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Energy dissipation in paper tearing as time-dependent phenomenon Sun, Bernard Ching-Huey 1967

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ENERGY  DISSIPATION  IN PAPER T E A R I N G  AS TIME-DEPENDENT  PHENOMENON  by  BERNARD  CHING-HUEY  B . S c . A . NATIONAL TAIWAN Taiwan,  A THESIS  The R e p u b l i c  SUBMITTED  FOR  MASTER  in  UNIVERSITY  of China,  IN P A R T I A L  REQUIREMENTS  SUN  1960  F U L F I L M E N T OF THE  THE DEGREE  OF  OF - F Q R € S W ¥  t h e Department of Forestry  We a c c e p t standard  this  THE  thesis  as c o n f o r m i n g  UNIVERSITY  OF B R I T I S H  September,  1967  to the required  COLUMBIA  In p r e s e n t i n g  for  thesis  an a d v a n c e d d e g r e e  that  the  Study.  thesis  Library  for  agree  at  in p a r t i a l  make i t  that  freely  or  representatives.  of  my w r i t t e n  this  thesis  for  may be g r a n t e d  for  permission.  Department The U n i v e r s i t y o f Br i t i s i r Col umb i a V a n c o u v e r 8, C a n a d a  It  is  financial  of  British  available  permission  purposes  by h.i-s  fulfilment  the U n i v e r s i t y of  scholarly  publication  without  shall  I further  Department  or  this  for  the  Columbia,  I  reference  and  extensive  by  the  requirements  copying  this  Head o f my  understood  gain  of  agree  shall  that  not  be  copying  allowed  i i ' ABSTRACT  The methods sector  nature has  In  reviewed.  variations It  that  calculate  the  net  available  for  tearing  sequently,  the  or  energy,  tearing  process. ing  energy  fect  jected  tear  by and  energy  rate  and  was  standard  samples. time  a quadratic tearing  energy tearing  process.  and  the  on  hence  can  model  residual  net  and  to  tester  paper and  model  kinetic  be  can  ener-  to  used  which  Con-  energy, rupture  relates examine  properties  to  is  energy.  i n the  model  used  be  residual  expended  time-dependent  devised  f o r measuring  From  an  relationship  equation.  and  the  concepts  sector,  mathematical  From  v e l o c i t i e s were  factors  based  test  tearthe  of paper  efsub-  stress.  A method  and  of  a mathematical  tester  portion  the  tear  tearing.  between  i s that  energy  dissipation  derived  the  paper,  paper  contributor  mathematical  of  difference  to tearing  distance  was  this  to v e l o c i t y ,  of tear  prime  energy  during  Furthermore,  kinetic  conservation,  energy  i n paper  i s shown  the  with  energy  tearing  The  t o be  agreement  p r i n c i p l e of  expressing gy  been  ballistic-type internal  i s considered  rupture. the  of  their  the  time  oscilloscope was this  measured  variation  trace, and  equation,  calculated at  required  tear  represented  sector  f o r computing any  the  to  instant  of  swing various the  i i i  Results methods paper  with  of plies  required paper  shown  that  are time-dependent,  varies  number  tear  have  was  also  i n that  t h e sample  torn  to tear  ballistic-type time  condition.  simultaneously,  a paper  sheet.  time-dependent,  tear  required  to  The h i g h e r  the longer  The  test  energy  increasing  was  tear  the the time  required  with  to  tear  decreasing  rate. It  was  found  ing  strength  not  always  exists  a n d number  hold,  between  Although inherent  that  t h e d i r e c t r e l a t i o n s h i p between of plies  but that  tearing  a constant  strength  properties  simultaneously  may  does  direct relationship  and t e a r i n g  the b a l l i s t i c - t y p e  specimen  torn  tear-  energy.  tear  test  i s time-dependent,  have  a profound  e f f e c t on  results. Test per have  grades  r e s u l t s with varying  confirmed  adequate.  that  an E l m e n d o r f  i n tearing t h e energy  tear  strength  tester  from  14  on f i v e t o 156  d i s s i p a t i o n concept i s  pa-  g/sheet  iv TABLE  OF CONTENTS Page  ABSTRACT TABLE  i  OF CONTENTS  LIST  OF T A B L E S  LIST  OF F I G U R E S  i  iv v i v i i  ACKNOWLEDGMENT  i x  INTRODUCTION  1  LITERATURE  4  REVIEW  Variation the Test  Caused  Principle  and Methods  7  Ballistic-type Tear Paper  Test  Tear  f o r Calibrating  Testers  Theory  Rheological  ENERGY D I S S I P A T I O N IN THE TEAR T E S T MATERIALS  by C o n d u c t o f  10 12  Properties AND RATE  15  OF TEAR 18  AND METHODS  27  Materials  27  Methods  28  Tearing  procedures  28  Experimental  design  29  Rate  measurement  29  of tear  Zero-swing tear measurement  distance-time  Handling fitting  and  Summary  o f data  36 curve 38  o f methods  41  Table  of Contents  (cont'd)  v  Page  DISCUSSION Curve  4 3  Fitting  Interpretation Review  44 of Results  of Present  (1)  Energy  (2)  Dynamic  (3)  Rate  Tear  dissipation property  of tear  limitation  '  Test  48 Knowledge  concept  o f paper  effect  of scale  57 57 59  and reading  59  CONCLUSIONS  61  REFERENCES  63  TABLES  AND  APPENDICES  FIGURES  66 99  vi LIST OF TABLES Page  TABLE 1. TABLE 2.  TABLE 3 .  TABLE 4 .  TABLE 5.  Parameters of the f i v e paper grades used i n the study  66  Average tear d i s t a n c e , time, v e l o c i t y , energy and strength values f o r f i v e paper grades  67  Comparison between three d i f f e r e n t methods of preparing conductive p l i e s used i n the study  74  Test of s i g n i f i c a n c e f o r part i a l regression c o e f f i c i e n t f o r the newsprint ten-ply specimen  75  Tear distance-time r e l a t i o n ships f o r f i v e paper grades with d i f f e r e n t number of p l i e s and r e p l i c a t i o n s  76  vii LIST  OF  FIGURES Page  FIGURE  FIGURE  FIGURE  FIGURE  FIGURE  FIGURE  FIGURE  FIGURE  1.  2.  3.  4.  5.  6.  7.  8.  Schematic diagram illustrating angles involved i n the b a l l i s t i c type tearing p r i n c i p l e  79  M o d e l c o n s i s t i n g of dashpots used t o i l stress-strain-time s h i p s for p o l y m e r i c  80  s p r i n g s and lustrate relationmaterials  Conversion of energies i n the b a l l i s t i c - t y p e tearing process  81  Schematic diagrams illustrating the conversion of energies i n , a. s e c t o r swing w i t h o u t s p e c i men, b . s e c t o r s w i n g when t e a r i n g a specimen  82  R e l a t i o n s h i p s between c r o s s machine-direction tearing strength and number o f p l i e s t o r n s i m u l t a n e o u s l y f o r f i v e paper grades  83  O s c i l l o s c o p e t r a c e s had w i t h f o u r d i f f e r e n t c o n d u c t i v e materials  84  P a t t e r n o f c o n d u c t i v e l i n e s used for t h e s t u d y . L i g h t and dark l i n e s a r e s i l v e r and g r a p h i t e conductive materials, respectively  86  Electrical circuit measuring the tear time r e l a t i o n s h i p  87  used  for  u s e d for distance-  FIGURE  9.  5et-up  the study  FIGURE  10.  Tear distance-time r e l a t i o n s h i p s f o r 3 0 - l b . n &. m b a g paper. Number o f p a p e r p l i e s i s m a r k e d on e a c h c u r v e  88  89  List  of Figures  viii  (cont'd)  Page FIGURE  11a.  FIGURE l i b .  FIGURE  11c.  FIGURE l i d .  FIGURE l i e .  FIGURE  FIGURE  FIGURE  12.  13.  14.  Tearing energy, distance and t i m e r e l a t i o n s h i p s f o r unglazed onion skin. B r o k e n and s o l i d l i n e s a r e number o f p a p e r p l i e s a n d t e a r distances, respectively  90  T e a r i n g energy, distance and t i m e relationships f o r newsprint. B r o k e n a n d s o l i d l i n e s a r e number of paper p l i e s and t e a r distances, respectively  91  Tearing energy, distance and t i m e r e l a t i o n s h i p s f o r 30 l b . n &, m bag p a p e r . B r o k e n and s o l i d l i n e s a r e number o f p a p e r p l i e s a n d t e a r distances, respectively ,  92  T e a r i n g energy, distance and t i m e r e l a t i o n s h i p s f o r I s l a n d 55.5 l b . wrapper. B r o k e n and s o l i d l i n e s a r e number o f p a p e r p l i e s a n d t e a r distances, respectively  93  Tearing energy, distance and t i m e r e l a t i o n s h i p s f o r p a r c e l wrap. B r o k e n a n d s o l i d l i n e s a r e number o f paper p l i e s and t e a r distances, respectively  94  Relationships between r e l a t i v e t e a r t i m e p e r u n i t p l y a n d number o f p l i e s t o r n s i m u l t a n e o u s l y  95  Relationships between t e a r i n g strength and t e a r i n g e n e r g y f o r f i v e paper grades  .....  96  T e a r i n g energy and d i s t a n c e r e l a t i o n s h i p s f o r unglazed onion skin. 97  FIGURE  15.  Relationships between t e a r i n g a c c e l e r a t i o n a n d number o f p l i e s torn simultaneously f o r f i v e paper grades  98  ix  ACKNOWLEDGMENT  The the  Faculty  offered with for  author  of Forestry,  special appreciation h i s guidance  Honorary  Officer,  planning  conduct  C o l u m b i a , who  of this  work;  Professor,  and e x p e r i m e n t a l  fessor,  Vancouver  planning  Wellwood,  phases and  Vancouver Forestry  t h e Department  technical  assistance  Acknowledgment Division  Laboratory,  of MacMillan  of Forestry  Assistant  Forest  and R u r a l  given  materials;  t o t h e Vancouver  the  Department  of Forestry  Biometrician,  Products  Limited, Forest  and to  Labora-  Development, f o r  to Island  and R u r a l  and t o t h e N a t i o n a l  Pro-  programming;  i n s e t t i n g up t e s t i n g  Bloedel,  t o Dr.  t h e Department o f  f o r computer  the  facilities  study;  f o rh e l p f u l suggestions  Vancouver  i s also  the  f o r help i n  a n d Mr. J . H e j j a s ,  Development,  Technician,  of this  and Re-  Laboratory,  Development,  a n d Mr. L . V a l g ,  t o M r s . H. F r o s e ,  R. K a b o s ,  of Forestry,  Products  and e x e c u t i o n  Products  and R u r a l  criticisms;  Forest  of Forestry,  Forest  made t o D r . L . B a c h ,  Faculty  and R u r a l  Professor,  Faculty  i s also  Professor,  of Forestry  initiation,  tory,  of British  t o D r . J . W. W i l s o n ,  acknowledgment  Assistant  Department  R.W.  during  t o members o f  of the thesis.  Grateful  search  during  h i s gratitude  University  h e l p f u l suggestions  writing  Mr.  acknowledges  facilities. Paper  Mills,  f o rsupplying Products  Development  Research Council  part of  Laboratory, f o r their o f Canada and  X  the  University  support  during  Last, tience, able  but  of  Columbia  f o r repeated  the  academic  programme.  not  least  Mrs.  devotion  years.  British  and  to  E.  encouragement  S.  Sun,  financial  f o r her  throughout  these  pamemor-  INTRODUCTION  Since method  advancement  about  measuring become  a  half  a  basic  widely  evaluations  test  procedures  tion  i n the  paper  and has  pulp  ting  effects  have  published  well  as  buted one on  to  of  has  not  dence. test  a  paper has  various  the  effect  obtained, been  which  method,  some  an  property.  test  valuable  This  for  tear  in  method  quality  has  con-  studies.  Adoption  of  important  point  communica-  of  standard  industry. been  tear  done  test  and  i n the  past  variables. instrument  mechanism,  standards. of  number  has  been  Yet, of  on  evalua-  Several  authors  calibration,  which there  plies  continue  p o i n t i n g to  by to  contri-  still  remains  torn  simultaneously  experimental doubt  the  value  number  as  have  treated theoretically  explained  authors  continually  proved  u s e f u l means  satisfactorily  Hence,  internal  i t has  f i n d i n g s on  recommended  variable, value  as  tearing principles the  ago,  research  work  their  Elmendorf  strength  l e d to  and  Considerable  the  century  accepted  trol  of  of  but evi-  of  this  plies  ef-  fect . It results is  has  been  increase  increased.  constantly reported as  This  the  number  positive  negative  r e l a t i o n s h i p ) may  strength  value,  when  of  that  plies  torn  relationship  be  compared  1  so  large  with  Elmendorf  as  results  test  simultaneously  (in a to  tear  few  double  obtained  cases, the  tear  under  2 rigorous conditions. or given  Previous  i n v e s t i g a t o r s have reasoned  evidence to e x p l a i n t h i s d e v i a t i o n i n terms of  several test factors. out" or "fan-out"  These i n c l u d e , clamp design,  of the p l i e s , f r i c t i o n a l binding  "spreadbetween  the torn paper edges, s t i f f n e s s d i f f e r e n c e s , basis weight and  the degree of s p l i t t i n g or  "skinning".  Recent advances i n paper t e s t i n g have emphasized s t r e s s - s t r a i n r e l a t i o n s h i p , and behavior has  been s t u d i e d .  been concentrated  i n some few  t e a r i n g r e s i s t a n c e has  cases r h e o l o g i c a l  Most work i n these f i e l d s  on t e n s i l e s t r e n g t h .  the  has  Sometimes paper  been examined under constant  r a t e of  s t r a i n or s t r e s s . The explained  a p p l i c a t i o n of r h e o l o g i c a l p r i n c i p l e s to paper some p r o p e r t i e s which were unknown i n the  In a s i m i l a r way,  t i o n about the t e s t not revealed  neglected  very obvious and or ignored  some basic informa-  previously.  b a s i c phenomena, which have been  i n paper tear t e s t i n g , are the time r e -  quired to f r a c t u r e the piece and curs.  past.  examination of the Elmendorf tear t e s t  method i n r h e o l o g i c a l terms could provide  Two  has  the r a t e at which t h i s  oc-  Researchers agree that the t o t a l time r e q u i r e d i s r e l a -  t i v e l y short compared to other more, one  can  paper strength t e s t s .  Further-  e a s i l y d i s t i n g u i s h a time d i f f e r e n c e between  t e a r i n g a s i n g l e paper ply and over the same d i s t a n c e .  The  ten p l i e s of the same m a t e r i a l  longer time required to tear  3 the ten p l i e s demonstrates the slower r a t e e f f e c t . ing  Combin-  t h i s simple f a c t with the understanding that energy f o r  the t e s t instrument o r i g i n a t e s from p o s i t i o n of the  tester  sector pendulum, suggests that a change i n t e a r i n g time changes tear r a t e , and  consequently changes the work done i n  tearing which a f f e c t s the f i n a l t e s t r e s u l t s . From t h i s i t i s p o s s i b l e to hypothesize t h a t : (1) The  time r e q u i r e d  to tear paper v a r i e s with sample con-  d i t i o n , which can be generalized  i n terms of tear r e -  sistance; (2)  Work r e q u i r e d to tear paper i s time-dependent, and i n creases with decreasing  (3) The  increase  r a t e of t e a r ;  i n tear strength  value as number of p l i e s  torn simultaneously i s increased time and  slower rate of t e a r , a c o n d i t i o n which  more energy; ( 4 ) Results  r e s u l t s from the  longer  requires  and  obtained from b a l l i s t i c tear t e s t e r s , such as  the  Elmendorf instrument, are time-dependent. Thereby, the present study was time-dependent behavior of the with one  standard instrument.  designed to i n v e s t i g a t e  paper tear t e s t as i t occurs  LITERATURE  bles to  Numerous  studies  have  on  tearing  strength.  paper  o r i g i n a t e from  characteristics and  pulp  during  fiber  the  of  wood,  papermaking  fiber  density  terms,  weakening processes, degree  of  the  length, the  fiber  microfibril cellulose  polymerization  on  These  well  to  pulping  as  beating  kinds  and  The  of  this  field  has  been  discussed  Casey  (6),  Dinwoodie  (10)  and  only  indirectly  literature  reviewed these and  by  factors  are  not  Since strength the  described of  further the  with  method  are  has  report  Elmendorf  been  much  theoretical  ballistic-type  refining  the  degree  factors. and  recently  (21).  All  present  study  discussed.  first the  or  hydrophylic  Rydholm the  or  of  important  r e l a t e d to  other  resultant  of  hemicelluloses  been  morphological  distribution  are  introduced  or  exposure,  as  known  pulping  i t has  fiber  varia-  chemical  fragmentation and  well  residual lignin  and  thickness  o r i e n t a t i o n and  amount  are  those  example,  strength,  of  during  as  various  polymerization,  and  variables  wall  of  effect  physical  an  cell  fiber due  as  As  ratio  the  occurring  processes,  individual  of  done  changes  processes.  that  characters,  been  morphological,  purification  reported  of  REVIEW  tear  on  determining  t e s t e r was  discussed.  aspects  of  testers.  4  the  paper  published  tearing in  1920  (11),  Carson  and  Snyder  design  and  calibration  Clark  (7)  used  Thwing  (5)  and  5  Marx-Elmendorf method  for  was  sheets, values  a  so over  the and  internal  forces  was  tearing  internal to  the  area  on  that  tearing  force  expressed  as  each  resistance on  the  R  =  internal  W  =  work  simple  which  each  distance  he  observed  of  approximately  which  sample constant  range. d e f i n i t i o n s and  with  paper The  equally  arose  as  was  equal  the  of  the  or  line.  described  group of  opposite  accepted as  a  resultant  in  force  widely  zone  to  a  c a l i b r a t i o n method,  zone.  either  tear  and  tearing  done  paper  d e f i n i t i o n has  Elmendorf  the  from  applied  yields  resultant  describe  tester  tearing  reviewed  acting  to  these  directions..  