Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

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

Item Metadata

Download

Media
831-UBC_1967_A6_7 S95.pdf [ 5.97MB ]
Metadata
JSON: 831-1.0104518.json
JSON-LD: 831-1.0104518-ld.json
RDF/XML (Pretty): 831-1.0104518-rdf.xml
RDF/JSON: 831-1.0104518-rdf.json
Turtle: 831-1.0104518-turtle.txt
N-Triples: 831-1.0104518-rdf-ntriples.txt
Original Record: 831-1.0104518-source.json
Full Text
831-1.0104518-fulltext.txt
Citation
831-1.0104518.ris

Full Text

ENERGY DISSIPATION IN PAPER TEARING AS TIME-DEPENDENT PHENOMENON by BERNARD CHING-HUEY SUN B.Sc.A. NATIONAL TAIWAN UNIVERSITY Tai w a n , The R e p u b l i c o f C h i n a , 1960 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF - F Q R € S W ¥ i n t h e Department o f F o r e s t r y We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1967 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h.i-s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . 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, Canada D e p a r t m e n t i i ' ABSTRACT The n a t u r e of b a l l i s t i c - t y p e i n t e r n a l paper t e a r t e s t methods has been r e v i e w e d . The k i n e t i c e n e rgy of t h e t e s t e r s e c t o r i s c o n s i d e r e d t o be t h e prime c o n t r i b u t o r t o paper r u p t u r e . In agreement w i t h energy d i s s i p a t i o n c o n c e p t s and t h e p r i n c i p l e of energy c o n s e r v a t i o n , a m a t h e m a t i c a l model e x p r e s s i n g t e a r i n g energy was d e r i v e d based on k i n e t i c e n e r -gy v a r i a t i o n s i n paper d u r i n g t e a r i n g . I t i s shown t h a t t h i s m a t h e m a t i c a l model can be used t o c a l c u l a t e t h e net energy o f t h e t e s t e r s e c t o r , which i s a v a i l a b l e for t e a r i n g p a p e r , and t h e r e s i d u a l e n e r g y . Con-s e q u e n t l y , t h e d i f f e r e n c e between net and r e s i d u a l e n e r g y , or t e a r i n g e n e r g y , i s t h a t p o r t i o n expended i n t h e r u p t u r e p r o c e s s . F u r t h e r m o r e , t h e m a t h e m a t i c a l model r e l a t e s t e a r -i n g energy t o v e l o c i t y , hence can be used t o examine t h e e f -f e c t o f t e a r r a t e and t i m e - d e p e n d e n t p r o p e r t i e s o f p a p e r sub-j e c t e d t o t e a r i n g s t r e s s . A method was d e v i s e d f o r m e a s u r i n g t h e t i m e r e q u i r e d t o t e a r s t a n d a r d s a m p l e s . From an o s c i l l o s c o p e t r a c e , t h e t e a r 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 was measured and r e p r e s e n t e d by a q u a d r a t i c e q u a t i o n . From t h i s e q u a t i o n , s e c t o r swing and t e a r i n g v e l o c i t i e s were c a l c u l a t e d f o r computing v a r i o u s e n e r gy f a c t o r s and t h e i r v a r i a t i o n a t any i n s t a n t of t h e t e a r i n g p r o c e s s . i i i R e s u l t s have shown t h a t b a l l i s t i c - t y p e t e a r t e s t methods a r e t i m e - d e p e n d e n t , i n t h a t t i m e r e q u i r e d t o t e a r p a p e r v a r i e s w i t h t h e sample c o n d i t i o n . The h i g h e r t h e 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 , t h e l o n g e r was t h e t i m e r e q u i r e d t o t e a r a p a p e r s h e e t . The energy r e q u i r e d t o t e a r p a p e r was a l s o t i m e - d e p e n d e n t , i n c r e a s i n g w i t h d e c r e a s i n g t e a r r a t e . I t was f o u n d t h a t t h e d i r e c t r e l a t i o n s h i p between t e a r -i n g s t r e n g t h 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 does not always h o l d , but t h a t a c o n s t a n t d i r e c t r e l a t i o n s h i p e x i s t s between t e a r i n g s t r e n g t h and t e a r i n g e n e r g y . A l t h o u g h t h e b a l l i s t i c - t y p e t e a r t e s t i s t i m e - d e p e n d e n t , i n h e r e n t specimen p r o p e r t i e s may have a p r o f o u n d e f f e c t on r e s u l t s . T e s t r e s u l t s w i t h an E l m e n d o r f t e a r t e s t e r on f i v e p a -pe r g r a d e s v a r y i n g i n t e a r i n g s t r e n g t h from 14 t o 156 g / s h e e t have c o n f i r m e d t h a t t h e energy d i s s i p a t i o n c o n c e p t i s a d e q u a t e . i v TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGMENT i x INTRODUCTION 1 LITERATURE REVIEW 4 V a r i a t i o n Caused by Conduct o f t h e T e s t 7 P r i n c i p l e and Methods f o r C a l i b r a t i n g B a l l i s t i c - t y p e T e a r T e s t e r s 10 T e a r T e s t T h e o r y 12 Paper R h e o l o g i c a l P r o p e r t i e s 15 ENERGY DISSIPATION AND RATE OF TEAR IN THE TEAR TEST 18 MATERIALS AND METHODS 27 M a t e r i a l s 27 Methods 28 T e a r i n g p r o c e d u r e s 28 E x p e r i m e n t a l d e s i g n 29 Ra t e o f t e a r measurement 29 Z e r o - s w i n g t e a r d i s t a n c e - t i m e measurement 36 H a n d l i n g o f d a t a and c u r v e f i t t i n g 38 Summary o f methods 41 T a b l e o f C o n t e n t s ( c o n t ' d ) v Page DISCUSSION 4 3 ' Cur v e F i t t i n g 44 I n t e r p r e t a t i o n o f R e s u l t s 48 Review o f P r e s e n t T e a r T e s t Knowledge 57 (1) E n ergy d i s s i p a t i o n c o n c e p t 57 ( 2 ) Dynamic p r o p e r t y o f paper 59 (3) Rate o f t e a r e f f e c t and l i m i t a t i o n o f s c a l e r e a d i n g 59 CONCLUSIONS 61 REFERENCES 63 TABLES AND FIGURES 66 APPENDICES 99 v i LIST OF TABLES Page TABLE 1. Parameters of the f i v e paper grades used in the study 66 TABLE 2. Average tear distance, time, velocity, energy and strength values for five paper grades 67 TABLE 3 . Comparison between three d i f -ferent methods of preparing conductive plies used in the study 74 TABLE 4. Test of significance for par-t i a l regression coe f f i c i e n t for the newsprint ten-ply specimen 75 TABLE 5. Tear distance-time r e l a t i o n -ships for five paper grades with different number of plies and replications 76 v i i LIST OF FIGURES Page FIGURE 1. S c h e m a t i c d i a g r a m i l l u s t r a t i n g a n g l e s i n v o l v e d i n t h e b a l l i s t i c -t y p e t e a r i n g p r i n c i p l e 79 FIGURE 2. Model c o n s i s t i n g of 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 s t r e s s - s t r a i n - t i m e r e l a t i o n -s h i p s for p o l y m e r i c m a t e r i a l s 80 FIGURE 3. C o n v e r s i o n o f e n e r g i e s i n t h e b a l l i s t i c - t y p e t e a r i n g p r o c e s s 81 FIGURE 4. S c h e m a t i c diagrams i l l u s t r a t i n g t h e 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 swing w i t h o u t s p e c i -men, b. s e c t o r swing when t e a r -i n g a specimen 82 FIGURE 5. R e l a t i o n s h i p s between c r o s s ma-c h i n e - d i r e c t i o n t e a r i n g s t r e n g t h 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 g r a d e s 83 FIGURE 6. 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 ma-t e r i a l s 84 FIGURE 7. 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 c o n d u c t i v e m a t e r i a l s , r e s p e c -t i v e l y 86 FIGURE 8. E l e c t r i c a l c i r c u i t used for m e a s u r i n g t h e t e a r d i s t a n c e -t i m e r e l a t i o n s h i p 87 FIGURE 9. 5e t - u p used for t h e s t u d y 88 FIGURE 10. T e a r d i s t a n c e - 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 bag p a p e r . Number o f paper p l i e s i s marked on each c u r v e 89 L i s t o f F i g u r e s ( c o n t ' d ) v i i i Page FIGURE 11a. FIGURE l i b . FIGURE 11c. FIGURE l i d . 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 u n g l a z e d o n i o n s k i n . B roken and 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 d i s t a n c e s , r e s p e c t i v e l y 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 n e w s p r i n t . Broken and 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 , r e s p e c t i v e l y 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 30 l b . n &, m bag p a p e r . Broken and 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 d i s t a n c e s , r e s p e c t i v e l y , 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 I s l a n d 55.5 l b . wrapper. Broken and 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 d i s t a n c e s , r e s p e c t i v e l y 90 91 92 93 FIGURE l i e . FIGURE 12. FIGURE 13. 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 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 paper p l i e s and t e a r d i s t a n c e s , r e s p e c t i v e l y 94 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 t i m e p e r u n i t p l y and num-ber 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 R e l a t i o n s h i p s between t e a r i n g s t r e n g t h and t e a r i n g e n e rgy f o r f i v e paper g r a d e s ..... 96 FIGURE 14. 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 u n g l a z e d o n i o n s k i n . 97 FIGURE 15. R e l a t i o n s h i p s between t e a r i n g a c c e l e r a t i o n 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 p a p e r g r a d e s 98 i x ACKNOWLEDGMENT The a u t h o r acknowledges h i s g r a t i t u d e t o members o f t h e F a c u l t y o f F o r e s t r y , U n i v e r s i t y o f B r i t i s h C o l u m b i a , who o f f e r e d h e l p f u l s u g g e s t i o n s d u r i n g c o n d u c t o f t h i s work; w i t h s p e c i a l a p p r e c i a t i o n t o Dr. J . W. W i l s o n , P r o f e s s o r , f o r h i s g u i d a n c e d u r i n g p l a n n i n g and e x p e r i m e n t a l phases and w r i t i n g o f t h e t h e s i s . G r a t e f u l acknowledgment i s a l s o made t o Dr. L. Bach, H o n o r a r y A s s i s t a n t P r o f e s s o r , F a c u l t y o f F o r e s t r y , and Re-s e a r c h O f f i c e r , V a n c o u v e r F o r e s t P r o d u c t s L a b o r a t o r y , t h e Department o f F o r e s t r y and R u r a l Development, f o r h e l p i n i n i t i a t i o n , p l a n n i n g and e x e c u t i o n o f t h i s s t u d y ; t o Dr. R.W. Wellwood, P r o f e s s o r , and Mr. L. V a l g , A s s i s t a n t P r o -f e s s o r , F a c u l t y o f F o r e s t r y , and Mr. J . H e j j a s , B i o m e t r i c i a n , V a n c o u v e r F o r e s t P r o d u c t s L a b o r a t o r y , t h e Department o f F o r e s t r y and R u r a l Development, f o r h e l p f u l s u g g e s t i o n s and c r i t i c i s m s ; t o Mrs. H. F r o s e , f o r computer programming; t o Mr. R. Kabos, T e c h n i c i a n , V a n c o u v e r F o r e s t P r o d u c t s L a b o r a -t o r y , t h e Department o f F o r e s t r y and R u r a l Development, f o r t e c h n i c a l a s s i s t a n c e i n s e t t i n g up t e s t i n g f a c i l i t i e s . Acknowledgment i s a l s o g i v e n t o I s l a n d P aper M i l l s , D i v i s i o n o f M a c M i l l a n B l o e d e l , L i m i t e d , f o r s u p p l y i n g p a r t o f t h e m a t e r i a l s ; t o t h e V a n c o u v e r F o r e s t P r o d u c t s L a b o r a t o r y , t h e Department o f F o r e s t r y and R u r a l Development f o r t h e i r f a c i l i t i e s and t o t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada and X t h e U n i v e r s i t y o f B r i t i s h Columbia f o r r e p e a t e d f i n a n c i a l s u p p o r t d u r i n g t h e academic programme. L a s t , but not l e a s t t o Mrs. E. S. Sun, f o r her pa-t i e n c e , d e v o t i o n and encouragement t h r o u g h o u t t h e s e memor-a b l e y e a r s . INTRODUCTION S i n c e advancement o f t h e E l m e n d o r f i n t e r n a l t e a r t e s t method about h a l f a c e n t u r y ago, i t has p r o v e d v a l u a b l e i n m e a s u r i n g a b a s i c paper s t r e n g t h p r o p e r t y . T h i s method has become w i d e l y a c c e p t e d as a u s e f u l means f o r q u a l i t y c o n -t r o l e v a l u a t i o n s and r e s e a r c h s t u d i e s . A d o p t i o n o f s t a n d a r d t e s t p r o c e d u r e s has l e d t o an i m p o r t a n t p o i n t o f communica-t i o n i n t h e p u l p and paper i n d u s t r y . C o n s i d e r a b l e work has been done i n t h e p a s t on e v a l u a -t i n g e f f e c t s o f v a r i o u s t e a r t e s t v a r i a b l e s . S e v e r a l a u t h o r s have p u b l i s h e d t h e i r f i n d i n g s on i n s t r u m e n t c a l i b r a t i o n , as w e l l as t e a r i n g p r i n c i p l e s and mechanism, which have c o n t r i -b u t e d t o t h e recommended s t a n d a r d s . Y e t , t h e r e s t i l l r e m a i n s one v a r i a b l e , t h e e f f e c t o f 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 on v a l u e o b t a i n e d , which has been t r e a t e d t h e o r e t i c a l l y but has not been s a t i s f a c t o r i l y e x p l a i n e d by e x p e r i m e n t a l e v i -d e nce. Hence, some a u t h o r s c o n t i n u e t o doubt v a l u e o f t h i s t e s t method, c o n t i n u a l l y p o i n t i n g t o t h e number o f p l i e s e f -f e c t . I t has been c o n s t a n t l y r e p o r t e d t h a t E l m e n d o r f t e a r t e s t r e s u l t s i n c r e a s e as t h e 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 i s i n c r e a s e d . T h i s p o s i t i v e r e l a t i o n s h i p ( i n a few c a s e s , n e g a t i v e r e l a t i o n s h i p ) may be so l a r g e as t o d o u b l e t h e t e a r s t r e n g t h v a l u e , when compared w i t h r e s u l t s o b t a i n e d under 1 2 rigorous conditions. Previous investigators have reasoned or given evidence to explain t h i s deviation in terms of several test factors. These include, clamp design, "spread-out" or "fan-out" of the p l i e s , f r i c t i o n a l binding between the torn paper edges, s t i f f n e s s differences, basis weight and the degree of s p l i t t i n g or "skinning". Recent advances i n paper testing have emphasized the stres s - s t r a i n relationship, and in some few cases rheological behavior has been studied. Most work in these f i e l d s has been concentrated on t e n s i l e strength. Sometimes paper tearing resistance has been examined under constant rate of strain or stress. The application of rheological principles to paper has explained some