l l cases, were  shown  i s of  involved i n  prestress levels  plane  least  straight  i n obtaining results was  are  shown,  the b e s t - f i t This  of  to the  method  For a l l cases  error  plotted  method  line  this  line.  are  with  the  of  constant  straight  r a t e dependence, the  x-t  theory,  of straight  lb/in .  static  2  x-t  2.4.1,  velocity.  independent  strain  above procedure using  a constant  5 percent.  and  i n Section  velocity  from  than  According  the propagation  the experimental  the m a t e r i a l exhibits  strain  points  i s less  traces.  44.  point.  of  must l i e  the best  and  level  from  plane  waves  fitted  This  has e l a p s e d  discussed  the slope  17,400  theory,  plane.  to a given  i n the  constant  and  2  the transient  oscilloscope  If  which  with  In a d d i t i o n ,  for transcient  same m a g n i t u d e a s  transient  plotted  of experimental  through  recording  theory  by  x-t  the impact  propagate  i s given  points.  lines  variation  line.  and F i g u r e  experimental as  level  level  o f 13,600  squares  t  rises  from  independent  strain  straight  x-t  x  must  i n the  the time strain  independent  i f the m a t e r i a l obeys r a t e  constant  along  until  increment  Consequently,  plotted  by p l o t t i n g  initiation e  the s t r a i n - r a t e  lines.  constant  l i ealong  curved  78 lines. of  One  the  using  velocities best  are  of  velocity  that  bar  Stuart  the  43  the  be  the  elastic  From  concluded  strain-rate have  bar  also  can  the  increments  i s known,  ponding  to  each  is  p o s s i b l e to  to  plot  does  elastic other  wave  front,  with  Sternglass  to  the  not  rate  theory predict  investigation,  front  propagates  based this i t with  i s governed  as  velocity  the  by  Bell  of  the  discontinbus yielding  and  by  Stein  plastic which  tests. velocity  dynamic  increment,  the  show  level  the the  independent  wave  points,  propagates  Furthermore,  propagation the  with  that  in this  elastic  slope  figures  with  result  curve  the  the  strain  material behaviour  observed  attributed  the  rate  theory.  in quasi-static Once  then  although  the  observed  their  be Propagation  Both  obtained  also  observed  velocity,  be  was  attributed  results  the  as  prestress level,  This  independent  44.  indicate  stress-strain  that  shown,  front  Results  from  propagates  were  material since  quasi-static  may  occurs  They  Figure  front  can  accuracy  experimental  decreases  wave  static  the  the  theory.  calculated  and  results  velocity.  of  phenomenon.  wave  the  [8].  dependence  [17]  the  independent'  through  propagation  Similar  of  material behaviour  level,  drawn  prestress levels.  elastic  on  strain  line  of  and  regardless  and  results,  in Figure  velocity.  three  therefore, that within  strain-rate  each  shown  increases, bar  a  straight  also  that  conclude,  experimental  explained  of  can  may  the  be  tangent  of  modulus  calculated  dynamic  various  using  (loading)  strain  ^—  , corres-  (2.4.4).  stress-strain  It  79 curve  as shown i n F i g u r e 45.  stress-strain  relation  I n F i g u r e 46,. t h e q u a s i - s t a t i c  obtained  with  the dynamic s t r e s s - s t r a i n  rate  of  test  results.  6 x 10  strain-rate  3  /sec  from curves  was u s e d  Although  independent  r a t e independent  plastic is  ison  distinct  theory  from  stress-strain  curve.'  experimentally reach  the'quasi-static  closely [41].  plane.  the q u a s i -  end o f a r o d , then the  the time  a loading-unloading  at different  taken  f o r the s t r a i n t o  positions  along  the r o d .  In  loading-unloading  o f 17,400 l b / i n  b o u n d a r y , g i v e n by  t = \p (x)  2  , i s shown.  ,  resembles  p r e d i c t e d by Skobeev [36 ] ,: and e a r l i e r  Furthermore,  compar-  T h i s b o u n d a r y may be o b t a i n e d  for a prestress level  the form  curve  shown t h a t i f s t r a i n  theory predicts  by p l o t t i n g  shape o f t h i s  confirm  o n e , and d i r e c t  phenomena w i t h  47, t h e e x p e r i m e n t a l l y o b s e r v e d  boundary, The  x-t  a maximum v a l u e  Figure  curves  i s irrelevant.  independent  boundary i n the  These r e s u l t s  b u t t h e dynamic s t r e s s - s t r a i n  c o n t i n u o u s l y a t the impact  strain-rate  strain  dynamic s t r e s s - s t r a i n  I n S e c t i o n 2.4.3, i t was decreases  A  i s appropriate f o r describing  o f p l a s t i c wave p r o p a g a t i o n  .static  superimposed.  i n o b t a i n i n g the q u a s i - s t a t i c  theory,;the  wave p r o p a g a t i o n ,  clearly  s p e c i m e n i s shown  t h e m a t e r i a l behaves a c c o r d i n g t o the  a r e w e i l above the. q u a s i - s t a t i c that  a similar  t h e boundary o b t a i n e d  satisfies  by L e e  the f o l l o w -  ing conditions:  4» '-(x)  Limit x-*°°  *.'.(x)  >  l / c  0  (5.4.1)  (5.4.2)  The  prime i n d i c a t e s d i f f e r e n t i a t i o n  to  x  ..  results  These c o n d i t i o n s are given  boundary that  by  Skobeev.  The  across  Section  the  2.4.3  with  and  [25]  shape o f  the  fact  the  boundary.  maximum s t r a i n  from the  l o a d i n g - u n l o a d i n g •boundary u s i n g  lb/in  the  , i s described  2  u s i n g , the  the  obtained  four other  with  i n Appendix E.  observed  value.  obtained  the  specimen, a higher  hammer w e i g h t o f observed  with  11,000 l b / i n at  increments  indicated material later,  2  2.69  with  shown i n  has  been reached  This  a p r e s t r e s s of  The  the  used  Throughout of  l b s . were u s e d .  2.25  the  f e e t but  - 20,000 l b / i n  P r e s t r e s s l e v e l was  t h a t t h e r e was under r e p e a t e d  no  1000  dynamic  loading.  small e f f e c t  on  the  increment of  1000  lb/in  2  tests  on  the  same  lb/in  2  .  i n the  range  increased  Initial  A detailed  tests  obtained  i n the  in this  the  investigation the m a t e r i a l  This work-hardening w i l l  results  confirm  a p p r e c i a b l e work-hardening of  impacts.  the  T r a n s i e n t waves were  prestress levels  approximately  of  obtained  to  fourteen d i f f e r e n t  of  17,400  value c a l c u l a t e d  (No.3) was  first.  .  pro-  used.  hammer h e i g h t  2  at  be c a l c u l a t e d  (2.4.17).  h o w e v e r , showed some w o r k - h a r d e n i n g o f  repeated  I t was  S i m i l a r r e s u l t s were  Another copper specimen results  changes  boundary i s w i t h i n 4 p e r c e n t  prestress level  the  3a/9t  a t t h a t p o s i t i o n may  loading-unloading  experimentally for  results  conclusion  loading-unloading  that  x  using  history  Bodner's  section  cedure,  , strain  the  theoretical  loading-unloading  loading-unloading  that after  the  respect  shape o f t h e  Rakhmatulin's view of the  sign  function with  i n agreement w i t h  substantiates Clifton  boundary i s i n c o m p a t i b l e  of the  have  under  a  specimen s i n c e  p r e s t r e s s f o r each  done  the  consecutive  81 t e s t was n o t l a r g e enough t o l o a d t h e m a t e r i a l above t h e maximum stress  (dynamic p l u s s t a t i c )  Oscilloscope are  on a  Sample p l o t s  x-t  plot  Constant  p l a n e were s t r a i g h t  lines  dynamic  with d i f f e r e n t  levels  plotted  obtained with  curves  this  uation  o f t h e waves o b s e r v e d  increases,  boundaries  exhibited  A l l tests similar  Sternglass  infinity  obtained There  f o r t h e two s p e c i m e n s . amplitude, f o r  i s shown i n F i g u r e 53.  are asymptotic,  results  [8] a l s o . o b s e r v e d  investigation.  values  observed  as d i s t a n c e  Attenx  u s i n g t h e two s p e c i m e n s  attenuation of the s t r a i n  and S t u a r t  experimental asymptotic  levels  t o a v a l u e o f s t r a i n w h i c h depends on t h e s t a t i c  prestress.  