to  the  definition  resistance  two  of  portions This  interfor  opposed of  the  paper  definition is  as:  of The  the  tearing  exerted to  of  resistance  the  be  another  working  (8)  resistance,  adjacent  where:  Watson  proposed  assuming  nal  ordinary  tear  to  instrument  tearing  acting  examples  factor  proposed  the  as  dynamic  adjustment  that  forces  By  (3)  upon  Cohen the  correction  Bergea  based  testers  calibrating a  determined value.  tear  in  tearing  having  been  resistance,  a  In  addition  to  this  defined  internal  tearing  certain  adopted  Poller-Elmendorf  across  in tear  and a  piece  length,  d.  c a l i b r a t i o n of  Thwing-  testers.  definition, Mallett  and  resistance  of  in  terms  Marx work  (19) done  in  6 t e a r i n g the paper d i v i d e d by the t e a r i n g distance as:  Clark  (7) has shown, however, that the Marx-Elmendorf tear  t e s t e r was graduated according Cottrall  to Equation ClH»  (9) s t u d i e d the mechanism of t e a r i n g and i t s  r e l a t i o n s h i p with  other paper strength parameters,  that the tear t e s t i t s e l f i s of l i t t l e  or no value  concluding and might  even be harmful, i f i n c l u d e d as part of paper s p e c i f i c a t i o n s . His explanation  i s that burst, t e n s i l e strength and f o l d i n g  endurance might e a s i l y be s a c r i f i c e d by putting too much emphasis  on o b t a i n i n g high t e a r i n g s t r e n g t h .  In f a c t , some  paper grades r e c e i v e more emphasis on burst and t e n s i l e s t r e n g t h than t e a r r e s i s t a n c e .  For other papers, t e a r  tance i s p r e f e r r e d over burst and t e n s i l e s t r e n g t h . these same three strength p r o p e r t i e s are considered  resis-  Otherwise, equally  important f o r a l a r g e range of paper grades. In order to account f o r both t e a r i n g and burst  strengths,  Fanselow and Fanselow (12) suggested using the product of both values  as an index f o r e v a l u a t i n g f i b r e and p a r t i c u l a r l y  q u a l i t i e s developed during r e f i n i n g .  They i n d i c a t e d advan-  tages of c h a r a c t e r i z i n g pulp strength p o t e n t i a l , removing comp l i c a t i o n s i n v o l v e d i n comparing pulps with c i e s i n developing  divergent  tenden-  burst or t e n s i l e strength versus r e t a i n i n g  r e s i s t a n c e to t e a r , e v a l u a t i n g performance c h a r a c t e r i s t i c s of d i f f e r e n t r e f i n e r s , e v a l u a t i n g pulp strength  and other  7 purposes. have  By  similar  appeared  Variation.:  i n the  Caused  Operation on  results.  number  of  has  been  (7,  8,  ties  a  of  adjusted  the  as  to  torn  profound that  In a  single  acy,  effect  to  remain and  on  over  the  case  29).  scale  the  (29)  grams  methods  should  with  the  a scale  sheets  be  upon  be  properthe  of  plies  number  tear tester  provide  they  has  doubled  a  reading working  suggested  too  of  recommended  large a  not  can  range.  that  too  plies  values  number  They  were  of  specify  value.  disadvantage,  variable  test  dependent  working  allows  results  a  number  tear  Elmendorf  to  but  effects  relationship  showed  test  plies  as  The  values  i n the  this  long  important  the  affecting  seemingly  final  forty  very  Varying  within  of  has  low to  range.  using  for  accur-  give  scale  30. and  variations  simultaneously of  as  i n no  Swartout value  to  values"  Test  Standard  Minor  the  overcome  sheet  while  readings  top  tester  simultaneously  twenty  order  and  paper.  a d j u s t i n g number  between  28  negative,  acceptable  Winterbottom plies  24,  or  "beater  includes several,  c o n s t a n t l y as  17,  the  test  together.  particular  so  tearing  specimen  reported 16,  Conduct,' o f  torn  positive  of  range  plies  14,  either  The  various  trade.  by  of  calculations,  Setterholm due  to  i n terms  specimen,  (24)  increasing of  explained the  "spread-out"  bulking  i n the  cause  number of  clamp  the and  of  of  test  plies  sheet  at  deviation  torn the from  a  a  straight  that  variation  constant thinner  in  the  line  projected due  that  ones,  spreading and  line  at  even the  projected  Increasing sheets  different  the  and  tear  by  the  higher  of  The  line  to  They  plies  tear  in  values  the  deviate  were  found  was  tendency  securing  slit  placing  tear  grades  not than  for  clamps,  from  associated  of  degree  with  plies  but  separators  value,  concluded  tester,  when  initial  by  the  number  number  density.  failure  increased  they  top  had  slit.  a  straight  with  in-  plies.  bulk  paper  specimens, value  f o r the  increased  accompanied  similar  specimen  of  initial  i n c r e a s i n g the  at  from  number  the  t h i c k e r papers  tendency  creasing  to  from  but of  2-mil  the  between  increase  spreading. and  3-mil  By  individual in  aluminum  r e l a t i o n s h i p between  torn  i s not  r e l a t e s to  the  type  inherent of  was  comparing  that  an  bulk  foil tear  feature  material  of  being  tested. Wink plies, clamp  as  and  Van  well  as  design. 10%  number  plies  variation tions such the  were as  could  of  and  tearing  as  examined  clamping  that  in  cause  upon  described  splitting  nature  found  variation  depended  (28)  different  They  introduce of  Eperen  the  test  the  effect  methods  method  results,  of  variation.  the  type  of  due  to  i s further  i n the of  changing  nature  sheets.  complicated  by  of  variaof  tearing  Change the  the  the  degree  These  of  could  varying  The  material.  change  number  clamping  while  100%  "spreading-out"  by  of  in  degree  of  9 interfibre  bonding.  of  tearing  by  in  the  the  usual  the  Jones between basis  was  and  tear  and  caused  by by  beating.  They  creasing  basis  behavior  higher than basis  In (27)  and  Van  non  plies  Elmendorf  ment  by  the  torn  plies  a  are  scale  be  reading  torn  value  of  pulp in  and  also  and  and  for  of  with  the  confirmed  by  in-  differ-  mentioned to  Relationship  as  increase  degree  tendency  Gallay  idea  rigidity.  i s the  basis  that  split  between results  of  (17),  most  tear  He  Wahlberg factor  suggested  important  and  that  phenome-  weight.  eliminating standard tear  together  within  of  stiffness  they  higher  strength  (14).  Marx-Elmendorf to  the  Rate  explanation  a  flexural  increasing of  failures  simultaneously  r e l a t i o n s h i p between  simultaneously, and  was  stiffness  purpose  of  sheets.  Jones  linear  torn  Instead,  have  Akker  as  nature  positive relationship  splitting.  an  papers.  the  because  changed For  to  the  tear  tearing  increase  not  weight  the  plies  type  that  was  den  with  questioned  between  paper  sheets  expressed  occurs  of  splitting  contrast  stiffness  al  and  and  of  weight,  their  basis  emphasized  this  degree  weight  weight  number  weight  tester  reported  ratio  concluded  lower  Hardacker  (the  the  compared  method.  (17)  basis  of  basis  the  Gallay  factor  affected  ent  test  they  tear  applications  tear  weight)  mainly  Elmendorf  paper  Elmendorf  Accordingly,  certain  the  effect  procedures  testers in  order  defined  of for  number  Thwing-  specify  that  to  the  give  limits.  of  severinstru-  Cohen  and  10  Watson  (8) presumed  ensure  a fairly  sidered  that  nificant  that  uniform  these  these rate  limits  influence  on t e a r  The importance  by  members  of the Institute  Cohen  and Watson  values er  widths,  ues. by  when  This  wider  specimens  stresses edges,  this  was  a more  caused also  to affect  produced  to higher  studied  could  higher  sig-  by number considered  tear  tear  but  significantly  bending  values.  on  50 ± 1 2 mm,  great-  higher  resistance  used  i n this  referred  val-  offered  increase  and Watson  or  line  (8).  They  friction  between  torn  values.  tear study  tester  bearing.  t h e same  Ballistic-type  such  consists  t o as s e c t o r )  i s attached  jaw h a v i n g  fracture  d i f f e r e n t d i s t r i b u t i o n of  for Calibrating  of a ball  t h e specimen  of the tear  by Cohen  cause  tear  ballistic-type  b y means  another  have  con-  Chemistry (16).  i s within  o f non-symmetry  and Methods  (hereafter post  a s 1 0 0 mm,  and c o n s e q u e n t l y  giving  strument  rate  no s i g n i f i c a n t e f f e c t  variation  has been  that  Principle Testers  A  (8) f o u n d  related  effect  concluded  of  was  reported  further  that  rate  o f Paper  They  to  specimens.  The in  such  than  of tear  has been  width  of tear.  value  plies.  width  are introduced  and t h e t e a r  of  Sample  limits  as t h e E l m e n d o r f i n -  of a sector  suspended  from  pendulum a  A jaw f o r h o l d i n g  t o the stationary function  Tear  post,  i s attached  stationary one  part  while  to the right  II radial  edge  of  Moving tion  the  the  the  the  sector  mass  its  center  of  complete cally,  of  one  the  tearing  is  the  as  net  tearing  scale in  a  amount the  the  is  angle  enables  During  in  is the  the  energy  the  tear  [l],  average  angle  ( Q^  which  allowed  to  theoreti-  Fig.  1)  However,  friction air  at  the  should very  the  ball  resistance. which  available  portion  net  to  expres-  do  work  the  the  required  a to  point  angle  specimen  net  energy  specimen. tear  a  the still  expresses has  and  work Based  certain  a  for  the  of  and  ( © 3 )  the  as  on  part  specimen  represented of  zero  available  operation,  This  after  energy  the  tearing  between  resistance force  or  fracturing  system  difference  overcome  of  illustrate  in  and  locating  obtained.  The  that  angle,  ( © 3 )  failed.  the  swing  in  energy  by  energy  function  is  release.  smaller  Con-  through  to  posi-  mass.  position,  graphically  friction  potential  a  sector  employed  absorbed  absorbed  residual  is  are  as  swing  specimen.  specimen  net  energy  of  the  before  slightly  the  scale.  as  pointer  paper  instrument  smaller  presented  of  distance  i t s raised  ( a l l angles  energy  Determining tearing  from  If  free  center  changes  vertical  raised.  (9^)  of  sector  energy  the  same m a g n i t u d e  result:  energy  of  half-swing  angle  bearing,and  on  and  i t s equilibrium the  potential  mass  amounts  These  from  raises  p r i n c i p l e are  the  small  ses  sector  simultaneously  sequently,  be  sector.  been residual  done on  to Equation  distance  12  through scale gy  a specimen  i s fixed  that  the  on t h e s e c t o r  has been  tearing  c a n be c a l c u l a t e d .  used  force  In p r a c t i c e ,  to represent  to tear  t o be r e a d  t h e amount  a specimen. directly  a  This  cosine of  scale  ener-  enables  i n grams  from t h e  by u s i n g  68.8  tester. Most tear  tear  testers  distance.  definition, practical  The  2 X  are calibrated  distance  68.8  cm  over  = 137.6  cm.  68.8  required  i n tearing  i s directly  ing  distance,  and t h a t  the force  t h e number  of plies  torn  tear  a booklet  of sixteen  from  the scale  when  calculating  i • tearxng  3  where  Tear  tear the  strength  « force  =  Test  Theory  Very  little  test  theory.  tearing  elemental  zone  forces  long.  across  basis.  attention Brecht  as e x t e n s i v e , involved  as p r o d u c i n g  related  read  Results of plies  are  used f o r  conditions i s : reading  r l_3J  of plies  given  instead  of force  -  u  i n grams p e r  t o development  (4) i n 1934  stress  related  1  sheet.  of  considered  o f as a p o i n t .  i n the tearing  a moment  tear-  cm.  these  i s expressed  and Imset  to  i s to  16 X a v e r a g e scale a r —T. T~-  has been  that  the practice  number  (26) under  force  assuming  i s linearly  4.3  i s by  i t i s not  The e q u a t i o n  number  tearing  By  as  acts  proportional  required  a different  3  the average  sidered  sheets  tearing  tearing  cm  simultaneously,  to the sixteen-sheet  . Average  Apparently,  a specimen  work  adjusted  the force  to tear  the  to  which  cm  were to a  The conreference  13  point.  The t e a r i n g  this  moment  ence  point  theory  affect  to the line  zone size  that  of the tearing  of  out weaknesses  during  steady  ism  Paper  fractional fibre  rupture they  Based to  fibres to  rupture  and s h e e t  and e n e r g y  two  i n the  (16) have  theory  arising  in  between t h e  a uniform d i s t r i b u t i o n  zone.  This  of the Institute  i s based  on t h e m e c h a n -  d i s s i p a t i o n , i . e . , energy i s dissipated  t h e energy  within  expended  exthe  i s trans-  parts:  network,  p u l l i n g i n d i v i d u a l f i b r e s out o f  and  caused  by s t r e s s i n g  in tensile  individual fibres  until  failure.  of s t r e s s - s t r a i n diagrams versus  t h e network, a fibre  This  strength.  Chemistry  by members  o f paper  work,  and f o r c e  force.  extensibility  and Imset  and assumes  adopted  the r e f e r -  concentration  the forces  a specimen,  drag  on a n a l y s i s  from  was  between  and t e a r i n g  the tearing  a sheet  into  work  break  failure,  zone  (16) i n 1944.  failure  mainly  the (2)  theory  In t e a r i n g  formed (1)  over  i n tearing  sheet.  length  with  failure  Chemistry  of tear  pended  tear  forces  Another of  of stress  of the Brecht  to deal  by d i v i d i n g  of the tearing  o f t h e I n s t i t u t e o f Paper  i t attempts  fibres  fibre  calculated distance  of action  influence  a n d how  Members pointed  i s then  by t h e p e r p e n d i c u l a r  emphasized  tearing  force  they  i s very  displacement concluded  much  less  curves  that  than  of f i b r e s  forpulling  t h e work  that  stretched  required  required to  14 extract  i t unbroken  required  to  rupture  drag  an  intact  out This  theory  associated and  with  burst,  factor  the  fibre  has  been  is  processes,  the  paper and  commercial  earliest  heating  stages,  strength  Initial curve  early  i n the stages of  pulled  intact  fibre  bonding  result work the  i n the  of  per  by  the  the  and  explanation  drag  force  needed  to  required been  to  is  to  to  as  general,  mesh.  Since  greater the  Tearing  sheet  account  strength  Hardwood  pulps  values.  is a  of  amount  the  rapid i n in  very  them in of  the are  the  strength  rupturing  effect  fail-  inter-  fractional  the  the  the  proceeds  change  fibre  time  i s reduced. for  the  decrease  fewer  than  in  tearing  This  intrinsic  with  maximum  beating  the  im-  the  increase  component  and  an  increase  there  per  increasing of  preparation  tearing  reach  that  increases  tear  used  pulp  phenomena  strength-beating  work  treatment.  fibre  In  slight  to  However,  lowering  beating  has  the  considered  range.  starts  theory  by  be  further beating.  tighter  i s caused  individual  energy  the  pulp  endurance  After  tearing  fibres  from  may  refining  beating  beating.  ruptured  mechanism  that  some  strength.  beating  fractional of  number  of  further  i s explained  crease  ure  periods  rise  than  Among  coniferous  p r o g r e s s i v e l y with  before  although  explain  folding  differently.  longer  to  beating  behaves  require  greater  used  strength  decreases  mesh,  strength.  strength  within  fibre  fibre.  affecting  tensile  beating  a  tearing  papermaking  portant  from  as  a  drag work,  This  same  of  fibre  15 length drag  and  work  strength increases  stronger  the  forces.  This,  intact  fibre in  the  mesh  Giertz  and  Helle  members  tested  i t with  were i n  general  slight  modification  fect  of  They  also  fibre the  strain  importance  the  for  have  out  ruptured  another  distance  the  on  importance Brecht  to  be  pulled  tearing  Paper  forces.  the  theory  Chemistry  but to  include  sides  of  the  the  strength  and  Imset  (4)  with  of  sheet  extensibility  in  ef-  failure.  fibre  agree  They  recommended  of  and  and  experiments.  term  both  the  rupturing  reviewed  theory,  fractional  and  to  by  laboratory  the  the  length  resistance  recently  of  adding  some  fibre  I n s t i t u t e of  with  since  m o r e members  than  a series  agreement  pointed  length,  (13)  by  the  provides  rather  of  strength,  increasing  higher  turn,  the  by  tearing  with  from  adopted have  on  and  concerning  determining  tearing  strength.  