properties which were unknown in the past. In a similar way, examination of the Elmendorf tear test method in rheological terms could provide some basic informa-tion about the test not revealed previously. Two very obvious and basic phenomena, which have been neglected or ignored in paper tear testing, are the time re-quired to fracture the piece and the rate at which this oc-curs. Researchers agree that the t o t a l time required i s r e l a -t i v e l y short compared to other paper strength tests. Further-more, one can easily distinguish a time difference between tearing a single paper ply and ten plies of the same material over the same distance. The longer time required to tear 3 the ten plies demonstrates the slower rate e f f e c t . Combin-ing this simple fact with the understanding that energy for the test instrument originates from position of the tester sector pendulum, suggests that a change in tearing time changes tear rate, and consequently changes the work done in tearing which affects the f i n a l test r e s u l t s . From this i t i s possible to hypothesize that: (1) The time required to tear paper varies with sample con-dit i o n , which can be generalized in terms of tear re-sistance; (2) Work required to tear paper i s time-dependent, and i n -creases with decreasing rate of tear; (3) The increase in tear strength value as number of plies torn simultaneously i s increased results from the longer time and slower rate of tear, a condition which requires more energy; and ( 4 ) Results obtained from b a l l i s t i c tear testers, such as the Elmendorf instrument, are time-dependent. Thereby, the present study was designed to investigate time-dependent behavior of the paper tear test as i t occurs with one standard instrument. LITERATURE REVIEW Numerous s t u d i e s have been done on t h e e f f e c t o f v a r i a -b l e s on paper t e a r i n g s t r e n g t h . These v a r i a b l e s a r e known t o o r i g i n a t e from t h e m o r p h o l o g i c a l , p h y s i c a l and c h e m i c a l c h a r a c t e r i s t i c s o f wood, changes o c c u r r i n g d u r i n g p u l p i n g and p u l p p u r i f i c a t i o n p r o c e s s e s , as w e l l as t h o s e i n t r o d u c e d d u r i n g papermaking p r o c e s s e s . As an example, i t has been r e p o r t e d t h a t f i b e r l e n g t h , c e l l w a l l t h i c k n e s s or o t h e r f i b e r d e n s i t y t e r m s , t h e r a t i o o f v a r i o u s f i b e r m o r p h o l o g i c a l c h a r a c t e r s , i n d i v i d u a l f i b e r s t r e n g t h , f r a g m e n t a t i o n or weakening o f t h e f i b e r due t o p u l p i n g and b e a t i n g o r r e f i n i n g p r o c e s s e s , m i c r o f i b r i l o r i e n t a t i o n and e x p o s u r e , r e s u l t a n t d e g r e e o f c e l l u l o s e p o l y m e r i z a t i o n , d i s t r i b u t i o n o f t h e d e g r e e o f p o l y m e r i z a t i o n and amount as w e l l as k i n d s o f h y d r o p h y l i c h e m i c e l l u l o s e s and r e s i d u a l l i g n i n a r e i m p o r t a n t f a c t o r s . The l i t e r a t u r e o f t h i s f i e l d has been d i s c u s s e d and r e c e n t l y r e v i e w e d by Casey ( 6 ) , D i n woodie (10) and Rydholm ( 2 1 ) . A l l t h e s e f a c t o r s a r e o n l y i n d i r e c t l y r e l a t e d t o t h e p r e s e n t s t u d y and a r e not f u r t h e r d i s c u s s e d . S i n c e t h e f i r s t r e p o r t on d e t e r m i n i n g paper t e a r i n g s t r e n g t h w i t h t h e E l m e n d o r f t e s t e r was p u b l i s h e d i n 1920 ( 1 1 ) , t h e method has been much d i s c u s s e d . C a r s o n and Snyder (5) d e s c r i b e d t h e o r e t i c a l a s p e c t s o f t h e d e s i g n and c a l i b r a t i o n o f b a l l i s t i c - t y p e t e a r t e s t e r s . C l a r k (7) used Thwing and 4 5 M a r x - E l m e n d o r f t e a r t e s t e r s as examples t o d e s c r i b e a s i m p l e method f o r c a l i b r a t i n g a dynamic t e a r t e s t e r from which he d e t e r m i n e d a c o r r e c t i o n f a c t o r t o be a p p l i e d t o each o b s e r v e d v a l u e . B e r g e a (3) p r o p o s e d a n o t h e r c a l i b r a t i o n method, which was b ased upon a d j u s t m e n t of t h e t e a r i n g d i s t a n c e o f sample s h e e t s , so t h a t t h e i n s t r u m e n t y i e l d s a p p r o x i m a t e l y c o n s t a n t v a l u e s o v e r t h e o r d i n a r y w o r k i n g r a n g e . Cohen and Watson (8) r e v i e w e d d e f i n i t i o n s and d e s c r i b e d t h e 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 w i t h paper as a group o f f o r c e s a c t i n g on t h e t e a r i n g zone. The r e s u l t a n t o f t h e s e f o r c e s was p r o p o s e d as a c t i n g e q u a l l y i n o p p o s i t e d i r e c t i o n s . . By assuming t h a t each r e s u l t a n t f o r c e was e q u a l t o t h e 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 , t h e w i d e l y a c c e p t e d d e f i n i t i o n f o r 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 a r o s e as t h e r e s i s t a n c e opposed t o t h e f o r c e e x e r t e d on e i t h e r o f t h e two p o r t i o n s o f t h e paper a r e a a d j a c e n t t o t h e t e a r zone or l i n e . T h i s d e f i n i t i o n i s e x p r e s s e d a s : where: R = 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 , and W = work done i n t e a r i n g a c r o s s a p i e c e o f paper h a v i n g a c e r t a i n l e n g t h , d. The d e f i n i t i o n has been a d o p t e d i n c a l i b r a t i o n o f Thwing-E l m e n d o r f and P o l l e r - E l m e n d o r f t e a r t e s t e r s . In a d d i t i o n t o t h i s d e f i n i t i o n , M a l l e t t and Marx (19) d e f i n e d 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 i n terms o f work done i n 6 tearing the paper divided by the tearing distance as: Clark (7) has shown, however, that the Marx-Elmendorf tear tester was graduated according to Equation ClH» C o t t r a l l (9) studied the mechanism of tearing and i t s relationship with other paper strength parameters, concluding that the tear test i t s e l f i s of l i t t l e or no value and might even be harmful, i f included as part of paper specif i c a t i o n s . His explanation i s that burst, te n s i l e strength and folding endurance might easily be s a c r i f i c e d by putting too much em-phasis on obtaining high tearing strength. In fact, some paper grades receive more emphasis on burst and t e n s i l e strength than tear resistance. For other papers, tear r e s i s -tance i s preferred over burst and tens i l e strength. Otherwise, these same three strength properties are considered equally important for a large range of paper grades. In order to account for both tearing and burst strengths, Fanselow and Fanselow (12) suggested using the product of both values as an index for evaluating f i b r e and pa r t i c u l a r l y qualities developed during r e f i n i n g . They indicated advan-tages of characterizing pulp strength potential, removing com-plications involved in comparing pulps with divergent tenden-cies in developing burst or te n s i l e strength versus retaining resistance to tear, evaluating performance characteristics of different refiners , evaluating pulp strength and other 7 p u r p o s e s . By s i m i l a r c a l c u l a t i o n s , v a r i o u s " b e a t e r v a l u e s " have a p p e a r e d i n t h e t r a d e . V a r i a t i o n . : Caused by Conduct,' o f t h e T e s t O p e r a t i o n o f t h e t e a r i n g t e s t has v e r y i m p o r t a n t e f f e c t s on r e s u l t s . The specimen i n c l u d e s s e v e r a l , but a v a r i a b l e number o f p l i e s t o r n t o g e t h e r . V a r y i n g t h e number o f p l i e s has been r e p o r t e d c o n s t a n t l y as a f f e c t i n g t e a r t e s t v a l u e s (7, 8, 14, 16, 17, 24, 28 and 2 9 ) . The r e l a t i o n s h i p can be e i t h e r p o s i t i v e o r n e g a t i v e , s e e m i n g l y dependent upon p r o p e r -t i e s o f a p a r t i c u l a r p a p e r . S t a n d a r d methods s p e c i f y t h e r a n g e o f a c c e p t a b l e t e s t e r s c a l e v a l u e s w i t h number o f p l i e s a d j u s t e d so as t o r e m a i n w i t h i n t h e w o r k i n g r a n g e . W i n t e r b o t t o m and M i n o r (29) showed t h a t t h e 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 i n t h e E l m e n d o r f t e a r t e s t e r has a p r o f o u n d e f f e c t on t h e f i n a l t e s t v a l u e . They recommended t h a t a d j u s t i n g number o f p l i e s t o p r o v i d e a s c a l e r e a d i n g between twenty t o f o r t y grams a l l o w s t o o l a r g e a w o r k i n g r a n g e . In o r d e r t o overcome t h i s d i s a d v a n t a g e , t h e y s u g g e s t e d u s i n g a s i n g l e s h e e t as l o n g as r e s u l t s were not t o o low f o r a c c u r -acy, w h i l e i n no c a s e s h o u l d s h e e t s be d o u b l e d t o g i v e s c a l e r e a d i n g s o v e r 30. S w a r t o u t and S e t t e r h o l m (24) e x p l a i n e d c a u s e o f t e s t v a l u e v a r i a t i o n s due t o i n c r e a s i n g t h e 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 i n terms o f " s p r e a d - o u t " o f t h e s h e e t at t h e t o p o f t h e s p e c i m e n , b u l k i n g i n t h e clamp and d e v i a t i o n from a a s t r a i g h t l i n e p r o j e c t e d from t h e i n i t i a l s l i t . They f o u n d t h a t v a r i a t i o n due t o i n c r e a s i n g t h e number of p l i e s was not c o n s t a n t i n t h a t t h i c k e r p a p e r s had h i g h e r t e a r v a l u e s t h a n t h i n n e r ones, even at s i m i l a r d e n s i t y . The t e n d e n c y f o r s p r e a d i n g at t h e s p e c imen t o p when s e c u r i n g i n t h e clamps, and t h e t e n d e n c y f o r t h e f a i l u r e l i n e t o d e v i a t e from a s t r a i g h t l i n e p r o j e c t e d from t h e i n i t i a l s l i t were a s s o c i a t e d w i t h i n -c r e a s i n g number o f p l i e s . I n c r e a s i n g b u l k by p l a c i n g s e p a r a t o r s between i n d i v i d u a l s h e e t s i n c r e a s e d t h e t e a r v a l u e , but t h e i n c r e a s e i n b u l k was accompanied by i n c r e a s e d d e g r e e o f s p r e a d i n g . By c o m p a r i n g d i f f e r e n t p a p e r g r a d e s w i t h 2 - m i l and 3 - m i l aluminum f o i l s p e c i m e n s , t h e y c o n c l u d e d t h a t r e l a t i o n s h i p between t e a r v a l u e and number o f p l i e s t o r n i s not an i n h e r e n t f e a t u r e o f t h e t e a r t e s t e r , but r e l a t e s t o t h e t y p e o f m a t e r i a l b e i n g t e s t e d . Wink and Van E p e r e n (28) examined t h e e f f e c t o f number o f p l i e s , as w e l l as d i f f e r e n t c l a m p i n g methods by c h a n g i n g t h e clamp d e s i g n . They f o u n d t h a t t h e method o f c l a m p i n g c o u l d i n t r o d u c e 10% v a r i a t i o n i n t e s t r e s u l t s , w h i l e v a r y i n g t h e number o f p l i e s c o u l d c a u s e 100% v a r i a t i o n . The d e g r e e o f v a r i a t i o n depended upon t h e t y p e o f m a t e r i a l . These v a r i a -t i o n s were d e s c r i b e d as due t o change i n t h e n a t u r e o f t e a r i n g s uch as s p l i t t i n g and " s p r e a d i n g - o u t " o f s h e e t s . Change i n t h e n a t u r e o f t e a r i n g i s f u r t h e r c o m p l i c a t e d by t h e d e g r e e o f 9 i n t e r f i b r e b o n d i n g . A c c o r d i n g l y , t h e y compared t h e n a t u r e o f t e a r i n g by t h e E l m e n d o r f t e a r t e s t e r w i t h t e a r f a i l u r e s i n t h e u s u a l paper a p p l i c a t i o n s and q u e s t i o n e d t h e v a l u e o f t h e E l m e n d o r f t e a r t e s t method. Jones and G a l l a y (17) r e p o r t e d t h e p o s i t i v e r e l a t i o n s h i p between t e a r f a c t o r ( t h e r a t i o between t e a r i n g s t r e n g t h and b a s i s w e i g h t ) and t h e 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 as m a i n l y c a u s e d by d e g r e e o f paper s p l i t t i n g . R a te o f i n c r e a s e was a f f e c t e d by b a s i s w e i g h t , t y p e o f p u l p and d e g r e e o f b e a t i n g . They c o n c l u d e d t h a t i n c r e a s e i n s t i f f n e s s w i t h i n -c r e a s i n g b a s i s w e i ght was not an e x p l a n a t i o n f o r t h e d i f f e r -ent b e h a v i o r o f t h e i r p a p e r s . I n s t e a d , t h e y m entioned t h a t h i g h e r b a s i s w e i g h t s h e e t s have a h i g h e r t e n d e n c y t o s p l i t t h a n t h e l o w e r b a s i s w e i ght s h e e t s . R e l a t i o n s h i p between b a s i s w e i g h t and s p l i t t i n g was a l s o c o n f i r m e d by r e s u l t s o f H a r d a c k e r and Van den Akker ( 1 4 ) . In c o n t r a s t t o t h e Jones and G a l l a y i d e a ( 1 7 ) , Wahlberg (27) emphasized a l i n e a r r e l a t i o n s h i p between t e a r f a c t o r and s t i f f n e s s e x p r e s s e d as f l e x u r a l r i g i d i t y . He s u g g e s t e d t h a t t h i s o c c u r s b ecause s t i f f n e s s i s t h e most i m p o r t a n t phenome-non changed by i n c r e a s i n g b a s i s w e i g h t . F o r t h e p u r p o s e o f e l i m i n a t i n g t h e e f f e c t o f 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 , s t a n d a r d p r o c e d u r e s f o r Thwing-E l m e n d o r f and M a r x - E l m e n d o r f t e a r t e s t e r s s p e c i f y t h a t s e v e r -a l p l i e s a r e t o be t o r n t o g e t h e r i n o r d e r t o g i v e t h e i n s t r u -ment s c a l e r e a d i n g w i t h i n c e r t a i n d e f i n e d l i m i t s . Cohen and 10 Watson (8) presumed t h a t t h e s e l i m i t s a r e i n t r o d u c e d t o e n s u r e a f a i r l y u n i f o r m r a t e o f t e a r . They f u r t h e r c o n -s i d e r e d t h a t t h e s e l i m i t s and t h e t e a r r a t e have a more s i g -n i f i c a n t i n f l u e n c e on t e a r v a l u e than t h a t c a u s e d by number o f p l i e s . The i m p o r t a n c e o f t e a r r a t e was a l s o c o