theory,  prestress levels,  This  specimen.  a r e shown i n F i g u r e 52.  c l o s e agreement o f a l l r e s u l t s  different  tested.  specimen a r e i n  O b s e r v e d d e c a y o f t h e maximum s t r a i n two  tested.  f o r a few p r e s t r e s s  Loading-unloading  prestress levels  were  i n the x - t  obtained using the other  stress-strain  2  a r e shown i n F i g u r e 49  f o ra l l prestress levels  the r e s u l t s  a r e shown i n F i g u r e 51.  is  strain  that the r e s u l t s  agreement w i t h Typical  f o r a l l the prestress levels  test.  lb/in  Constant.strain levels  f o r two p r e s t r e s s l e v e l s  F i g u r e 50.  indicates  i n the previous  t r a c e s o f waves f o r a p r e s t r e s s o f 11,600  shown i n F i g u r e 48a and 48b.  plotted  and  value reached  similar  According  amplitude  which corresponds  amplitude  as  to strain-rate  f o rcontinuously decreasing strain  t h e maximum s t r a i n  results  i n their  The mechanism w h i c h g o v e r n s t h e  o f maximum s t r a i n  i s not clear.  amplitude.  i s asymptotic  t o t h e dynamic y i e l d  x  goes t o  independent  a t t h e impact end, to a value of s t r a i n  stress.  I f the asymptotic  value  o f 320  prestress yield  u  levels  stress  experimental  and 220  with  the e l a s t i c  results  bar v e l o c i t y .  be n e c e s s a r y  history  described with  these  f o r t h e two  t o the dynamic  strain  bar v e l o c i t y .  The  levels  observed  ( F i g u r e 51) show t h a t t h e two s t r a i n with  a much s m a l l e r v e l o c i t y  Further experimental  unloading  at that section  i n Appendix E.  at a position  Calculated values results.  will  x  c a n t h e n be  Table  , the  u s i n g t h e method  3 shows  Results indicate  good a g r e e m e n t b e t w e e n c a l c u l a t e d  than the  investigation  c a n be c a l c u l a t e d  f o r t h e two s p e c i m e n s .  levels  phenomenon.  has o c c u r r e d  experimentally observed  parison  , obtained  both  to e x p l a i n the observed  After strain  in/in  shown i n F i g u r e 53, c o r r e s p o n d s  question propagated  elastic  p  o f the m a t e r i a l , then  must p r o p a g a t e  in  in/in  compared  this  com-  that there i s  and e x p e r i m e n t a l l y  observed  results. Dynamic prestress Due  levels  strain  identical. first  strain  this  in/in  strain  The d y n a m i c  s p e c i m e n s were  i n hammer h e i g h t , waves  five  compared.  recorded  a t the  f o r t h e two s p e c i m e n s were n o t  T h i s c a n be s e e n  by c o m p a r i n g t h e r e s p o n s e ' o f  ;  gauge shown i n F i g u r e 41a w i t h  ,  Below a s t r a i n  responses  level,  level  t h a t of the  of  the first  approximately  o f t h e two gauges a r e s i m i l a r .  Above  however, t h e shape o f t h e wave i s d i f f e r e n t .  stress-strain  curves  55 and F i g u r e 56) a g r e e . c l o s e l y in/in  obtained using  gauge p o s i t i o n  gauge i n F i g u r e 54. u  curves  i n t h e two c o p p e r  t o the d i f f e r e n c e  first  240  stress-strain  and d e v i a t e above t h i s  f o r t h e two s p e c i m e n s ( F i g u r e up t o a s t r a i n  value  of s t r a i n .  level  o f 240 p  One may  there-  fore, the  conclude  that  s h a p e o f t h e wave a t t h e i m p a c t e n d .  when p r o p a g a t i o n in  o f an i n c r e m e n t a l  soft electrolytic  stress-strain of  t h e d y n a m i c s t r e s s - s t r a i n c u r v e depends on  of the material  t h e wave a t t h e i m p a c t e n d .  a strain one this  material  of loading.  total  stress  percent  the  of the t o t a l curves  be s m a l l .  obtain  between d y n a m i c o v e r - s t r e s s  This  o f 4-6  t h e p e r c e n t a g e v a r i a t i o n between  them  p u r p o s e s , one c o u l d  s t r e s s - s t r a i n r e l a t i o n which w i l l o v e r a range o f r a t e s .  describe  This, was  i n this investigation.  Results  f r o m t h e two s p e c i m e n s have shown t h a t , r e -  of the prestress  the e l a s t i c  the material  level,  bar v e l o c i t y .  p l a s t i c wave f r o n t s  propagate  F u r t h e r m o r e , dynamic  behaviour  obeys t h e s t r a i n - r a t e independent t h e o r y ,  dynamic s t r e s s - s t r a i n curves curve.  and  f o r t h e two s p e c i m e n a r e s u p e r i m p o s e d on  attempted  static  Hence,  i n d i c a t e s t h a t i f t h e dynamic  not  the  i s altered.  stress.  response of the m a t e r i a l  of  d e p e n d on t h e shape  56) a r e o f t h e o r d e r  the  with  of loading  Hence, f o r e n g i n e e r i n g  an a p p r o x i m a t e  gardless  wiil  55 and F i g u r e  q u a s i - s t a t i c curve  will  s h a p e o f t h e dynamic  p r e d i c t i t s response under d i f f e r e n t  Differences  (Figure  stress-strain  accurately  a s i n g l e dynamic c o n s t i t u t i v e e q u a t i o n f o r  which w i l l  rates  that  I n a d d i t i o n , when t h e shape o f  wave i s c h a n g e d t h e r a t e  cannot o b t a i n  indicates  wave i s a n a l y s e d  copper, the exact  relation  This  lying well  with  above t h e q u a s i -  84 5.4.3  P l a s t i c L o a d i n g Waves Impact L o a d i n g : The  five  repeated  impacts  gation  in this  gauges  above  gauges  were  were  used  are  of  the  this  Transient  by  platform.  Although  the  test, of  a  of  dynamic  static  the  only  two and  first  prestress  of  of  after  These  i n shape  four  and the  propa-  a l l four  third  gauges  material. lb/in  different traces  o f wave  twenty-  strain  of  14,300  four  under wave  responses  the  F i g u r e 58.  a p p r e c i a b l e change  mounting  response  gauges  (No.4)  was  2  impacts  show  after  a  that number  impacts.  the  first  number. number times  There  impacts,  required gauge isia  which  understood  trend  tic  velocity.  e =  200  The  in/in  i s the  0  same,  until  increases  against  This  shows  wave  fronts  against  impact  effect  can of  with  number. at  impact, better  propagation The  i n a l l cases  significantly  be  The  first  reached  number.  propagation of  the  the  fifteenth  velocity  impact  at  impact  t o be  the  of  behaviour.  levels  level  propagates of  elastic  about  off.  the  irrespective  given  strain  against  velocity  with  of  plotted  plotted  which  ,  times  are  reach  also  level  levels e =  to  times  F i g u r e 60  level  position  given  curves  from  strain  are  for a  earlier  the  arrival  i n accordance  for strain  various strain  u  time  position  the  the  gauge  arrival  progressively after  F i g u r e 59, strain  The of  strain  of  investigated.  outputs  Repeated  specimen  observed  i n F i g u r e 57  i s no  copper  Under  was  recorded,  In at  a  was  specimen  Responses  shown  there  of  f o r studying the  Throughout used.  response  i n Copper  the the  at  wave the  strain first  front, elaslevel few  impacts, This  and then  indicates  becomes  that  there  the  first  ing  i s reached  without  as  the limiting  value  the  few i m p a c t s ,  elastic  constant  f o rhigher  i s noticeable  but a limit  becoming  of the propagation O n e may,  hardening  of the material  different  from  ditions.  Similar  the behaviour  dynamic  observed  r e s u l t s were  also  during  o f work-harden-  elastic,  inasmuch  velocity i s less  therefore,  under  numbers.  work-hardening  t o the degree  the material  velocity.  impact  conclude  condition  under  work-  i s quite  static  obtained  that  than  loading  by K o l s k y  con-  and  Douch [ 2 1 ] .  5.4.4  Unloading  Waves i n C o p p e r :  Propagation which  was  above  and below  the  statically  responses  copper  fourth feet  of four  traces.  