Paper  Rheological  Recent  Properties  studies  stress-strain-time ing  of  about  the test  elastic  and  nature  on  paper  strength  r e l a t i o n s h i p have of  methods.  paper  strength,  Paper  has  time-dependent  been  flow  properties provided as  well  shown  based  new  information  exhibit  properties.  It  load-elongation  ( s t r e s s - s t r a i n ) curve.  is  rate-dependent,  i . e . , at  breaking which  or  elongation  results in  a  decreases smaller  and  high  rate  breaking  rupturing  work.  of  load The  both  displays  non-linear time  the  understand-  as  to  on  This  a curve  loading, increases time-dependent  16  behavior a  of  Schopper  onto  paper  has  tensile  paper  been  tester  illustrated and  by  relationship  (1,  instead  Special  20,  22),  strength. examined  been  try  (16),  Cohen  by  and  of  well,  but  The as  supporting contrast that  contributes  a  also  to  using  weights  to  stress-strain  strength  tensile  regarding Paper  proper-  strength  tearing  Chemistry  support  of  rate  claimed  that  caused  start  creasing  load  reported  by  by  have  their  theory  and  I n s t i t u t e of Balodis  detailed  rate  on  assumed loading to  the  the need  tearing  Anderson  no  the  tear  (2)  Paper  as  an  discussion  tear  test  no  testers  evidence  energy  has  values  Chemis-  been  has  has  been  been  ad-  idea.  tearing, as  of  unfortunately  the of  ballistic-type  (8)  but  this to  with  tear  speciality  phenomenon  order  done  further  the  paper  given  been  Watson  effect  (15)  been  members  given.  indicated  various  examined  I n s t i t u t e of  mechanism  phenomenon,  In  has  the  dissipation  vanced  (20),  above.  indicated  mentioned  has  relationship in  failure  has  of  have  unidimensional  little  Members  mentioned  of  attention  but  the  The  hanging  Ranee  specimens.  Frequentlyj investigators  ties.  by  and  tear was  rate  not  by  tear  load  need  progresses. (1),  basic  distance  increase  followed  Falk  the  conventional  load-tear to  effect,  The who  for  Higgins factor  test.  (15) which  Higgins  relationship to  a  maximum  gradually  same  that  in  de-  phenomenon  suggested  was  the  was  ddcrease elastic  i n load energy  requirement  during  the  resulted  increase  from  i n tear  the  release  distance.  ENERGY  DISSIPATION  Recent tention study on  of  the  tory  on  advances paper  i n f l u e n c e of to  final  time  with  various  gations  also  reveal that  as  conducted  at  have  applied to  been  serves  increased  models  consisting  (1)  (2)  (3)  of  springs  Basically  dead  strain.  in Use  studies  loads  in  stress  relaxa-  viscoelastic  materials,  have  in  and  investi-  technologists.  involved  his-  Some  of  polymeric  dashpots  the  de-  Network  been  used  to  stress-strain-  i t i s necessary  to  differen-  between:  Elastic  deformation  applied  load),  Viscoelastic when  Flow  deformation  and  (3) of  (immediate  deformation  able  havior  and  to  attention i s  occur  many  at-  stress-strain  failure.  can  paper  mathematical analyses  exterior (1)  a t t e n t i o n of  relationships.  tiate  constant  involved  under  to  TEST  strength  example,  rates  TEAR  refocuses  has  the  w o r k - t o - f a i l u r e i n creep  tests  time  on  For  rupture  THE  rupture  This  factors  strain  IN  science  with  rupture.  testing  visualize  TEAR  phenomena.  on  models,  OF  concern  focused  tion  RATE  in materials  from  pre-rupture  prior  AND  the  (time  response  in  dependent,  a p p l i e d f o r c e i s removed), (not  recoverable  upon  phase  but  with  an  recover-  and  removal  of  the  force). seem  paper  to  be  predominant  i s concerned  where  ( F i g . 2). 18  the  mechanical  be-  19 Fundamental  s t u d i e s of  paper  stress-strain-time  relationships  knowledge  strength  to  on  examine  test the  paper  adequacy  requirements. limiting  bitrarily The the  The  variables  assigned  pulp  and  a  means  to  TAPPI  is  making  up  a  torn  together,  near  40.  No  been  tear  of  by  Cohen  Paper  rate  reasoned  on  number  plies.  In  increases serving son  the  r e q u i r e s the  tance.  with  these  the two  paper  has  test  tear  an  longer  total  time but  number  grades  of  ar-  to  plies  of  plies  scale  in  members mention  concept  widely  with  that torn  limitation of  the  can  the effect  be  tear  easily two  different  higher  the  which,  reading  for this  t e a r through  number  the  tear  whole test  simultaneously.  disallows meaningful  different  used  requirement  required to  with  one  widely  arbitrary  This  the  restrictions  been  of  According  simply  values.  of  have  some  resistance.  ( 8 ) , and  i t i s reported  number  standard  activities  one  (16),  time  and  helped  research  instrument  Watson  Apparently,  have  because  been  related  certain  same s o u r c e ,  addition,  between  and  provided  methods  d i s c u s s i n g reasons  estimating the  from  plies  give  Chemistry  tear  specimens of  will  a  such  foundation.  (26),  with  upon  also  standards  tester  ts-64  literature  found.  Institute  sound  tear  specimen  when  of  i n prevalent  i n d u s t r y and  T414  only  i s important  f o r evaluating sheet  Standard  not  testing  latter  without  paper  have  based  p r o p e r t i e s , but  present  ballistic-type  as  has  of  strength  of dis-  value Ob-  compari-  characteristics.  20 T e n s i l e t e s t i n g has shown that paper e x h i b i t s e l a s t i c and time-dependent flow p r o p e r t i e s . s t r e s s - s t r a i n behavior from t e s t s conducted r a t e of s t r a i n can  Non-linear with constant  indicate:  (1)  Non-linear e l a s t i c  (2)  Linear v i s c o e l a s t i c behavior, or  (3)  Non—linear v i s c o e l a s t i c  The  both  behavior,  behavior.  s t r e s s - s t r a i n curve can be d i v i d e d i n t o e l a s t i c and  y i e l d regions.  In the e l a s t i c region  ing to Hooke's law.  paper performs  A f t e r reaching the e l a s t i c l i m i t ,  post-  accordthe  curve begins to represent a p l a s t i c region by d e v i a t i o n from the almost gion.  s t r a i g h t l i n e e s t a b l i s h e d w i t h i n the e l a s t i c r e -  The m a t e r i a l begins to e x h i b i t flow.  or rate-dependent. t i o n decreases  T h i s flow i s time  At high r a t e of l o a d i n g , breaking  and the breaking load i n c r e a s e s .  Total  elongaenergy  r e q u i r e d to produce f a i l u r e i s also a f f e c t e d by r a t e of l o a d ing. The breaking or f a i l u r e energy  d i f f e r e n c e between f r a c -  t i o n a l drag work and f i b r e rupture has been adopted of theory e x p l a i n i n g some paper t e a r phenomena. energy  as determined  i n support  Breaking  from the s t r e s s - s t r a i n curve i s much a f -  f e c t e d by the time-dependent p l a s t i c flow r e g i o n . The  energy  s t o r e d i n the t e a r t e s t e r i s obtained by  r a i s i n g the s e c t o r center of mass. swing at the time of t e a r i n g .  The s e c t o r i s allowed to  The p o t e n t i a l energy  (P.E.)  21 of the sector i s transformed to k i n e t i c energy  (K.E.).  k i n e t i c energy i s capable of doing work, overcoming at the sector bearing, p o i n t e r  This  friction  bearing and a i r r e s i s t a n c e ,  as w e l l as tear through a specimen i f the r e s i d u a l from the s e c t o r i s l a r g e r than the specimen  force  resistance.  Otherwise, the specimen w i l l not be torn through and the sector  w i l l be stopped.  of the t e s t e r  When t e a r i n g a specimen within  (the Dynamic Tear Tester  capacity  designed i n A u s t r a l i a  has three d i f f e r e n t , interchangeable weight sectors  to pro-  vide d i f f e r e n t c a p a c i t i e s ) part of the energy i s expended to tear the paper and to overcome f r i c t i o n s , energy w i l l be l e f t  as r e s i d u a l energy.  dual energies are p a r t i a l l y  and part of the  Since net and  resi-  k i n e t i c energy, they are highly a f -  f e c t e d by the t e a r i n g v e l o c i t y and consequently are time-dependent . If energy d i s s i p a t i o n phenomenon i s expressed i n mathematical  form, t h i s can be used to evaluate:  (1) the amount of energy d i s s i p a t e d i n t e a r i n g , (2) the amount of r e s i d u a l energy, and (3) the increase is  i n energy required  as t e a r i n g  distance  increased.  If the energy d i s s i p a t i o n concept i s c o r r e c t , then the energy expended to tear a specimen  can be c a l c u l a t e d from the  d i f f e r e n c e between net energy and r e s i d u a l energy i n the form of k i n e t i c energy.  Results  c a l c u l a t e d from t h i s r e l a t i o n s h i p  22 would  then  be u s e f u l  cept,  as r e g a r d s t e a r  i n examining rate  the energy  effects  dissipation  and o t h e r f a c t o r s  of  conpaper  tearing. The no  energy  ther and  principle i s lost,  form.  but that  Thereby,  c a n be  o f energy  the t o t a l  ^^ ^.  ~~ ^ k i n  energy  "total  =  0  ^pot  +  energy  kinetic  of a  energy  one  o f a system  that  t o ano-  i s constant  form: t3  ^converted  +  4  system,  1 =  describes  i t i s c o n v e r t e d from  expressed i n the following ^tot  where:  of conservation  2  =  ,  ^pot P o t e n t i a l e n e r g y = mgy, E , , = enerqy c o n v e r t e d t o forms converted =  3 J f  m = sector g = y  The trated is  by  raised  tential  gravitational  acceleration,  displacement of sector  (tangential)  energy  from  energy  velocity  conversion  diagram  i n a tearing  (Fig. 3).  i t s lowest i s maximum,  shows  position and t h i s  upon  sector  i s converted t o other energy  t o do  releasing  This  paper  the sector.  tearing  (^-^  ear  j_ g)« n  center  o f mass,  and  at time t .  energy  and  E. . , kin  mass,  - vertical  v =  other than  p r o c e s s may that  when  be  the  illussector  to the top position, i s converted into The  kinetic  forms,  energy  friction  pokinetic  of the  (E„ . . . ) fraction  23.  A shorter the  i s required  b e g i n n i n g and f i n i s h  done, to  time  and l o n g e r t i m e  tear  a paper  of tearing  i s required  i s allowed  specimen  (Fig. 4a)  contrast  to a frictionless  velocity  (v^) at time  The t o t a l  following  E  where:  form  total  (hereafter  which  energy  = -§-  v  )  m v  t o swing  referred  energy  without  i s expended  with  gravitational  ( E q u a t i o n C43  components  tearing  a  zero-swing position  ) take the  i  +  m  9»i  ^  +  forms  ( v  l' l t  E ^  )  5  of f r i c t i o n  zero-swing  velocity  as a  function  and t i m e .  r e p r e s e n t net energy f o rtearing the  at time t ^ . tearing  slower  tangential  Then,  The s e c t o r ,  velocity  i n a tearing  at time  ( F i g . 4b)  specimens.  i s dissipated.  form  work i s  t o as z e r o - s w i n g i n  i t swings  i n E q u a t i o n T5j  energy  the  as more  system),  various  =  two t e r m s  When  y^.  no t e a r i n g  i s d i r e c t l y and i n d i r e c t l y a v a i l a b l e  specimen  between  at time t ^ :  ^( ]_>  first  when  t ^ at v e r t i c a l  of  The  the distance  specimen.  When t h e s e c t o r  (y^).  to travel  when  a part  therefore,  of the kinetic swings  (v£) a t g r a v i t a t i o n a l  swing  the total  a test  specimen  energy  with  position  components  i s torn  a  take  distance  24  L  total  C6H  i-mv? + mgy, + -ftv-.t-) + T.E. 2 J <•  F  ~  2  1  where: J ( v , t ) = various forms of f r i c t i o n as a f u n c t i o n of 2  2  t e a r i n g v e l o c i t y and time. Tearing energy  (T.E.) i s that part of the net energy which has  been d i s s i p a t e d i n t e a r i n g a specimen. There are two kinds of f r i c t i o n f o r c e s i n the system, namely, s l i d i n g  and r o l l i n g  f r i c t i o n of the bearing, and  viscous f r i c t i o n between the s e c t o r and a i r . rolling  The s l i d i n g and  f r i c t i o n f o r c e i s assumed independent of v e l o c i t y and  hence c o n t r i b u t e s no d i f f e r e n c e between zero-swing and t e a r ing  swing.  The viscous f r i c t i o n f o r c e i s d i r e c t l y  proportion-  a l to the v e l o c i t y , when v e l o c i t y i s not too high (23). ballistic-type  In  t e a r t e s t e r s the \MocLty i s considered low, t h e r e -  f o r e , viscous f r i c t i o n d i f f e r e n c e between the zero-swing and t e a r i n g swing can be considered as n e g l i g i b l e . ^(v^t.^ Since Equations  Hence:  = y(v ,t ) 2  2  \_ 5J and rj6J are equal according t o the  law of conservation of energy, they may be w r i t t e n as:  .i_mv^ + mgy + ^ ( v , t ) + T.E. = — | - ^ m v  1  2  2  +  9y  m  2  1  +^(v ,t ) 1  1  LjU  2  and transposed as: T.E. = ~ | - ^ " -|- 2 m v  Equation  mV  +  m  9  [8j then becomes:  y  i  "  m  9  y  i  +  f  (  v  l  ,  t  l  )  "f  ( v  2' 2 t  )  M  25  i 7 _L_mv  9 -i _L_mv  2  T.E. The  =  mass  2  (m)  contributing  11  under  t h e same  can  be c a l c u l a t e d  y^, v a r i e s  c a n be  but t e a r i n g  term  squared  velo-  (v-^) f o r s e c t o r instrument  velocity  the specimen.  zero-swing  a certain  c a n be f i t t e d  distance  derivative  =  operated  (v£) f o r s e c -  Therefore,  velocity  by f i n d i n g  and  T.E.  tearing  the time (At)  (AL), i . e . , rate  distance t o express  -(tear  tear  distance  of  tear.  as a  liminary tests best expressed  [^1  time)  the instantaneous  j^ioj  of Equation  acceleration  where:  between  f o r one s p e c i f i c  with  i f both  definition,  L  The o n l y  of time:  tear  the  2  velocity  c a n be c a l c u l a t e d  to tear  equation  By  r[9]  2  measured.  Velocities  function  v 2» )  1  i s a constant.  conditions,  position  required  2  The z e r o - s w i n g  y^ i s a constant  velocity  2  2-  2  tor  An  2  2  t o T.E. i s t h e d i f f e r e n c e  ( v ^ - v^)-  position  2  =  of the sector  2 cities  2  _ Li _ m (/ v  2  , while  i n the zero-swing  showed t h a t i n a second  velocity will  the second and t e a r i n g  be t h e  first  derivative i s swing.  Pre-  t h e r e g r e s s i o n f u n c t i o n c a n be degree or quadratic form:  2 = a + bt + c t L = tear  distance,  t  time,  a,  = tear  b, and c = c o n s t a n t s  or regression  coefficients.  26 The f i r s t d L  dt  d e r i v a t i v e of Equation  =  i s velocity (v):  fl23  b + 2ct  The second d e r i v a t i v e of Equation Q l l ^ i s a c c e l e r a t i o n :  By using  t h i s mathematical model, both r a t e of tear and  t e a r i n g energy may  be r e l a t e d to time, enabling  time e f f e c t s i n the tear t e s t or time-dependent regards paper t e a r i n g  strength.  evaluation  of  phenomenon as  MATERIALS  AND  METHODS  Materials  Tear to  the  this  distance  tear  tearing  by  tear  of  paper  tearing  a  series  treating Five  tearing  of  a  single  and  wrap  ly)  bag  (All  grade  and  to  156  basis  g/sheet  order refer  paper  weight  bag  to  of to  listed  to  as  and  27  of  to  better  properties  cover  tearing  skin,  wrapper  skin,  wrap,  24-in.  wide  onion  onion  values 1  a  55.5-lb.  parcel  sheets,  i n Table  measurements.  purpose  unglazed  Island  direction tearing  are  displaying  ways.  increasing 500  same  time-dependent  papers  adjustment  included  wrapper  the  of  rate.  this  selected  paper,  of  evaluate  meet  various  referred  55.5-lb.  the  same t e a r  related  specifically,  expression  Specimens  was  were  directly  More  another  would  They  (hereafter  weights  Cross-machine 14  in  is  minor  in  papers  n &. m  paper,  arranged  with  range.  are  Consequently,  range  pulp  commercial  30-lb.  print,  tearing.  concerned  newsprint, parcel  show t h e study  time  method.  results.  this  strength  strength  test  function  test  should  purpose  phenomena wide  tear  distance-time  strength  The  than  i t s required  ballistic-type  ballistic-type  a  and  news-  respective-  strength. X  36-in.).  ranging  together  with  from caliper  28  Methods Tearing  procedures  A Thwing-Albert in  this  TAPPI and  study.  ±  within long  T  3.5°F  reached.  402  Papers  were  T  414  ts-64  adjustment  adjustment  of  paper  tightly a  read  (26)  table  base from  was of  of  are  an  the  to  equipment as  sections.  by  that The  any  6.3  cutter.  only the tear  which  was  Test  The  are  regards  cross-mawas  movement  adjusted to  and  and  i n turn  and  bolted clamped  of  the  results  average  tearing  plotted  The  TAPPI  as  tester  scale  [j3]] .  segments  procedures  sector.  2  cm  direction.  the  tear  storage  X  paper  possible  Equation  to  humidity  f o l l o w e d , except  and  i n Table  used  c o n d i t i o n s were  of  in Fig.  data  7.6  testing  sheet,  swing  was  relative  shear-type  tested.  tear-  strength  versus  num-  5. distance-time data scope.  means.  