n s i d e r e d by members o f t h e I n s t i t u t e o f Paper C h e m i s t r y ( 1 6 ) . Sample w i d t h has been r e p o r t e d t o a f f e c t t e a r v a l u e s . Cohen and Watson (8) f o u n d no s i g n i f i c a n t e f f e c t on t e a r v a l u e s when w i d t h v a r i a t i o n i s w i t h i n 50 ± 12 mm, but g r e a t -er w i d t h s , s u c h as 100 mm, p r o d u c e d s i g n i f i c a n t l y h i g h e r v a l -ues. T h i s was r e l a t e d t o h i g h e r b e n d i n g r e s i s t a n c e o f f e r e d by w i d e r s p e c i m e n s . The e f f e c t o f non-symmetry o f t h e t e a r f r a c t u r e o r l i n e i n s p e c i m e n s has been s t u d i e d by Cohen and Watson ( 8 ) . They c o n c l u d e d t h a t t h i s c o u l d c a u s e d i f f e r e n t d i s t r i b u t i o n o f s t r e s s e s and c o n s e q u e n t l y i n c r e a s e f r i c t i o n between t o r n edges, g i v i n g h i g h e r t e a r v a l u e s . P r i n c i p l e and Methods f o r C a l i b r a t i n g B a l l i s t i c - t y p e T e a r  T e s t e r s A b a l l i s t i c - t y p e t e a r t e s t e r such as t h e E l m e n d o r f i n -s t r u m e n t used i n t h i s s t u d y c o n s i s t s o f a s e c t o r pendulum ( h e r e a f t e r r e f e r r e d t o as s e c t o r ) suspended from a s t a t i o n a r y p o s t by means o f a b a l l b e a r i n g . A jaw f o r h o l d i n g one p a r t o f t h e specimen i s a t t a c h e d t o t h e s t a t i o n a r y p o s t , w h i l e a n o t h e r jaw h a v i n g t h e same f u n c t i o n i s a t t a c h e d t o t h e r i g h t I I r a d i a l edge o f t h e s e c t o r . Moving t h e s e c t o r from i t s e q u i l i b r i u m f r e e swing p o s i -t i o n s i m u l t a n e o u s l y r a i s e s t h e s e c t o r c e n t e r o f mass. Con-t h e s e c t o r mass and o f t h e v e r t i c a l d i s t a n c e t h r o u g h which i t s c e n t e r o f mass i s r a i s e d . I f t h e s e c t o r i s a l l o w e d t o c o m p l e t e one h a l f - s w i n g from i t s r a i s e d p o s i t i o n , t h e o r e t i -c a l l y , t h e a n g l e (9^) ( a l l a n g l e s employed t o i l l u s t r a t e t h e t e a r i n g p r i n c i p l e a r e p r e s e n t e d g r a p h i c a l l y i n F i g . 1) s h o u l d be o f t h e same magnitude as b e f o r e r e l e a s e . However, v e r y s m a l l amounts of energy a r e a b s o r b e d by f r i c t i o n a t t h e b a l l b e a r i n g , a n d a s s c a l e p o i n t e r f r i c t i o n and a i r r e s i s t a n c e . T h e s e r e s u l t : i n a s l i g h t l y s m a l l e r a n g l e ( Q^ which e x p r e s -s e s t h e net amount o f p o t e n t i a l e n e rgy a v a i l a b l e t o do work on t e a r i n g t h e paper s p e c i m e n . D e t e r m i n i n g t h e swing a n g l e , or net e n e rgy a v a i l a b l e f o r t e a r i n g t h e s p e c i m e n e n a b l e s l o c a t i n g t h e z e r o p o i n t on t h e i n s t r u m e n t s c a l e . D u r i n g t h e t e a r i n g o p e r a t i o n , p a r t o f t h e net e n e rgy i s a b s o r b e d i n f r a c t u r i n g t h e specimen and a s t i l l s m a l l e r a n g l e ( © 3 ) i s o b t a i n e d . T h i s a n g l e ( © 3 ) e x p r e s s e s r e s i d u a l e n e r g y i n t h e s y s t e m a f t e r t h e specimen has been f a i l e d . The d i f f e r e n c e between t h e net energy and r e s i d u a l e n e r gy i s t h a t energy p o r t i o n r e p r e s e n t e d as work done t o overcome t h e t e a r r e s i s t a n c e o f a s p e c i m e n . Based on E q u a t i o n [ l ] , t h e a v e r a g e f o r c e r e q u i r e d t o t e a r a c e r t a i n d i s t a n c e s e q u e n t l y , t h e p o t e n t i a l e n e r g y changes as a f u n c t i o n o f 12 t h r o u g h a specimen can be c a l c u l a t e d . In p r a c t i c e , a c o s i n e s c a l e i s f i x e d on t h e s e c t o r t o r e p r e s e n t t h e amount o f e n e r -gy t h a t has been used t o t e a r a s p e c i m e n . T h i s s c a l e e n a b l e s t h e t e a r i n g f o r c e t o be r e a d d i r e c t l y i n grams from t h e t e s t e r . Most t e a r t e s t e r s a r e c a l i b r a t e d by u s i n g 68.8 cm as t e a r d i s t a n c e . The d i s t a n c e o v e r which t h e f o r c e a c t s i s by d e f i n i t i o n , 2 X 68.8 cm = 137.6 cm. A p p a r e n t l y , i t i s not p r a c t i c a l t o t e a r a specimen 68.8 cm l o n g . By assuming t h a t t h e work r e q u i r e d i n t e a r i n g i s d i r e c t l y p r o p o r t i o n a l t o t e a r -i n g d i s t a n c e , and t h a t t h e f o r c e r e q u i r e d i s l i n e a r l y r e l a t e d t o t h e 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 , t h e p r a c t i c e i s t o t e a r a b o o k l e t o f s i x t e e n s h e e t s a c r o s s 4.3 cm. R e s u l t s r e a d from t h e s c a l e when t e a r i n g a d i f f e r e n t number o f p l i e s a r e a d j u s t e d t o t h e s i x t e e n - s h e e t b a s i s . The e q u a t i o n used f o r c a l c u l a t i n g t e a r i n g s t r e n g t h (26) under t h e s e c o n d i t i o n s i s : . i • « 16 X a v e r a g e s c a l e r e a d i n g r Average t e a r x n g f o r c e = r a—T. T~- l _ 3 J 3 3 number o f p l i e s u -1 where t h e a v e r a g e t e a r i n g f o r c e i s e x p r e s s e d i n grams per s h e e t . T e a r T e s t T h e o r y V e r y l i t t l e a t t e n t i o n has been g i v e n t o development o f t e a r t e s t t h e o r y . B r e c h t and Imset (4) i n 1934 c o n s i d e r e d t h e t e a r i n g zone as e x t e n s i v e , i n s t e a d o f as a p o i n t . The e l e m e n t a l f o r c e s i n v o l v e d i n t h e t e a r i n g s t r e s s were c o n -s i d e r e d as p r o d u c i n g a moment o f f o r c e r e l a t e d t o a r e f e r e n c e 13 p o i n t . The t e a r i n g f o r c e i s t h e n c a l c u l a t e d by d i v i d i n g t h i s moment by t h e p e r p e n d i c u l a r d i s t a n c e between t h e r e f e r -ence p o i n t t o t h e l i n e o f a c t i o n o f t h e t e a r i n g f o r c e . T h i s t h e o r y emphasized i n f l u e n c e o f s t r e s s c o n c e n t r a t i o n i n t h e t e a r i n g zone and how f i b r e l e n g t h and s h e e t e x t e n s i b i l i t y a f f e c t s i z e o f t h e t e a r i n g zone and t e a r i n g s t r e n g t h . Members o f t h e I n s t i t u t e o f Paper C h e m i s t r y (16) have p o i n t e d out weaknesses o f t h e B r e c h t and Imset t h e o r y i n t h a t i t a t t e m p t s t o d e a l w i t h t h e f o r c e s a r i s i n g between t h e f i b r e s d u r i n g t e a r f a i l u r e and assumes a u n i f o r m d i s t r i b u t i o n o f s t e a d y f o r c e s o v e r t h e t e a r i n g zone. A n o t h e r t h e o r y was a d o p t e d by members o f t h e I n s t i t u t e o f Paper C h e m i s t r y (16) i n 1944. T h i s i s based on t h e mechan-ism o f t e a r f a i l u r e and e n e r g y d i s s i p a t i o n , i . e . , e n e rgy ex-pended i n t e a r i n g a s h e e t o f paper i s d i s s i p a t e d w i t h i n t h e s h e e t . In t e a r i n g a s p e c i m e n , t h e energy expended i s t r a n s -formed m a i n l y i n t o two p a r t s : (1) f r a c t i o n a l d r a g work, 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 th e f i b r e network, and (2) r u p t u r e work c a u s e d by s t r e s s i n g i n d i v i d u a l f i b r e s u n t i l t h e y break i n t e n s i l e f a i l u r e . B ased on a n a l y s i s o f s t r e s s - s t r a i n diagrams o f f i b r e s s t r e t c h e d t o f a i l u r e , and f o r c e v e r s u s d i s p l a c e m e n t c u r v e s f o r p u l l i n g f i b r e s from t h e network, t h e y c o n c l u d e d t h a t t h e work r e q u i r e d t o r u p t u r e a f i b r e i s v e r y much l e s s t h a n t h a t r e q u i r e d t o 14 e x t r a c t i t unbroken from t h e f i b r e mesh, a l t h o u g h t h e f o r c e r e q u i r e d t o r u p t u r e a f i b r e i s g r e a t e r t han t h a t needed t o d r a g out an i n t a c t f i b r e . T h i s t h e o r y has been used t o e x p l a i n some phenomena a s s o c i a t e d w i t h t e a r i n g s t r e n g t h . Among p u l p p r e p a r a t i o n and papermaking p r o c e s s e s , b e a t i n g may be c o n s i d e r e d an im-p o r t a n t f a c t o r a f f e c t i n g paper s t r e n g t h . In g e n e r a l , t h e b u r s t , t e n s i l e s t r e n g t h and f o l d i n g e n d u r a n c e i n c r e a s e w i t h b e a t i n g w i t h i n t h e c o m m e r c i a l r e f i n i n g r a n g e . T e a r i n g s t r e n g t h behaves d i f f e r e n t l y . A f t e r s l i g h t i n c r e a s e i n t h e e a r l i e s t h e a t i n g s t a g e s , c o n i f e r o u s p u l p t e a r i n g s t r e n g t h d e c r e a s e s p r o g r e s s i v e l y w i t h f u r t h e r b e a t i n g . Hardwood p u l p s r e q u i r e l o n g e r p e r i o d s o f b e a t i n g t o r e a c h maximum t e a r i n g s t r e n g t h b e f o r e f u r t h e r b e a t i n g s t a r t s t o d e c r e a s e v a l u e s . I n i t i a l r i s e i n t h e t e a r i n g s t r e n g t h - b e a t i n g t i m e c u r v e i s e x p l a i n e d by t h e t h e o r y t h a t t h e r e i s a r a p i d i n -c r e a s e i n t h e f r a c t i o n a l d r a g work per component i n t h e v e r y e a r l y s t a g e s o f b e a t i n g . However, as b e a t i n g p r o c e e d s t h e number o f r u p t u r e d f i b r e s i n c r e a s e s and fewer o f them a r e p u l l e d i n t a c t from t h e t i g h t e r mesh. T h i s change i n t h e f a i l -u r e mechanism i s c a u s e d by i n c r e a s i n g t h e amount o f i n t e r -f i b r e b o n d i n g and l o w e r i n g o f i n t r i n s i c f i b r e s t r e n g t h as a r e s u l t o f t h e b e a t i n g t r e a t m e n t . S i n c e t h e f r a c t i o n a l d r a g work per i n d i v i d u a l f i b r e i s g r e a t e r t h a n t h e r u p t u r i n g work, t h e e n ergy r e q u i r e d t o t e a r t h e s h e e t i s r e d u c e d . T h i s same e x p l a n a t i o n has been used t o a c c o u n t f o r t h e e f f e c t o f f i b r e 15 l e n g t h and s t r e n g t h on t e a r i n g s t r e n g t h , s i n c e t h e f r a c t i o n a l d r a g work i n c r e a s e s w i t h i n c r e a s i n g f i b r e l e n g t h and t h e s t r o n g e r t h e f i b r e t h e h i g h e r t h e r e s i s t a n c e t o r u p t u r i n g f o r c e s . T h i s , i n t u r n , p r o v i d e s more members t o be p u l l e d i n t a c t from t h e mesh r a t h e r t h a n r u p t u r e d by t e a r i n g f o r c e s . G i e r t z and H e l l e (13) have r e c e n t l y r e v i e w e d t h e t h e o r y a d o p t e d by members o f t h e I n s t i t u t e o f Paper C h e m i s t r y and have t e s t e d i t w i t h a s e r i e s o f l a b o r a t o r y e x p e r i m e n t s . They were i n g e n e r a l agreement w i t h t h e t h e o r y , but recommended s l i g h t m o d i f i c a t i o n by a d d i n g a n o t h e r term t o i n c l u d e t h e e f -f e c t o f s t r a i n f o r some d i s t a n c e on b o t h s i d e s o f t h e f a i l u r e . They a l s o p o i n t e d out t h e i m p o r t a n c e o f f i b r e s t r e n g t h and f i b r e l e n g t h , and a g r e e w i t h B r e c h t and Imset (4) c o n c e r n i n g t h e i m p o r t a n c e o f s h e e t e x t e n s i b i l i t y i n d e t e r m i n i n g t e a r i n g s t r e n g t h . Paper R h e o l o g i c a l P r o p e r t i e s R e cent s t u d i e s on paper s t r e n g t h p r o p e r t i e s based on t h e s t r e s s - s t r a i n - t i m e r e l a t i o n s h i p have p r o v i d e d new u n d e r s t a n d -i n g o f t h e n a t u r e o f paper s t r e n g t h , as w e l l as i n f o r m a t i o n about t e s t methods. Paper has been shown t o e x h i b i t b o t h e l a s t i c and t i m e - d e p e n d e n t f l o w p r o p e r t i e s . I t d i s p l a y s a n o n - l i n e a r l o a d - e l o n g a t i o n ( s t r e s s - s t r a i n ) c u r v e . T h i s c u r v e i s t i m e or r a t e - d e p e n d e n t , i . e . , a t h i g h r a t e o f l o a d i n g , b r e a k i n g e l o n g a t i o n d e c r e a s e s and b r e a k i n g l o a d i n c r e a s e s which r e s u l t s i n a s m a l l e r r u p t u r i n g work. The t i m e - d e p e n d e n t 16 b e h a v i o r o f p a p e r has been i l l u s t r a t e d by Ranee ( 2 0 ) , u s i n g a S c h o p p e r t e n s i l e t e s t e r and by h a n g i n g v a r i o u s w e i g h t s onto paper s p e c i m e n s . F r e q u e n t l y j i n v e s t i g a t o r s have examined t h e s t r e s s - s t r a i n r e l a t i o n s h i p i n s t e a d o f u n i d i m e n s i o n a l paper s t r e n g t h p r o p e r -t i e s . S p e c i a l a t t e n t i o n has been g i v e n t o t e n s i l e s t r e n g t h (1, 20, 2 2 ) , but l i t t l e has been done r e g a r d i n g t e a r i n g s t r e n g t h . Members o f t h e I n s t i t u t e o f Paper C h e m i s t r y have examined t h e r e l a t i o n s h i p i n f u r t h e r s u p p o r t o f t h e i r t h e o r y m e n t i o n e d above. The f a i l u r e mechanism w i t h b a l l i s t i c - t y p e t e a r t e s t e r s has been i n d i c a t e d by members o f t h e I n s t i t u t e o f Paper Chemis-t r y ( 1 6 ) , Cohen and Watson (8) and B a l o d i s (2) as an energy d i s s i p a t i o n phenomenon, but no d e t a i l e d d i s c u s s i o n has been g i v e n . The e f f e c t o f t e a r r a t e on t e a r t e s t v a l u e s has been m e n t i o n e d as w e l l , but u n f o r t u n a t e l y no e v i d e n c e has been ad -v a n c e d s u p p o r t i n g t h i s i d e a . In c o n t r a s t t o t h e assumed t e a r r a t e e f f e c t , H i g g i n s (15) i n d i c a t e d t h a t r a t e o f l o a d i n g was not t h e b a s i c f a c t o r which c o n t r i b u t e s s p e c i a l i t y t o t h e c o n v e n t i o n a l t e a r t e s t . H i g g i n s (15) a l s o c l a i m e d t h a t t h e l o a d - t e a r d i s t a n c e r e l a t i o n s h i p was a phenomenon c a u s e d by need t o i n c r e a s e l o a d t o a maximum i n o r d e r t o s t a r t t e a r i n g , f o l l o w e d by need f o r g r a d u a l l y de-c r e a s i n g l o a d as t e a r i n g p r o g r e s s e s . The same phenomenon was r e p o r t e d by Anderson and F a l k ( 1 ) , who s u g g e s t e d t h a t t h e d d c r e a s e i n l o a d r e q u i r e m e n t r e s u l t e d from t h e r e l e a s e e l a s t i c e n e r gy d u r i n g t h e i n c r e a s e i n t e a r d i s t a n c e . ENERGY DISSIPATION AND RATE OF TEAR IN THE TEAR TEST Recent advances i n m a t e r i a l s s c i e n c e r e f o c u s e s a t -t e n t i o n on p a p e r from c o n c e r n w i t h r u p t u r e s t r e n g t h t o t h e s t u d y o f p r e - r u p t u r e phenomena. T h i s has i n v o l v e d s t u d i e s on t h e i n f l u e n c e o f t i m e f a c t o r s on t h e s t r e s s - s t r a i n h i s -t o r y p r i o r t o f i n a l r u p t u r e . F o r example, a t t e n t i o n i s f o c u s e d on w o r k - t o - f a i l u r e i n c r e e p under dead l o a d s and i n t e s t i n g w i t h v a r i o u s s t r a i n r a t e s t o f a i l u r e . Some i n v e s t i -g a t i o n s a l s o r e v e a l t h a t r u p t u r e can o c c u r i n s t r e s s r e l a x a -t i o n t e s t s c o n d u c t e d a t c o n s t a n t s t r a i n . Use o f v i s c o e l a s t i c models, as have been a p p l i e d t o many p o l y m e r i c m a t e r i a l s , de-s e r v e s i n c r e a s e d a t t e n t i o n o f p a p e r t e c h n o l o g i s t s . Network models 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 have been used t o v i s u a l i z e mathematical a n a l y s e s i n v o l v e d i n s t r e s s - s t r a i n -t i m e r e l a t i o n s h i p s . B a s i c a l l y i t i s n e c e s s a r y t o d i f f e r e n -t i a t e between: (1) E l a s t i c d e f o r m a t i o n (immediate r e s p o n s e i n phase w i t h an a p p l i e d l o a d ) , (2) V i s c o e l a s t i c d e f o r m a t i o n ( t i m e dependent, but r e c o v e r -a b l e when t h e a p p l i e d f o r c e i s removed), and (3) Flow d e f o r m a t i o n ( n o t r e c o v e r a b l e upon r e m o v a l o f t h e e x t e r i o r f o r c e ) . (1) and (3) seem t o be predominant where t h e m e c h a n i c a l be-h a v i o r o f paper i s c o n c e r n e d ( F i g . 2 ) . 18 19 Fundamental s t u d i e s o f paper s t r e n g t h based upon such s t r e s s - s t r a i n - t i m e r e l a t i o n s h i p s have not o n l y p r o v i d e d knowledge on paper s t r e n g t h p r o p e r t i e s , but a l s o have h e l p e d t o examine adequacy o f p r e s e n t t e s t i n g methods and s t a n d a r d t e s t r e q u i r e m e n t s . The l a t t e r i s i m p o r t a n t b e c a u s e some o f t h e l i m i t i n g v a r i a b l e s i n p r e v a l e n t s t a n d a r d s have been a r -b i t r a r i l y a s s i g n e d w i t h o u t sound f o u n d a t i o n . The b a l l i s t i c - t y p e t e a r t e s t e r has been w i d e l y used i n t h e p u l p and paper i n d u s t r y and r e l a t e d r e s e a r c h a c t i v i t i e s as a means f o r e v a l u a t i n g s h e e t t e a r r e s i s t a n c e . 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 ( 2 6 ) , one a r b i t r a r y r e q u i r e m e n t i s making up a specimen w i t h a c e r t a i n number o f p l i e s w h i c h , when t o r n t o g e t h e r , w i l l g i v e an i n s t r u m e n t s c a l e r e a d i n g n e a r 40. No l i t e r a t u r e d i s c u s s i n g r e a s o n s f o r t h i s l i m i t a t i o n has been f o u n d . Cohen and Watson ( 8 ) , and members o f t h e I n s t i t u t e o f Paper C h e m i s t r y ( 1 6 ) , s i m p l y m e ntion t h e e f f e c t o f t e a r r a t e on t e a r t e s t v a l u e s . T h i s c o n c e p t can be e a s i l y r e a s o n e d by e s t i m a t i n g t h e t o t a l t i m e r e q u i r e d t o t e a r two s p e c i m e n s from t h e same s o u r c e , but w i t h w i d e l y d i f f e r e n t number o f p l i e s . A p p a r e n t l y , t h e one w i t h h i g h e r number o f p l i e s r e q u i r e s t h e l o n g e r t i m e t o t e a r t h r o u g h t h e whole d i s -t a n c e . In a d d i t i o n , i t i s r e p o r t e d t h a t t h e t e a r t e s t v a l u e i n c r e a s e s w i t h t h e 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 . Ob-s e r v i n g t h e s e two r e s t r i c t i o n s d i s a l l o w s m e a n i n g f u l c o m p a r i -son between paper gr a d e s o f d i f f e r e n t c h a r a c t e r i s t i c s . 20 Tensile testing has shown that paper exhibits both e l a s t i c and time-dependent flow properties. Non-linear stress - s t r a i n behavior from tests conducted with constant rate of strain can indicate: (1) Non-linear e l a s t i c behavior, (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 behavior. The stress-strain curve can be divided into e l a s t i c and post-y i e l d regions. In the e l a s t i c region paper performs accord-ing to Hooke's law. After reaching the e l a s t i c l i m i t , the curve begins to represent a p l a s t i c region by deviation from the almost straight l i n e established within the e l a s t i c re-gion. The material begins to exhibit flow. This flow i s time or rate-dependent. At high rate of loading, breaking elonga-tion decreases and the breaking load increases. Total energy required to produce f a i l u r e i s also affected by rate of load-ing. The breaking or f a i l u r e energy difference between f r a c -t i o n a l drag work and f i b r e rupture has been adopted in support of theory explaining some paper tear phenomena. Breaking energy as determined from the stress-strain curve i s much af-fected by the time-dependent p l a s t i c flow region. The energy stored in the tear tester i s obtained by raising the sector center of mass. The sector i s allowed to swing at the time of tearing. The potential energy (P.E.) 21 of the sector i s transformed to kinetic energy (K.E.). This kinetic energy i s capable of doing work, overcoming f r i c t i o n at the sector bearing, pointer bearing and a i r resistance, as well as tear through a specimen i f the residual force from the sector i s larger than the specimen resistance. Otherwise, the specimen w i l l not be torn through and the sec-tor w i l l be stopped. When tearing a specimen within capacity of the tester (the Dynamic Tear Tester designed in Australia has three different, interchangeable weight sectors to pro-vide different capacities) part of the energy i s expended to tear the paper and to overcome f r i c t i o n s , and part of the energy w i l l be l e f t as residual energy. Since net and r e s i -dual energies are p a r t i a l l y kinetic energy, they are highly af-fected by the tearing velocity and consequently are time-de-pendent . If energy dissipation phenomenon i s expressed in mathe-matical form, this can be used to evaluate: (1) the amount of energy dissipated in tearing, (2) the amount of residual energy, and (3) the increase in energy required as tearing distance i s increased. If the energy dissipation concept i s correct, then the energy expended to tear a specimen can be calculated from the difference between net energy and residual energy in the form of kinetic energy. Results calculated from this relationship 22 would t h e n be u s e f u l i n e x a m i n i n g t h e e n e r g y d i s s i p a t i o n c o n -c e p t , as r e g a r d s t e a r r a t e e f f e c t s and o t h e r f a c t o r s o f paper t e a r i n g . The p r i n c i p l e o f c o n s e r v a t i o n o f e n e r g y d e s c r i b e s t h a t no e nergy i s l o s t , but t h a t i t i s c o n v e r t e d from one t o ano-t h e r form. T h e r e b y , t h e t o t a l e n e r g y o f a system i s c o n s t a n t and can be e x p r e s s e d i n t h e f o l l o w i n g f o r m : ^ t o t ~~ ^ k i n + ^ p o t + ^ c o n v e r t e d t43 where: ^^ 0^. = " t o t a l energy o f a system, 1 2 = k i n e t i c e n e rgy = , ^ p o t = P o t e n t i a l e n e r g y = mgy, E , , = en e r q y c o n v e r t e d t o forms o t h e r than E. . , c o n v e r t e d 3 J f k i n m = s e c t o r mass, g = g r a v i t a t i o n a l a c c e l e r a t i o n , y - v e r t i c a l d i s p l a c e m e n t o f s e c t o r c e n t e r o f mass, and v = ( t a n g e n t i a l ) v e l o c i t y a t t i m e t . The e n e r g y c o n v e r s i o n i n a t e a r i n g p r o c e s s may be i l l u s -t r a t e d by d i a g r a m ( F i g . 3 ) . T h i s shows t h a t when t h e s e c t o r i s r a i s e d from i t s l o w e s t p o s i t i o n t o t h e t o p p o s i t i o n , po-t e n t i a l energy i s maximum, and t h i s i s c o n v e r t e d i n t o k i n e t i c e n e r g y upon r e l e a s i n g t h e s e c t o r . The k i n e t i c e n e rgy o f t h e s e c t o r i s c o n v e r t e d t o o t h e r energy f o r m s , f r i c t i o n (E„ . . . ) f r a c t i o n and t o do paper t e a r i n g (^-^ e a rj_ ng)« 23. A s h o r t e r t i m e i s r e q u i r e d t o t r a v e l t h e d i s t a n c e between t h e b e g i n n i n g and f i n i s h o f t e a r i n g when no t e a r i n g work i s done, and l o n g e r t i m e i s r e q u i r e d as more energy i s expended t o t e a r a paper s p e c i m e n . When t h e s e c t o r i s a l l o w e d t o swing w i t h o u t t e a r i n g a specimen ( F i g . 4 a ) ( h e r e a f t e r r e f e r r e d t o as z e r o - s w i n g i n c o n t r a s t t o a f r i c t i o n l e s s s y s t e m ) , i t swings w i t h z e r o - s w i n g v e l o c i t y (v^) a t t i m e t ^ at v e r t i c a l g r a v i t a t i o n a l p o s i t i o n ( y ^ ) . The t o t a l e n e r g y components ( E q u a t i o n C43 ) t a k e t h e f o l l o w i n g form at t i m e t ^ : E = - § - m v i + m 9 » i + ^ ( v l ' t l ) E 5 ^ t o t a l where: ^ ( v ] _ > ) = v a r i o u s forms o f f r i c t i o n as a f u n c t i o n o f z e r o - s w i n g v e l o c i t y and t i m e . The f i r s t two terms i n E q u a t i o n T5j r e p r e s e n t n et energy which 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 f o r t e a r i n g t h e specimen a t t i m e t ^ . When t e a r i n g specimens. ( F i g . 4 b ) a p a r t o f t h e k i n e t i c e n e r g y i s d i s s i p a t e d . The s e c t o r , t h e r e f o r e , swings w i t h s l o w e r t a n g e n t i a l v e l o c i t y (v£) at g r a v i t a t i o n a l p o s i t i o n y ^ . Then, i n a t e a r i n g swing t h e t o t a l energy components take t h e form a t t i m e when a t e s t specimen i s t o r n a d i s t a n c e 24 F i-mv? + mgy, + -ftv-.t-) + T.E. C 6 H L t o t a l ~ 2 2 1 J <• where: J ( v 2 , t 2 ) = various forms of f r i c t i o n as a function of tearing velocity and time. Tearing energy (T.E.) i s that part of the net energy which has been dissipated in tearing a specimen. There are two kinds of f r i c t i o n forces in 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 sector and a i r . The s l i d i n g and r o l l i n g f r i c t i o n force i s assumed independent of velocity and hence contributes no difference between zero-swing and tear-ing swing. The viscous f r i c t i o n force i s d i r e c t l y proportion-al to the velocity, when velocity i s not too high (23). In b a l l i s t i c - t y p e tear testers the \MocLty i s considered low, there-fore, viscous f r i c t i o n difference between the zero-swing and tearing swing can be considered as negligible. Hence: ^ ( v ^ t . ^ = y ( v 2 , t 2 ) Since Equations \_ 5J and rj6J are equal according to the law of conservation of energy, they may be written as: .i_mv^ 2 2 + mgy1+ ^ ( v 2 , t 2 ) + T.E. = —|- m v^ + m 9 y 1 + ^ ( v 1 , t 1 ) L j U and transposed as: T.E. = ~|- m v^ " -|- m V2 + m 9 y i " m 9 y i + f ( v l , t l ) " f ( v 2 ' t 2 ) M Equation [8j then becomes: 25 i 7 i 9 i / 2 2» r T.E. = _L_mv2 - _L_mv2 = _L_m(v 2 - v 2 ) [9] 2 1 2 2 2 1 2  1  2 2 The mass (m) o f t h e s e c t o r i s a c o n s t a n t . The o n l y t e r m c o n t r i b u t i n g t o T.E. i s t h e d i f f e r e n c e between s q u a r e d v e l o -2 2 c i t i e s ( v ^ - v^)- The z e r o - s w i n g v e l o c i t y (v-^) f o r s e c t o r p o s i t i o n y^ i s a c o n s t a n t f o r one s p e c i f i c i n s t r u m e n t o p e r a t e d under t h e same c o n d i t i o n s , but t e a r i n g v e l o c i t y (v£) f o r s e c -t o r p o s i t i o n y^, v a r i e s w i t h t h e s p e c i m e n . T h e r e f o r e , T.E. can be c a l c u l a t e d i f bo t h z e r o - s w i n g v e l o c i t y and t e a r i n g v e l o c i t y can be measured. V e l o c i t i e s can be c a l c u l a t e d by f i n d i n g t h e t i m e ( A t ) r e q u i r e d t o t e a r a c e r t a i n d i s t a n c e ( A L ) , i . e . , r a t e o f t e a r . An e q u a t i o n can be f i t t e d t o e x p r e s s t e a r d i s t a n c e as a f u n c t i o n o f t i m e : t e a r d i s t a n c e = - ( t e a r t i m e ) [^1 By d e f i n i t i o n , t h e i n s t a n t a n e o u s v e l o c i t y w i l l be t h e f i r s t d e r i v a t i v e o f E q u a t i o n j^ ioj , w h i l e t h e s e c o n d d e r i v a t i v e i s t h e a c c e l e r a t i o n i n t h e z e r o - s w i n g and t e a r i n g s w i n g . P r e -l i m i n a r y t e s t s showed t h a t t h e r e g r e s s i o n f u n c t i o n can be b e s t e x p r e s s e d i n a s e c o n d d e g r e e or q u a d r a t i c f o r m : 2 L = a + b t + c t where: L = t e a r d i s t a n c e , t = t e a r t i m e , a, b, and c = c o n s t a n t s or r e g r e s s i o n c o e f f i c i e n t s . 