waves  i n that  strain  they  show  are  stress by  observing  the platform made  on  levels  on a  elastic  second and  61.  height  waves  A hammer were  used  resemble  similar attenuation  o f 2.25  i n obtaining  the elastic  loading  o f t h e maximum  amplitude.  tension  ient  to  i n copper,  of the f i r s t ,  o f 1.4 l b s .  The u n l o a d i n g  waves  investigated  t e s t s were  i n Figure  weight  Propagation in  was  mounted below  Initial  a r e shown  strain  i n tension  and t h e responses  a n d a hammer  these  point,  gauges  (No.5).  waves  gauges  prestressed  the yield  specimen  unloading  of compressive  waves shown  above  of unloading  the yield  point  waves  was  recorded  with  a prestress  i n Figure  62.  In obtaining  also  i n a rod prestressed investigated.  level these  o f 11,600 traces  a  Trans-  lb/in  2  hammer  86 height It  of  2.25  feet  i s seen t h a t  similar  to the  and  a hammer w e i g h t o f  e l a s t i c waves shown i n F i g u r e  hammer w e i g h t u s e d , t h e  in  62  Similar with  are  results  different  increments of  l a r g e r than the were o b t a i n e d  prestress 1000  Result compressive stressed  waves i n an  5.5  from t h i s  unstressed  decay of  ated  strain  at  the  first  holding  the  hammer h e i g h t  for  aluminum a r e  8 feet,  i n the the  amount, b u t  of  of  amplitude  gauge by  over the 15  under dynamic  increased  elastic  in .  2  that pre-  unloading  l o a d i n g when t h e  a l l three  in/in  was  gener-  hammer w e i g h t  feet.  Results  levels  of  and  shown  30  attenuation  Over a d i s t a n c e  attenuate  there  i s an  by  a  attenuation  percent  load  of  negligable of  i n aluminum.  s p e c i m e n o b e y s Hooke's  total  three  specimen.  steel  steel  For  y.  the  2.0  i n c o p p e r and mild  670  Steel,  waves f o r t h e  63.  materials.  same d i s t a n c e  the  of  different  different  percent  indicates that  observed  18,000 l b / i n  strain  at  2024S-0  shows t h e  three  waves  in Mild  varying  constant  f o r the  63  the  waves.  in statically  t o the  are  waves shown  (No.5) have shown  L o a d i n g Waves  s t r a i n waves i n m i l d  approximately This  similar  amplitudes  a strain  observed  to a value  i n v e s t i g a t e d i s shown i n F i g u r e  Figure  the  to  used.  rod.  specimens t e s t e d ,  the  of  P r e s t r e s s was  waves, p r o p a g a t i n g  Comparison of E l a s t i c A l u m i n u m and C o p p e r  materials  were  Due  e l a s t i c unloading  specimeh  are  61.  f o r a l l unloading  up  2  i n tension,  The  amplitude  levels.  lb/in  unloading  rod  lbs.  t h e waves o b s e r v e d u n d e r t h e s e c o n d i t i o n s  increased Figure  2.69  i s below the  law  static  87  yield not of  point  of  material.  obey Hooke's law the  internal  observed  i n these  i n terms of To  , with  of  similar  two  damping phenomenon and  described  t^E/p  under  C o p p e r and  the  a linear  compare t h e  the  materials  was  materials  the  The  were u s e d  16,300  of  5.6  of  12,200  ft/sec,  mass  balance.  steel,  the  given  c o p p e r and ft/sec  experimentally  ft/sec,  by  density weighing  For by  by  the  the  re-  manu-  were r e a d i l y a v a i l a b l e .  16,800  and  be  16,800  and  aluminum  are  respectively. obtained  ft/sec  values  for  the  metals.  A c c e l e r a t i o n of To  impact, orded meter.  the  on  an One  study  the  triggering  the  Platform motion of  o s c i l l o s c o p e using of  the  D.C. the  copper  The  platform.  hammer, t h e  the  acceleration-time  investigation. of  ft/sec  3 percent  within  16,600  three  11,900  ft/sec,  These, a r e  for mild  the  a chemical  values  can  given  accurately  since these values  c a l c u l a t e d values  velocity  determined  modulus, however, the  some  model.  observed value,  quired  that  behaviour  visco-elastic  of  facturers  shown  do  analysis  i s c a u s e d by  material  a known l e n g t h elastic  s p e c i m e n on  Fourier  aluminum has  t h e o r e t i c a l bar  experimentally  three  aluminum, h o w e v e r ,  conditions.  o b s e r v e d waves i n c o p p e r and  attenuation  p  the  top  trace  platform  h i s t o r y of a  B & K  specimens  When t h e voltage  the  of  hammer  platform  was  piezoelectric  (No.4)  Figure  platform  the  after  64  was  used  for  shows t h e  i s i n contact  rec-  accelerothis  acceleration  with  the  d i f f e r e n c e b e t w e e n them, u s e d  for  o s c i l l o s c o p e s , i s reduced  to  zero.  This t r i g g e r -  88 v o l t a g e i s d i s p l a y e d i n the bottom t r a c e o f F i g u r e 6 4 .  ing  sudden r i s e indicates time. the  of the t r i g g e r i n g  t h a t t h e hammer s e p a r a t e s  To o b s e r v e  test  procedure,  time.  the  T h i s was  tact with t  The r e s u l t s  sec  after  the accelerometer  of  t h e hammer, i n d i c a t e s  T h i s .secondary  the  occurs  static  t = 0 . 5 6 m.  output,  The e x a c t  Figure  64.  uation  of strain  rise  t o the second  impact  wave.  i n many o f t h e o s c i l l o time  when t h e  second  o f s t r a i n waves r e c o r d e d  simultaneously  amplitude  along the r o d .  Figure  recorded  o f t h e p l a t f o r m shown i n  drawn i n F i g u r e 67 show t h e a t t e n recorded  i n F i g u r e 66.  of the f i r s t  using the a r r i v a l  velocity.  with  s t r a i n waves were  gauges w h i c h were  the a c c e l e r a t i o n lines  and  . ;  of the s t r a i n  the l o c a t i o n  calculated  con-  and a g a i n a t  o f a secondary  of the platform, e l a s t i c  The f u l l  of  t r a c e s a r e shown  A l s o the sharp  corresponding  compare t h e a m p l i t u d e  simultaneously with  propagation  impact.  the propagation  at four position  graph,  t h e sweep s p e e d  d e p e n d s on t h e hammer h e i g h t , hammer w e i g h t ,  66 shows t h e r e s p o n s e s  this  sec  l o a d i n g wave i s o b s e r v e d  acceleration  recorded  length  p r e s t r e s s on t h e r o d . To  the  by i n c r e a s i n g  a longer  show t h a t t h e hammer comes i n t o  scope' t r a c e s shown e a r l i e r . impact  shown i n F i g u r e 6 4  over  and t h e o b s e r v e d  the f i r s t  in  was  and-recorded  accomplished  the platform at  = 0 . 8 1 m.  t = 0 . 1 5 m. s e c  the p l a t f o r m at t h i s  under which t h e r e s u l t s  recording oscilloscope, Figure 6 5 .  in  from  at  the motion of the p l a t f o r m f o r a longer p e r i o d  were o b t a i n e d , was r e p e a t e d of  voltage observed  The  time  gauge f r o m o f t h e wave  The maximum v e l o c i t y  In p l o t t i n g the p l a t f o r m front  and i t s  o f t h e p l a t f o r m was  89 calculated  by i n t e g r a t i n g  Under. e l a s t i c iated  by  :  conditions strain  e = c /v.  Using  0  amplitude,  the a c c e l e r a t i o n  corresponding  was c a l c u l a t e d .  F i g u r e 67.. A s t r a i n  by  e x t r a p o l a t i n g the observed  be  i n reasonably  be  perfectly  extends  over  while  a finite  value.  i n fact  = c /v 0  i s shown  i s obtained amplitudes  i s 8 percent  T h i s may be c o n s i d e r e d t o  The p l a t f o r m i s assumed t o i t i s not.  A l s o , the g r i p  l e n g t h , and some d e f o r m a t i o n  imen w o u l d o c c u r w i t h i n t h e g r i p . E  u in/in  y in/in  value  strain  of'the platform,  attenuation of s t r a i n  good a g r e e m e n t .  rigid,  Of 1320  The c a l c u l a t e d  the experimental  velocity, are re- •  v a l u e o f 1430  amplitude  end.  (Figure 64).  e x p r e s s i o n , t h e maximum  t o maximum v e l o c i t y  in  higher than  and p a r t i c l e  this  The c a l c u l a t e d  back t o t h e impact  trace  Furthermore,  o f the spec-  the formula  i s s u b j e c t t o some m o d i f i c a t i o n b e c a u s e t h e c o p p e r  specimen i s n o t p e r f e c t l y  elastic.  