f o r simultaneous as  a  prevent  photographic  well  2%  standard  plies,  tester  same t i m e ,  oscilloscope  recorded  time,  plies  ±  equilibrium  tear  plywood  presented  paper  50  closely  was  during the  on  strength according to  At on  number  a -y-in.  rigid  results ber  until  by  tester  conditioned according at  into  tearing  r e p r e s e n t e d machine  direction  onto  instrument  ing  (25)  cut  always  instrument  were  m-49  allowable variation  Standard  onto  were  temperature  dimension  chine  Elmendorf  A l l papers  Standard  73  Co.  Traces Design  measurement  handling w i l l  be  were  and of  were  obtained  permanently  arrangement  tear  distance  described i n  of and  subsequent  29 Experimental  As  design  stated,  five  kinds  study.  Each  paper  samples  with  d i f f e r e n t number  plies as  depended  related  were  as  upon  proper  between  of plies.  strength  5 and 75. consecutive  The  assignments  GRADE  NUMBER  Onion  Skin  1 0 , 2 0 , 4 0 , 6 0 , 7 0 , 80  OF P L I E S  (TREATMENT)  Newsprint  5, 1 0 , 1 5 , 2 0 , 25  Bag  2,  6, 1 0 , 1 4 , 1 8 , 22  2,  4,  6,  8  1,  3,  5,  7  paper  number  test  wrapper  wrap analysis f o r both  number  strength  replication  number  showed tearing  was f u r t h e r  r e s u l t s which  tearing  Rate  tearing  w e r e made b e t w e e n  t h e numbers  o f number o f  PAPER  replication  quired  reading  of several  Choice  sheet  i nthe  follows:  Statistical  of  examined  preparation  of plies.  scale  spacings  by v a r y i n g  Parcel  ther  were  i n d i v i d u a l paper  equal  55.5-lb.  a  required  to the sector  Approximately specimens  type  o f paper  showed,  and t i m e  Precise  three  strength  confirmed with  r e p l i c a t i o n s were  and t e a r  This  by f u r -  9 5 % p r o b a b i l i t y a t ± 5% that  the required  f o r a few c a s e s  one and  time.  as adequate  mean v a l u e s ,  was o n e , e x c e p t  r e p l i c a t i o n s between  of tear  that  which r e -  three.  measurement measurement  of the tear  distance-time  relationship  30 is  the center  are a  usually  torn  over  ballistic-type  oscilloscope events  vals  of time.  over  longer  With  time  voltage  scope  with  When  to tear  oscilloscope the.current since  units,  plies This  by  a s an  can measure short  inter-  c a n b e made to tear  was f o u n d range  rate  such  over  required  across  t o range  well  f i t s the  t o time  regulated,  c a n be r e c o r d e d useful  i n a system changes  i n analysis  sheet  passing  through  the sheet  shown  of a resistor  These  (Appendix  through  x,y c o o r d i n a t e s  can pass  of  a  area.  increases  t h e sheet  al-  voltage  through  1) and t h i s  locate  with  oscillo-  as p a r t  creating  on t h e o s c i l l o s c o p e .  to tear  time  simultaneously  on t h e o s c i l l o s c o p e  proceeds  coordi-  and c a l c u l a t i o n .  to i t s cross-sectional  the y-axis  required  sheet  that  by  i n tear  i s connected  this  of current  can r e g i s -  on a s t o r a g e  tearing  tearing  the x-axis.  oscilloscope  circuit,  t h e amount  along  564 s t o r a g e  conductive  difference  as  paper  respect  i s proportional  the time  tear  measurements  The time  properly  voltage  ly,  high  accurately  accuracy  out data  an  registered  plug-in  seconds.  distance  an e l e c t r i c a l l y  conductor  proper  Inc. type  f o r reading  change,  lower  t o 0.4  If  ters  with  of several  change  methods.  respect  with  Papers  capacity.  A Tektronix  nate  time  of the study.  Electronic instruments,  intervals.  0.08 s e c o n d s  oscilloscope  ter  short  part  down t o m i c r o - s e c o n d s  4.3 cm s p e c i m e n  from  very  tester.  equipped  time  a  of the experimental  a  This as  c a n be  Simultaneousi s registered  tearing  distance  31 at  any  specific  time  allowing  subsequent  tear  rate  calcula-  tions. Ordinary trical  papers  conductivity.  incorporate  a  c i r c u i t , as  part  Several principle  was  material.  are  discussed  (1)  of  to  study,  obtained  A major  cimen factor  under  specimen  a  and  low  i t is as  a  elec-  useful part  studied.  specimen  disadvantages  paper  curve  continuous  to  of  The  sheet  a  of  general  as  conduct-  several  was  paper test, may  has  a  methods  by  resistance  method  be  requires  a  constant,  i f the  tear  by  over  A  of  this  mar-  on  the  times  against  voltage  cutting.  were a  tear  variation This  pro-  time. i s that  no  elaborate  disadvantage resistance  correction. but  test  the  curve  curve  measuring  reached  needed.  need  on  corresponding  different tear  which not  this  are  the  voltage-time  and  calibration scale of  available  voltage-time  obtained  distance  is  fitted w e l l  distances  advantage  itself  operated  were  one  conductivity  preparations  conductive  modify  calibrating this  voltage  material  methods  conductive  Tearing  certain tear a  conditions  have  specimen.  or  gave  distance-voltage  vided  content  below.  tests  by  such  material,  Advantages  oscilloscope.  when  the  add  uniform  and  moisture  Under  conductive  Electrically Its  low  conductivity  ive  ket.  at  is  will  vary  i s that than  The  the  the spe-  correction  with  total  time-dependent.  32 (2)  Thin  negligible tivity, spect This  tearing  however,  to  change  small  scope  aluminum  Metal  deposited  which  is  a  was  needed layer  results in in  Its  very  little  cross-sectional change  particles a  on  lowers  tried  homogeneous  good  change  area  tearing  as  accuracy  and  electrical  voltage  For  example,  ing  one  7.6  surface  accessory given  poor  by  two 6.3  layer  deposited  rough  paper  surface.  median  Electrical line  results  in  an  showed  for  up  in  electron  has  conduc-  with  re-  proceeds.  reading  oscillo-  paper  may  be  attempt  to  there of  this  strength  (55.5  g/sheet)  of  specimens  introduce frequently  disadvantage  perforations the  on  tear-path  is  Further, by  was  that  a  the  This  without  Too  thin  layers  a  re-  needed  for  prepar-  uniformity  of  the  microscopically  and  in  along  the  One.,  set  tearing  the  treated  tearing conductive  c a r e f u l work  conductive  specimen  me-  temperature  resistance.  i t requires  whole  evaporator,  perforated  combination  electrically of  was  can  desirable.  difference  g/sheet)  metal  i s not  was  no  surface.  high  tear  (55.3  The  The  reduce  strength  paper.  vacuum  thicker  exposure  paper  a  other  which  affected  conductivity  or  paper  properties.  while  specimen.  a  microscopy.  immediately.  minutes  that  using  temperature,  cm  on  silver  by  change  high  metal  (4)  of  conductivity,  at  about cm  deposited  paper  and  time  be  layer  drastically  provided long  may  uniform a  basic  could  quired  of  essentially  strength.  voltage  Theoretically,  thod  is  traces.  (3)  be  foil  paper  booklet  to  and  does  not  33  follow the  exactly  the direction of perforations  (Fig. each  step  tion  on t h e c u r v e  distance.  Graphite  a rather  aluminum paper its  surface  disadvantage.  onto  a member  tear  resistance  terial  into  obtained (Fig.  very  cause  factory (6)  effect  has r e a c h e d time  by  a  projec-  booklet  4,  but, w i t h  from  accuracy  line  to a but not  directly  a foreign  A stepwise vertical  step  tear  inaccurate. t o form  ma-  curve i s lines  time.  t o the specimen  i s required  with  the variable  i n reading  a 6B p e n c i l  4,  graphite  avoids  better  and sometimes  wide  lines  by i n c l u d i n g  o f the specimen.  since i t  i n comparison  as Method  applying  introduced  consuming  material,  ladder-like graphite  increases  a thick,  The end o f  surface  This a  i s be-  satis-  conductor. As r e p o r t e d  be u s e d  commercial has  conductivity  Furthermore,  as i n Method  quite  tear  as a c o n d u c t i v e  t h e same a d v a n t a g e  of graphite  tearing  curve  x-axis.  be u s e d  t h e body  time  reading  o f t h e specimen  6b), which  Transfer  enables  Applying gave  of perforations. when  low e l e c t r i c a l  foil.  a stepwide  indicates  This  may  produced  spacings  t o the time-base  (5) has  perforations  6a), r e f l e c t i n g  certain  may  in  center p l y . Introducing  is  introduced  been  by A n d e r s s o n  as c o n d u c t i v e silver  found  paint  and F a l k  material.  used  t o be a good  Instead  f o r drawing conductor  (1), conducting ink of using  electric  that  c a n be  graphite,  circuits transferred  34  easily ness  and  of  cross-lines  drafting finer fine  reproducibly to  pen.  tear  to  I t was  be  missed.  reducing overall  the  step  voltage  All noted.  Methods  6,  ver  paint  was  used  two  main  of  each  vertical  line  Thereafter, phite  and  addition  5  causing  and  less  reduced which  a  provided  one  in coarser also  using  using  not  and  was  to  allowed  6  are  adopted  combined  a l l the  which  by  vertical  as  too  or  have  several this  accurate  sensitivity  accompanied  by  small  was  in  line  was  made t h i c k  good  conductivity.  of  and  was  of  the  cross-lines. The  A  continuity  applied to  each  as  give  Manipulating of  one  Sil-  ensure c o n d u c t i v i t y .  enough  height  both  F i g . 6d.  a l l the  to  as  3.  advantages  adopted.  This  a d j u s t i n g the  i n Table  cross-lines  voltmeter main  disadvantages,  shown  connect  coarseness  checked  and  compared  results  draw  lines  another  also  advantages  yielding  line  (Fig. 7).  tinuity  Yet,  did  ( F i g . 6c)  have  method  medium  curves.  cross-lines  and  coarse-  d e s i r e d by  cross-lines  discontinuities,  curves,  4,  and  to  finer  stepwise  height  5  fine  that  The  change.  Methods final  as  resulted  s i x methods  The  controlled  Coarse  They  distance-time  surface.  found  introduced  disadvantage. tear  be  distance-time  lines  steps  can  a specimen  good  the  step  gra-  on  con-  graphite the  curve. Distances ator.  For  between  these  cross-lines  experiments  tear  are  determined  distances  of  by  0.2,  the 0.5,  oper1.0,  35 1.5,  2.0, 2.5, 3.0, 3.5, 4.0  cross-line ruler  was  drawn  and d r a f t i n g  by hand  pen.  a n d 4.3 cm w e r e with  The f i n a l  was l o c a t e d e x a c t l y a t t h e t e a r except  the last  position also  one, w i t h  i n order  some m a t h e m a t i c a l At  testing,  (hereafter the  many  ter.  This  including Voltage  the  battery,  alligator The  8)  anode  make  left  cross-line above, cm  reading.  and t h e r e b y  o f paper  with  p l y i s connected  I ti s  represent  conductive ply)  lines  to a circuit  i n Appendix  of razor stiff,  are connected  one o f  at t h e specimen  a r e l e d out from  thin,  material  becomes  box and o s c i l l o s c o p e  a r e shown  of being  Wires  cen-  ( F i g . 8)  ( F i g . 8 and 9 ) .  1.  T h e two main  t h e specimen t o  blade.  Razor.blades  available  to the razor  and v e r y  blades  good  by two  clips. wire  l e a d i n g from  of a battery  at least which  t h e s e c t o r clamp  and s u s p e n d e d  freely  was c o n n e c t e d i n the a i r  p o s s i b l e r e s i s t a n c e due t o t h e w i r i n g  Standard  sector  time  spacings  by two h a l f - p i e c e s  t o reduce  TAPPI  a final  and i s p l a c e d  decade  conductive  conductors.  the  t o be t o r n  advantages  straight-  l o c a t e d a t t h e 4.3  t o as t h e c o n d u c t i v e  conductive  circuit  have  the sheet  calculations  vertical  beginning  line  o f each  Each  function.  referred  plies  edge  of a  d i s t a n c e mentioned  to f a c i l i t a t e  p o s s i b l e t o vary  the help  used.  T 414 t s - 6 4 20 c o m p l e t e  engages  of a pencil  line  (26) r e q u i r e s t h a t oscillations  the sector stop, l o c a t e d one i n c h  before  (Fig.  arrangement.  the sector t h e edge  no l o n g e r  passes  to the right  to  should  of the to the  of the  36 edge the it  of  the  a i r , the no  this  passed  arrangement on  The short  the  A slight  was  mounted  down  cess  the  of  the  ductive eously steps  could  was  then  the  line,  a  step  can  be  can  be  collected data  Zero-swing  can tear  According  a  specimen  paper  significant  test  To  do  stop  this, in  such  both The  procedures.  errors in  for starting  started  the  the tear  the a  and  results.  tear  test  micro-switch a way  that  tearing  pro-  distance-time  oscilloscope.  known  total  current  passing  i s reduced,  i f ten  and  be  on  ten  used  voltage  A  polaroid  permanent  the  sets to  the  distance through  changes  and  distance-time  conductive of  lines  tear  record  are  breaks  a  the  con-  simultan-  diagram. painted  distance-time  description  for calculating  distance-time to  serious  certain  according  then  Thereby,  relatively  strength  recording. on  is  a  obtained ply,  in  before  line.  c o n t r i b u t i n g no  sector  i s registered  conductive  freely  traces.  the  grid  pencil  for easier reading  tearing across  sheet  as  wire  oscillations  the  cause  stop  stored  taken  50  of  i s necessary  oscilloscope  the  These  tear other  sector  oscilloscope  conductive  left  simultaneously.  the  curve  When  to  underneath  pressing  than  the  method.  with  system  oscilloscope  was  suspending  i s considered  variation  and  picture  the  required  synchronous  stepwise  to  compared  time  and  By  basic test  time  when  stop.  s e c t o r made m o r e  longer  effect  A  sector  i n Method  tear  Ten on  data 4.  rate.  measurement  »  two  velocity  measurements  are  37 required y^,  for  namely,  v2»  calculating tearing  zero-swing  Tearing  mentioned  v e l o c i t y , v^,  v e l o c i t y can  above.  measured  by  to  simulate  zero-swing  contributes velocity. to  the  which  Cellophane,  characteristic  ing  seem  with  applied the  the as  tearing low  be  tear  an  lower  resistance,  paper.  Perhaps  tive  a  simultaneously  as  Another  with  ladder-like  tear  force  sector  a  tear  responds  similar  tear  their value  o r i g i n a t i n g from material, silver  as  paper  scale.  This  and  lines,  registered  method  the  perforated  paint  is  true  tear this  no  assumed  of  a  is  started  purpose.  number  the  tear to  obtained  suggests  force  foil  conclusion  tracing  to  extremely  tear  aluminum  gave  paint.was  This  of  keep-  Contrary  with  and (15),  In  curves  paper.  type  coherence  homogeneity  samples.  and  hence,  of  silver  d i f f e r e n t l y to  to  convenient  sheet  degree  material  e f f e c t with  (24)  between  for  cannot  zero-swing  the  distance-time  had  method  to  conductive  non-fibrous  Setterholm  relationship  tested.  the  the  and  once for  to  resistance,  molecular  cellophane  velocity,  i t is  one  i t s high  (6),  resistance  v e l o c i t y than  bending  and  forces  l i n e s to  as  of  procedure,  behavior,  cellophane,  5wartout  tearing  position  velocity  practice,  tear  ideal material  regular  spaced  expected  displayed that  to  by  low  because  tearing  according  s i g n i f i c a n t adjustment  concentrated  would  extremely  sector  zero-swing In  behavior  at  and  measured  same m e t h o d .  has  no  be  Theoretically,  be  material  energy  than  led  on of  low  the  posi-  plies  material paper,  torn  being  prepared  same r e s p o n s e  resistance  closely  on  the  approximate  to  38 the  true  zero-swing  response y^.  f o r the  Certainly  zero-swing ample, and  From  data  each  distance-time  for  It time  a second  A least  squares  mathematical gression  Some s e t s equations.  tear  tear  analysis  that  the  (Table  bt + was  Second  ct  Statistical  position  deriving For  sector  ex-  base  used.  For  of the  data  were  the  tear  example,  five  papers,  relationships  bag  existed  third  to  according  tear  relationship  ( F i g . 10).  on  f i t the r e l a t i o n s h i p c u r v e s were  an  were  IBM  7044  studied  expression  comparisons  degree  to  2  degree  degree  (L) r e l a t e s  quadratic  used  programmed  second  4)  to the be  rate.  distance  or  although s t a t i s t i c a l  stances  for  distance-time  Similar  o f measurements  The  might  f o r one  degree  method  analysis,  sector  paper.  = a +  form.  fixed  of tear  F i g . 10.  kinds of  L  Li's  as  at  measuring  accuracy i s required.  eye  sets  relationship  i s observed that  ( t ) as  quate,  ten  for  fitting  for defining  i s plotted  a l l five  curve  adopted  b-B d e v i s e d  scale  electric  test,  made a v a i l a b l e  could  perforated  and  was  calculation  i f greater  b e f o r e an  of  and  (V^)  other systems  velocity  passed  paper,  velocity  a properly  Handling  velocity  term  was  was  fitted  by  as  to  re-  computer. as  third  proved  showed  the  procedures  I n f e r e n c e (18)  was  done t o  t o be  ade-  i n some i n -  significant.  