26 The f i r s t derivative of Equation i s velocity (v): d L = b + 2ct f l23 dt The second derivative of Equation Qll^ i s acceleration: By using t h i s mathematical model, both rate of tear and tearing energy may be related to time, enabling evaluation of time effects i n the tear test or time-dependent phenomenon as regards paper tearing strength. MATERIALS AND METHODS M a t e r i a l s T e a r d i s t a n c e and i t s r e q u i r e d t i m e a r e d i r e c t l y r e l a t e d t o t h e b a l l i s t i c - t y p e t e a r t e s t method. More s p e c i f i c a l l y , t h i s t e a r d i s t a n c e - t i m e f u n c t i o n i s a n o t h e r e x p r e s s i o n o f b a l l i s t i c - t y p e t e a r t e s t r e s u l t s . Specimens o f t h e same t e a r i n g s t r e n g t h s h o u l d show t h e same t e a r r a t e . The p u r p o s e o f t h i s s t u d y was t o e v a l u a t e t i m e - d e p e n d e n t phenomena o f p a p e r t e a r i n g . C o n s e q u e n t l y , p a p e r s d i s p l a y i n g a wide t e a r i n g s t r e n g t h r a n g e would meet t h i s p u r p o s e b e t t e r t h a n a s e r i e s c o n c e r n e d w i t h minor a d j u s t m e n t o f p r o p e r t i e s by t r e a t i n g a s i n g l e p u l p i n v a r i o u s ways. F i v e c o m m e r c i a l p a p e r s were s e l e c t e d t o c o v e r a wide t e a r i n g s t r e n g t h r a n g e . They i n c l u d e d u n g l a z e d o n i o n s k i n , n e w s p r i n t , 3 0 - l b . n &. m bag p a p e r , I s l a n d 5 5 . 5 - l b . wrapper and p a r c e l wrap ( h e r e a f t e r r e f e r r e d t o as o n i o n s k i n , news-p r i n t , bag p a p e r , 5 5 . 5 - l b . wrapper and p a r c e l wrap, r e s p e c t i v e -l y ) a r r a n g e d i n t h e o r d e r o f i n c r e a s i n g t e a r i n g s t r e n g t h . ( A l l grade w e i g h t s r e f e r t o 500 s h e e t s , 2 4 - i n . X 3 6 - i n . ) . C r o s s - m a c h i n e p a p e r d i r e c t i o n t e a r i n g v a l u e s r a n g i n g from 14 t o 156 g / s h e e t a r e l i s t e d i n T a b l e 1 t o g e t h e r w i t h c a l i p e r and b a s i s w e i g h t measurements. 27 28 Methods T e a r i n g p r o c e d u r e s A T h w i n g - A l b e r t Co. E l m e n d o r f t e a r i n g t e s t e r was used i n t h i s s t u d y . A l l p a p e r s were c o n d i t i o n e d a c c o r d i n g t o TAPPI S t a n d a r d T 402 m-49 (25) at 50 ± 2% r e l a t i v e h u m i d i t y and 73 ± 3.5°F t e m p e r a t u r e u n t i l e q u i l i b r i u m c o n d i t i o n s were r e a c h e d . P a p e r s were c u t i n t o s t a n d a r d 7.6 X 6.3 cm segments w i t h i n a l l o w a b l e v a r i a t i o n by a s h e a r - t y p e paper c u t t e r . The l o n g d i m e n s i o n always r e p r e s e n t e d machine d i r e c t i o n . TAPPI S t a n d a r d T 414 t s - 6 4 (26) on t e a r t e s t i n g p r o c e d u r e s and i n s t r u m e n t a d j u s t m e n t was c l o s e l y f o l l o w e d , e x c e p t as r e g a r d s a d j u s t m e n t o f number o f p l i e s , and t h a t o n l y t h e c r o s s - m a -c h i n e paper d i r e c t i o n was t e s t e d . The t e a r t e s t e r was b o l t e d t i g h t l y o n t o a - y - i n . plywood s h e e t , which was i n t u r n clamped ont o a r i g i d t a b l e t o p r e v e n t any p o s s i b l e movement o f t h e i n s t r u m e n t base d u r i n g swing o f t h e s e c t o r . T e s t r e s u l t s were r e a d from t h e t e s t e r s c a l e and a d j u s t e d t o a v e r a g e t e a r -i n g s t r e n g t h a c c o r d i n g t o E q u a t i o n [j3]] . The t e a r i n g s t r e n g t h r e s u l t s a r e p r e s e n t e d i n T a b l e 2 and a r e p l o t t e d v e r s u s num-b e r o f p a p e r p l i e s i n F i g . 5. At t h e same t i m e , t e a r d i s t a n c e - t i m e d a t a were o b t a i n e d on an o s c i l l o s c o p e s t o r a g e s c o p e . T r a c e s were p e r m a n e n t l y r e c o r d e d by p h o t o g r a p h i c means. D e s i g n and arrangement o f equipment f o r s i m u l t a n e o u s measurement o f t e a r d i s t a n c e and t i m e , as w e l l as d a t a h a n d l i n g w i l l be d e s c r i b e d i n s u b s e q u e n t s e c t i o n s . 29 E x p e r i m e n t a l d e s i g n As s t a t e d , f i v e k i n d s o f paper were examined i n t h e s t u d y . Each p a p e r t y p e r e q u i r e d p r e p a r a t i o n o f s e v e r a l samples w i t h d i f f e r e n t number o f p l i e s . C h o i c e o f number o f p l i e s depended upon i n d i v i d u a l paper s h e e t t e a r i n g s t r e n g t h as r e l a t e d t o t h e s e c t o r s c a l e r e a d i n g between 5 and 75. A p p r o x i m a t e l y e q u a l s p a c i n g s were made between c o n s e c u t i v e s p e c i m e n s by v a r y i n g t h e numbers o f p l i e s . The a s s i g n m e n t s were as f o l l o w s : PAPER GRADE NUMBER OF PLIES (TREATMENT) Onion S k i n 10, 20, 40, 60, 70, 80 N e w s p r i n t 5, 10, 15, 20, 25 Bag p a p e r 2, 6, 10, 14, 18, 22 5 5 . 5 - l b . wrapper 2, 4, 6, 8 P a r c e l wrap 1, 3, 5, 7 S t a t i s t i c a l a n a l y s i s showed t h a t t h r e e r e p l i c a t i o n s were a p r o p e r number f o r b o t h t e a r i n g s t r e n g t h and t e a r t i m e . T h i s r e p l i c a t i o n number was f u r t h e r c o n f i r m e d as adeq u a t e by f u r -t h e r t e s t r e s u l t s which showed, w i t h 95% p r o b a b i l i t y a t ± 5% o f t e a r i n g s t r e n g t h and t i m e mean v a l u e s , t h a t t h e r e q u i r e d r e p l i c a t i o n number was one, e x c e p t f o r a few c a s e s which r e -q u i r e d r e p l i c a t i o n s between one and t h r e e . R a te o f t e a r measurement P r e c i s e measurement o f t h e t e a r d i s t a n c e - t i m e r e l a t i o n s h i p 30 i s t h e c e n t e r o f t h e e x p e r i m e n t a l p a r t o f t h e s t u d y . P a p e r s a r e u s u a l l y t o r n o v e r v e r y s h o r t t i m e w i t h h i g h t e a r r a t e by a b a l l i s t i c - t y p e t e s t e r . E l e c t r o n i c i n s t r u m e n t s , such as an o s c i l l o s c o p e e q u i p p e d w i t h p r o p e r p l u g - i n u n i t s , can measure t i m e e v e n t s down t o m i c r o - s e c o n d s a c c u r a t e l y o v e r s h o r t i n t e r -v a l s o f t i m e . With l o w e r a c c u r a c y measurements can be made ov e r l o n g e r t i m e i n t e r v a l s . The t i m e r e q u i r e d t o t e a r a c r o s s a 4.3 cm specimen o f s e v e r a l p a p e r p l i e s was f o u n d t o r a n g e from 0.08 s e c o n d s t o 0.4 s e c o n d s . T h i s r a n g e w e l l f i t s t h e o s c i l l o s c o p e c a p a c i t y . A T e k t r o n i x I n c . t y p e 564 s t o r a g e o s c i l l o s c o p e can r e g i s -t e r v o l t a g e change w i t h r e s p e c t t o t i m e i n a system by c o o r d i -n a t e methods. When p r o p e r l y r e g u l a t e d , changes i n t e a r t i m e w i t h r e s p e c t t o t e a r d i s t a n c e can be r e c o r d e d on a s t o r a g e o s c i l l o -s c o p e f o r r e a d i n g out d a t a u s e f u l i n a n a l y s i s and c a l c u l a t i o n . I f an e l e c t r i c a l l y c o n d u c t i v e s h e e t i s c o n n e c t e d as p a r t o f an o s c i l l o s c o p e c i r c u i t , t e a r i n g t h i s s h e e t s i m u l t a n e o u s l y a l -t e r s t h e . c u r r e n t p a s s i n g t h r o u g h t h e s h e e t c r e a t i n g a v o l t a g e change, s i n c e t h e amount o f c u r r e n t t h a t can pass t h r o u g h a c o n d u c t o r i s p r o p o r t i o n a l t o i t s c r o s s - s e c t i o n a l a r e a . T h i s v o l t a g e d i f f e r e n c e shown on t h e o s c i l l o s c o p e i n c r e a s e s as t e a r i n g o f a r e s i s t o r p r o c e e d s (Appendix 1) and t h i s can be r e g i s t e r e d a l o n g t h e y - a x i s on t h e o s c i l l o s c o p e . S i m u l t a n e o u s -l y , t h e t i m e r e q u i r e d t o t e a r t h r o u g h t h e s h e e t i s r e g i s t e r e d as t h e x - a x i s . T h e s e x,y c o o r d i n a t e s l o c a t e t e a r i n g d i s t a n c e 31 a t any s p e c i f i c t i m e a l l o w i n g s u b s e q u e n t t e a r r a t e c a l c u l a -t i o n s . O r d i n a r y p a p e r s a t low m o i s t u r e c o n t e n t have low e l e c -t r i c a l c o n d u c t i v i t y . Under such c o n d i t i o n s i t i s u s e f u l t o i n c o r p o r a t e a c o n d u c t i v e m a t e r i a l , o p e r a t e d as a p a r t o f a c i r c u i t , as p a r t o f t h e s p e c i m e n . S e v e r a l c o n d u c t i v i t y methods were s t u d i e d . The g e n e r a l p r i n c i p l e was t o add o r m o d i f y one specimen s h e e t as c o n d u c t -i v e m a t e r i a l . A d v a n t a g e s and d i s a d v a n t a g e s o f s e v e r a l methods a r e d i s c u s s e d below. (1) E l e c t r i c a l l y c o n d u c t i v e paper i s a v a i l a b l e on t h e mar-k e t . I t s u n i f o r m c o n d u c t i v i t y fitted w e l l t h e need o f t h i s s t u d y , and t e s t s gave a c o n t i n u o u s v o l t a g e - t i m e c u r v e on t h e o s c i l l o s c o p e . T e a r i n g d i s t a n c e s and c o r r e s p o n d i n g t i m e s were o b t a i n e d by c a l i b r a t i n g t h i s v o l t a g e - t i m e c u r v e a g a i n s t a t e a r d i s t a n c e - v o l t a g e c u r v e o b t a i n e d by m e a s u r i n g v o l t a g e v a r i a t i o n when c e r t a i n t e a r d i s t a n c e was r e a c h e d by c u t t i n g . T h i s p r o -v i d e d a v o l t a g e c a l i b r a t i o n s c a l e o v e r t i m e . A major a d v a n t a g e o f t h i s method i s t h a t no e l a b o r a t e m a t e r i a l p r e p a r a t i o n s a r e needed. A d i s a d v a n t a g e i s t h a t t h e c o n d u c t i v e p a p e r has a d i f f e r e n t t e a r r e s i s t a n c e t h a n t h e s p e -cimen under t e s t , which r e q u i r e s a c o r r e c t i o n . The c o r r e c t i o n f a c t o r i t s e l f may not be c o n s t a n t , but w i l l v a r y w i t h t o t a l s p ecimen r e s i s t a n c e i f t h e t e a r t e s t i s t i m e - d e p e n d e n t . 32 (2) T h i n aluminum f o i l i s e s s e n t i a l l y homogeneous and has n e g l i g i b l e t e a r i n g s t r e n g t h . I t s v e r y good e l e c t r i c a l c o n d u c -t i v i t y , however, r e s u l t s i n l i t t l e v o l t a g e change w i t h r e -s p e c t t o change i n c r o s s - s e c t i o n a l a r e a as t e a r i n g p r o c e e d s . T h i s s m a l l v o l t a g e change l o w e r s a c c u r a c y i n r e a d i n g o s c i l l o -s c o p e t r a c e s . (3) M e t a l p a r t i c l e s may be d e p o s i t e d on a p a p e r s u r f a c e . T h e o r e t i c a l l y , a u n i f o r m l a y e r o f s i l v e r o r o t h e r m e t a l can be d e p o s i t e d on a p a p e r s u r f a c e by u s i n g a vacuum e v a p o r a t o r , w hich i s a b a s i c a c c e s s o r y f o r e l e c t r o n m i c r o s c o p y . T h i s me-t h o d was t r i e d and g i v e n up i m m e d i a t e l y . The h i g h t e m p e r a t u r e needed c o u l d d r a s t i c a l l y change paper p r o p e r t i e s . Too t h i n a l a y e r p r o v i d e d p oor c o n d u c t i v i t y , w h i l e t h i c k e r l a y e r s r e -q u i r e d l o n g t i m e a t h i g h t e m p e r a t u r e , which i s not d e s i r a b l e . F o r example, about two m i n u t e s e x p o s u r e was needed f o r p r e p a r -i n g one 7.6 cm by 6.3 cm s p e c i m e n . F u r t h e r , u n i f o r m i t y o f t h e m e t a l l a y e r d e p o s i t e d may be a f f e c t e d by t h e m i c r o s c o p i c a l l y r ough paper s u r f a c e . (4) E l e c t r i c a l c o n d u c t i v i t y paper was p e r f o r a t e d a l o n g t h e median l i n e i n an attempt t o r e d u c e t e a r r e s i s t a n c e . One., s e t o f r e s u l t s showed t h a t t h e r e was no d i f f e r e n c e i n t e a r i n g s t r e n g t h (55.3 g / s h e e t ) o f t h i s c o m b i n a t i o n and t h e t e a r i n g s t r e n g t h (55.5 g / s h e e t ) o f s p e c i m e n s w i t h o u t t r e a t e d c o n d u c t i v e p a p e r . The d i s a d v a n t a g e i s t h a t i t r e q u i r e s c a r e f u l work t o i n t r o d u c e p e r f o r a t i o n s on e l e c t r i c a l l y c o n d u c t i v e p a p e r and f r e q u e n t l y t h e t e a r - p a t h o f a whole specimen b o o k l e t does not 33 f o l l o w e x a c t l y t h e d i r e c t i o n o f p e r f o r a t i o n s i n t r o d u c e d i n t h e c e n t e r p l y . I n t r o d u c i n g p e r f o r a t i o n s p r o d u c e d a s t e p w i d e c u r v e ( F i g . 6 a ) , r e f l e c t i n g s p a c i n g s o f p e r f o r a t i o n s . The end o f each s t e p on t h e c u r v e i n d i c a t e s when t e a r i n g has r e a c h e d a c e r t a i n d i s t a n c e . T h i s e n a b l e s r e a d i n g t e a r t i m e by p r o j e c -t i o n t o t h e t i m e - b a s e x - a x i s . (5) G r a p h i t e may be used as a c o n d u c t i v e m a t e r i a l , s i n c e i t has a r a t h e r low e l e c t r i c a l c o n d u c t i v i t y i n c o m p a r i s o n w i t h aluminum f o i l . A p p l y i n g l a d d e r - l i k e g r a p h i t e l i n e s t o a paper s u r f a c e gave t h e same a d v a n t a g e as Method 4, but not i t s d i s a d v a n t a g e . F u r t h e r m o r e , a p p l y i n g g r a p h i t e d i r e c t l y o n t o a member o f t h e specimen b o o k l e t a v o i d s t h e v a r i a b l e t e a r r e s i s t a n c e e f f e c t i n t r o d u c e d by i n c l u d i n g a f o r e i g n ma-t e r i a l i n t o t h e body o f t h e s p e c i m e n . A s t e p w i s e c u r v e i s o b t a i n e d as i n Method 4, but, w i t h b e t t e r v e r t i c a l s t e p l i n e s ( F i g . 6 b ) , which i n c r e a s e s a c c u r a c y i n r e a d i n g t e a r t i m e . T r a n s f e r o f g r a p h i t e from a 6B p e n c i l t o t h e specimen s u r f a c e i s v e r y t i m e consuming and sometimes i n a c c u r a t e . T h i s i s be-c a u s e q u i t e a t h i c k , wide l i n e i s r e q u i r e d t o form a s a t i s -f a c t o r y c o n d u c t o r . (6) As r e p o r t e d by A n d e r s s o n and F a l k ( 1 ) , c o n d u c t i n g i n k may be used as c o n d u c t i v e m a t e r i a l . I n s t e a d o f u s i n g g r a p h i t e , c o m m e r c i a l s i l v e r p a i n t used f o r d r a w i n g e l e c t r i c c i r c u i t s has been f o u n d t o be a good c o n d u c t o r t h a t can be t r a n s f e r r e d 34 e a s i l y and r e p r o d u c i b l y t o a specimen s u r f a c e . The c o a r s e -ness o f c r o s s - l i n e s can be c o n t r o l l e d as d e s i r e d by u s i n g a d r a f t i n g pen. I t was f o u n d t h a t f i n e r c r o s s - l i n e s p r o v i d e d f i n e r t e a r d i s t a n c e - t i m e s t e p w i s e c u r v e s . Y e t , u s i n g t o o f i n e l i n e s i n t r o d u c e d d i s c o n t i n u i t i e s , c a u s i n g one o r s e v e r a l s t e p s t o be m i s s e d . C o a r s e c r o s s - l i n e s d i d not have t h i s d i s a d v a n t a g e . They r e s u l t e d i n c o a r s e r and l e s s a c c u r a t e t e a r d i s t a n c e - t i m e c u r v e s , and a l s o r e d u c e d s e n s i t i v i t y by r e d u c i n g t h e s t e p h e i g h t ( F i g . 