S U M  0 F  MA  R Y  C 0 N C L US  ,,,  I O N  S  SUMMARY OF  1.  Apparatus useful  and  for further  i n metal 2.  Test in  3.  used  i n v e s t i g a t i o n s of  justify  neglecting  elastic  steel,  loading  5.  a l u m i n u m and  of  mild  A l u m i n u m and  c o p p e r do  not  A  of  even i f the  the  strain  15  describes  loading  of e i g h t  inch  and  f a c t o r and  the  w h i c h does n o t  material, theory.  of  mild  static  i s below the  yield  law.  1  Hooke's Law  can  700  feet.  m.  The  30  under static  be  in/in  in  dynamic yield  percent  amplitude in  attenuation  f o r aluminum,  aluminum  f a c t o r , which  amplitude,  for  0.00398 p e r  Higher  frequency  described  the  using  copper  components t r a v e l  components.  c o p p e r and exceed  copper  inch.  p h a s e v e l o c i t y i n aluminum and  lower frequency  M a t e r i a l behaviour of loading  the  obeys H o o k e s  i n c o p p e r and  frequency dependent. than  obey  exceed  d e c a y o f maximum s t r a i n  0.00140 p e r  faster 8.  the  Attenuation are  inertia effect  12,200 f t / s e c i n  steel  approximately  percent  over a distance  7.  lateral  material.  wave o f  attenuates  is  the  which does not  material,  conditions  6.  propagation  copper r e s p e c t i v e l y .  the  point  s t r e s s wave  are  wave f r o n t s p r o p a g a t e w i t h v e l o c i t i e s  Under dynamic l o a d i n g point  investigation  equation of motion .  16,600. f t / s e c , 16,800 f t / s e c , and  4.  in this  rods.  conditions  the  The  instrumentation  CONCLUSIONS  aluminum, u n d e r static  yield  dynamic  point  of  a linear visco-elastic  the  91 9.  Irrespective copper  10.  In  11.  Under  i s well dynamic  13.  the  loading  quasi-static  i n the  independent  plastic  predicted  Calculations  based  on  Under  the  independent  of  different  corresponding  relation  for  curve. region  copper  i n copper  obeys  the  Skobeev  specimens  [36].  experimentally observed are  impact,  loading-  compatible with  copper  from  the  under  exhibits  the  some  work-  propagating i n a  to  u n l o a d i n g waves.  into  aluminum which  stress dynamic  strain plastic  of  the  loading  material  is  under  loading.  prestressed  elastic  r e p e a t e d dynamic  hardening  statically  under  velocity.  theory.  copper  static  U n l o a d i n g waves  static  in  effects.  Work-hardening  Annealed  by  i n copper  r e p e a t e d dynamic  front  theory.  shape  boundaries  wave  stress-strain  the  quite  17.  above  dynamic  plastic  bar  resemble  hardening  16.  the  elastic  loading-unloading boundaries  strain-rate  15.  the  the  Observed  unloading  14.  prestress  with  regions,  strain-rrate 12.  static  propagates  plastic  copper  of  the  does  not  relation loading.  copper  specimen,  plastic  region,  possess  exhibits  a  are  smooth  unstable  which  is  similar  quasi-  behaviour  '  B I B L I O G R A P H Y  BIBLIOGRAPHY  1.  D o n n e l l , L.H. "Lonqitudinal ASME 52 , p . 153 ( 1 9 3 0 ) .  Wave T r a n s m i s s i o n , " T r a n s a c t i o n s  2.  T a y l o r , G.TImpact Load,"  3.  K a r m a n , T . V . , D u w e z , P. Deformation i n Solids," pp. 987-994, (1950).  4.  R a k h m a t u l i n , K.A. "On t h e T h e o r y P r i k l , M a k . 9, p . 91 ( 1 9 4 5 ) .  5.  W h i t e , M.P., G r i f f s , L . "The P e r m a n e n t S t r e s s i n a U n i f o r m Bar due t o L o n g i t u d i n a l Impact," J o u r n a l o f A p p l i e d Mechanics , V o l . 1 4 , p . 337 ( 1 9 4 7 ) .  6.  Duwez, P . E . , C l a r k , D.S. "An E x p e r i m e n t a l S t u d y o f t h e P r o p a g a t i o n o f P l a s t i c Deformation under C o n d i t i o n o f L o n g i t u d i n a l Impact," Proceedings American Society o f T e s t i n g M a t e r i a l s , V o l . 47, p p . 502-532 (1947).  7.  B e l l , J . F . " P r o p a g a t i o n o f P l a s t i c Waves i n P r e s t r e s s e d B a r s , " U . S . N a v y C o n t r a c t N6-ONR 2 4 3 , T a s k o r d e r V I I I ( N R - 0 3 5 - 2 1 5 ) , T e c h n i c a l R e p o r t No. 5, ( 1 9 5 1 ) .  8.  S t e r n g l a s s , D.A. S t u a r t . "An E x p e r i m e n t a l S t u d y o f t h e Propagation o f Transient Longitudinal Deformation i n E l a s t i c - P l a s t i c M e d i a , " J o u r n a l o f A p p l i e d M e c h a n i c s , V o l . 20, pp. 427-434, (1953).  9.  M a l v e r n , L . E . "The P r o p a g a t i o n o f L o n g i t u d i n a l Waves o f P l a s t i c Deformation i n Bars o f M a t e r i a l E x h i b i t i n g a S t r a i n Rate E f f e c t , " J o u r n a l o f Applied Mechanics, V o l . 18, pp. 203-208, (1951). ~~  " T h e P l a s t i c Wave i n a W i r e E x t e n d e d b y a n B r i t i s h O f f i c i a l R e p o r t RC 329 ( 1 9 4 2 ) . "On t h e P r o p a g a t i o n o f P l a s t i c J o u r n a l o f A p p l i e d P h y s i c s 21,  of Unloading  Waves,"  10.  S o k o l o v s k y , V.V. "The P r o p a g a t i o n o f E l a s t i c V i s c o u s P l a s t i c Waves i n B a r s , " P r i k l . Mat. Mech., V o l . 12, pp. 261-280, (1948).  11.  E f r o n , L. " L o n g i t u d i n a l P l a s t i c Wave P r o p a g a t i o n i n A n n e a l e d Aluminum B a r s , " Ph.D. d i s s e r t a t i o n , M i c h i g a n S t a t e U n i v e r s i t y , T e c h n i c a l R e p o r t N o . 1, G r a n t 6 - 2 4 8 9 8 , N a t i o n a l S c i e n c e F o u n d a t i o n , (1964) .  12.  C a m p b e l l , N.R. " D e t e r m i n a t i o n o f Dynamic S t r e s s - S t r a i n C u r v e s f r o m S t r a i n Waves i n L o n g B a r s , " Proceedings Society E x p e r i m e n t a l S t r e s s A n a l y s i s , V o l u m e 1 0 , N o . 1, p p . 1 1 3 -  124 , (1952).  93 13.  B e l l , J . F . "The I n i t i a t i o n o f F i n i t e A m p l i t u d e Waves i n A n n e a l e d M e t a l s , " S t r e s s Waves i n A n e l a s t i c S o l i d s , ( E d . H. K o l s k y and W. P r a g e r ) S p r i n g e r V e r l a g , p p . 166-182,  (1964).  14.  . "The Dynamic P l a s t i c i t y o f M e t a l s a t H i g h R a t e s an E x p e r i m e n t a l G e n e r a l i z a t i o n , " B e h a v i o u r o f M a t e r i a l s u n d e r Dynamic L o a d i n g (ED. N . J . H u f f i n g t o n , J r . ) ASME, p . 19, ( 1 9 6 5 ) .  1.5.  • "Normal I n c i d e n c e i n t h e D e t e r m i n a t i o n o f L a r g e S t r a i n through the use o f D i f f r a c t i o n G r a t i n g s , " Proceedings 3rd U n i t e d S t a t e s N a t i o n a l Congress o f A p p l i e d Mechanics, ASME, p p . 489-493, ( 1 9 5 3 ) .  16. A l t e r , B.E.K. and C u r t i s , C.W. " E f f e c t o f S t r a i n R a t e on the P r o p a g a t i o n o f T r a n s i e n t L o n g i t u d i n a l Deformations i n E l a s t i c P l a s t i c Media," J o u r n a l o f A p p l i e d Mechanics, V o l . 20, p p . 427-434, (1953)":—' : ; :  17.  ;  B e l l , T . F . and S t e i n , A. "The I n c r e m e n t a l L o a d i n g Wave i n the P r e - S t r e s s e d P l a s t i c F i e l d , " J o u r n a l Mecanique, V o l . I , No. 4, p p . 101-117, ( 1 9 6 2 ) .  18. T a y l o r , G . I . "The Use o f F l a t e n e d P r o j e c t i l e s f o r D e t e r m i n i n g Dynamic Y i e l d S t r e s s , " Proceedings of the Royal S o c i e t y , L o n d o n , V o l . A 199, p . 289, ( 1 9 4 8 ) . 19. V o l t e r r a , A. " A l c u n i R i s u l t a t i d i Prove Dinamiche S u i Material," L a R i v . d e l Nuovo C i m . , V o l . 4, p p . 1-2 8, (1948). : ; !  20. K o l s k y , H. "An I n v e s t i g a t i o n o f t h e M e c h a n i c a l P r o p e r t i e s of M a t e r i a l s a t Very High Rates o f Loading," Proceedings Physical Society, V o l . 62B, pp. 676-700, (1949l~i 21. K o l s k y , H. and Douch, L . S . " E x p e r i m e n t a l S t u d i e s i n P l a s t i c Wave P r o p a g a t i o n , " J o u r n a l o f M e c h a n i c s and P h y s i c s o f S o l i d s , V o l . 