to  degree  Further  described  find  in  reasons f o r  39  the  unexpected When  ted  by  calculating  each  term,  always  found  1.5  3.5  to  rendered  significance  X 10  the  the .  the  difference,  1  to  difference,  1  t o 10  dual  very 20  best  argument  for this  the  prised  several  grades  respectively),  cept  that  skin  paper.  study.  ways. used a  test  m  ,  solve  between  will  were  of  the  i t  was  order  residual  of  variance  sensitive.  used  was  a single  their  be  given  measured. (6,  5,  6,  paper,  was  a total  the  When  4  and  second  of  76  have  v a l u e s from equation. utilizes  been the This  resi-  degree Further  Discussion.  paper 4  grade  of  each  f o r 70  tests  and  com-  treatments f o r  number  used  third  small  55.5-lb. wrapper  and  and  f o r zero-swing  Each  analyzed at ten  which  extremely  i n the  different  only  degree  of measurements.  f o r zero-swing  gave  second  relationships  f o r the  average  approach,  small  contribu-  variance,  extremely  a l l sets  replication  test  squares  was  concluded that  expressions could  First,  second  were  These  Each  to  be  n e w s p r i n t , bag  one  Final  m  treatments  wrap,  of  term.  small multiple correlation coefficient _5 X 10 , and s t a n d a r d e r r o r o f e s t i m a t e _2  choice  skin,  sum  variance  together with  onion  degree  residual  extremely  distance-time  paper  replications  , and  ratio  describes  tear  five  , t  i t may  equation  The  3  X 10  expressions,  variance,  2  This  considering  degree  regression  residual  variance  third  the  t, t  that  of the  and  parcel  plies.  Three  treatment,  ex-  plies  f o r the  of  onion  entire  positions. found three  by  of  three  replications  method  a l l three  any  i s no  may  better  individual  sets  be  than of  40  data The  by t h e l e a s t disadvantage  mate  three  t o f i t an  different  method  correlation the t h i r d  be  calculated first  method  directly  of For  study, sion  to a  used  and  an  and  also  by  the  their of  into  third  multiple  can  of these  tear  Advantages  of  contribute  an i n d i v i d u a l  has  in  estimate.  measurements  measurement  can  i t s correspond-  average  that  dis-  can r e s u l t  replication  result.  a l lthe o r i g i n a l  distance-time  errors  t e a r time  as t h e f i n a l  result  with  f o r each  tear  a smaller  equa-  standard  estimate. these  into  reasons  were  energy  Equation constant  [9]  specimen t h e mass  for a particular  f o r t h e main  plies,  three  which  were  energy.  c a n be (m)  averaged By  such  to  find  means, according  i n Equation  sector, a l l tearing  tear-  equation  calculated  term  regres-  corresponding  of the regression  and r e s i d u a l  Since  adopted  o f paper  velocities  f o r each  was  Substituting  derivative  tearing  •  method  number  fitted.  the f i r s t  three  this  a certain  s e c t o r net energy  tearing  and s t a n d a r d  esti-  s e t of data  developed  5 together  equation,  to the f i n a l  i.e.,for  provided  Equations  substituting  of  (in a l l tests,  replications  tear velocity  are that  equations  times  by  c a n be  for tear  error  the  method,  equation.  error  individual  Three  i n Table  derivative  velocities  tion  equations.  standard  times  each  equation.  coefficients  By  this  Third,  are presented  to f i t a single  i s the higher  (L) i s f i x e d ) .  used  ing  here  method  due t o d i s p e r s i o n o f t e a r  tance be  squares  energy  [9]  i s  values  41  presented removing per each  a r e made a f r a c t i o n the sector  p l y c a n be specimen  presented 4.3  cm  further  Summary  the  specimen  Tearing  required  These  to rate  with  booklets  an  of  o f each  booklet  required energy f o r  energy  to tear  tearing  as  across  energies  are  energy  tear.  connected  through  pre-assigned  beginning Tear  allowed  distances. the tear  tearing  distance-time  oscilloscope  and were  tures.  test  Each  cross-lines  this  at  as p a r t  of times  had  of which  o f an  required  distance-time  mabeen was  arrangement,  A synchronous  electric to  tear  mechanism  was  relationship,  started.  permanently  measurements which  By  relationships  as done  plies  p l y of the test  paint  the specimen  of  Positioned  v e r t i c a l - l i n e s , one  measurement  to record  as p a p e r  a single  the c r o s s - l i n e s .  which  numbers  tester.  silver  edges,  p l y rendered  incorporated  different  tear  was  ladder-like  At i t s o u t e r  center  with  Elmendorf  circuit  time  the tearing  of p l i e s .  sheet.  energy  by  i n F i g . 1 1 a t o l i e t o show t e a r i n g  respect  t o which  graphite, the  dividing  energy  o f rmethods  center  added.  by  tearing  Tearing  2 i s t h e energy  a single  with  factor.  i t s number  i n Table  tested  terial  by  presented  Test were  obtained  through  variation  mass  of t o t a l  here  were  recorded  provided  read  were  from  registered as p o l a r o i d  on  an  pic-  ten tear  distance-  a polaroid  picture.  42  These was  d a t a were  calculated  fitted from  as s e c o n d  the f i r s t  degree  equations.  derivative  Velocity  of the second  degree  equation. The on  an i m a g i n e d  sheet for  zero-swing time  of perforated  tearing Three  sample:  test  terms  sheet  was  tracing  to tear  simulated  paper  treated  through  distances  by t e a r i n g  a  single  i n t h e same way  as  specimens.  velocity  t h e s e were  according in  blank  required  v a l u e s arose from averaged  for calculating  to Equation £ 9 J . of f r a c t i o n a l  replications  Tearing  energy  energy  of  tearing was  each  energy  then  per p l y f o r comparison  presented between  materials. For tester tearing  comparison  sector  scale  strength  purposes, and  r e a d i n g s were  adjusted  according  taken  to the standard  t o TAPPI  from  the  average  S t a n d a r d T 414  ts-64 (26).  DISCUSSION  The has:  background  been  reviewed  According pended  of factors i n former  a sheet  sheet.  In t h e present  derived  f o r demonstrating  testers.  This  of  tester  a tear  dual the by  energy  ing  energy  tear  test  culation.  o f paper  the nature  c a n be u s e d which  which  i s that  the tearing  by u s i n g Since  thus  means  providing  proceeds, energy the  associated  and t e a r  part  Furthermore,  tear  with  of tear  t h e paper and e n e r g y  rate  tear  new  for caland  data  will  test.  with  of tear clarify F o r ex-  v a r i a t i o n as  information  tear-  of the  the effect  t h e r e l a t i o n s h i p between  provides  this  distance test  and  absorbed  nature  tear  f o r evaluating  progress  rate,  from  resi-  friction,  of t h e energy  for relating  examination  and e x p l o r i n g  tearing  tearing  on c o n d u c t  of  test. Tearing  to  this  tear  work,  v e l o c i t y as t h e b a s i s  time,  showing  of doing  and o v e r c o m i n g  process.  tear  the  has been  paper  exists  ample,  model  t o the dynamic  an o p p o r t u n i t y  some p r o b l e m s  within  ex-  t o c a l c u l a t e the net energy  velocity i s calculated  Hopefully,  energy  i s related  time,  rate.  results  of ballistic-type  i s capable  paper  concept  i s dissipated  a mathematical  tearing  energy  during  d i s s i p a t i o n concept,  study  sector  after  tearing paper  model  test  sections.  t o the energy  i n tearing  affecting tear  tear  v e l o c i t y i s an e x p r e s s i o n  across  a certain  distance. 43  of the time  In order  required  t o measure t h e  44 distance-time  relationship,  veloped.  e f f e c t i v e n e s s and  method  The  i s demonstrated  replications. measurement tion  In  most  be  by  Curve  by  of  applying  device  done  which  hand  the  the  small  i t has  basic  regulates  of  shown the  de-  this  variation  been  be  had  that  between  only  specimen  one  condi-  method  paint  could  lines  distances  more  be  with  further some  mecha-  precisely  than  can  drawing.  Fitting i s no  distance-time  data,  use  ther  discussion.  form  f o r these  or  cubic  to  express in  data,  the  first  If  used,  the  affects  needs  of  the  as  of  on  a linear,  equations  relationship.  evaluation  t e a r i n g energy  equation  first  types  is calculated  L  i n measuring  measurements  distance-time  derivative  a linear  these  problem  i . e . , expression  the  velocity  of  A l l three  tear  Since  technical  i s , decision i s required  degree,however,  velocity, the  That  equation.  tribution.  is  very  silver  there  to  to  reproducibility  for representing  Although  ence  had  study.  Reproducibility  nical  the  cases,  i s necessary  under  improved  by  a p r e c i s e method  by  of  substituting of  any  the  fur-  best  quadratic  The  calculation  equation  some  can  the  tear  be  used  differ-  energy  dis-  i s based  on  tear  time  degree.  as:  = a +  bt  derivative,  velocity,  will  be  a  constant  in-  45  with  no  acceleration,  constant If  so  v e l o c i t y and a  quadratic  used,  its first  sector  residual  equation L  is  the  =  swings  tearing  and  tears  with  a  velocity.  as:  a +  bt  +  derivative,  c t  Trl]  2  velocity, will  have  a  linear  form dL dt with  a  constant  If  a  cubic  used,  b  +  2ct  bt  +  ct  C123  acceleration  equation L  is  =  its first  =  as:  a +  +  2  dt  Cl6~|  3  derivative,  velocity, will  dL_ dt  2ct  be  of  parabolic  form,  with  =  b  Using cause  of ing  +  3dt  Vll j  2  U  J  acceleration  - $ ! - • •  ent  +  a l l three  difficulty  forms tearing energy  of or  in  2  forms  c  +  6  among  comparisons.  equation nature  originates  refer of  the  from  d  to  [ " J  t  data  of  This  i s because  basic  applied the  single  differences  force.  sector,  a  so  In  study the in  fact,  a l l the  would  differthe  nature  a l l tear-  applied  46  force  will  havior.  b e t h e same,  Hence,  measurements the  same  can l o g i c a l l y  tear  menon  i s slightly  smaller  Therefore,  relationship. f o r the last  than  paper  part  of tear  men  high  tear  resistance,  Preliminary bag paper  plies  depth, curve  decreasing the  curve  This  linear limited tion  tear  the tear  tests  of plies,  of Fig. that  distance-time  v e l o c i t y becomes more  r e l a t i o n s h i p can always  order  over  75  to avoid  with  10 w i t h  pheno-  when a a  25  speci-  plies  other  s t a r t i n g with  3.5  cm  final 0.5 t o  the effect of  and more into  pronounced  a concave  be i d e n t i f i e d  the complication study  of the tear  i s not necessary,  this  and  form.  with  sector  o r 80.  relationship, this capacity  approaching  curve  data  of  r e l a t i o n s h i p o f t h e 25  After  d i r e c t i o n s t a r t s t o change  linear  increment  o r more p r o p e r l y  test  form.  expressed The  confirmed  distance-time as p a r t  be  at a state  torn.  i s of linear  readings In  of  resistance i n -  distance-time  I t c a n be o b s e r v e d  tear  linear  scale  by e q u a t i o n s  tests.  tear  was  i s plotted  results. cm  be-  distance-time  the net energy  Preliminary  number  with  paper  i s torn  high  3.5  be e x p r e s s e d  i f the total  with  test  tear  f o r various  specimen  of  a s common  r e l a t i o n s h i p s could  equations  i s true  sector.  linear  only  distance-time  relationship  the  to proceed  the r e p l i c a t e d zero-swing  d i f f e r e n t degree  crement  tear  degree.  The as  causing  was  tester.  because  caused  designed  t o use only  Actually,  the i n i t i a l  by u s i n g  part  this  the a  considera-  of the tear  47  distance-time expression A  curve  ignores  regression  i s always this  of  curvilinear  form  and  a  linear  fact.  a n a l y s i s program  eliminating least  important 2  variables s u c c e s s i v e l y s h o w e d ways  the  most  tionship. linear  Therefore,  specimen  cave  with  and  matically  part  should  not  The  25  good  a  plies  x-axis. cubic  exist  within  by  second  showed  the  of  trials,  with  but  i n some reasons  significant  data  t  , as a l -  the  to  rela-  use  for this. third  be  the  by  quadratic  A  cubic  only  terms  mathe-  a  for  limi-  study.  t  for  the  ( F i g . 10)  correct for 3  possessed  con-  relationship  forms  discovered  to  design  collected  term,  This  form,  a  majority  , could  that an  best  analyses  f u r t h e r a n a l y s i s was  I t was  degree  case.  Statistical  degree  paper  expressed  using  present  the  expres-  beginning  best  of  equations.  equations.  bag  curvilinear  the  third  an  experimental  case  as  the  the  Hence,  as  as  for this  at  can  The:  relationship  a  above,  relationships  easily  that  As  cubic  example  this  range  degree  quadratic  the  another  These  distance-time  expressed  find  as  avoided  observed  to  incorrect  curvilinear  equation.  be  retained  from  i t appears  can  the  is a  sector capacity.  tear  using  data.  purposely  of  of  distance-time  the  by  study  study  term,  s i n g l e variable r e p r e s e n t i n g  possibility  then  facing  this  degree  i t is definitely  reflexes twice,  linear,  ted  is a  for tear  curve  second  form.  There sion  important  the  be  conducted  a l l  equations  extremely  48 small  residual variance  residual  variance  0.1 t o 8.5 X 10  made t h e v a r i a n c e  _2  .  ratio  This  test  level  of  extremely  sensi-  tive. Table test  4 contains  an a n a l y s i s  s i g n i f i c a n c e of the p a r t i a l  measurements  on a 1 0 - p l y  of variance regression  newsprint  model  used t o  coefficient f o r  specimen.  This  analysis  3 showed  that  the t  term  should  be r e t a i n e d .  Variance  ratio  3  of  the t  level.  term  When  was  6.610,  comparing  which  this  was s i g n i f i c a n t  variance  ratio  a t t h e 5%  to the t o t a l 2  variance t  ratio  variance  16,739.9,  contributed  Furthermore, tion  value  only  the multiple 2  containing  including t, t  about  3 , and t  , the  1/2500 o f t h e t o t a l  correlation coefficient 3  t, t  , and t  was 0 . 9 9 9 9 4 ,  which  than  t h e 0 . 9 9 9 8 7 f o r an e q u a t i o n  terms.  The d i f f e r e n c e o f standard  variance.  f o r an was  equa-  only  _5 7 X 10  larger  containing  2 only  t and t  estimate 22.5  mm  between  these  o f t h e average  two e q u a t i o n s tear  distance.  indistinguishable  i n the graphic 3  be  the t  concluded  quadratic  that  equation  may  was  term be u s e d  _2  6.66 X 10  Such  form.  error of  mm  difference  Consequently,  i s not necessary, t o express  from was  i t  can  hence t h e  a l l tear  distance-  time r e l a t i o n s h i p s . Interpretation of Results The lated in  zero-swing  from  Equation  energy  the f i r s t [12J  and paper  .  v e l o c i t y and t e a r i n g derivatives From  tearing  these  v e l o c i t y were  of Equation  [llj,  two v e l o c i t i e s ,  r e s i d u a l energy  were  as  calcugiven  the sector net calculated.  The  49  difference the  these  number o f p l i e s  energy  these  instant ten  sets  each  time  have  tearing been  were  across  tear  included.  together tional  with  cross  These average  Tearing distance further times  between tear  figures  those r e q u i r e d without  a paper  tearing  tearing  distance,  swing  distance.  time  t o swing  time  from  cm  distimes  velocity,  with  and  respect  kind  conventheir  as t i m e  load  t h e same  time  tear  differences  to a certain  and t h a t r e distance  i s obtained  zero-swing  to tear  o f paper are  The r e l a t i v e  t o swing  specimen  The v a l u e  tearing  2.  f o r each  over  distance  per p l y values  strength  are defined  a paper  of  a n d 4.3  time,  i n Table  Although  measured f o r  total  energy  tearing  f o r the sector  specimen.  tearing  corresponding  i n F i g . 11a t o l i e .  i n these  been  Only  and t h e i r  average  tear  by  p e r p l y a t any  a t any p o i n t  per p l y v a r i a t i o n s  f o r the sector  tearing  tear  presented  distance  quired  ing  energy  have  2 . 0 , 3.0, 4.0  are presented  and r e l a t i v e  used  data  machine-direction  identification  energy  calculated.  tear  divided  c a n be c a l c u l a t e d .  energy  specimens  when  i s the total  tearing  or time  energies p e r p l y a t 0.2, 1.0, tances  terms,  study.  