6c) which accompanied s m a l l o v e r a l l v o l t a g e change. A l l s i x methods have a d v a n t a g e s and d i s a d v a n t a g e s , as n o t e d . Methods 4, 5 and 6 a r e compared i n T a b l e 3. The f i n a l method a d o p t e d combined a d v a n t a g e s o f b o t h Methods 5 and 6, y i e l d i n g r e s u l t s as shown i n F i g . 6d. S i l -v e r p a i n t was used t o draw a l l t h e c r o s s - l i n e s and one o f t h e two main v e r t i c a l l i n e s which c o n n e c t a l l t h e c r o s s - l i n e s . A f i n e t o medium l i n e c o a r s e n e s s was a d o p t e d . The c o n t i n u i t y o f each l i n e was c h e c k e d by v o l t m e t e r t o e n s u r e c o n d u c t i v i t y . T h e r e a f t e r , a n o t h e r v e r t i c a l main l i n e was a p p l i e d as g r a -p h i t e ( F i g . 7 ) . T h i s was made t h i c k enough t o g i v e good c o n -t i n u i t y and a l s o good c o n d u c t i v i t y . M a n i p u l a t i n g t h e g r a p h i t e a d d i t i o n a l l o w e d a d j u s t i n g t h e h e i g h t o f each s t e p on t h e c u r v e . D i s t a n c e s between c r o s s - l i n e s a r e d e t e r m i n e d by t h e o p e r -a t o r . F o r t h e s e e x p e r i m e n t s t e a r d i s t a n c e s o f 0.2, 0.5, 1.0, 35 1.5, 2.0, 2.5, 3.0, 3.5, 4.0 and 4.3 cm were u s e d . Each c r o s s - l i n e was drawn by hand w i t h t h e h e l p o f a s t r a i g h t -r u l e r and d r a f t i n g pen. The f i n a l edge o f each c r o s s - l i n e was l o c a t e d e x a c t l y at t h e t e a r d i s t a n c e m e n t i o n e d above, e x c e p t t h e l a s t one, w i t h b e g i n n i n g l o c a t e d a t t h e 4.3 cm p o s i t i o n i n o r d e r t o f a c i l i t a t e a f i n a l t i m e r e a d i n g . I t i s a l s o p o s s i b l e t o v a r y l i n e s p a c i n g s and t h e r e b y r e p r e s e n t some m a t h e m a t i c a l f u n c t i o n . At t e s t i n g , t h e s h e e t o f paper w i t h c o n d u c t i v e m a t e r i a l ( h e r e a f t e r r e f e r r e d t o as t h e c o n d u c t i v e p l y ) becomes one o f t h e many p l i e s t o be t o r n and i s p l a c e d a t t h e specimen c e n -t e r . T h i s c o n d u c t i v e p l y i s c o n n e c t e d t o a c i r c u i t ( F i g . 8) i n c l u d i n g b a t t e r y , decade box and o s c i l l o s c o p e ( F i g . 8 and 9 ) . V o l t a g e c a l c u l a t i o n s a r e shown i n Appendix 1. The two main v e r t i c a l c o n d u c t i v e l i n e s a r e l e d out from t h e specimen t o t h e c i r c u i t by two h a l f - p i e c e s o f r a z o r b l a d e . R a z o r . b l a d e s have a d v a n t a g e s o f b e i n g t h i n , s t i f f , a v a i l a b l e and v e r y good c o n d u c t o r s . W i r e s a r e c o n n e c t e d t o t h e r a z o r b l a d e s by two a l l i g a t o r c l i p s . The w i r e l e a d i n g from t h e s e c t o r clamp was c o n n e c t e d t o t h e anode o f a b a t t e r y and suspended f r e e l y i n t h e a i r ( F i g . 8) t o r e d u c e 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 a r r a n g e m e n t . TAPPI S t a n d a r d T 414 t s - 6 4 (26) r e q u i r e s t h a t t h e s e c t o r s h o u l d make a t l e a s t 20 c o m p l e t e o s c i l l a t i o n s b e f o r e t h e edge o f t h e s e c t o r which engages t h e s e c t o r s t o p , no l o n g e r p a s s e s t o t h e l e f t o f a p e n c i l l i n e l o c a t e d one i n c h t o t h e r i g h t o f t h e 36 edge o f t h e s e c t o r s t o p . By s u s p e n d i n g t h e w i r e f r e e l y i n t h e a i r , t h e s e c t o r made more t h a n 50 o s c i l l a t i o n s b e f o r e i t no l o n g e r p a s s e d t o t h e l e f t o f t h e p e n c i l l i n e . T h e r e b y , t h i s arrangement i s c o n s i d e r e d as c o n t r i b u t i n g no s i g n i f i c a n t e f f e c t on t h e b a s i c t e s t method. The t i m e r e q u i r e d t o t e a r a specimen i s r e l a t i v e l y s h o r t when compared w i t h o t h e r paper s t r e n g t h t e s t p r o c e d u r e s . A s l i g h t t i m e v a r i a t i o n c o u l d c a u s e s e r i o u s e r r o r s i n r e s u l t s . A s y n c h r o n o u s s y s t e m i s n e c e s s a r y f o r s t a r t i n g t h e t e a r t e s t and o s c i l l o s c o p e s i m u l t a n e o u s l y . To do t h i s , a m i c r o - s w i t c h was mounted u n d e r n e a t h t h e s e c t o r s t o p i n such a way t h a t p r e s s i n g down t h e s e c t o r s t o p s t a r t e d b o t h t h e t e a r i n g p r o -c e s s and t h e o s c i l l o s c o p e r e c o r d i n g . The t e a r d i s t a n c e - t i m e s t e p w i s e c u r v e was s t o r e d on t h e o s c i l l o s c o p e . A p o l a r o i d p i c t u r e was t h e n t a k e n f o r e a s i e r r e a d i n g and permanent r e c o r d o f t h e o s c i l l o s c o p e t r a c e s . When t e a r i n g a c r o s s a c e r t a i n known d i s t a n c e b r e a k s a c o n d u c t i v e l i n e , t h e t o t a l c u r r e n t p a s s i n g t h r o u g h t h e c o n -d u c t i v e s h e e t g r i d i s r e d u c e d , v o l t a g e changes and s i m u l t a n -e o u s l y a s t e p i s r e g i s t e r e d on t h e d i s t a n c e - t i m e d i a g r a m . Ten s t e p s can be o b t a i n e d i f t e n c o n d u c t i v e l i n e s a r e p a i n t e d on t h e c o n d u c t i v e p l y , and t e n s e t s o f t e a r d i s t a n c e - t i m e d a t a can be c o l l e c t e d a c c o r d i n g t o t h e d e s c r i p t i o n i n Method 4. T hese d a t a can t h e n be used f o r c a l c u l a t i n g t e a r r a t e . Z e r o - s w i n g t e a r d i s t a n c e - t i m e measurement A c c o r d i n g t o » two v e l o c i t y measurements a r e 37 r e q u i r e d f o r c a l c u l a t i n g t e a r i n g energy a t s e c t o r p o s i t i o n y^ , namely, z e r o - s w i n g v e l o c i t y , v^, and t e a r i n g v e l o c i t y , v2» T e a r i n g v e l o c i t y can be measured a c c o r d i n g t o t h e method m e n t i o n e d above. T h e o r e t i c a l l y , z e r o - s w i n g v e l o c i t y c a n n o t be measured by t h e same method. In p r a c t i c e , i t i s c o n v e n i e n t t o s i m u l a t e z e r o - s w i n g b e h a v i o r by t e a r i n g one s h e e t o f a m a t e r i a l which has e x t r e m e l y low t e a r r e s i s t a n c e , and hence, c o n t r i b u t e s no s i g n i f i c a n t a d j u s t m e n t t o t h e t r u e z e r o - s w i n g v e l o c i t y . C e l l o p h a n e , b e c a u s e o f i t s h i g h d e g r e e o f c o h e r e n c e t o c o n c e n t r a t e d t e a r i n g f o r c e s ( 6 ) , m o l e c u l a r homogeneity and c h a r a c t e r i s t i c low t e a r r e s i s t a n c e once t e a r i s s t a r t e d ( 1 5 ) , would seem t o be an i d e a l m a t e r i a l f o r t h i s p u r p o s e . In keep-i n g w i t h t h e r e g u l a r p r o c e d u r e , c o n d u c t i v e s i l v e r p a i n t . w a s a p p l i e d as s p a c e d l i n e s t o c e l l o p h a n e s a m p l e s . C o n t r a r y t o t h e e x p e c t e d b e h a v i o r , t h e t e a r d i s t a n c e - t i m e c u r v e s o b t a i n e d d i s p l a y e d l o w e r v e l o c i t y t h a n had f o r p a p e r . T h i s s u g g e s t s t h a t c e l l o p h a n e , as a n o n - f i b r o u s m a t e r i a l w i t h e x t r e m e l y low b e n d i n g r e s i s t a n c e , r e s p o n d s d i f f e r e n t l y t o t e a r f o r c e t h an p a p e r . P e r h a p s a s i m i l a r e f f e c t w i t h aluminum f o i l l e d 5 w a r t o u t and S e t t e r h o l m (24) t o t h e i r c o n c l u s i o n on t h e p o s i -t i v e r e l a t i o n s h i p between t e a r v a l u e 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 as o r i g i n a t i n g from t h e t y p e o f m a t e r i a l b e i n g t e s t e d . A n o t h e r m a t e r i a l , p e r f o r a t e d t r a c i n g p a p e r , p r e p a r e d w i t h l a d d e r - l i k e s i l v e r p a i n t l i n e s , gave t h e same r e s p o n s e t o t e a r f o r c e as p a p e r and r e g i s t e r e d no t e a r r e s i s t a n c e on t h e s e c t o r s c a l e . T h i s method i s assumed t o c l o s e l y a p p r o x i m a t e 38 t h e t r u e z e r o - s w i n g v e l o c i t y and was a d o p t e d f o r m e a s u r i n g r e s p o n s e f o r t h e v e l o c i t y (V^) c a l c u l a t i o n a t s e c t o r p o s i t i o n y^ . C e r t a i n l y o t h e r systems c o u l d b-B d e v i s e d f o r d e r i v i n g z e r o - s w i n g v e l o c i t y i f g r e a t e r a c c u r a c y i s r e q u i r e d . F o r ex-ample, a p r o p e r l y p e r f o r a t e d s c a l e f i x e d t o t h e s e c t o r base and p a s s e d b e f o r e an e l e c t r i c eye might be u s e d . H a n d l i n g o f d a t a and c u r v e f i t t i n g From each t e s t , t e n s e t s o f t e a r d i s t a n c e - t i m e d a t a were made a v a i l a b l e f o r d e f i n i n g t e a r r a t e . F o r example, t h e t e a r d i s t a n c e - t i m e r e l a t i o n s h i p f o r one o f t h e f i v e p a p e r s , bag p a p e r , i s p l o t t e d as F i g . 10. S i m i l a r r e l a t i o n s h i p s e x i s t e d f o r a l l f i v e k i n d s o f p a p e r . I t i s o b s e r v e d t h a t t e a r d i s t a n c e (L) r e l a t e s t o t e a r t i m e ( t ) as a s e c o n d d e g r e e o r q u a d r a t i c r e l a t i o n s h i p ( F i g . 1 0 ) . A l e a s t s q u a r e s method was used t o f i t t h e r e l a t i o n s h i p as t o m a t h e m a t i c a l f o r m . Second d e g r e e c u r v e s were f i t t e d by r e -g r e s s i o n a n a l y s i s , programmed on an IBM 7044 computer. Some s e t s o f measurements were s t u d i e d as t h i r d d e g r e e e q u a t i o n s . The s e c o n d d e g r e e e x p r e s s i o n was p r o v e d t o be ade-q u a t e , a l t h o u g h s t a t i s t i c a l c o m p a r i s o n s showed i n some i n -s t a n c e s t h a t t h e t h i r d d e g r e e term was s i g n i f i c a n t . F u r t h e r a n a l y s i s ( T a b l e 4) a c c o r d i n g t o t h e p r o c e d u r e s d e s c r i b e d i n L i ' s S t a t i s t i c a l I n f e r e n c e (18) was done t o f i n d r e a s o n s f o r L = a + b t + c t 2 39 t h e u n e x p e c t e d s i g n i f i c a n c e o f t h e t h i r d d e g r e e t e r m . When c a l c u l a t i n g t h e r e g r e s s i o n sum o f s q u a r e s c o n t r i b u -2 3 t e d by each term, t , t , t , and r e s i d u a l v a r i a n c e , i t was always f o u n d t h a t t h e r e s i d u a l v a r i a n c e was o f t h e o r d e r o f 1.5 t o 3.5 X 10 . T h i s e x t r e m e l y s m a l l r e s i d u a l v a r i a n c e r e n d e r e d t h e v a r i a n c e r a t i o t e s t e x t r e m e l y s e n s i t i v e . When c o n s i d e r i n g t h e v e r y s m a l l m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t _5 d i f f e r e n c e , 1 t o 20 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 d i f f e r e n c e , 1 t o 10 X 10 m m , between s e c o n d d e g r e e and t h i r d d e g r e e e x p r e s s i o n s , t o g e t h e r w i t h t h e i r e x t r e m e l y s m a l l r e s i -d u a l v a r i a n c e , i t may be c o n c l u d e d t h a t t h e s e c o n d d e g r e e e q u a t i o n b e s t d e s c r i b e s a l l s e t s o f measurements. F u r t h e r argument f o r t h i s c h o i c e w i l l be g i v e n i n t h e D i s c u s s i o n . The t e a r d i s t a n c e - t i m e r e l a t i o n s h i p s f o r z e r o - s w i n g and t h e f i v e p a p e r g r a d e s were measured. Each paper grade com-p r i s e d s e v e r a l t r e a t m e n t s (6, 5, 6, 4 and 4 t r e a t m e n t s f o r o n i o n s k i n , n e w s p r i n t , bag p a p e r , 5 5 . 5 - l b . wrapper and p a r c e l wrap, r e s p e c t i v e l y ) , f o r t h e d i f f e r e n t number o f p l i e s . T h r e e r e p l i c a t i o n s were used f o r z e r o - s w i n g and each t r e a t m e n t , ex-c e p t t h a t one r e p l i c a t i o n o n l y was used f o r 70 p l i e s o f o n i o n s k i n p a p e r . These gave a t o t a l o f 76 t e s t s f o r t h e e n t i r e s t u d y . Each t e s t was a n a l y z e d a t t e n p o s i t i o n s . F i n a l e x p r e s s i o n s c o u l d have been f o u n d by any o f t h r e e ways. F i r s t , a v e r a g e v a l u e s from t h e t h r e e r e p l i c a t i o n s may be used t o s o l v e a s i n g l e e q u a t i o n . T h i s method i s no b e t t e r t h a n a s e c o n d a p p r o a c h , which u t i l i z e s a l l t h r e e i n d i v i d u a l s e t s o f 40 d a t a by t h e l e a s t s q u a r e s method t o f i t a s i n g l e e q u a t i o n . The d i s a d v a n t a g e h e r e i s t h e h i g h e r s t a n d a r d e r r o r o f e s t i -mate due t o d i s p e r s i o n o f t e a r t i m e s ( i n a l l t e s t s , t e a r d i s -t a n c e (L) i s f i x e d ) . T h i r d , each i n d i v i d u a l s e t o f d a t a can be used t o f i t an e q u a t i o n . T h r e e r e p l i c a t i o n s can r e s u l t i n t h r e e d i f f e r e n t e q u a t i o n s . E q u a t i o n s d e v e l o p e d by t h e t h i r d method a r e p r e s e n t e d i n T a b l e 5 t o g e t h e r w i t h t h e i r m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t s and s t a n d a r d e r r o r s o f e s t i m a t e . By t h e t h i r d method, t e a r v e l o c i t y f o r each r e p l i c a t i o n can be c a l c u l a t e d by s u b s t i t u t i n g t e a r t i m e i n t o i t s c o r r e s p o n d -i n g f i r s t d e r i v a t i v e e q u a t i o n , and an a v e r a g e o f t h e s e t e a r v e l o c i t i e s can be used as t h e f i n a l r e s u l t . A dvantages o f t h i s method a r e t h a t a l l t h e o r i g i n a l measurements c o n t r i b u t e d i r e c t l y t o t h e f i n a l r e s u l t and a l s o t h a t an i n d i v i d u a l equa-t i o n f o r t e a r d i s t a n c e - t i m e measurement has a s m a l l e r s t a n d a r d e r r o r o f e s t i m a t e . F o r t h e s e r e a s o n s t h i s method was a d o p t e d f o r t h e main s t u d y , i . e . , f o r a c e r t a i n number o f pape r p l i e s , t h r e e r e g r e s -s i o n e q u a t i o n s were f i t t e d . S u b s t i t u t i n g c o r r e s p o n d i n g t e a r -t i m e s i n t o t h e f i r s t d e r i v a t i v e o f t h e r e g r e s s i o n e q u a t i o n p r o v i d e d t h r e e t e a r i n g v e l o c i t i e s which were a v e r a g e d t o f i n d t h e s e c t o r net energy and r e s i d u a l e n e r g y . By s u c h means, t e a r i n g e n e r g y f o r each s pecimen can be c a l c u l a t e d a c c o r d i n g t o E q u a t i o n [9] • S i n c e t h e mass (m) term i n E q u a t i o n [9] i s a c o n s t a n t f o r a p a r t i c u l a r s e c t o r , a l l t e a r i n g energy v a l u e s 41 p r e s e n t e d a r e made a f r a c t i o n o f t o t a l t e a r i n g e n e r g y by r e m o v i n g t h e s e c t o r mass f a c t o r . T e a r i n g energy r e q u i r e d per p l y can be o b t a i n e d by d i v i d i n g t h e t e a r i n g e n e r g y f o r each specimen by i t s number o f p l i e s . T e a r i n g e n e r g y as p r e s e n t e d i n T a b l e 2 i s t h e energy r e q u i r e d t o t e a r a c r o s s 4.3 cm t h r o u g h a s i n g l e s h e e t . T h e s e t e a r i n g e n e r g i e s a r e f u r t h e r p r e s e n t e d i n F i g . 