10, p p . 195-223, ( 1 9 6 2 ) . ~~ 22. B i a n c h i , G. "Some E x p e r i m e n t a l and T h e o r e t i c a l S t u d i e s on t h e P r o p a g a t i o n o f L o n g i t u d i n a l P l a s t i c Waves i n a S t r a i n R a t e Dependent M a t e r i a l , " S t r e s s Waves i n A n e l a s t i c S o l i d s , ( E d s . H. K o l s k y a n d W. P r a g a r ) , S p r i n g e r V e r l a g , pp. 166-182, (1964). 23. M a l v e r n , L . E . " E x p e r i m e n t a l S t u d i e s o f S t r a i n Rate E f f e c t s and P l a s t i c Wave P r o p a g a t i o n i n A n n e a l e d Aluminum," Behavi o u r o f M a t e r i a l s U n d e r Dynamic L o a d i n g , ( E d . N . J . H u f f i n g t o n , J o u r n a l ) , ASME, p p . 81-92, ( 1 9 6 5 ) .  94 24.  A n d e r s o n , C . J . " E x p e r i m e n t a l S t u d y o f S t r a i n Wave P r o p a g a t i o n i n Quarter-hard E l e c t r o l y t i c Copper Bars," M.A.Sc. T h e s i s , U n i v e r s i t y o f B r i t i s h Columbia, August,. (1962).  25.  C l i f t o n , R . J . a n d B o d n e r , S.R. "An A n a l y s i s o f L o n g i t u d i n a l E l a s t i c - P l a s t i c Pulse Propagation," Journal of Applied Mechanics, V o l . 3 3 , T r a n s a c t i o n s , ASME, V o l . 88, S e r i e s E , pp. 248-255, (1966).  26.  B o d n e r , S.R. a n d C l i f t o n , R . J . "An E x p e r i m e n t a l Investig a t i o n o f E l a s t i c - p l a s t i c P u l s e P r o p a g a t i o n i n Aluminum Rods," J o u r n a l o f A p p l i e d Mechanics, Transaction,ASME, Vol. 89, S e r i e s E , pp. 91-99, (1967).  27.. S c h u l t z , A . B . " M a t e r i a l B e h a v i o u r i n W i r e s o f 1100 A l u m i n u m S u b j e c t e d t o T r a n s v e r s e Impact," A p p l i e d Mechanics Confere n c e , A S M E , P a p e r N o . 6 8-APM-16, (196 8 ) . ~ 2 8 . - K e n . i g , M . J . a n d D i l l o n , W. J r . " S h o c k W a v e s P r o d u c e d b y S m a l l S t r e s s Increments i n Annealed Aluminum," Journal o f A p p l i e d M e c h a n i c s , V o l . 3 3 , T r a n s a c t i o n s , A S M E , V o l . 8, S e r i e s E , pp. 907-916, (1966). 29.  Dillon,O.W. S r . "Waves i n B a r s o f M e c h a n i c a l l y U n s t a b l e Materials," J o u r n a l o f A p p l i e d M e c h a n i c s , V o l . 33, T r a n . s a c t i o n s , ASME, V o l . 88, S e r i e s E , p p . 2 6 7 - 2 7 4 , (1966) v  30.  A b r a m s o n , H.N., P l a s s , N.H. a n d R i p p e r g e r , E . A . "Stress Wave P r o p a g a t i o n i n R o d s a n d B e a m s , " Advances i n A p p l i e d Mechanics, V o l . 5, A c a d e m i c P r e s s I n c ~ ( 1 9 5 8 ) .  31.  L e e , E.H. " T h e T h e o r y o f Wave P r o p a g a t i o n i n A n e l a s t i c Materials," I n t e r n a t i o n a l S y m p o s i u m o n S t r e s s Wave P r o p agation i n Materials, ( E d . b y N. D a v i d s ) , p p . 1 9 9 - 2 2 8 , (1960) .  32.  Craggs, Sneddon  33.  H o p k i n s , H.G. "Dynamic N o n e l a s t i c D e f o r m a t i o n o f M e t a l s , " A p p l i e d M e c h a n i c s R e v i e w s , V o l . 14, p p . 417, (1961).  34.  K o l s k y , H". " E x p e r i m e n t a l S t u d i e s i n S t r e s s Wave P r o p a g a t i o n , " P r o c e e d i n g s o f t h e F i f t h U.S. N a t i o n a l C o n g r e s s o f A p p l i e d M e c h a n i c s , ASME, p p . 2 1 - 3 6 , (1966).  35.  Symonds, Progress Division  J.N.O. Progress i n S o l i d Mechanics, a n d R. H i l l ) , V o l . I I , p p . 1 4 3 - 1 9 7 ,  ( E d . by I.N. (1961).  P . S . , T i n g , T . C . T . a n d R o b i n s o n , D.N. "Survey i n i n P l a s t i c Wave P r o p a g a t i o n i n S o l i d Bodies," o f E n g i n e e r i n g R e p o r t , Brown U n i v e r s i t y , ( J u n e 1 9 6 7 ) .  95 36.  S k o b e e v , A.M. "On t h e T h e o r y o f U n l o a d i n g W a v e s , " Applied Mathematics and Mechanics, PMM, V o l . 2 6 , p p . 1 6 0 5 - 1 6 1 5 , (1963) .  37.  K o l s k y , H. S t r e s s Waves Publication, (1963).  38.  B l a n d , D.E. a n d L e e , E.H. "On t h e D e t e r m i n a t i o n o f a V i s c o - e l a s t i c Model f o r Stress A n a l y s i s o f P l a s t i c s , " Journal o f Applied Mechanics, V o l . 23, pp. 416-420, (1956).  39.  M o r r i s o n , J.A. "Wave P r o p a g a t i o n i n Rods o f V b i g t M a t e r i a l and V i s c o - e l a s t i c M a t e r i a l s w i t h T h r e e P a r a m e t e r M o d e l , " Quarterly Applied Mathematics, V o l . 14, pp. 153-169, (1953).  40.  D i l l o n , O.W. S r . "The Response o f P r e s t r e s s e d Aluminum," International Journal of Engineering S c i e n c e , V o l . 2, pp. 327-339, (1964).  41.  L e e , E.H. "A B o u n d a r y V a l u e P r o b l e m i n t h e T h e o r y Wave P r o p a g a t i o n , " Quarterly Applied Mathematics, pp. 335-346, (1952).  i n Solids,  Second  edition,  Dover  of Plastic V o l . 10,  A P P E N D I X  A  APPENDIX  A  A TYPICAL REPRESENTATION OF A  LINEAR  VISCO-ELASTIC MODEL In  studying  the response•of.a  m o d e l , two t y p e s o f model a r e u s e d . generalized choosing  Voigt  m o d e l and t h e g e n e r a l i z e d M a x w e l l m o d e l . By  one f o r m o f t h e m o d e l r e p r e s e n t a t i o n identical  is  shown b e l o w  visco-elastic  They a r e known as t h e  t h e model p a r a m e t e r s t o s a t i s f y  an  linear  manner  to the other.  El  E  particular  conditions,  c a n be'made t o b e h a v e i n  The g e n e r a l i z e d  Voigt  model  n  2  n  Figure A l  The t o t a l and  may  strain  e  be w r i t t e n  ei  i s t h e sum o f t h e s t r a i n  o f each element  as  +  e  2  +  e3  +  (1)  The  corresponding  element  stresses  the f i r s t  force  across  across  2,-  n)  across  each  are a l l equal,  o = ai = o  In  (k = 1,  = a  2  = o  3  element the f o r c e the dashpot i s  the element  across  o = E e i  + ni  x  the s p r i n g  ni 9ei/9t  i s then given  (2)  .  The  is  total  E J E J  , and  force  a  by  -^-^  (3)  at  Applying Section  the F o u r i e r 2.3,  and  transformation E J  solving for  to  (3). as d e s c r i b e d  leads  in  to  °i £1  For  the k —  =  element,  e  k  =  Ei  (4) may  using may  ( 5 ) , (1) and  be w r i t t e n  as  2,  (4) ioon i  be w r i t t e n  as  °k E  where k = l ,  -  k  ( 5 )  ~  ia,n  k  3 - - - - - - - -  (2) t h e r e l a t i o n  n .  between s t r e s s and  strain  \  n  e  Comparing  =  (6) w i t h  /— » k=l  j  -—K  (6)  a  K  (2.3.1) and u s i n g  (2.3.4) i t f o l l o w s  that  n J*  =  Ji+ i J  2  =  (7)  ~  k=l  E  -iojn.  K  K  Therefore, n Jl  =  T"* k=l  E  E  2  k  (8) +  u, n 2  2  n ton. j  2  =  5^ k=l  For positive.  a l lreal positive  _ J S E + to n k K 2  2  to ,  ( 9 )  2  Ji  and  J  2  are  A P P E N D I X  B  APPENDIX  TO DETERMINE  THE NATURE OF THE FUNCTION and f  « (OJ)  From  B  (o>)  S e c t i o n 2.3  2<*f  f -<* 2  If  « (to)  =  a  2  =  pw J  2  (2.3.8)  =  pto Ji  (2.3.9)  2  2  + ajw + a u  0  + a oo  2  2  3  +-  3  —  and  ' f (to) • . =  where  b +b ( o 1  0  J  a. and b . ( i = l , l l  +  b  2, —  a j  2  2  n)  + b  3  w  3  + - -  are r e a l  numbers,  then  «f  =;  a b 0  + mtaobj+ajbo)  0  to ( a b + a b + a b i + a b  +  f -« 2  2  =  (b  3  2 0  -a  2a a )] G  ^ [  2  1  2 0  2  2  3  G  0  ) + oi(2b b -2a a ) + 0  1  0  + t o [ 2 b b + 2b!b 3  0  2  1  2  0  )+.