distance-time  specimen,  could  total  distance  of tear  test  i n this  methods,  of tear  two e n e r g y  i n t h e specimen,  per p l y sought  By  or  between  at  by  when subtract-  corresponding  50  There five ply  a  one  materials specimen  greater all  is  of  sector  were  never of  scale  newsprint,  1-ply  parcel  high  wrap, low  to  behavior  which  were  was  the It  at  each  of  paper  below 14.5 bag  tearing  lower  from  Figures  11a,  for  onion  number  of  tear  plies  Increase more  sample  the was  expression  rate  of  when  only  a  low. of  and Tear  number  the  also  did  the  of  short  plies  not  to  and  time  can  within  allow  series  these  lowest  total  the  increased. per  required  of  l i d  energy as  time  energy  kind  and  in  tearing  of  considered  they  discussion.  newsprint  resistance  one  a  hence  specimens,  tear  be  of  increased  was  the  force,  variation  tearing  and  having  further  total  paper  skin,  relatively  Since  other  same  when t e a r  at  specimens  l i b , for  total  resistance of  torn  had  wrapper  advancing  from  simultaneously  the  paper  specimens,  i s , the  sheet  onion  These were  that  readings  55.5-lb  velocity.  skin,  That  single  torn  in  pronounced  through  a  10-ply  contributing  a l l showed  scale  2-ply  other  differently  ply.  for  to  tearing  kind  exact  velocity  completely  data  to  10.0  High  the  required  The  paper,  from  per  15.  and  excluded  energy  having  from  are  tearing  specimens  specimens  resistance,  wrapper  lowest  note  specimens  55.5-lb.  other  the  the  to  tear  behaved  as  that  a l l  important  ply  for  same  with  also  different  torn  l i e ) , namely,  respectively.  respond  appearing  is  2-ply  velocities.  specimens the  ply  7.5,  5-ply  comparatively  behaved  reading  10.5,  to  plies.  lowest  11.7,  phenomenon  ( F i g . 11a  number the  common  ply  to  tear  the as  was  paper another  paper.  The  51  increase as  i n the  tearing One  reports Also  rate  time  was  of the the  shown  of  advantages  their  r e l a t i o n s h i p with  Onion  skin  The at  the  a  55.5-lb.  then  was  because  one  disproportionately  cm/sec  compared  tance. dual  This  energy  as  slower and  higher tearing  the  tearing  energy  smaller  and  time  tearing  tear  the  distancea.  test.  pattern per  ply  was  cm  tear  disresi-  Consequently,  shown  same p a t t e r n  at  12.42  o f 4:-ply s p e c i m e n s  the  l i d ) .  torn  altered  in Fig.  v a l u e s o f 4;-ply s p e c i m e n s  rendering F i g . l i d with  de-  calculated  energy.  as  that,  (Fig.  0.2  in less  relationship  energy  in  example, at  and  energy-time  increased  cm/sec  distance.  profiles  energy  For  resistance.  any  same t e a r i n g  tearing  decreased  i s that i t  energy  throughout  14.08  energy  time  tear  over  at c e r t a i n  higher tearing  average  the  tearing  velocity resulted  i n turn  tear  velocity.  and  the  Otherwise,  to  ply  t h r e e 4:-ply s p e c i m e n s  slower  t o 13.67  the  tearing  the  per  used  presented another  tear,  of  method  the  distance  wrapper  increased  time  showed  tear  beginning of the  creased This  any  the  required  tear  energy  higher paper  to l i e are  newsprint  at  of  energy  i n F i g . 11a  relationship  tearing  p r o l o n g e d by  tearing  and  total  as  l i d .  were  F i g . 11a  and l i b . These total  tearing  required the  three cases  same,  to  energy  tear  the  ( F i g . 11a,  per  through  longer time  l i b and  ply i s directly a paper. required  When  l i d ) demonstrate related  the  to tear  to  the  incident through  that  time  force  i s  a sheet  of  52  paper  of  d e f i n i t e length  Conversely, energy of  is  Bag  is  tween per In  paper  ply the  time  to  values  In  as  per  tear  energy  have  energy  ply  and time  then was  for  the from  is  as  sheet  slower.  more  when  total  the  rate  in  Fig.  involved  is  unknown.  multaneously.  of  paper  bag  plies  two a  parts.  certain  total and  The  distance  tearing  tear  until  be-  energy  time  increased.  failure, of  first  plies  total  as  number  to  increase  as  number  occurred  in  measuring  errors  paper.  such  pointed  The  to  continued  Otherwise,  smooth  v a r i a t i o n with  tearing  into  shown  and  tear of  increased.  provided  been  v a r i a t i o n , as  i n d i v i d u a l specimens  ty  (Fig.  tear  described  region,  started  bag  the  between  this  experimental  in  occur  of  single  tear  decreased  inherent  has  of  which  be  It  a  divided  number  region,  calculated  tearing  be  cm.  energy  not  can  2.0  unusual  tearing  be  another  beginning  increased  and No  would  presented  increased,  plies  through  the  second  tearing  tear  pattern  from  1.0  to  rate  slower.  This  part  that  r e l a t i o n s h i p can  required  tear  11c.  the  means  sample.  The  out  a  that  strength paper  with  curves.  respect  to  tearing  The  time  energy  replications different  i s thought  p a r t i c u l a r paper  negative  and  number  showed  this  of  to  proper-  relationship plies  behavior  may  torn s i in  part  5). Another  paper,  parcel  wrap,  presented  an  irregular  tear-  53  ing It  energy was  distribution  found  irregular spots  that  distribution  over  some  distribution When lowest 4.3  a  the  ply  plies  are  of  paper  higher  number  tion,  this  finding test.  being  The  dissipation been  and  for  of  E q u a t i o n £ 9^]  is  based  of  into  energy  is  required  also  tendencies.  formation,  bundles  and  irregular  the  tear  versus,  times to  plies,  for  confirms unit  as  torn  simultaneously.  sheet  of  time-dependent be  five  expected,  are  can  of a l l  per  this  tear  number  tear  the  the  numbers  to  confirms  for  different  obtained  finding  tests.  required  containing  thin  fibre  a l l strength  grade)  plotted  This  was  of  for  designed  paper  the  in  paper In  addi-  beha/ior  described  and  gravitational  energy  obtained  to by  do  work  paper  tear  tear  kinetic  as  rate  evaluating  energy  This  concept  strength  Development  energy  in  tester  sector  gravi-  In  the  course  is  transformed  energy.  potential in  the  tearing  tester.  tearing  between  to  properties.  evaluating  calculating  energy  according  tearing  Elmendorf  transformation  the  kinetic  and  accepted  potential  tearing,  of  paper  sheets  plies  poor  Such  relationship.:;is  study  calibrating  on  each of  very  (excluding  Furthermore,  concept  widely  tational  times  special  fibre  sheet.  usefulness  12).  of  and  no  time-dependent.  present  has  is  fibres the  by,  (Fig.  when  tear  had  for  divided  time  of  wrap  of  curvilinear  tear  parcel  booklets  longer  the  with  of  relative  that  of  lie)  affect  specimens  positive  kinds  parts  can  cm-through  of  this  (Fig.  energy  tearing velocity  paper. at  this  study  Kinetic  instantaneous  54  time. the  By t h e d e f i n i t i o n o f v e l o c i t y , t h e r e  time-dependent  energy lated  i s itself as i n t h i s  energy be  terms,  confirmed between  study  as  with  skin,  concept  tearing  newsprint  by s h o w i n g  energy  curvilinear relationship  men  resistance.  Tear tance, time of  was  directly  i . e . , the higher  required  paper,  plies. time  time  to tear  tear  Hence,  through  to tear  of p l i e s .  This  ply  specimens.,  f o r each  when be  more  concluded  that  more  The  results  necessary  through  total  longer  tearing  should  grade)  tearing  time,  total  with  bag paper  to slightly  modify  with  This by  speci-  resis-  one  with  a given  number  energy higher  and p a r c e l  num-  a l l lowest  as a c u r v i l i n e a r  rela-  per p l y i s rate,  of  more  higher  (excluding  energy  the kind  paper  with  or at a lower  tearing  a specimen  Within  specimens...  i n F i g . 12  time.  the longer  directly with  wrapper  specimen  resistance  that  paper  over  when  kinetic  relationship  affected  to test  increased  i s shown  tearing  quired  is  Since  positive  a specimen.  i t c a n be s a i d  ber  tionship.  calcu-  and  and 5 5 . 5 - l b .  i s also  related  the tear  resistance  i s required  two  energy  p e r p l y and t e a r  positive tear  kinetic  kinetic  energy  between  about  well.  onion  t h e above  The t e a r i n g  i t i s also  doubt  and hence  i s the difference  therefore  total  of velocity  time-dependent.  time-dependent Results  property  i s no  required  i t can  then  per p l y i s r e number wrap  the conclusion  of  show  plies. that i t  described  above.  55  That  i s , while  more  tearing  higher ent  tearing of  The  required  Taking  distribution resistance, variation  known  creased eously  Bag  was  fications (29),  that  number  onion  paper  of energy  distance. more  (Fig. 5). t h e same  the conclusion  torn  simultaneously  tearing  energy  a s shown  The l o n g e r energy  between  For un-  torn  tear  de-  simultani s shown f o r These and  and  i n the  tested.  relationship for  i s l i n e a r i n that  thei n -  per p l y i s a l i n e a r function the tear  required  distance,  per p l y .  modi-  Minor  value  i s not i n h e r e n t being  given  first  tendency  distance  i n F i g . 14  dissipated  effect.  tear  energy a  (Fig. 11c).  of material  and t e a r  to tear  of Winterbottom  relationship  fibre  tearing  of plies  material  energy.  irregular  o f bag paper  T h e same  can  direct relation-  caused  required  strength  amount  the i r r e g u l a r  s h o w e d t h e same  but i n the type  tearing  skin  crease  with  was  a s t h e number  the positive  tester, The  the  increased  sheet  inher-  which  alters tearing this  when  In f a c t ,  factor  t h e same . t o t a l  time  also  increased  of plies  also  some  i s the total Any  and  sheet  effects.  study  f o r example,  the tearing  reaffirm  profound  confirmed  caused  what  paper  energy  simultaneously,  i n t h e paper  therefore  then  tearing  tear  noted  a single  a specimen.  13  wrap  time-dependent,  to tear  i n this  i n Fig.  parcel  reasons  have  resistance  no m a t t e r  specimen.  may  to tear  tear  shown  i s still  are torn  calculated  specimen  ship.  tear  of plies  energy  results  energy  i s required  properties  energy  alter  energy  number  sheet  tearing  of  proportionally,  This  tearing  56 energy  p e r p l y and t e a r  affected higher is  t h e number  required  fewer  more  tear  The  tearing  t h e same  distance  that  onion  tearing  distance-time, Equation  Cll]  When  plications  versus  paper,  a linear 55.5-lb.  15).  Onion  skin  plies  before  changing  be n o t e d  that  almost  t h e same  point,  sector  all  data  Linear plies  relationships means  was  and  specimen because other  disin  Fig.  of i t s specimens  velocity,  the average of p l i e s  obtained  2c  (Equation  of the  tear  of three r e -  torn  simultan-  f o r newsprint,  up t o a b o u t  40  t o a c u r v i l i n e a r form  (Fig. 15).  It  which  with  lines  between  proportional  intersected  represented  small tear  decrease  bag  wrap, r e s p e c t i v e l y ( F i g . line  velocity.  were measured  energy  skin  a straight  a l l five  zero-swing  higher  and as a r e s u l t  derivative  number  and p a r c e l  maintained  should  the  .  relationship wrapper  to  plies.  o f each  plotted  per p l y  compared  d i f f e r e n t l y from  second  was  The  energy  having  specimen,  f r o m .the  eously,  obtained  of  rate,  The t e a r i n g  this  behaved  number  specimens  at slower  i s further  simultaneously.  t h e more  energy.  acceleration  ) , was  torn  be b e c a u s e  demonstrates  higher  relationship  of the ID-ply  resistance,  comprising  Hl3^]  may  tearing  relationship  further  low  This  through  o f p l i e s were t o r n  required tance  of plies  of plies,  to tear  plies.  number  14  by t h e number  distance  This  the acceleration  presents  range  the y-axis  evidence  of experimental  acceleration i n tearing  of that error.  a n d number  velocity  at  with  of  57  respect  to increase  weakest  member  change 40  may  (24).  specimen when  effect  bending  on  the distribution  can ness  respect  be u s e d  due have  Otherwise,  there  at  about  plies  by S w a r t o u t  i s large.  and  hence  due t o serious  displacement  some  slow  had Setter-  t o be more This  t o produce  relationship  t o number that,  serious  down  between  of p l i e s when  friction  of outer plies  within  the t o t a l  between  effect  tearing  effect  would  on t e a r i n g  ve-  acceler-  a certain specimen  or specimen  range  thickplies,  i n bending  energy  be a c u r v i l i n e a r  tearing  neighboring  and i n c r e a s e  i n t h e number o f p l i e s  a significant  The  acceleration.  i s not too l a r g e ,  to increase  tested.  t o o many  i s expected  of stresses,  t o reason,  displacement  was t h e  displacement of outer p l i e s  tearing  negative linear  with  skin  to curvilinear  specimens-with  c a n be e x p e c t e d  by r e d u c i n g  The  linear  gross thickness  outer plies  Onion  kinds of paper  as d e m o n s t r a t e d  when  of  locity  from  In a d d i t i o n ,  specimen  ation  the five  be b e c a u s e  same b u l k  holm  among  i n relationship  plies  the  i n number o f p l i e s .  resistance bulk  do n o t  requirements.  instead  of linear r e -  lationship .  Review (1)  o f Present Tear  Energy The  brating  dissipation  energy  Test  concept  dissipation  the Elmendorf  Knowledge  tear  concept tester.  has been Members  adopted of the  for cali-  Institute  58  of  Paper  paper  Chemistry  tearing The  methods  and  according  Kinetic  energy  ing  energy energy  pended  in  as  amount  the  then  of  further  an a  as  ply  lowest  ply  specimens  of  figure  that  higher  energy  direct  evidence  tearing tear  is a  some  for  of  tearing  requirement  measure  resistance  their  of  the  a  is  the  paper  paper,  of  energy  ex-  considered  to  The the  should  plies,  total  single  All  been  the  excluded Positive  paper.  All  accompanied sheet.  prevalent  energy  are  tearing  plies.  of  energy  strength  of  have  kinds  hence  against  tear-  behavior.  present  amount  of  always a  This  strength  the  paper  tearing  of  to  a l l five  resi-  sheet.  number  number  strength  that  sheet of  for  be  the  special  are  and  Tearing  to  of  of  according  respect  kind  for  confirmed  i t can  tearing  respect  each  amount  energy.  corresponding  energy  energy.  within  Hence,  with  obtained  particular  or  measured  5 with 13  the  study  dissipation.  net  tearing  paper;  is  this  energy  of  tearing  because  higher  a  of  well. to  the  were  showed  strength  explain  of  sector  dissipated  Fig.  per  force.  expression  Fig.  in  of  between  strength  in  plotted  relationships  i t to  model  calculate  energy  concept  shown  this  concept  sheet  energy  the  to  proportionally  results,  from  the  used  tearing  dissipation relate  to  tearing  prevalent  applied  mathematical  difference  are is  have  phenomena.  derived  dual  (16)  This tearing  dissipated  i t is  by  in  a measure  incident  tearing  of  59 (2)  Dynamic In  in  property  order  t o determine  the conventional  paper  which  of paper  reflected the response  (15) e s t a b l i s h e d  al  factors  tearing  test  good  a n d maximum  at a rate  (Division of Forest  cluded  that  As  the rate  basically  rate  tear  It effect  nounced men,  loading,  conventionvery  slow  ( 2 2 ) . He  a con-  factor  which  tear  test.  has shown, t h e c o n v e n t i o n a l  tear  test i s  rate  or rate  used.  Higgins  of strain.  a n d specimen  strain  i s considered  of loading.  and t h i s  This was  study  used  study  The  arose  very  used  a  not considered.  to relate better  slow high The  t o t h e conven-  test.  of tear  effect  and l i m i t a t i o n  has been  proved  i n this  on t h e f i n a l  illustrate  during  of  i n the conventional  conclusion  tional  of  not the basic  Higgins  approach  Rate  was  between  of loading  rate  rheometer  of tear  at constant  property  o f 0.64 mm/min. by u s i n g  by r a t e  latter  (3)  measured  Products)  t h e d i f f e r e n t approaches  tearing  t o a high  affected  difference from  study  load  of loading  the speciality  this  resistance,measured  c o r r e l a t i o n between  of strain  D.F.P.  contributes  tear  way, was b a s i c a l l y a d y n a m i c  Higgins tear  whether  that  when  o r when  test  value.  study  of scale that  Figures  the v a r i a t i o n of tearing  longer  time  specimen  i s required  tear  resistance  tear  reading rate  h a s an  11a, l i b , and l i d energy  to tear  i s less  through  i s large.  a  I t has  prospecialso  60  been  found  that  less  pronounced  more  than  50.  variation when  the  Hence,  sector  t o have  a specimen  around  or  than  TAPPI  Standard T  reading  40,  tearing  Specimens tear  values.  behavior that to  using  single  should  not  tear  be  than  study  kinds of  r e a d i n g s above results  are  which  as  long  are  If results, correct,  adopted,  resistance  Hence,  and  Minor  these and  tearing  i s not  scale  range  energy  stable.  