11a t o l i e t o show t e a r i n g e n e rgy v a r i a t i o n w i t h r e s p e c t t o r a t e o f t e a r . Summary o f rmethods T e s t s p e c imen b o o k l e t s w i t h d i f f e r e n t numbers o f p l i e s were t e s t e d w i t h an E l m e n d o r f t e a r t e s t e r . P o s i t i o n e d a t t h e c e n t e r o f each b o o k l e t was a s i n g l e p l y o f t h e t e s t ma-t e r i a l t o which l a d d e r - l i k e s i l v e r p a i n t c r o s s - l i n e s had been added. At i t s o u t e r edges, v e r t i c a l - l i n e s , one of which was g r a p h i t e , c o n n e c t e d t h e c r o s s - l i n e s . By t h i s a r r angement, t h e c e n t e r p l y r e n d e r e d t h e specimen as p a r t o f an e l e c t r i c c i r c u i t which a l l o w e d measurement o f t i m e s r e q u i r e d t o t e a r t h r o u g h p r e - a s s i g n e d d i s t a n c e s . A s y n c h r o n o u s mechanism was i n c o r p o r a t e d t o r e c o r d t h e t e a r d i s t a n c e - t i m e r e l a t i o n s h i p , b e g i n n i n g as pape r t e a r i n g s t a r t e d . T e a r d i s t a n c e - t i m e r e l a t i o n s h i p s were r e g i s t e r e d on an o s c i l l o s c o p e and were p e r m a n e n t l y r e c o r d e d as p o l a r o i d p i c -t u r e s . Each t e s t as done h e r e p r o v i d e d t e n t e a r d i s t a n c e -t i m e measurements which were r e a d from a p o l a r o i d p i c t u r e . 42 T h e s e d a t a were f i t t e d as s e c o n d d e g r e e e q u a t i o n s . V e l o c i t y was c a l c u l a t e d from t h e f i r s t d e r i v a t i v e o f t h e s e c o n d d e g r e e e q u a t i o n . The z e r o - s w i n g t i m e r e q u i r e d t o t e a r t h r o u g h d i s t a n c e s on an i m a g i n e d b l a n k s h e e t was s i m u l a t e d by t e a r i n g a s i n g l e s h e e t o f p e r f o r a t e d t r a c i n g p a p e r t r e a t e d i n t h e same way as f o r t e a r i n g t e s t s p e c i m e n s . T h r e e v e l o c i t y v a l u e s a r o s e from r e p l i c a t i o n s o f each sample: t h e s e were a v e r a g e d f o r c a l c u l a t i n g t e a r i n g e n e rgy a c c o r d i n g t o E q u a t i o n £ 9 J . T e a r i n g e n ergy was t h e n p r e s e n t e d i n terms o f f r a c t i o n a l e n e r g y p e r p l y f o r c o m p a r i s o n between m a t e r i a l s . F o r c o m p a r i s o n p u r p o s e s , r e a d i n g s were t a k e n from t h e t e s t e r s e c t o r s c a l e and a d j u s t e d t o t h e s t a n d a r d a v e r a g e t e a r i n g s t r e n g t h a c c o r d i n g t o TAPPI S t a n d a r d T 414 t s - 6 4 ( 2 6 ) . DISCUSSION The b a c k g r o u n d o f f a c t o r s a f f e c t i n g t e a r t e s t r e s u l t s h a s : been r e v i e w e d i n f o r m e r s e c t i o n s . A c c o r d i n g t o t h e e n e r g y d i s s i p a t i o n c o n c e p t , e n e r g y ex-pended i n t e a r i n g a s h e e t o f paper i s d i s s i p a t e d w i t h i n t h e s h e e t . In t h e p r e s e n t s t u d y a m a t h e m a t i c a l model has been d e r i v e d f o r d e m o n s t r a t i n g t h e n a t u r e o f b a l l i s t i c - t y p e t e a r t e s t e r s . T h i s model can be used t o c a l c u l a t e t h e net energy o f a t e a r t e s t e r s e c t o r w hich i s c a p a b l e o f d o i n g work, r e s i -d u a l e n e rgy a f t e r t e a r i n g p a p e r and o v e r c o m i n g f r i c t i o n , and t h e t e a r i n g e n e r g y which i s t h a t p a r t o f t h e energy a b s o r b e d by p a p e r d u r i n g t h e t e a r i n g p r o c e s s . F u r t h e r m o r e , t h i s t e a r -i n g e n e r gy c o n c e p t i s r e l a t e d t o t h e dynamic n a t u r e o f t h e t e a r t e s t by u s i n g p a p e r t e a r v e l o c i t y as t h e b a s i s f o r c a l -c u l a t i o n . S i n c e v e l o c i t y i s c a l c u l a t e d from d i s t a n c e and t i m e , an o p p o r t u n i t y e x i s t s f o r r e l a t i n g t e a r t e s t d a t a w i t h t i m e , t h u s p r o v i d i n g means f o r e v a l u a t i n g t h e e f f e c t o f t e a r r a t e . H o p e f u l l y , t h i s e x a m i n a t i o n o f t e a r r a t e w i l l c l a r i f y some problems a s s o c i a t e d w i t h t h e paper t e a r t e s t . F o r ex-ample, showing t e a r p r o g r e s s and energy v a r i a t i o n as t e a r i n g p r o c e e d s , and e x p l o r i n g t h e r e l a t i o n s h i p between t e a r i n g e n e r g y and t e a r r a t e , p r o v i d e s new i n f o r m a t i o n on c o n d u c t o f t h e t e s t . T e a r i n g v e l o c i t y i s an e x p r e s s i o n o f t h e t i m e r e q u i r e d t o t e a r a c r o s s a c e r t a i n d i s t a n c e . In o r d e r t o measure t h e 43 44 d i s t a n c e - t i m e r e l a t i o n s h i p , a p r e c i s e method had t o be de-v e l o p e d . The e f f e c t i v e n e s s and r e p r o d u c i b i l i t y o f t h i s method i s d e m o n s t r a t e d by t h e v e r y s m a l l v a r i a t i o n had between r e p l i c a t i o n s . In most c a s e s , i t has been shown t h a t o n l y one measurement i s n e c e s s a r y f o r r e p r e s e n t i n g t h e specimen c o n d i -t i o n under s t u d y . R e p r o d u c i b i l i t y o f t h e b a s i c method c o u l d be f u r t h e r i m p r o v e d by a p p l y i n g t h e s i l v e r p a i n t l i n e s w i t h some mecha-n i c a l d e v i c e which r e g u l a t e s d i s t a n c e s more p r e c i s e l y t h a n can be done by hand d r a w i n g . Curve F i t t i n g A l t h o u g h t h e r e i s no t e c h n i c a l p r o b l e m i n m e a s u r i n g t e a r d i s t a n c e - t i m e d a t a , use o f t h e s e measurements needs some f u r -t h e r d i s c u s s i o n . T h a t i s , d e c i s i o n i s r e q u i r e d on t h e b e s t form f o r t h e s e d a t a , i . e . , e x p r e s s i o n as a l i n e a r , q u a d r a t i c o r c u b i c e q u a t i o n . A l l t h r e e t y p e s o f e q u a t i o n s can be used t o e x p r e s s t h e t e a r d i s t a n c e - t i m e r e l a t i o n s h i p . The d i f f e r -ence i n degree,however, a f f e c t s e v a l u a t i o n o f t h e energy d i s -t r i b u t i o n . S i n c e t h e t e a r i n g energy c a l c u l a t i o n i s based on v e l o c i t y , v e l o c i t y i s c a l c u l a t e d by s u b s t i t u t i n g t e a r t i m e i n -t o t h e f i r s t d e r i v a t i v e o f t h e e q u a t i o n o f any d e g r e e . I f a l i n e a r e q u a t i o n a s : i s u s e d , t h e f i r s t d e r i v a t i v e , v e l o c i t y , w i l l be a c o n s t a n t L = a + b t 45 w i t h no a c c e l e r a t i o n , so t h e s e c t o r swings and t e a r s w i t h a c o n s t a n t v e l o c i t y and r e s i d u a l t e a r i n g v e l o c i t y . I f a q u a d r a t i c e q u a t i o n a s : L = a + b t + c t 2 T r l ] i s u s e d , i t s f i r s t d e r i v a t i v e , v e l o c i t y , w i l l have a l i n e a r f orm dL = b + 2 c t C 1 23 dt w i t h a c o n s t a n t a c c e l e r a t i o n I f a c u b i c e q u a t i o n a s : L = a + bt + c t 2 + d t 3 Cl6~| i s u s e d , i t s f i r s t d e r i v a t i v e , v e l o c i t y , w i l l be o f p a r a b o l i c f o r m , 2 dL_ = b + 2 c t + 3dt V l l j dt U J w i t h a c c e l e r a t i o n - $ ! - • • 2 c + 6 d t [ " J U s i n g a l l t h r e e forms among d a t a o f a s i n g l e s t u d y would c a u s e d i f f i c u l t y i n c o m p a r i s o n s . T h i s i s b e c a u s e t h e d i f f e r -ent forms o f e q u a t i o n r e f e r t o b a s i c d i f f e r e n c e s i n t h e n a t u r e o f t e a r i n g o r n a t u r e of t h e a p p l i e d f o r c e . In f a c t , a l l t e a r -i n g e n e r gy o r i g i n a t e s from t h e s e c t o r , so a l l t h e a p p l i e d 46 f o r c e w i l l be t h e same, c a u s i n g t e a r t o p r o c e e d as common be-h a v i o r . Hence, t h e r e p l i c a t e d z e r o - s w i n g t e a r d i s t a n c e - t i m e measurements can l o g i c a l l y o n l y be e x p r e s s e d by e q u a t i o n s o f t h e same d e g r e e . The t e a r d i s t a n c e - t i m e r e l a t i o n s h i p s c o u l d be e x p r e s s e d as d i f f e r e n t d e g r e e e q u a t i o n s f o r v a r i o u s t e s t s . The l i n e a r r e l a t i o n s h i p i s t r u e i f t h e t o t a l paper t e a r r e s i s t a n c e i n -crement i s s l i g h t l y s m a l l e r t h a n t h e net e n e r g y i n c r e m e n t o f t h e s e c t o r . T h e r e f o r e , p a p e r i s t o r n at a s t a t e a p p r o a c h i n g l i n e a r r e l a t i o n s h i p . P r e l i m i n a r y t e s t s c o n f i r m e d t h i s pheno-menon f o r t h e l a s t p a r t o f t e a r d i s t a n c e - t i m e c u r v e when a specimen w i t h h i g h number o f p l i e s , o r more p r o p e r l y a s p e c i -men w i t h h i g h t e a r r e s i s t a n c e , was t o r n . P r e l i m i n a r y t e a r d i s t a n c e - t i m e t e s t d a t a w i t h 25 p l i e s o f bag paper i s p l o t t e d as p a r t o f F i g . 10 w i t h o t h e r f i n a l t e s t r e s u l t s . I t can be o b s e r v e d t h a t s t a r t i n g w i t h 0.5 t o 3.5 cm d e p t h , t h e t e a r d i s t a n c e - t i m e r e l a t i o n s h i p o f t h e 25 p l i e s c u r v e i s o f l i n e a r f o r m . A f t e r 3.5 cm t h e e f f e c t o f d e c r e a s i n g t e a r v e l o c i t y becomes more and more pro n o u n c e d and t h e c u r v e d i r e c t i o n s t a r t s t o change i n t o a c o n c a v e f o r m . T h i s l i n e a r r e l a t i o n s h i p can always be i d e n t i f i e d w i t h s e c t o r s c a l e r e a d i n g s o v e r 75 or 80. In o r d e r t o a v o i d t h e c o m p l i c a t i o n c a u s e d by u s i n g t h e l i n e a r r e l a t i o n s h i p , t h i s s t u d y was d e s i g n e d t o use o n l y a l i m i t e d c a p a c i t y o f t h e t e a r t e s t e r . A c t u a l l y , t h i s c o n s i d e r a -t i o n i s not n e c e s s a r y , b e c a u s e t h e i n i t i a l p a r t o f t h e t e a r 47 d i s t a n c e - t i m e c u r v e i s always o f c u r v i l i n e a r form and a l i n e a r e x p r e s s i o n i g n o r e s t h i s f a c t . A r e g r e s s i o n a n a l y s i s program e l i m i n a t i n g l e a s t i m p o r t a n t 2 variables s u c c e s s i v e l y showed t h e s e c o n d d e g r e e term, t , as a l -ways t h e most i m p o r t a n t s i n g l e variable r e p r e s e n t i n g t h e r e l a -t i o n s h i p . T h e r e f o r e , i t i s d e f i n i t e l y i n c o r r e c t t o use t h e l i n e a r f orm. T h e r e i s a p o s s i b i l i t y o f u s i n g t h e c u b i c as an e x p r e s -s i o n f o r t e a r d i s t a n c e - t i m e d a t a . As above, t h e bag p a p e r specimen w i t h 25 p l i e s i s a good example f o r t h i s c a s e . T h i s c u r v e r e f l e x e s t w i c e , from c u r v i l i n e a r a t t h e b e g i n n i n g t o l i n e a r , and t h e n i t a p p e a r s as a n o t h e r c u r v i l i n e a r form, c o n -c a v e f a c i n g t h e x - a x i s . T h e s e d a t a can be b e s t e x p r e s s e d mathe-m a t i c a l l y by a c u b i c e q u a t i o n . T h e : e x p e r i m e n t a l d e s i g n f o r t h i s s t u d y p u r p o s e l y a v o i d e d t h i s c a s e by u s i n g o n l y a l i m i -t e d p a r t o f s e c t o r c a p a c i t y . Hence, t h e c u b i c r e l a t i o n s h i p s h o u l d not e x i s t w i t h i n r a n g e o f t h e p r e s e n t s t u d y . The t e a r d i s t a n c e - t i m e r e l a t i o n s h i p s c o l l e c t e d f o r t h e s t u d y can be o b s e r v e d e a s i l y as q u a d r a t i c forms ( F i g . 