----  3  0  ( a b + a jb + a b ) +  2 w  3  2  ] +  1  2 w  (1)  [bi  + 2b b -(a 0  2  2 1  +  (2a a +2a! a ) ] + 0  3  2  (2)  100 (10)  From  and (11)  i n Appendix  A  P  N  V V +U  k=l  n  ton  =  (4) k=l.  substituting  n  k E  n  C  k  2 k  a  + t o  n  d  2  n  2 k  choosing  to  forJ  and J  such  that  k  2  (to _k)  (4)  =  E  < 1 ,  the expression  1  c a n be e x p a n d e d i n a power  to = o  n  =y~^^~ E,  k=l  using  to  in  given  by  (3)  and  about the p o i n t  and may be w r i t t e n as  n  J!  series  2  K  E, k=l  4  n 2  K  (2.3.8) t h e c o e f f i c i e n t  + a,-  ^  k=l  'E,  +""  (5)  K  of the l i k e  powers o f (1) c a n be  equated  with  ( 5 ) and t h e r e s u l t i n g  e q u a t i o n s may be w r i t t e n  as  a b  0  = 0  (7)  aobi+a^o  = 0  (8)  =0  (9)  0  a b2+a b +a bo 0  1  1  2  a b +a b +a2b +a b 0  3  1  2  1  3  = |  0  k=l a bi +a b3+a b +a3bi+ai b  0  = 0  aobs+axbjj+aaba+aabz+ai^b^asbo  = -  0  t  1  2  2  t  the c o e f f i c i e n t  (2.3.9)  equated  with  k  E  (11) \  2  (  ( 5 ) and t h e r e s u l t i n g  )  of  (2)  c a n be  e q u a t i o n may be w r i t t e n as  -a  2 0  2  E  o f t h e l i k e powers  b  1  k=l k  1  using  (10)  —j  2 0  2 (bobi-aoa-!)  = 0  (13)  = 0  (14) n  b  2 1  +2b b -(a 0  2  2  1  +2a a ) 0  2  2b bo+2bibo-(2a a +aia ) 0  0  3  2  =  p  -Lk = l E, k  =0  (16)  n 2b b Q  l +  +2b b +b 1  3  2 2  -(2a  &^ + 2& a- +& l  i  l 2  )  =  0  2b b 0  3  +  2b bi +2b b -(2a  Solving  1  +  2  equation  3  (7) -  a + 2a ai + 2a a ) 3  1  (18)  t  2  3  (15)  -2—.  g  k  4  k = l E, k. = 0  (17)  (18)  w i t h t h e r e s t r i c t i o n t h a t a^ and  2, ient  a^  3, - - n  and  are a l l r e a l ,  the values  of the  i n t e r m s o f t h e model p a r a m e t e r s a r e  a  = ai = b  G  b l 555 ( p  = a  2  coefficgiven  = bit = 0  3  2kI= l — k  )1/2  E  a  =f  P  2  n ^  n  Z _  k=l  b  ai+  The  series for  11  3  f (OJ) =  /  E,k  (a  2 2  b  x  2  Yl  k=l  E,  )  =  (OJ)  (oo) =  = — 2b!  k  and  a co 2  2  f(oj) r e d u c e t o  + a^co  bxoj + b o j 3  3  4  + a oj 6  + b oj 5  5  5  + - - - - - -  + - - - - - -  (19)  (20)  103 The and Jl  above e x p r e s s i o n s f  show t h a t  i s an o d d f u n c t i o n o f  i s positive  positive  represent  or negative  two waves,  propagating  x  ,  co .  «  and  «  to ,  F u r t h e r m o r e , (3) shows t h a t  and  f ,  simultaneously.  For the s o l u t i o n  side of  t h e r e f o r e must The  one t r a v e l l i n g t o t h e r i g h t  t o the l e f t .  a meaningful r e s u l t , positive  i s an e v e n f u n c t i o n o f  f o r a l l to , h e n c e t h e r i g h t hand  (2.3.8) i s a l w a y s p o s i t i v e . either  «  be  choice of signs and t h e  (2.3.10)  other  to y i e l d  f o r a wave t r a v e l l i n g -in t h e d i r e c t i o n f  must  be  positive.  of  A P P E N D I X  P R E - T E S T  C  R E S U L T S  F i g u r e CI  Load c e l l  calibration  curves  105  PIECE NO.  ALCAN  NORANDA COPPER  ALUMINUM  24S  SOFT 2024S-0 'F  1  1  2024S-F  SCALE  2024S-T4  •F' SCALE  1  25  25  26.5 64.5 64.5 66  2 1 27  26  26.5 62.5 64.5 63.5 .78  3  25.5 26.5 27.5 66.5 66.5 66  4  26.5 27  27  B  1  7 7 . 5 78  SCALE  78  'B' SCALE  75  75  76  78.5 78. 5  76  75  76  78. 5 77.5 77. 5  77  79  77  76  74  75  75  75  73  74  76  77  76  76  77  78. 5 78.5 78. 5  74  76  75  78  76  77  75  63.5 65.5 64.5 78  78.5 7 8 . 5  27  26.5 27  65  65.5 65  77. 5 78  6  26  27  26  65  66  78  .7  27  27  28  63.5 64  65  ELECTROLYTIC  78  76.5 77. 5  63.5 77. 5 77.5 77  8  28.5 27  9  2 8.5 28.5 26  65.5 65.5 65  10  29  63  64.5 63.5 77. 5 78.5 78  74  76  74  11  26.5 26  65  65  75  75  76  12  27.5 65.5 65.5 66  29.5 28 26  78.5 78. 5  66  78  78  29  27.5 27.5 65.5 65.5 66  78  77.5 78. 5  78  78  79  13  27.5  30  29  64.5 65  65  77  77  77  75  75  76  14  26  26  27.5 65.5 66  65  78. 5 79  78. 5  75  77  IT  15  29.5 35  30  66  77  77  73  73  73  16  29  29.5 28.5 64  77. 5  74  75  74  17  30.5  30  30.5 66 . 5 66 .5 66 . 5 78  79  78. 5  74  76  76  18  33  33  32 . 5 65.5 66  77.5 79. 5  72  74  74  19  33  36  36.5 68.5 66 .5 66  20  44  41  41  TABLE NO. 1  65.5 65  65.5 64  66.5 64  VARIATION  65  78  77. 5 78  79  78  78. 5 79  79  76  76  78  65.5 77. 5 77  78  72  75  74  OF ROCKWELL HARDNESS NUMBER  8  2  o  2  *  2  a  6-  o  A  O A  o A  O  Piece  N o IS  A  Wece  No  7  o A O A  o A  O  ^  E z 10.6 KlO  6  Ibt./  to.  2  o  A O A  *  Figure  C2  c 3 . . 12 t *10 in./ i n .  Stress-strain  i  n  7  curves  f o r 2024S-T4  16  aluminum  20  107 6  6  A O  O  A  A  A O  O  o A  32  o A O A O  24 L  A  A  I  Piece  No.  2  Piece  No. 1 7  o  CO I  A  b  O  16  A O  8 A O  4  F i g u r e C3  6xl0 in./in. 3  Stress-strain  8  curves  12  f o r 2024S-F  aluminum  16  A A  o  ro • O  A  8  A  o  o  o  O  O  o  o  8  C  IA3  /  1  o  Piece  No. 4  A  Piece  No. 17  16  2  0 x 10 in./in.  F i g u r e C4  Stress-strain  curves  for  2024S-0  aluminum  r g  2  o  o  o  o  o  P»ce  No. 15  A  Piece  No. 3  A  2'  2 o  A  E=  17.0x10 |bs./in?6  8  F i g u r e C5  Stress-strain  1  6  curves  6x10  3  in/in.24  for soft  electrolytic  32  copper  A P P E N D I X  D  APPENDIX WAVE PROPAGATION  D  IN A THREE-PARAMETER  VISCO-ELASTIC MODEL For be  any l i n e a r  calculated  using  visco-elastic  model,  (2.3.8) a n d ( 2 . 3 . 9 ) .  «  and  For a three  model, however, a c l o s e d form  solution,  parameters,  T h i s model was u s e d  etically c.  may be o b t a i n e d .  predict  parameter  <*  t o theor-  and p h a s e  T h e t h r e e m o d e l p a r a m e t e r s were o b t a i n e d by c u r v e values of  may  i n t e r m s o f t h e model  the attenuation factor  of experimentally observed  f  J  1  and J  velocity fitting  2  l  E  r\2  Figure Dl  The  a  constitutive  equation  (E + E j ) + r i | f 2  2  Combining  =  for this  EiE  (1) w i t h e q u a t i o n  equation i s  2  model may be w r i t t e n a s ,  e + n Eill 2  o f m o t i o n , t h e wave  (1)  propagation  I l l  n p  9 E  3 e 3  2  + Ej+  E  3t  The  solution  2  H  2  _  p  i  3 e , 3x 3t  E!+E  2  E E ._ E!+E  3  3t  3  2 p  L  2  2  3 u 2  2  =  3x  2  o f (2) may be w r i t t e n as  e  = A exp i  [-oot+ (f+i«)x]  (3)  E }E2 Let  E  (2)  0  2  ^2  ='  and Ei+E  C  Substituting  c,  =  2  — E 2  (3) i n t o  the d i f f e r e n t i a l  equation  (2) and  simplifying  _  P »  E  2  2  i  {  2E^E  2 OJ  -c  2  1/2 }  £ - 7  X  E  {  2 2  +E  l+oo ? 2  E i  c *  2  E i+E  +  •  1 + 00  p 00 2E  2  +E  1  2  oo £ • (4)  C  2  2  L  l+oo £  u ? 2  2  £  1/2  2  } _  2  Ex+E oo ?  2  2  l+u C 2  (5) 2  and (6)  ai/f  For Ei, ity  any g i v e n E  2  and  frequency n  2  OJ ,  and t h e t h r e e model  the attenuation  c a n be c a l c u l a t e d  factor  using expressions  «  parameters  and p h a s e  ( 4 ) , (5) and ( 6 ) .  veloc-  A P P E N D I X  E  APPENDIX DETERMINING  E  STRAIN AT A POINT IN THE UNLOADING DOMAIN USING THE LOADING-UNLOADING BOUNDARY  The  l o a d i n g - u n l o a d i n g boundary obtained f o r a copper  specimen i s shown i n F i g u r e E l . observed  Maximum s t r a i n  amplitude  a t d i f f e r e n t s e c t i o n s o f the rod i s a l s o shown i n  this figure.  The method d e s c r i b e d i n S e c t i o n 5.4.2  was used  to f i n d p r o p a g a t i o n v e l o c i t y of the s t r a i n increments c o r r e s p o n d i n g dynamic s t r e s s - s t r a i n curve. shown i n F i g u r e E2.  P(x, t )  i s sought.  with slopes boundary  x-t  plane where the s t r a i n  Through t h i s p o i n t , two s t r a i g h t  ± l/c  Q  t = i> (x)  are drawn. at  P (x ,t ) 1  1  lines  These l i n e s i n t e r s e c t the and  1  e x p r e s s i o n f o r s t r a i n a t any p o i n t Maximum s t r a i n s c o r r e s p o n d i n g t o from F i g u r e E l  These r e s u l t s are  (Figure E l ) r e p r e s e n t s a p o i n t  i n the u n l o a d i n g domain o f the history  and the  P  P (x , t ) . 2  2  2  i s g i v e n by  P, P  and  l f  P  2  The  (2.4.16).  are o b t a i n e d  and the c o r r e s p o n d i n g s t r e s s from F i g u r e E2.  The p a r t i c l e v e l o c i t i e s  V j and  v  are g i v e n by  2  (1)  These i n t e g r a l s were e v a l u a t e d by c a l c u l a t i n g the  c(e)  versus  The p o i n t The  e  curve  (Figure E2) u s i n g a  P(x, t) choosen i s  corresponding points  Pi and  P  2  the area under planimeter  P (30 ,0.89)  are  113  PxUiftx)  =  P!