of  tearing  i t i s not  (29)  two  from  adequate different  suggested  give  values  interpretation and  in  i s different  d i d not  the  test  stable.  that  o b t a i n e d from  Winterbottom  since  range  0 t o 15  at  not  a wide  showed  and  reading  recommended  i s because  covered  15.  as  40  becomes  standard  a sector  is still  paper  Winterbottom  samples  the  time  i s around  f o r the  give  variation  with  reading  This  r e a d i n g s between  f o r accuracy. study  will  (26).  in this  behaviors.  present  low  test  rather  ts-64  A l l five scale  scale  which  energy  used  scale  compare  tearing  low  at  with  50,  414  energy  i t i s preferable  method  more  in tearing  of  too the  Minor's suggestion variation  over  the  CONCLUSIONS  The these 1.  following  tearing  conclusions  energy  The energy  required  p l y when  total  tearing  total  time  longing 2.  Longer  time  was  required torn  number o f  time-dependent,  properties,  may  profoundly  The c o n v e n t i o n a l  tance  more  when  reduced  tearing  were  when  by  pro-  higher  and more  a specimen  to tear  torn  particularly effected  o f a second increase  affected  sheet  simultaneously,  some  internal tear  i s increased  further  per unit  t o be a p a r a b o l i c  The l i n e a r  Variation i n  affected  and more e n e r g y  number o f p l i e s  terms  slower.  e f f e c t was  required  material  in  needed p e r  num-  energy  having  plies.  t h e energy  found  was  was  time.  p e r p l y when  have  was  This  required  was  5.  short.  was  higher  results of  of the study  energy  p e r p l y was  were  was  4.  energy  rate  of plies  Although  papers  More t e a r i n g  ber  higher 3.  tear  to tear  the tear  was  from  studies.  time-dependent. unit  c a n be drawn  specimen  required  simultaneously,  when inherent  sheet d i s c o n t i n u i t i e s , tearing  energy  which  results.  relationship  c a n be  expressed  equation.  in total  tearing  i s positive.  b y number  was  a  distance-time  form  degree  through  energy  The r a t e  of plies  torn  as t e a r  dis-  of increase i s simultaneously.  62  The  linear  number  of p l i e s  between and  relationship torn  neighbouring  increase  of p l i e s  energy  requirements.  The  positive  tearing energy From  findings  sector no  case  and  test  scale  considered  due  found  concept  energy  treat  reading  around  o r more  scale  friction  of outer  plies in  tearing  conventional  proves  that  i t i s suggested  should  acceptable.  that  the  i s adequate.  study,  sector  and  to increase  between  method  should  implies  s i g n i f i c a n t l y change  tearing  of t h i s  acceleration  displacement  resistance  relationship  dissipation  standard  plies,  d i d not  strength  tear  simultaneously  i n bending  number  between  a specimen  readings  than  50,  lower  that  the  giving and  than  a  that 15  be  in  REFERENCES  1.  A n d e r s s o n , • . a n d 0. F a l k . 1 9 6 6 . Spontaneous crack formation i n paper. Svensk P a p p e r s t i d . 69: 91-99.  2.  Balodis,  3.  Bergea,  4.  B r e c h t , W. a n d 0. I m s e t . 1 9 3 3 / 1 9 3 4 . The i n i t i a l t e a r and the through-tear resistance of papers. Z e l l s t o f f &. P a p i e r 1 3 : 5 6 4 - 5 6 7 ; 1 4 : 1 4 - 1 6 . ( N o t s e e n , f r o m reference 16).  5.  Carson,  6.  Casey,  J.P. 1960/1961. P u l p and P a p e r . C h e m i s t r y and C h e m i c a l T e c h n o l o g y . 2nd. e d . 3v. I n t e r s c i e n c e P u b l i s h e r s I n c . , New Y o r k , 2 1 1 3 p p .  7.  Clark,  J . D'A. 1 9 3 2 . C a l i b r a t i o n of Elmendorf tester. P a p e r T r a d e J . 9 4 ( 1 ) : 33-34.  8.  C o h e n , W.E. and A . J . W a t s o n . 1949. ternal tearing resistance. 244.  9.  Cottrall,  10.  D i n w o o d i e , J.M. 1 9 6 5 . The p h o l o g y and p a p e r ature. T a p p i 48:  11.  E l m e n d o r f , A. per. (Not  12.  Fanselow,  V. 1 9 6 3 . The s t r u c t u r e and p r o p e r t i e s o f p a p e r . XV. F r a c t u r e e n e r g y . A u s t . J . A p p l . S c i . 1 4 : 284-304. E. 1942. tester.  The c a l i b r a t i o n o f E l m e n d o r f ' s Svensk P a p p e r s t i d . 45: 71-73.  tearing  F . T . a n d L.W. S n y d e r . 1928. Increasing the capac i t y of the Elmendorf t e a r i n g t e s t e r . Paper T r a d e J . 8 6 ( 1 3 ) : 57-60. (Not seen, from r e f e r e n c e 8).  tearing  The measurement o f i n P r o c . A P P I T A 3: 2 1 2 -  L.G. 1932. Tearing r e s i s t a n c e of paper. Paper Maker B r i t i s h P a p e r T r a d e J . 8 3 ( 3 ) : T5127-132. (Not s e e n , from B.I.P.C. 2 ( 9 ) : 246. relationship properties: 440-447.  b e t w e e n f i b r e morA review of l i t e r -  1920. Testing the t e a r i n g strength of paP a p e r T r a d e J . 7 0 ( 1 6 ) : 213, 215, 217. seen, from r e f e r e n c e 1 6 ) .  J.R. and J . L . F a n s e l o w . I 9 6 0 . The use of t h e p r o d u c t o f b u r s t a n d t e a r v a l u e s a s an i n d e x t o f i b r e and r e f i n e r e v a l u a t i o n . T a p p i 43: 205215A. 63  References  64  (cont'd)  13.  G i e r t z , H.W. a n d T. H e l l e . I 9 6 0 . On t h e t e a r of paper. Norsk S k o g i n d . 14: 455-469. from B.I.P.C. 3 1 ( 7 ) : 1006.)  strength (Not seen,  14.  H a r d a c k e r , K.W. a n d J . A . V a n d e n A k k e r . 1 9 5 0 . Instrumentation s t u d i e s LIX. T e a r i n g s t r e n g t h o f paper I I . The B r e c h t - I m s e t t h r o u g h - t e a r t e s t e r . Tappi 3 3 ( 1 1 ) : 109-11BA.  15.  H i g g i n s , H.G. 1 9 5 8 . T h e s t r u c t u r e a n d p r o p e r t i e s o f p a per X. Some c r i t i c a l p r o b l e m s . A p p i t a 1 2 : 1 - 2 4 .  16.  I n s t i t u t e o f Paper C h e m i s t r y . 1944. I n s t r u m e n t a t i o n s t u d i e s XLVI. Tearing strength of paper. Paper Trade J . 1 1 8 ( 5 ) : 13-16.  17.  Jones,  18.  L i , J.C.R. 1 9 6 4 . S t a t i s t i c a l I n f e r e n c e . V o l . I I . Edwards B r o t h e r s I n c . , Ann A r b o r , p p . 1 7 6 - 1 8 1 .  19.  M a l l e t t , E . a n d R. M a r x . 1 9 2 3 / 1 9 2 4 . Calibration of the Elmendorf t e a r i n g t e s t e r . P a p e r Maker A s s o c . T e c h . S e c . P r o c . 4: 2 1 2 - 2 2 2 . ( N o t s e e n , f r o m r e f e r e n c e 8 ) .  20.  Ranee,  21.  R y d h o l m , 5.A. 1 9 6 5 . P u l p i n g P r o c e s s e s . Interscience P u b l i s h e r s I n c . , New Y o r k . 1 2 6 9 p p .  22.  S a n t e r , L . , H.G. H i g g i n s a n d J.W.P. N i c h o l l s . 1 9 5 5 . T h e s t r u c t u r e and p r o p e r t i e s o f paper I I . The D.F.P. R h e o m e t e r . A u s t . J . A p p l . S c i . 6: 1 9 7 - 2 0 7 .  23.  Sears,  24.  S w a r t o u t , J . T . a n d V . C . S e t t e r h o l m . 1 9 6 3 . E f f e c t o f number o f p l i e s on t h e t e a r r e s i s t a n c e o f p a p e r . U.S. D . A . f . P . L . R e s . N o t e No. F P L - 0 5 .  H.W.H. a n d W. G a l l a y . 1 9 5 2 . T e a r i n g s t r e n g t h a n d i t s r e l a t i o n s h i p t o basis weight. P u l p P a p e r Mag. Can. 53: 116-120.  H.F. 1 9 4 9 . Some new s t u d i e s i n t h e s t r e n g t h p r o p e r t i e s of paper. W o r l d ' s P a p e r T r a d e Rev. 131 ( 3 ) : T e c h . S u p . 1-7; ( 7 ) : T e c h . S u p . 9 - 1 3 .  F.W. a n d M.W. Addison-Wesley  Zemansky. 1952. C o l l e g e P h y s i c s . P u b l i s h i n g Co., Cambridge, pp. 22-23.  65  25.  T e c h n i c a l A s s o c i a t i o n o f P u l p and P a p e r I n d u s t r y . 1949. Conditioning p a p e r and p a p e r b o a r d f o r t e s t i n g . T 402 m-49. TAPPI S t a n d a r d s . The Technical A s s o c i a t i o n o f P u l p and P a p e r I n d u s t r y , New York.  26.  T e c h n i c a l A s s o c i a t i o n o f P u l p and P a p e r I n d u s t r y . I n t e r n a l t e a r i n g r e s i s t a n c e o f p a p e r . T 414 TAPPI S t a n d a r d s . The T e c h n i c a l A s s o c i a t i o n P u l p and P a p e r I n d u s t r y , New York.  27.  W a h l b e r g , T.K. 1953. S t u d i e s on t e a r i n g s t r e n g t h . Part I I . The i n f l u e n c e o f s t i f f n e s s . S v e n s k Papperstid, 56: 173-177.  28.  Wink,  29.  W i n t e r b o t t o m , M. and J . E . M i n o r . 1 9 3 7 . The t e s t f o r tearing resistance. Paper Ind. 1 8 ( 1 1 ) : 928-930. (Not s e e n , f r o m B . I . P . C . 7 ( 7 ) : 234.).  1964. ts-64, of  W.A. a n d R.H. Van E p e r e n . 1 9 6 3 . Does t h e E l m e n d o r f t e s t e r measure t e a r i n g s t r e n g t h ? Tappi 46: 323-325.  66  Table  Parameters  of the five  paper grades  CALIPER mil  PAPER GRADE  1  *  used  i nthe study  B A S I S WEIGHT g/m  **  2  ***  TEARING STRENGTH q/sheet  1.70  28.5  14.1  Newsprint  3.41  52.1  42.7  Bag  2.88  48.8  57.0  5.90  90.3  128.0  6.65  97.6  156.0  Onion  skin  paper  55.5-lb. Parcel  #  Data  wrapper  wrap  were  measured  according  Basis weight i s defined per square meter. Data Standard. T410 bs-61.  to  TAPPI  Standard  T411  m-44.  as t h e w e i g h t o f p a p e r i n grams were measured a c c o r d i n g t o TAPPI  *** Cross machine-direction tearing strengths a c c o r d i n g t o TAPPI S t a n d a r d . T414 t s - 6 4 .  measured  67  Table  2  Average t e a r d i s t a n c e , time, v e l o c i t y , energy s t r e n g t h v a l u e s f o r f i v e paper grades  and  AVERAGE PAPER GRADE  NO. OF PLIES  zeroswing  TEAR DISTANCE (cm)  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  TEAR TIME (sec)  0.078 0.098 0.127 0.150 0.171 0.190 0.209 0.226 0.243 0.254  ZERO-SWING VELOCITY (cm/sec)  SECTOR NET ENERGY (erg)  15.13  114.43  19.84  196.73  24.19  292,53  27.83  387.20  31.20 32.20  486.68 518.53  TEARING* STRENGTH (g/sheet)  0.00  AlTERTGT PAPER GRADE  NO. OF PLIES  TEAR DISTANCE (cm)  TEAR TIME (sec)  TEARING VELOCITY (cm/sec)  TEARING ENERGY (erg)  onion 0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3 These  0.084 0.104 0.133 0.157 0.179 0.200 0.220 0.239 0.255 0.264  are cross-maching  14.48  0.96  18.92  1.77  23.21  2.32  26.91  2.51  30.16 31.01  3.20 3.95  direction tearing  strength  TEARING STRENGTH (g/sheet)  Table  2  68  cont'd  AVERAGE  GRADE  NO. OF PLIES  onion skin  20  PAPER  40  60  70*  *result  TEAR DISTANCE (cm)  TEAR TIME (sec)  TEARING VELOCITY (cm/sec)  TEARING ENERGY (erg)  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.087 0.108 0.137 0.161 0.184 0.204 0.225 0.245 0.262 0.270  14.60  0.39  18.78  1.02  22.76  1.68  26.23  2.16  29.36 30.06  2.79 3.33  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.094 0.114 0.146 0.174 0.198 0.221 0.243 0.263 0.284 0.294  13.57  0.56  17.19  1.23  20.76  1.93  23.85  2.57  26.62 27.33  3.31 3.63  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.096 0.120 0.157 0.186 0.214 0.240 0.263 0.288 0.312 0.323  12.56  0.59  15.43  1.30  18.28  2.09  20.67  2.89  23.05 23.59  3.68 4.01  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.097 0.122 0.160 0.191 0.223 0.250 0.276 0.302 0.330 0.344  12.45  0.53  14.64  1.28  16.84  2.15  18.68  3.04  20.57 21.05  3.93 4.24  of a single  measurement  without  replication  TEARING STRENGTH (g/sheet)  11.80  13.00  14.00  14.30  69 Table  2  cont'd  AVERAGE  GRADE  NO. OF PLIES  onion skin  80  PAPER  newsprint  5  10  15  TEARING ENERGY (erg)  TEAR TIME (sec)  TEARING VELOCITY (cm/sec)  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.097 0.124 0.167 0.204 0.238 0.270 0.300 0.331 0.365 0.382  11.53  0.60  13.02  1.40  14.54  2.34  15.86  3.27  17.25 17.60  4.22 4.55  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.084 0.105 0.134 1.158 0.181 0.202 0.221 0.240 0.258 0.267  14.31  2.40  18.73  4.26  22.91  6.01  26.42  7.66  29.66 30.51  9.39 10.62  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.086 0.106 0.138 0.164 0.187 0.208 0.229 0.250 0.270 0.279  14.35  1.15  18.07  3.36  21.51  6.13  24.54  8.60  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.093 0.114 0.147 0.176 0.200 0.225 0.247 0.270 0.292 0.302  13.46  1.59  16.64  3.89  19.75  6.50  22.49  8.95  25.08 25.66  11.48 12.62  TEAR DISTANCE (cm)  27.43 28.09  11.05 12.39  TEARING STRENGTH (g/sheet)  15.30  33.60  41.60  42.70  70 Table  2  cont'd  AVERAGE PAPER  NO.  OF  GRADE  PLIES  newsprint  20  25  bag paper  TEARING ENERGY (erg)  TEAR TIME (sec)  TEARING VELOCITY (cm/sec)  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.094 0.118 0.153 0.185 0.213 0.240 0.264 0.290 0.315 0.328  12.98  1.51  15.29  3.99  17.69  6.80  19.72  9.64  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.102 0.130 0.168 0.204 0.237 0.268 0.297 0.326 0.356 0.372  11.74  1.82  13.51  4.22  15.34  7.00  16.95  9.74  18.51 18.93  12.62 13.57  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.083 0.102 0.130 0.155 0.177 0.198 0.217 0.236 0.254 0.263  14.86  2.04  19.00  8.14  23.11  12.70  26.59  16.90  29.76 30.51  21.91 26.50  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.087 0.107 0.138 0.163 0.186 0.208 0.227 0.248 0.267 0.277  14.65  1.18  18.81  4.54  21.94  8.65  TEAR DISTANCE (cm)  21.75 22.28  12.51 13.52  24.98  12.52  27.91 28.67  16.19 17.93  TEARING STRENGTH  45.10  46.60  60.00  58.20  71 Table  2  cont'd  AVERAGE PAPER GRADE bag paper  NO. OF PLIES  10  14  IB  22  TEAR DISTANCE (cm) (cm)  TEAR TIME ( ( sseecc) )  TEARING VELOCITY (cm/sec)  TEARING ENERGY ( eerrgg) ) (  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.089 0.110 0.142 0.169 0.194 0.217 0.238 0.260 0.281 0.293  14.23  1.32  17.35  4.61  20.46  8.32  23.08  12.09  25.62 26.32  15.84 17.22  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.091 0.113 0.147 0.177 0.203 0.229 0.253 0.275 0.298 0.312  13.53  1.64  16.15  4.73  18.78  8.29  21.10  11.75  23.22 23.87  15.51 16.69  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.095 0.119 0.155 0.190 0.220 0.246 0.273 0.301 0.326 0.343  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.096 0.123 0.163 0.201 0.236 0.268 0.300 0.333 0.365 0.387  12.63  1.93  14.62  4.99  16.78  8.43  18.53  11.97  20.29 20.86  15.60 16.72  12.24  1.80  13.20  4.98  14.25  8.68  15,17  12.37  TEARING STRENGTH (g/sheet)  57.00  57.00  57.60  „  16.11 16.44  16.22 17.43  59.40  Table  2  72  cont'd  AVERAGE PAPER GRADE 55.5-lb . wrapper  NO. OF PLIES 2  4  6  8  TEARING VELOCITY (cm/sec)  TEAR DISTANCE (cm)  TEAR TIME (sec)  TEARING ENERGY (erg)  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.080 0.101 0.132 0.158 0.182 0.202 0.223 0.241 0.259 0.270  13.32  12.85  17.88  18.42  22.35  21.34  25.92  25.68  29.12 30.06  31.30 33.31  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.084 0.108 0.140 0.167 0.192 0.214 0.235 0.255 0.277 0.289  13.39  6.21  17.16  12.38  20.68  19.67  23.60  27.18  26.41 27.23  34.50 36.94  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.092 0.115 0.149 0.180 0.207 0.231 0.254 0.278 0.300 0.315  13.17  4.62  15.98  11.52  18.77  19.40  21.09  27.46  23.33 24.06  35.74 38.17  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.097 0.123 0.161 0.195 0.226 0.254 0.281 0.308 0.334 0.352  12.26  4.91  14.30  11.82  16.39  19.78  18.16  27.80  19.87 20.46  36.15 38.65  TEARING STRENGTH (g/sheet)  116.00  122.00  127.10  133.10  Table  2  73  cont'd  AVERAGE PAPER GRADE parcel wrap  NO. OF PLIES  TEAR DISTANCE (cm) (cm)  TEAR TIME ( ( sseecc) )  TEARING VELOCITY ( ccrmn//sseecc))  TEARING ENERGY ( eerrgg) ) (  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.083 0.104 0.134 0.158 0.181 0.201 0.221 0.240 0.258 0.268  14.21  13.48  18.67  22.37  22.79  32.94  26.34  40.18  29.52 30.39  51.05 56.72  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.090 0.111 0.143 0.170 0.193 0.215 0.236 0.258 0.277 0.290  14.22  4.44  17.67  13.54  20.95  24.36  23.75  35.04  26.42 27.25  45.89 49.11  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.094 0.121 0.156 0.184 0.211 0.236 0.261 0.283 0.305 0.320  12.63  6.95  15.82  14.31  18.73  23.44  21.31  32.02  23.58 24.40  41.71 44.16  0.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.3  0.103 0.131 0.170 0.205 0.237 0.267 0.296 0.325 0.354 0.372  11.94  6.16  13.73  14.65  15.51  24.61  17.07  34.50  18.62 19.09  44.77 48.03  TEARING STRENGTH (g/sheet)  160.00  164.00  157.90  163.80  74  Table  3  Comparison between t h r e e d i f f e r e n t methods o f p r e p a r i n g conductive p l i e s used i n t h e study  METHOD  CONDUCTIVE MATERIAL  4  perforated t i v e paper  5  graphite  6  silver  Control  These  conduc-  lines  paint  no c o n d u c t i v e terial added  are cross  lines  PREPARATION TIME  SHARPNESS OF S T E P S  TEARING STRENGTH (g/sheet)  medium  worst  55.3  medium  55.5  shortest  best  54.4  ma-  machine-direction  55.5  tearing strengths  (n = 5 ) .  75  Table  4  Test of s i g n i f i c a n c e f o r p a r t i a l regression c o e f f i c i e n t f o r t h e newsprint t e n - p l y specimen  SUM  SOURCE  R e g r e s s i o n due t o t, t 2 , and t 3  (1)  (2)  Due t2, t3  t o t and ignoring  Addition to t3  Total  significant  SQUARES  D.F.  