(34,0.86)  P (x ,t )  =  P (100,  2  2  From F i g u r e at  these  2  E l and F i g u r e  E2  170 p  in/in  a  e  (xj)  =  230 y  in/in  a  (x)  =  m  and s t r e s s  sections are  =  e  1.37)  t h e maximum s t r a i n  e (x ) m  using  2  2  m  (  x  2 )  =  m  (  x  l )  =  a (x) m  350 y i n . i n  2250  2 5  =  50  lb/in  lb/in  2840  :  :  lb/in  2  (1)  vi-v  2  = 2c  c  - 16.5  y  in/in  (2)  114  T  T Prestress  17400 lbs/in . 2  s*f  (specimen no.2)  V^V  1.2  Loading> Unloading  SS' P(x,t>' .  Boundary  U,t  0.8  0  800  E  J 400  Max.Strain.  40  X  Inches  Figure  8  E l  0  120  Figure  E2  116 In o b t a i n i n g  (2), p a r t i c l e  t o be n e g a t i v e travelling values  E  V i and  v  2  s i n c e t h e l o a d i n g wave i s a t e n s i l e  i n the p o s i t i v e  x  direction.  are taken wave  S u b s t i t u t i n g these  i n (2.4.16),  e  The  velocities  (30, 0.89)  =  e x p e r i m e n t a l l y observed (30, 0.89) i s e q u a l  to  308  value  320  y  y  in/in  of s t r a i n in/in.  Figure  42 a t  Pesition SPECIMEN  of the  TYPE OF WAVE ! 11 r s T. -g'/!.; I •.j & form  approxim.  S I T I ;•"YKj OF OTHER GAUGES FROM ," RST GAUGE IN FEET  SEC OND i VHIRD !  FOURTH  Steel  Mild  Steel  2024S-T4 2024S-F 2024S-0 2024S-0 Copper N o . l C o p p e r No.2 C o p p e r No.2 C o p p e r No.3 C o p p e r No.4 C o p p e r No.5 C o p p e r Mo.5  2.0  ElasticLoading Unloading ElasticLoading ElasticLoading ElasticLoading PlasticLoaoxijy ElasticLoading ElasticLoading PlasticLoading PlasticLoading PlasticLoading ElasticUnloading PlasticUnloading  TABLE  2 :  ft.-Above  _  HEIGHT  WEIGHT  1  ~ 1 Mild  HAMMER SIXTH  FIFTH  1  THE  i  5 .0  8.0  2 .0 f t .  1.25  £b.-5eiuw  1 ,0  2 ,0  4.0  —  —  2 .0 f t . . 2.69  lbs.  1.75  ft,-Above  1 .0  3 .0  4.5  5.5  6.5  2 .0 f t . 2.69  lbs.  1.5  ft.-Above  0 .5  2. 5  4.5  —  —  2 .0 f t . 2.04  lbs.  2,0  ft.-Above  1 .75  2 .75  4.5  5.0  6.5  2 .25 f t . 1,40  lbs.  2,0  ft.-Above  1 .75  o ^.. 75  4.5  5.0  6.5  2 .25 f t . 2.69  lbs.  1.75  ft.-Above  2 .0  2 .5  5.25  7.25  7.5  2 .0 f t . 2.69  lbs.  1.75  ft.-Above  1 .25  1 .75  4.0  4.5  7.0  2 .0 f t . 2.69  lbs.  1.75  ft.-Above  1 .25  1 .75  4.0  4.5  7.0  2 .0 f t . 2.69  lbs.  1.75  ft.-Above  1 .5  2 .0  4.25  4.75  7.25  2 .25 f t . 2.69  lbs.  3.5  ft.-Above  0 .5  3 .0  3.5  —  2 .25 f t . 2 .69 l b s .  2.0  ft.-Below  1 .5  2 .0  3.0  —  2 .25 f t . 1.40  lbs.  2.0  ft.-Below  1 .5  2 .0  3.0  2 .25 f t . 2.69  lbs.  L o c a t i o n o f t h e s t r a i n gauges on t e s t s p e c i m e n s a n d t h e hammer h e i g h t and w e i g h t u s e d d u r i n g t e s t .  Specimen  Strain  Pre-Stress  No.  lb/in  £l ( x , t )  e  2  (x,t)  •e ( x , t )  2  E  %Error y in/in Diff. Calcu- Experilated mental Experi,  3  11,650  e (25.5,0.67)  e (100,1.29)  e (18,0.72)  474  450  + 5.3  3  11,650  e (31.0,0.72)  e (100,1.29)  e (24,0.760)  444  450  -1.0  3  11,650  e (55.0,0.92)  e(100,1.29)  E (51,0.95)  362  360  -0.5  3  11,650  e(60.0,0.97)  e(100,1.29)  £(57,0.99)  357  370  -3.5  3  13,400  e (22.0,0.67)  e(100,1.27)  E(18,0.705)  400  410  -2.4  3  13,400  e(27.5,0.72)  E (100,1.27)  E(24,0.745)  382  400  -4.5  2  15 ,500  e (27.0,0.68)  e (87,1.14)  E(24,0.700)  336  340  -1.1  2  15 ,500  e (32.0,0.73)  e(87,1.14)  E(30,0.745)  311  310  .0  2  17,400  e(29.0,0.81)  e (100,1.37)  E(24,0.85)  313  340  -7.9  2  17,400  e (34.0,0.86)  e (100,1.37)  E(30,0.89)  308  320  -3.7  N.B.  E  2  (x,t)  where  x  TABLE  e  (x ,t ) 2  i s i n inches  3  El (X,t)  2  and  t  i s i n m.  E  (Xxfti )  sees.  Comparison o f c a l c u l a t e d values of s t r a i n  and o b s e r v e d  118  F i g u r e 14  E l a s t i c l o a d i n g waves i n m i l d s t e e l w i t h a hammer weight of 2.69 l b s .  VERTICAL SCALE:  1 D i v . = .1 v o l t = 2 0 0 jj i n / i n ,  HORIZONTAL SCALE: 1 D i v . = .1 m. s e c .  Figure  15  E l a s t i c l o a d i n g waves i n m i l d s t e e l with a hammer weight of 3.32 l b s .  VERTICAL  SCALE:  HORIZONTAL SCALE: Striking  Figure  16  1 Div. =  .1 v o l t  1 Div. =  = 200  y in/in.  .1 m. s e c .  P l a t f o r m - A n v i l Gap =  .015 i n .  E l a s t i c l o a d i n g waves i n m i l d s t e e l with the a n v i l to d e c e l e r a t e the platform  Figure  17  Propagation loading  of s t r a i n  increments  i n mild  steel  under  elastic  o  VERTICAL  SCALE: 1 D i v . =  .1 v o l t  HORIZONTAL SCALE: 1 D i v . =  Figure  18  Elastic  unloading  = 20 0 y i n /  .05 m. s e c .  waves  i n mild  s  122  —  Loading  Wave  Unloading  0.2  0-4  +  Wave  0.6 L mil.sec.  L o a d i n g wave a p p r o x i m a t e l y  0.8  4'0" above t h e p l a t f o r m  U n l o a d i n g wave a p p r o x i m a t e l y  VERTICAL  SCALE:  3'3" b e l o w t h e p l a t f o r m  1 D i v . = .1 v o l t  = 2 00 y i n / i n .  HORIZONTAL SCALE: 1 D i v . = .1 m. s e c .  Figure  19  Comparison o f e l a s t i c l o a d i n g and i n g waves i n m i l d s t e e l  unload-  123  (a)  Responses of the f i r s t , the s i x t h gauges  VERTICAL SCALE: HORIZONTAL:  (b)  t h i r d , f o u r t h and  1 D i v . = .1 v o l t = 2 00 JJ i n / i n .  1 D i v . = .1 m. s e c .  Responses of the second and f i f t h  F i g u r e 20  gauges  E l a s t i c l o a d i n g waves i n 2024S-T4 aluminum  124  (a)  Responses o f the F i r s t , t h i r d , and s i x t h gauges  VERTICAL:  1 Div. =  HORIZONTAL SCALE: P l a t f o r m - A n v i l Gap  (b)  Figure  .1 v o l t 1 Div. = = .015".  = 200  fourth  y in/in.  .1 m. s e c .  R e s p o n s e s o f t h e s e c o n d and f i f t h  21  gauges  E l a s t i c l o a d i n g waves i n 2024S-T4 inum w i t h t h e a n v i l t o d e c e l e r a t e platform  alumthe  125  Figure  22  L o a d i n g - u n l o a d i n g boundary under l o a d i n g f o r 2024S-T4 aluminum  elastic  127  6000 c.p.s. 5000 4000 3000  2000 Experimental Best Least Square Fit -1000  B  400  X Inches G hange in phase  Figure  24  80  120  V a r i a t i o n o f a m p l i t u d e and p h a s e a n g l e w i t h d i s t a n c e o f t h e F o u r i e r components i n 2024S-T4 aluminum  Figure  25  Variation  of attenuation  factor  with  frequency  i n 2024S-T4  Aluminum  "Q  •  •  •  •  •  • l  •  Experimental  •  Theory (Three Parameter Model) E 12.5x10* Ibs./in E,45.5xl0^ „ T.9xl0 lbs.sec./in? 2  1  3  1000  Figure  26  Variation  Frequency  c.p.s  5000  3000  o f phase v e l o c i t y w i t h  frequency  i n 2024S-T4  aluminum  tvj  T  T  W5L  o  r, 0  0  ° O o  J i (<*>) ^1 ,  O  w  ]  0  ° 0 _  J  A  o  2  ^  o  1.5 o  o  o  o A &  i.ooi.  o  A  A  oo  A A  2  A  A  X  S  o  A O  i  1.0  A  A A A A 0.95  I  A A  A  A  *  A  0.5  o o  0.90  o  1  0.881 2000  Figure  27  4000 Frequency c.p.s.  e  V a r i a t i o n o f complex c o m p l i a n c e q u e n c y i n 2024S-T4 aluminum  with  fre-  131  VERTICAL  SCALE:  HORIZONTAL SCALE:  Figure  28  1. D i v . = 1 Div. =  E l a s t i c loading aluminum  .1 v o l t  = 200  y in/in.  .1 m. s e c .  waves i n 2024S-F  (a)  Responses of the f i r s t , and f i f t h gauges  VERTICAL SCALE: HORIZONTAL SCALE:  (b)  second, f o u r t h  1 D i v . = . 1 v o l t = 200 p i n / i n . 1 D i v . = . 1 m. s e c .  Responses of the t h i r d and f i f t h  F i g u r e 29  E l a s t i c l o a d i n g waves aluminum  in  gauges  2024S-0  and s i x t h gauge VERTICAL SCALE: HORIZONTAL SCALE:  1 D i v . = .1 v o l t = 200 p i n / i n . 1 D i v . = ,2 a, sec.  P r e s t r e s s = 7,200 l b / i n . 2  (b)  Figure  Responses of t h i r d and f i f t h gauges  30  P l a s t i c l o a d i n g waves i n 2024S-0, aluminum before workhardening  134  (a)  Responses o f f i r s t , s i x t h gauges  VERTICAL  SCALE:  second,  1 D i v . =.1  HORIZONTAL SCALE: 1 Div. = P r e s t r e s s = 7,200 l b / i n .  volt  fourth  =200  and  y in/in.  .2 m. s e c .  2  Figure  31  P l a s t i c l o a d i n g waves in 2024S-0 aluminum a f t e r w o r k h a r d e n i n g  !  I  0  I  .008  Figure  32  ,.  