MEAN  SQUARE  1,890.2742  3  630.0914  16,739.9**  1,890.0254  2  945.0127  25,106.6**  0.2488  1  0.2488  0.2259  6  0.0376  1,890.5000  9  due  Residual  significant  OF  a t 5% a t 1%  level level  6.6*  76 Table  5  Tear distance-time r e l a t i o n s h i p s for five p a p e r g r a d e s w i t h d i f f e r e n t numbers o f p l i e s and replications  PAPER GRADE zero swing  onion skin  newsprint  NO. OF PLIES  REGRESSION n =  REPLI-  a  CATION  EQUATIONS 10 b (T) c CT)  R*  SE  E  0  1  -6.3734  64.8480  524.552  0.99988  0.2504  0  2  -7.1554  77.4525  476.396  0.99987  0.2600  0  3  -7.6369  82.6602  460.313  0.99983  0.3030  10  1  -6.6601  65.1748  464.815  0.99989  0.2435  10  2  -6.9184  70.1631  450.124  0.99984  0.2952  10  3  -7.2590  66.7583  462.635  0.99996  0.1442  20  1  -7.2360  72.3810  418.749  0.99986  0.2782  20  2  -7.6428  71.7888  423.554  0.99989  0.2402  20  3  -8.0094  72.1515  427.279  0.99989  0.2420  40  1  -7.2567  65.2621  356.484  0.99998  0.1005  40  2  -7.6026  67.1902  351.563  0.99993  0.2004  40  3  -8.1862  81.3997  322.430  0.99993  0.1931  60  1  -8.3481  82.2631  238.983  0.99990  0.2284  60  2  -7.4723  77.4058  249.292  0.99988  0.2576  60  3  -8.0355  77.0044  240.137  0.99983  0.3024  70  1  -8.6412  90.6479  174.260  0.99985  0.2835  80  1  -7.9450  90.5993  120.855  0.99987  0.2639  80  2  -8.3707  96.0691  99.866  0.99964  0.4396  80  3  -8.9828  96.7433  100.187  0.99978  0.3424  5  1  -6.8220  69.4849  440.333  0.99991  0.2233  5  2  -7.1175  69.6755  441.026  0.99994  0.1824  5.  3  -7.0784  67.6351  443.757  0.99995  0.1704  10  1  -8.0350  92.1140  327.827  0.99972  0.3882  10  2  -8.0386  82.7237  355.897  0.999B7  0.2609  A l l R values 0.999.  are highly  significant  a t 1,% l e v e l , a l l  exceeding  Table  PAPER GRADE news print  bag paper  5  77  cont'd  NO. OF PLIES  R E G R E S S I O N EQUATIONS n = 10  REPLI-  R  SE  C  t  CATION  a  b (T)  10  3  - 7 . 3408  71.7685  383.236  0.99996  0.1535  15  1  - 7 . 6375  73.2894  314.879  0.99992  0.2099  15  2  - 8 . 2155  84.7818  271.433  0.99985  0.2860  15  3  - 8 . 2148  83.5157  289.588  0.99989  0.2469  20  1  - 8 . 8950  99.4156  184.703  0.99980  0.3262  20  2  - 8 . 8636  91.8325  197.206  0.99994  0.1855  20  3  -8.0376  85.4665  214.333  0.99992  0.2141  25  1  - 8 . 7982  92.2542  122.220  0.99985  0.2805  25  2  - 8 . 8402  88.0074  150.664  0.99992  0.2123  25  3  - 8 . 7746  90.6555  127.307  0.99987  0.2647  2  1  -8. 6775  95.8356  377.100  0.99993  0.1876  2  2  - 7 . 2409  74.8557  437.617  0.99993  0.1960  2  3  - 6 . 2072  60.3486  488.641  0.99995  0.1672  6  1  - 7 . 6722  80.4961  378.226  0.99991  0.2264  6  2  - 8 . 8059  91.8116  338.037  0.99987  0.2698  6  3  - 7 . 7453  74.4592  391.204  0.99990  0.2371  10  1  - 8 . 3260  97.2049  262.674  0.99982  0.3102  10  2  - 8 . 2116  82.9565  320.323  0.99997  0.1328  10  3  -8. 6947  87.5043  308.978  0.99991  0.2262  14  1  - 8 . 5775  91.9893  237.750  0.99994  0.1777  14  2  -9. 2317  103.4080  207.965  0.99994  0.1813  14  3  - 7 . 7378  83.0536  254.740  0.99990  0.2365  18  1  - 9 . 2403  98.5485  158.527  0.99988  0.2552  18  2  - 7 . 8305  93.1304  169.119  0.99964  0.4439  18  3  -8. 8837  93.0997  169.829  0.99983  0.3049  22  1  - 8 . 8406  104.3110  76.548  0.99970  0.4050  22  2  - 9 . 6838  114.3930  65.638  0.99973  0.3827  22  3  - 9 . 5859  106.7370  74.125  0.99990  0.2300  c  CT)  Table  5  PAPER GRADE 55.5lb. wrapper  parcel wrap  78  cont'd  NO. OF PLIES  REPLI-  R E G R E S S I O N EQUATIONS n = 10 b (T) c (T )  R  SE  F  t  CATION  a  2  1  -5.5841  63.2721  441.483  0.99976  0.3579  2  2  -6.3464  68.5575  424.470  0.99987  0.2676  2  3  -5.6081  55.9263  458.187  0.99994  0.1808  4  1  -7.3089  81.2928  333.736  0.99970  0.4017  4  2  -7.9274  88.5774  306.911  0.99958  0.4787  4  3  -6.1280  61.6303  372.181  0.99980  0.3258  6  1  -8.0230  84.3801  241.810  0.99978  0.3442  6  2  -8.2676  89.8457  238.169  0.99984  0.2904  6  3  -8.3716  86.7671  250.907  0.99984  0.2933  8  1  -8.4925  91.1334  154.389  0.99966  0.4305  8  2  -8.8739  92.5650  159.350  0.99991  0.2150  8  3  -8.5037  90.0730  169.091  0.99974  0.3745  1  1  -6.7607  59.2432  462.346  0.99996  0.1434  1  2  -8.0221  84.1428  398.174  0.99989  0.2473  1  3  -6.0168  65.8681  451.687  0.99996  0.1488  3  1  -8.3449  87.7845  314.328  0.99986  0.2711  3  2  -8.1078  80.8438  336.730  0.99984  0.2901  3  3  -8.4429  82.6506  326.161  0.99978  0.3441  5  1  -7.5136  75.7751  268.025  0.99981  0.3175  5  2  -8.9070  83.0102  247.169  0.99980  0.3249  5  3  -7.4411  72.7527  266.158  0.99966  0.4306  7  1  -9.1375  90.1060  130.316  0.99953  0.3561  7  2  -9.1205  88.9178  140.065  0.99980  0.3249  7  3  -9.3193  96.9541  128.945  0.99983  0.3209  d  Fig.  1.  Schematic diagram i l l u s t r a t i n g angles i n v o l v e d i n the b a l l i s t i c - t y p e t e a r i n g principle.  A n g l e s a r e d e f i n e d by t h e v e r t i c a l l i n e and t h e l i n c o n n e c t i n g t h e s e c t o r c e n t e r o f mass and a x i s o f r o tat ion. 0,  = i n i t i a l angle before a specimen,"  swinging  or t e a r i n g  Q^-  ~ angular displacement of sector center of mass r e p r e s e n t i n g s e c t o r net e n e r g y , and  0^  = angular displacement of s e c t o r center of mass r e p r e s e n t i n g s e c t o r r e s i d u a l e n e r g y  80  .  2.  Model c o n s i s t i n g o f s p r i n g s and d a s h p o t s used t o i l l u s t r a t e stress-strain-time relationships f o r polymeric materials.  elastic  springs  \  viscoelastic  dashpots flow  load  I time dependent  81  Fig.  3.  Conversion of energies type t e a r i n g process.  begin tearing top  in  the.ballistic-  finish tearing SECTOR  VERTICAL  POSITION  bottom  82  Fig.  4.  Schematic diagrams i l l u s t r a t i n g the c o n v e r s i o n o f e n e r g i e s i n , a, s e c t o r s w i n g w i t h o u t s p e c i m e n , b . s e c t o r s w i n g when t e a r i n g a s p e c i m e n .  mgy  mgy mgy =  •Tot  mgy mgy  "Tot  1 2 =  +  m  gy  +  1  ( j(v ,t ) 1  1  =  mgy  =  1 2  -IT!V/  + "Tot  = total  energy  of a  m  = sector  g  = gravitational  y  = vertical position before swing,  ^ l  = vertical  2 2  +  T.E.  system,  mass, acceleration, of the sector  center  displacement of the sector  o f mass  center  o f mass,  = tangential  velocity  of time  t-^, a t z e r o - s w i n g ,  =  velocity  at time  t y , a t t e 3 r i n g - s w i n g , and  tangential  |(v^,t^) '  &. j ( v 2 » t 2 )  = f r i c t i o n terms respectively^  i n case  a and b.,  83 Fig.  5.  R e l a t i o n s h i p s between c r o s s t e a r i n g s t r e n g t h a n d number multaneously f o r f i v e paper  machine-direction of plies torn s i grades.  A  onion skin  -e  newsprint bag paper  160 -. '" *  • .  :  • '•' -  • - . ..V .  MO  55-5-lb wrapper parcel wrap  -  '- -0-  hee  •f-  10  1 20  V.  100 X  1-  (D  Z  LU  80  -  or t— 60 o z or  <  LU  ^  o  -a-  — — g _ —  -e  a-  40 -  20  A i  0 0  . 5 newsprint,  10 bag paper,  '  |  10 NUMBER  i  i  : I  20 OF  &• • — A  . A —  & •  1  15 20 25 55-5-lb wropper, parcel wrap I  30 40 onion PLIES  A  TORN  '  50 60 skin  I  I  70  80  SIMULTANEOUSLY  84  Fig.  6.  O s c i l l o s c o p e t r a c e s had conductive materials.  with  four  a.  Perforated e l e c t r i c a l p a p e r as c e n t e r ply.  b.  Graphite conductive the c e n t e r ply.  different  conductivity  lines  applied  to  c.  d.  Silver conductive the c e n t e r p l y .  lines  applied  to  Combined s i l v e r and g r a p h i t e l i n e s a p p l i e d to the c e n t e r p l y . (see Fig. 7).  86  Fig.  7.  P a t t e r n of c o n d u c t i v e l i n e s used f o r the study. L i g h t and d a r k l i n e s a r e s i l v e r and graphite conductive materials, respectively.  Fig.  8.  E l e c t r i c a l c i r c u i t used f o r measuring distance-time relationship.  the  tear  88  Fig.  9.  Set-up  used  for  the  study  1.  Thwing-Albert Instrument 60-100 E l m e n d o r f t e a r i n g  2.  Tektronix, I n . , t y p e 564 storage o s c i l l o s c o p e equipped with 3B4 t i m e b a s e and 3A3 dual trace differential amplifier plug-in units.  3.  T y p e C-27 P o l a r o i d camera w i t h camera mounting frame a t t a c h e d the o s c i l l o s c o p e .  4.  Battery  eliminator.  5.  6-volt  6.  6-decade  7.  Voltmeter.  8.  Pilot  battery. resistance  box.  kit.  Co. No. tester,  to  TEAR  DISTANCE,  ( Cm) I-1  in  CD cr Ci 0) n U3 zr TJ n O c TJ  —i  CD Qi  l-i  a H-  ro 05 < h rtro • 01  •  n  ro  c 1 3 c+ 5" HCO 3 H CD  O H -n CD I—  1  TJ ru TJ ro  DJ  c+ HO  H •  CD TJ : r  HH-TJ CO ia U5 -tj H- o CO  3 fl) a i-i I •\  CD CT  a. • o Z3 P° 3  68  90 Fig.  11a.  Tearing energy, distance and t i m e relationships for unglazed onion s k i n . Broken and s o l i d l i n e s a r e number o f p a p e r p l i e s a n d t e a r distances, respectively.  LU  5  4  CO CO  < 2  or 3 h o io LU CO  e> or LU  z LU O  z or <  LU  J  I  I  2  4  6  RELATIVE  ;  TEAR  I  8  I  I  10  T I M E , ( I0" Sec) 2  12  14  91  Fig.  l i b . Tearing energy, distance and t i m e r e l a t i o n s h i p s for newsprint. B r o k e n a n d s o l i d l i n e s a r e number of paper p l i e s and t e a r d i s t a n c e s , respectively.  - 1 4  5  a. | 2  CO  to < S  I 0 |  CC  o  IGO  O tr  6  UJ  z  UJ  o z  4  or < H  2  2  4 RELATIVE  8 TEAR  TIME,  10 ( I0"  12 2  Sec)  14^  92  Fig.  11c.  T e a r i n g e n e r g y , d i s t a n c e and t i m e r e l a t i o n s h i p s f o r 3 0 - l b . n &. m b a g p a p e r . Broken and s o l i d l i n e s a r e number o f p a p e r p l i e s and t e a r distances, respectively.  2-0CM  '  O  /  /  /  /  z  or <  l-Ocm  0  2  4  RELATIVE  6 TEAR  8 TIME,  10 ( 10"  2  12 Sec)  14  93 Fig.  l i d .  Tearing energy, distance and time r e l a t i o n s h i p s f o r I s l a n d 55.5-lb. w r a p p e r . Broken and s o l i d l i n e s a r e number o f p a p e r p l i e s a n d t e a r distances, respectively.  4-3 cm — 40cm w  UJ  m  CO CO < S  or o iu  UJ (O  >• o or  UJ  1 0 cm  z  UJ  10  or < UJ  4 RELATIVE  6 TEAR  8 TIME,  10 I I0"  12 2  Sec)  14  94  Fig.  l i e .  Tearing energy, distance and t i m e relationships f o r p a r c e l wrap. Broken and s o l i d l i n e s a r e number o f p a p e r p l i e s a n d t e a r distances, respectively.  60  0I 0  t  1  2 4 RELATIVE  I  6 TEAR  1  8 TIME,  ;  —I  _  10 ( 1 0 ' Sec) 2  1  12  1  14  95 Fig.  12,  R e l a t i o n s h i p s between r e l a t i v e t e a r time p e r u n i t p l y a n d number o f p l i e s t o r n simultaneously . *  16  • — onion skin newsprint  u V) lO  /  I 4  --S  —  — X  —  bog —  paper  . 55-5-lb wrapper parcel wrap  12  r  / u 10 2  or  8  UJ  i-  UJ  > r-  <. _J UJ  D  B—  —  or  0  5 newsprint, 10 NUMBER  10 bag paper, 20 OF  30  15 20 55-5-lb wrapper, parcel  40 onion  PLIES  TORN  50 skin  60  70  25 wrap 80  SIMULTANEOUSLY  96 Fig.  13.  R e l a t i o n s h i p s between t e a r i n g t e a r i n g energy f o r f i v e paper  strength grades.  and  -.134  I65r  164  newsprint  16 - -£J  -132  bog paper — • 55-5-lb wrapper•  ^  15  163  ©.  7  -46  porcel wrap  • -59  x i-  z  LU or  or <  - 130  -45 «M62 o  +-  14  c  128  a.  jf  u  -44*1- s s :  * 161  o z  - 60  -47  onion skin  A-  u w  o  a.  160 •  a. u>  13  .O  126 7  5  c o c o  -43  a* c  57  12  IN  o» O  M IN  - 124  LU  -42 159  122  / 158  -41  - 56 -  I 3  11  4  u>  onion skin 12 ,I3 newsprint  +  14  16  17 18 paper 35 36 37 38 39 55-5-lb wrapper 43 45 47 49 parcel wrap TEARING ENERGY / SECTOR MASS / P L Y , bag  (Erg)  120  97  Fig.  14.  T e a r i n g e n e r g y and for unglazed onion  distance skin.  relationships  Fig.  15.  R e l a t i o n s h i p s b e t w e e n t e a r i n g a c c e l e r a t i o n and number p l i e s t o r n s i m u l t a n e o u s l y f o r f i v e paper grades.  of  1000  ~  800-  600-  400-  200  2  4 newsprint, 10  6  bag 20  NUMBER  30 o n i o n OF  14 16 55-5-lb  10 12 paper,  PLIES  40  18 20 wrapper, 50  s k i n T O R N  22 24 26 parcel wrap  60  SIMULTANEOUSLY  70  80  99 APPENDIX  1  E l e c t r i c a l C i r c u i t Used f o r t h e S t u d y in the Tearing Process. a.  Circuit  f o r measuring  the tear  and V o l t a g e  distance-time  Variation  relationship.  V v W ! r*  1  l  I VxX-Y J ['conductive ply "j i 1  Re b.  Trace  of the voltage  variation  TEAR 0-2  I  0  ;  I  on t h e  DISTANCE,  10  !  20  i  2  TEAR  oscilloscope.  (Cm)  3 0 4-0 4 3  —  •'  1  3  4  T I M E , ( ICT  1  Sec)  i  5  Appendix  c.  1  100  cont'd  Voltage  variation  tearing  process.  The  voltage  i n the  difference v  =  i R^  = =  (V  ) across  during  the  the  conductive  ply i s : ri9~i  e  L_  current (Amp) equivalent resistance,  the  r^  of  to  ply  iR  x where:  conductive  r^g  in parallel,  1  +  combination the  specimen  of  _ l resistors  conductive  piy. 1 R  A  6 volt  e  R  =  g  o  l  r  battery V  where:  =  1 r  i s used  =  6 =  i  +  +  1  2  r  i n the  (R  ....  +  a  circuit  10 (Appendix  R) e  r20^| 1_ _ l  r e s i s t a n c e chosen  to  resistance  conductive  of  l a ) , hence  the  be  equal  to  the  equivalent  ply.  Therefore, =  v  * This  i s the  equivalent Hence, ply as  V^  6  R  e  _ __ e  a  R  B  voltage  as  R  = R  lb.  volt  e before  , increases  tearing  torn.  i n Appendix  a  +  difference  resistance, increases  3  6R  i(R +R )  i s completely shown  i  Then,  tearing as  tearing  progresses, V^  starts.  reaches  until  is in the  i t s maximum  The progress. conductive of  6  volt,  101 APPENDIX  Inter-relationship and  Energy  tionship  Between  2  Tear  D i s t a n c e , Time,  i n the B a l l i s t i c - t y p e Tearing Process.  between  zero-swing  (L^)  distance  and t i m e  Velocity  The  rela-  (t^)  i s :  2 l  L  The  "  l  a  +  relationship L  = a  2  b  l * l  +  c  between + b t  2  2  l  t  tear + c  2  l distance  t  2  c  2 2  Both  E q u a t i o n s Q 1 1 a J a n d J ^ l l b J show  time  relationship The  time  v  2  ^ 0  {t^) i s : Cubl]  that  the-tear  distance-  zero-swing  velocity  ( v ^ ) and  i s curvilinear.  relationship  (t^)  2  ( l _ ) an,d t i m e  between  i s :  + 2c t  =  x  1  Th e r e l a t i o n s h i p  c ^ O  1  between  tearing  L ~3 12a  velocity  (v,,) a n d  time  (t ) i s : 2  v Both  2  = b  +  2  2c t 2  c 2  2  / °  E q u a t i o n s £ l 2 a ^ | a n d £ l 2 b 2 | show  relationship The  that  the  1  2  b  J  velocity-time  i s linear.  relationship (L^)  distance  C  between  zero-swing  c a n be o b t a i n e d f r o m  velocity  ( v ^ ) and  E q u a t i o n s £ l l a ~ J a n d QL2a,~J  where: v  becomes:  l  =  b  l  +  2^1  t 0 1 2  2 cont'd  Appendix  Equation £ 2lJ  Substituting L,  1  =  a,  +  1  102  b. (  l  V  2  1 V  -  l)+  b  c.  1  C l  into (  l " 2c  V  ^  1  E q u a t i o n £lla2j g i v e s :  2 =  a  +  2  l  b  l  v  1  2  b  2  2bivi  ~  V J L  + bj,  4c ^  1  4c  of  v  4c L  =  c  2  1  2  Transposition 2  +  1  4cj^  1  2  l  2  u  1  4c  "  M /  b  ±  1  [^222]  Equation  1  -  1  4a c 1  +  1  gives:  b ^  C  2  3  a  H  hence: v  =  x  By ship  the  v  2  v  2  =  2  tearing 4c l_ 2  2  the  The E  l  C  +  l  -  4a c  2  -  2  2  4a c 2  b  i t can  +  b  2  +  C23bD  2 x  be  (v )  shown  and  2  that  and  the  distance  (L ) 2  relationi s :  L7  2 2  b  v ">0) 2  2 4 a  U  L~24b3  2 2  v e l o c i t y - d i s t a n c e r e l a t i o n s h i p can of  the  be  expressed  hyperbolas,  net  sector  net  = residual  . , . resxdual  difference energy  energy i s : = ¥  U C  1 1  "  L  4 §  1 1 C  +  b  l  2  ^  )  energy i s :  = Imv 2 2  between (T.E.).  2  = km 2  sector  (4c_L22 net  and  -  4a-c_ 22  +  residual  b  as  Equations  C24a]. sector  E  4a  velocity  (where: v ^ > 0  Q 2 3 a 3 and  tearing  2  4c L  =  branchs  The  -  same a p p r o a c h ,  between  Hence,  The  ± J  ) \~26~] 2 1 — —I 2  energies  is  Appendix  T  2  , i _2 _ 2 1  r  _  km  1  (b "l  tearing  The  by  equation  (E ^.  E  sector  can  be  dual  km  E  obtained  T.E.  the E  =  km  2 x  4  that  -  2  a-^) 1*1  to  and  tear  time  distance  linearly.  (t^) relationship  E q u a t i o n L7i2a~J i n t o  2c.t  (b. + 1  )  1  the  can  net  be  energy  4b _c t J  1  +• 4 c  1  = km 2  (b  2 2  +  t  2 1  2 1  tear  (b„ 2  C  time  ~  =  2  (t ) 2  2  +  2  energy-time  4c  8  3  relationship the  resi-  ^  2c~t_) 2 2  4b c t  2  2  msJ  +  2  )  E q u a t i o n Ql2bH| i n t o  (^peg^jyai  2  2  1  substituting  km  jTb  2  )•  2  E  4(a c  >]  energy'and  = imv„ 2  . net  shows  linearly.  +  2 x  ^tearing  =  j_  equation  + Which  m v  by  =  +  2  relates  = km 2  2  (b  . , , residual  Hence,  2 2  energy  residual  energy  2  2  1  2  1 = ^  . = Imv net 2 1  The  net  2  - c l_ )  " *  2  2^  1 / 2  2  substituting  n e  =  2  1  energy  sector  obtained  i 2  |4(c L  + Hence,  103  cont'd  2  2 2  t  2 2  C  )  2 9  3  relationship i s :  . . . residual (\ Cl  b  2 2  )  2  tearing  + -  4 c  ( b ^ t ,  1 1  2 2  energy  t  1  -  b c t )  2 2*2' 0  W  0  0  3  M  2  2  relates  to  tear  time  curvi-  


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