f  •  O  in./m.  Quasi-static  , -  stress-strain  I -  0  2  L_ 4  curve  -  0  3  2  f o r 2024S-0 aluminum  136  (a)  Responses o f the f i r s t , f i f t h gauges  VERTICAL SCALE: HORIZONTAL SCALE:  • nm  second, f o u r t h and  1 D i v . = .1 v o l t = 2 0 0 p i n / i n . 1 D i v . = .1 m. s e c .  B Si MR UN  •  i  i  i  '^^^^* —1 i \1—1—1—f 1  1  1  \m 1 1 1 .Wi 1 I  ^  1 t 1 1 I ! I j  / 1  (b)  1  Responses of the t h i r d  F i g u r e 33  i  and s i x t h  i  gauges  E l a s t i c l o a d i n g waves i n copper (Specimen No. 1)  137  (a)  Responses of the f i r s t , s i x t h gauges  VERTICAL SCALE: HORIZONTAL SCALE:  t h i r d , f o u r t h and  1 D i v . = .1 v o l t = 2 00 y i n / i n . 1 D i v . = .1 m. s e c .  A n v i l - P l a t f o r m gap = .015 i n .  (b)  Figure  Responses of the second and f i f t h  34  gauges  E l a s t i c l o a d i n g wave i n copper with the a n v i l t o d e c e l e r a t e the p l a t f o r m (Specimen No. 2)  Figure  35  L o a d i n g - u n l o a d i n g boundary under l o a d i n g f o r c o p p e r (Specimen No.  elastic 2)  Figure  36  Fourier transform  of the  first  wave i n F i g u r e  34a  140  i  40  a.  40  b. Figure  37  l  X Inches Amplitude  80  120  decay  80 X Inches Change in p h a s e  120  V a r i a t i o n o f a m p l i t u d e and p h a s e a n g l e w i t h d i s t a n c e o f t h e F o u r i e r components i n c o p p e r  Experimental Theory  OO  IThree Parameter Model) E 16.6x 10 lbs./in . 6  1  E ij  1000  Figure  38  2  Frequency  2  10.2x10 „ 9 . 4 x 1 0 * lbs. sec./in! 7  c.p.s  Variation of attenuation  3000  f a c t o r with  5000  frequency  i n copper  •  11.6 u 4)  • •  •  •  •  •  •  • • •  •  •  o  Experimental  X O  Theory 11.2  ( Three Parameter Model) E 16.6 Ibs./in? 10 E 10.2 >, xlO 9.4 lbs. sec ./in? t  x  t%l  7  2  10.81 1000  Figure  39  Frequency  Variation  c.p.s.  5000  3000  o f phase v e l o c i t y w i t h  frequency  i n copper  6  x  10  3  143  6.25  6.20  L  2000  Figure  40  F  r  e  q  u  e  n  c  y  c > p  . . 4000 s  V a r i a t i o n o f complex quency i n copper  compliance with  fre-  144  (a)  Responses of f i r s t , f i f t h gauges  VERTICAL SCALE:  1 D i v . = .1 v o l t = 2 0 0 y i n / i n .  HORIZONTAL SCALE:  (b)  second, f o u r t h and  1 D i v . = .2 m. sec.  Responses of t h i r d and s i x t h  F i g u r e 41  gauges  P l a s t i c waves i n copper with a p r e s t r e s s of 13 400 l b / i n . (Specimen No.2) 2  f  145  (a)  Responses o f f i r s t , f i f t h gauges  VERTICAL  SCALE:  HORIZONTAL SCALE:  (b)  Figure  1 Div. =  . 1 volt  1 Div. =  Responses of t h i r d  42  second, f o u r t h  and  = 2 00 p i n / i n .  . 2 m. s e c .  and s i x t h  gauges  P l a s t i c waves i n c o p p e r w i t h a o f 17,400 l b / i n . (Specimen No. 2) 2  prestress  Figure  43  Propagation of s t r a i n increments i n the x - t plane p r e s t r e s s o f 13,400 l b / i n (Specimen No. 2) 2  f o r copper  with  I  I  0  Figure ..  I 40  44  I X  Inches  I 80  1  1— 120  P r o p a g a t i o n o f s t r a i n increments i n the x - t p l a n e f o r copper 2 wdrth-ra- p r e s t r e s s o f - 1 7 ,400- l b / i n (Specimen"- N l > •  Figure  45  Dynamic s t r e s s - s t r a i n (Specimen No. 2)  curves f o r copper  Elastic Slope  17400 lbs./in. 16  2  15200  CM .  13400  c  11600  -Q  CO I  o  Quasi. Static  <8  Dynamic Prestress  / 1  4  F i g u r e 46  C o m p a r i s o n o f Dynamic (Specimen No. 2)  £ xlO-^m'U. I«yin.  and Q u a s i - s t a t i c  10  8  stress-strain  curves  f o r copper  150  Figure  47  L o a d i n g - u n l o a d i n g boundary (Specimen No. 2)  i n copper  151  (a)  Responses of f i r s t , s i x t h gauges  VERTICAL SCALE: HORIZONTAL SCALE:  (b)  Figure  second, f o u r t h and  1 D i v . = . l v o l t = 200 p i n / i n . 1 D i v . = .2 m. s e c .  Responses of t h i r d and f i f t h  48  gauges  P l a s t i c waves i n copper with a p r e s t r e s s of 11,600 l b / i n . (Specimen No. 3 ) 2  Strain Increment mic. in./ in. o A  a • A  • V  •  O  > •  _J 40  I X  I—  Inches  12000 11820 11500 11010 10010 88 20 7820 7420 6820 6630 6260 5940  0 80 160 200 240 280 320 360 400 440 480 520 1_  80  Propagation of s t r a i n increments i n the x - t plane p r e s t r e s s o f 11 ,,600 „lb/in (Specimen No. 3) 2  Propagation Velocity tt./sec.  1—  120  f o r copper w i t h a ... . . .  Strain Increment mic. in./in. o • A  • • V  •  • O  •  40  X  Inches  12460 9920 8260 7040 6060 5350 5050 4510 3960 3620  0 40 80 120 160 200 240 320 400 480  120  80  Propagation of s t r a i n increments w i t h a p r e s t r e s s o f 18,800 l b / i n  Propagation Velocity ft./sec.  2  i n the x - t plane f o r copper ( S p e c i m e n N o . 3)  154  Figure  51  Dynamic s t r e s s - s t r a i n (Specimen No. 3)  curves f o r copper  155 T——  ,—I  1  1  1  1  1  1  I  I  I  L_  40  0  Figure  52  x  Inches  80  Loading-unloading (Specimen No. 3)  boundaries  120  in  copper  n  Figure  53  A t t e n u a t i o n o f amximum p l a s t i c wave a m p l i t u d e i n c o p p e r (Specimen No. 3)  strain  157  Figure  54  P l a s t i c waves i n c o p p e r w i t h a o f 13,400 l b / i n . (Specimen No. 3) 2  prestress  158  Figure  55  Comparison of dynamic s t r e s s - s t r a i n curves i n c o p p e r w i t h a p r e s t r e s s o f 13,400 l b / i n  2  159  Figure  56  Comparison o f dynamic s t r e s s - s t r a i n c u r v e s i n c o p p e r w i t h a p r e s t r e s s o f 15,200 l b / i n  2  160  lilt  t ) ti  •  I T  >k > i  (a)  Impact No. 2  VERTICAL SCALE:  1 D i v . =.1 v o l t = 200 y i n / i n .  HORIZONTAL SCALE:  1 D i v . = .2 m. s e c .  S t a t i c P r e s t r e s s = 14,300  (b)  F i g u r e  lb/in . 2  Impact N o . 6  57  P l a s t i c waves i nc o p p e r under r e p e a t e d i m p a c t (2nd a n d 6th i m p a c t s i n S p e c i m e n No. 4 )  161  (a)  Impact No. 8  VERTICAL SCALE:  1 D i v . = .1 v o l t = 200 u i n / i n .  HORIZONTAL SCALE:  1 D i v . = .2 m. sec.  S t a t i c P r e s t r e s s = 14,300 l b / i n . 2  (b)  Impact No.  F i g u r e 58  17  P l a s t i c waves i n copper under repeated impact (8th and 17th impacts i n Specimen No. 4)  162  Figure  59  R i s e t i m e o f s t r a i n wave i n c o p p e r a t f i r s t s t r a i n ' gauge p o s i t i o n (Specimen No. 4)  163  Figure  60  Propagation of s t r a i n increments i n copper u n d e r r e p e a t e d i m p a c t s (Specimen No. 4)  164  Responses of f i r s t ,  VERTICAL SCALE: HORIZONTAL SCALE:  F i g u r e 61  second  and f o u r t h gauges  1 D i v . = .1 v o l t = 200 y i n / i n . 1 D i v . = .1 m. sec.  E l a s t i c unloading waves i n copper (Specimen No. 5)  165  Responses o f f o u r  VERTICAL  SCALE:  HORIZONTAL SCALE:  Figure  62  gauges  1 Div. =  .1 v o l t  1 Div. =  = 200  y  in/in.  .1 m. s e c .  Unloading waves i n copper with a p r e s t r e s s of 11,600 l b / i n . (Specimen No. 5) 2  1  v  Mild Steel  •  Copper  o 800  "  r  Aluminum  L  E  400  4  F i g u r e 63  ^  X  Attenuation metals  inches  80  of elastic  waves i n t h r e e  167  VERTICAL SCALE:  1 D i v . = 760  ft/sec . 2  (ACCELEROMETER)  VERTICAL SCALE:  1 D i v . = 10  volts  (TRIGGER)  HORIZONTAL SCALE:  F i g u r e 64  1 Div. =  .05 m. s e c .  A c c e l e r a t i o n of the p l a t f o r m impact  after  VERTICAL SCALE:  1 D i v . = 760  ft/sec . 2  (ACCELEROMETER)  VERTICAL SCALE:  1 D i v . = 10  volts  (TRIGGER) HORIZONTAL SCALE:  Figure  65  Multiple  1 Div. =  impacts  .1 m. s e c .  o f t h e hammer  Figure  66  E l a s t i c waves i n (Specimen No. 4)  copper  170  1200  c \ c  s  800  Strain Gauges 400  Accelerometer  40  Figure  67  v  A  . inches  80  120  C o m p a r i s o n o f maximum s t r a i n a m p l i t u d e r e c o r d e d by t h e a c c e l e r o m e t e r w i t h t h e s t r a i n gauge r e a d i n g s  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0104174/manifest

Comment

Related Items