10) b e s t e x p r e s s e d by s e c o n d d e g r e e e q u a t i o n s . S t a t i s t i c a l a n a l y s e s showed t h e q u a d r a t i c r e l a t i o n s h i p as c o r r e c t f o r a m a j o r i t y 3 o f t h e t r i a l s , but t h a t a t h i r d d e g r e e term, t , c o u l d be r e t a i n e d i n some e q u a t i o n s . A f u r t h e r a n a l y s i s was c o n d u c t e d t o f i n d r e a s o n s f o r t h i s . I t was d i s c o v e r e d t h a t a l l e q u a t i o n s w i t h s i g n i f i c a n t t h i r d d e g r e e terms p o s s e s s e d an e x t r e m e l y 48 _2 s m a l l r e s i d u a l v a r i a n c e 0.1 t o 8.5 X 10 . T h i s l e v e l o f r e s i d u a l v a r i a n c e made t h e v a r i a n c e r a t i o t e s t e x t r e m e l y s e n s i -t i v e . T a b l e 4 c o n t a i n s an a n a l y s i s o f v a r i a n c e model used t o t e s t s i g n i f i c a n c e o f t h e p a r t i a l r e g r e s s i o n c o e f f i c i e n t f o r measurements on a 1 0 - p l y n e w s p r i n t s p e c i m e n . T h i s a n a l y s i s 3 showed t h a t t h e t term s h o u l d be r e t a i n e d . V a r i a n c e r a t i o 3 o f t h e t term was 6.610, which was s i g n i f i c a n t a t t h e 5% l e v e l . When co m p a r i n g t h i s v a r i a n c e r a t i o t o t h e t o t a l 2 3 v a r i a n c e r a t i o v a l u e 16,739.9, i n c l u d i n g t , t , and t , t h e t v a r i a n c e c o n t r i b u t e d o n l y about 1/2500 o f t h e t o t a l v a r i a n c e . F u r t h e r m o r e , t h e m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t f o r an equa-2 3 t i o n c o n t a i n i n g t , t , and t was 0.99994, which was o n l y _5 7 X 10 l a r g e r t h a n t h e 0.99987 f o r an e q u a t i o n c o n t a i n i n g 2 o n l y t and t t e r m s . The d i f f e r e n c e o f s t a n d a r d e r r o r o f _ 2 e s t i m a t e between t h e s e two e q u a t i o n s was 6.66 X 10 mm from 22.5 mm o f t h e a v e r a g e t e a r d i s t a n c e . Such d i f f e r e n c e was i n d i s t i n g u i s h a b l e i n t h e g r a p h i c f o r m . C o n s e q u e n t l y , i t can 3 be c o n c l u d e d t h a t t h e t te r m i s not n e c e s s a r y , hence t h e q u a d r a t i c e q u a t i o n may be used t o e x p r e s s a l l t e a r d i s t a n c e -t i m e r e l a t i o n s h i p s . I n t e r p r e t a t i o n o f R e s u l t s The z e r o - s w i n g v e l o c i t y and t e a r i n g v e l o c i t y were c a l c u -l a t e d from t h e f i r s t d e r i v a t i v e s o f E q u a t i o n [ l l j , as g i v e n i n E q u a t i o n [12J . From t h e s e two v e l o c i t i e s , t h e s e c t o r n et ene r g y and paper t e a r i n g r e s i d u a l e n e rgy were c a l c u l a t e d . The 49 d i f f e r e n c e between t h e s e two energy t e r m s , when d i v i d e d by t h e number o f p l i e s i n t h e spe c i m e n , i s t h e t o t a l t e a r i n g e n e r g y p e r p l y sought i n t h i s s t u d y . By t h e s e methods, t o t a l t e a r i n g energy p e r p l y a t any i n s t a n t o f t e a r d i s t a n c e o r t i m e can be c a l c u l a t e d . A l t h o u g h t e n s e t s o f t e a r d i s t a n c e - t i m e d a t a have been measured f o r each t e s t s p e c i m e n , t e a r i n g energy at any p o i n t o f d i s t a n c e o r t i m e c o u l d have been c a l c u l a t e d . O n l y t o t a l t e a r i n g energies p e r p l y a t 0.2, 1.0, 2.0, 3.0, 4.0 and 4.3 cm d i s -t a n c e s a c r o s s t e a r s p e c i m e n s and t h e i r c o r r e s p o n d i n g t i m e s were i n c l u d e d . T h e s e a v e r a g e t e a r i n g energy p e r p l y v a l u e s t o g e t h e r w i t h a v e r a g e t e a r d i s t a n c e , t i m e , v e l o c i t y , c o n v e n -t i o n a l c r o s s m a c h i n e - d i r e c t i o n t e a r i n g s t r e n g t h and t h e i r i d e n t i f i c a t i o n a r e p r e s e n t e d i n T a b l e 2. T e a r i n g e n e r g y p e r p l y v a r i a t i o n s w i t h r e s p e c t t o t e a r d i s t a n c e and r e l a t i v e t e a r t i m e f o r each k i n d o f paper a r e f u r t h e r p r e s e n t e d i n F i g . 11a t o l i e . The r e l a t i v e t e a r t i m e s used i n t h e s e f i g u r e s a r e d e f i n e d as t i m e d i f f e r e n c e s between those r e q u i r e d f o r t h e s e c t o r t o swing t o a c e r t a i n t e a r d i s t a n c e w i t h o u t a p a p e r specimen l o a d and t h a t r e -q u i r e d f o r t h e s e c t o r t o swing o v e r t h e same d i s t a n c e when t e a r i n g a paper s p e c i m e n . The v a l u e i s o b t a i n e d by s u b t r a c t -i n g t e a r i n g swing t i m e from z e r o - s w i n g t i m e a t c o r r e s p o n d i n g t e a r d i s t a n c e . 50 T h e r e i s one common phenomenon a p p e a r i n g w i t h a l l t h e f i v e m a t e r i a l s ( F i g . 11a t o l i e ) , namely, t h a t t h e l o w e s t p l y specimen n e v e r behaved t h e same as o t h e r s p e c i m e n s h a v i n g g r e a t e r number o f p l i e s . I t i s a l s o i m p o r t a n t t o n o t e t h a t a l l o f t h e l o w e s t p l y s pecimens from each k i n d o f p a p e r had a s e c t o r s c a l e r e a d i n g below 15. The e x a c t s c a l e r e a d i n g s were 11.7, 10.5, 7.5, 14.5 and 10.0 f o r 1 0 - p l y o n i o n s k i n , 5 - p l y n e w s p r i n t , 2 - p l y bag p a p e r , 2 - p l y 5 5 . 5 - l b wrapper and 1 - p l y p a r c e l wrap, r e s p e c t i v e l y . T hese s p e c i m e n s , h a v i n g c o m p a r a t i v e l y low t e a r r e s i s t a n c e , were t o r n at r e l a t i v e l y h i g h v e l o c i t i e s . High t e a r i n g v e l o c i t y d i d not a l l o w t h e s p e c i m e n s t o r e s p o n d c o m p l e t e l y t o t h e a d v a n c i n g f o r c e , hence t h e b e h a v i o r was d i f f e r e n t from o t h e r s p e c i m e n s o f a s e r i e s which were t o r n at l o w e r t e a r i n g v e l o c i t y . S i n c e t h e s e l o w e s t p l y s p e c i m e n s behaved d i f f e r e n t l y from o t h e r s p e c i m e n s , t h e y a r e e x c l u d e d from t h e d a t a c o n t r i b u t i n g t o f u r t h e r d i s c u s s i o n . F i g u r e s 11a, f o r o n i o n s k i n , l i b , f o r n e w s p r i n t and l i d f o r 5 5 . 5 - l b . wrapper a l l showed t h e same v a r i a t i o n i n t o t a l t e a r i n g e n e rgy per p l y . T h a t i s , t h e t o t a l t e a r i n g e n e r g y r e q u i r e d t o t e a r a s i n g l e s h e e t o f p a p e r i n c r e a s e d as t h e 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 and t e a r t i m e i n c r e a s e d . I n c r e a s e i n t h e r a t e of t o t a l t e a r i n g e n e rgy p e r p l y was more p r o n o u n c e d when o n l y a s h o r t t i m e was r e q u i r e d t o t e a r t h r o u g h t h e p a p e r and a l s o when t e a r r e s i s t a n c e o f t h e p a p e r sample was low. T e a r r e s i s t a n c e can be c o n s i d e r e d as a n o t h e r e x p r e s s i o n o f number of p l i e s w i t h i n one k i n d o f p a p e r . The 51 i n c r e a s e i n t h e r a t e o f t o t a l t e a r i n g e n e rgy per p l y d e c r e a s e d as t e a r i n g t i m e was p r o l o n g e d by h i g h e r p a p e r t e a r r e s i s t a n c e . One o f t h e a d v a n t a g e s o f t h e method used i s t h a t i t r e p o r t s t h e t e a r i n g e n e rgy r e q u i r e d t o t e a r o v e r any d i s t a n c e . A l s o shown i n F i g . 11a t o l i e a r e t e a r i n g energy p r o f i l e s and t h e i r r e l a t i o n s h i p w i t h t e a r t i m e a t c e r t a i n t e a r d i s t a n c e a . Onion s k i n and n e w s p r i n t showed t h e same t e a r i n g e n e r g y - t i m e r e l a t i o n s h i p at any t e a r d i s t a n c e t h r o u g h o u t t h e t e s t . The 5 5 . 5 - l b . wrapper p r e s e n t e d a n o t h e r p a t t e r n i n t h a t , a t t h e b e g i n n i n g o f t h e t e a r , t h e t e a r i n g e n e rgy per p l y de-c r e a s e d t h e n i n c r e a s e d as t e a r i n g t i m e i n c r e a s e d ( F i g . l i d ) . T h i s was b e c a u s e one o f t h e t h r e e 4:-ply specimens was t o r n a t a d i s p r o p o r t i o n a t e l y s l o w e r v e l o c i t y . F o r example, 12.42 cm/sec compared t o 13.67 and 14.08 cm/sec a t 0.2 cm t e a r d i s -t a n c e . T h i s s l o w e r v e l o c i t y r e s u l t e d i n l e s s c a l c u l a t e d r e s i -d u a l e n ergy and i n t u r n h i g h e r t e a r i n g e n e r g y . C o n s e q u e n t l y , t h e a v e r a g e h i g h e r t e a r i n g e n e rgy o f 4:-ply specimens a l t e r e d t h e t e a r i n g e n e rgy and t i m e r e l a t i o n s h i p as shown i n F i g . l i d . O t h e r w i s e , t h e t e a r i n g e n e r g y v a l u e s o f 4;-ply specimens were s m a l l e r r e n d e r i n g F i g . l i d w i t h t h e same p a t t e r n as F i g . 11a and l i b . These t h r e e c a s e s ( F i g . 11a, l i b and l i d ) d e m o n s t r a t e t h a t t o t a l t e a r i n g e n e r g y p e r p l y i s d i r e c t l y r e l a t e d t o t h e t i m e r e q u i r e d t o t e a r t h r o u g h a p a p e r . When t h e i n c i d e n t f o r c e i s t h e same, t h e l o n g e r t i m e r e q u i r e d t o t e a r t h r o u g h a s h e e t o f 52 p a p e r o f d e f i n i t e l e n g t h means t h a t r a t e of t e a r i s s l o w e r . C o n v e r s e l y , t h e r e l a t i o n s h i p can be d e s c r i b e d as more t o t a l e n e r g y r e q u i r e d t o t e a r t h r o u g h a s i n g l e s h e e t when t h e r a t e o f t e a r i s s l o w e r . Bag paper p r e s e n t e d a n o t h e r v a r i a t i o n , as shown i n F i g . 11c. T h i s p a t t e r n can be d i v i d e d i n t o two p a r t s . The f i r s t p a r t i s from t h e b e g i n n i n g o f t e a r t o a c e r t a i n d i s t a n c e be-tween 1.0 t o 2.0 cm. In t h i s r e g i o n , t o t a l t e a r i n g e n e rgy per p l y i n c r e a s e d as number o f p l i e s and t e a r t i m e i n c r e a s e d . In t h e s e c o n d r e g i o n , which c o n t i n u e d u n t i l f a i l u r e , t o t a l t e a r i n g energy p e r p l y d e c r e a s e d as number o f p l i e s and t e a r t i m e i n c r e a s e d , and t h e n s t a r t e d t o i n c r e a s e as number o f p l i e s and t e a r t i m e was i n c r e a s e d . No u n u s u a l e x p e r i m e n t a l e r r o r s o c c u r r e d i n m e a s u r i n g t e a r i n g e n e rgy f o r t h e bag p a p e r . O t h e r w i s e , t e a r i n g e n e rgy v a l u e s c a l c u l a t e d from i n d i v i d u a l s p e c i m e n s w i t h r e p l i c a t i o n s would not have p r o v i d e d such smooth c u r v e s . The d i f f e r e n t t e a r i n g e n e r g y v a r i a t i o n w i t h r e s p e c t t o time i s t h o u g h t t o be i n h e r e n t i n t h e p a p e r sample. The p a r t i c u l a r p a p e r p r o p e r -t y i n v o l v e d i s unknown. I t has been p o i n t e d out t h a t a n e g a t i v e r e l a t i o n s h i p may o c c u r between t e a r i n g s t r e n g t h 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 . The bag p a p e r showed t h i s b e h a v i o r i n p a r t ( F i g . 5 ) . A n o t h e r p a p e r , p a r c e l wrap, p r e s e n t e d an i r r e g u l a r t e a r -53 i n g e n e r g y d i s t r i b u t i o n ( F i g . l i e ) w i t h no s p e c i a l t e n d e n c i e s . I t was f o u n d t h a t t h i s p a r c e l wrap had v e r y poor f o r m a t i o n , i r r e g u l a r d i s t r i b u t i o n o f f i b r e s and f i b r e b u n d l e s and t h i n s p o t s o v e r some p a r t s o f t h e s h e e t . Such i r r e g u l a r f i b r e d i s t r i b u t i o n can a f f e c t u s e f u l n e s s o f a l l s t r e n g t h t e s t s . When t h e r e l a t i v e t i m e s ( e x c l u d i n g t h e t e a r t i m e s f o r t h e l o w e s t p l y s p e c i m e n s f o r each paper grade) r e q u i r e d t o t e a r 4.3 cm-through b o o k l e t s o f s h e e t s c o n t a i n i n g d i f f e r e n t numbers o f p l i e s a r e d i v i d e d by, and p l o t t e d v e r s u s , number o f p l i e s , a p o s i t i v e c u r v i l i n e a r r e l a t i o n s h i p . : ; i s o b t a i n e d f o r a l l f i v e k i n d s o f p a p e r ( F i g . 1 2 ) . T h i s f i n d i n g c o n f i r m s as e x p e c t e d , t h a t l o n g e r t i m e i s r e q u i r e d t o t e a r p e r u n i t s h e e t o f p a p e r when h i g h e r number o f p l i e s a r e t o r n s i m u l t a n e o u s l y . In a d d i -t i o n , t h i s f i n d i n g a l s o c o n f i r m s t h e t i m e - d e p e n d e n t beha/ior o f t h e t e a r t e s t . F u r t h e r m o r e , t h i s can be d e s c r i b e d as r a t e o f t e a r b e i n g t i m e - d e p e n d e n t . The p r e s e n t s t u d y was d e s i g n e d a c c o r d i n g t o t h e energy d i s s i p a t i o n c o n c e p t o f p a p e r t e a r i n g p r o p e r t i e s . T h i s c o n c e p t has been w i d e l y a c c e p t e d i n e v a l u a t i n g p a p e r t e a r i n g s t r e n g t h and f o r c a l i b r a t i n g t h e E l m e n d o r f t e a r t e s t e r . Development o f E q u a t i o n £ 9^] f o r c a l c u l a t i n g t e a r i n g energy i n t h i s s t u d y i s based on t r a n s f o r m a t i o n between t e a r t e s t e r s e c t o r g r a v i -t a t i o n a l p o t e n t i a l e n e r g y and k i n e t i c e n e r g y . In t h e c o u r s e o f t e a r i n g , t h e g r a v i t a t i o n a l p o t e n t i a l e n e r gy i s t r a n s f o r m e d i n t o k i n e t i c energy t o do work i n t e a r i n g p a p e r . K i n e t i c e n e r g y i s o b t a i n e d by e v a l u a t i n g v e l o c i t y at i n s t a n t a n e o u s 54 t i m e . 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 i s no doubt about t h e t i m e - d e p e n d e n t p r o p e r t y o f v e l o c i t y and hence k i n e t i c e n e r g y i s i t s e l f t i m e - d e p e n d e n t . The t e a r i n g e n e rgy c a l c u -l a t e d as i n t h i s s t u d y i s t h e d i f f e r e n c e between two k i n e t i c e n e r g y t e r m s , t h e r e f o r e i t i s a l s o k i n e t i c e n e rgy and s h o u l d be t i m e - d e p e n d e n t as w e l l . R e s u l t s w i t h o n i o n s k i n , n e w s p r i n t and 5 5 . 5 - l b . wrapper c o n f i r m e d t h e above c o n c e p t by showing p o s i t i v e r e l a t i o n s h i p between t o t a l t e a r i n g e n e r g y p e r p l y and t e a r t i m e . T h i s p o s i t i v e c u r v i l i n e a r r e l a t i o n s h i p i s a l s o a f f e c t e d by s p e c i -men t e a r r e s i s t a n c e . T e a r t i m e was d i r e c t l y r e l a t e d t o t e s t specimen r e s i s -t a n c e , i . e . , t h e h i g h e r t h e t e a r r e s i s t a n c e t h e l o n g e r t h e t i m e r e q u i r e d t o t e a r t h r o u g h a s p e c i m e n . W i t h i n one k i n d o f p a p e r , t e a r r e s i s t a n c e i n c r e a s e d d i r e c t l y w i t h number o f p l i e s . Hence, i t can be s a i d t h a t w i t h a g i v e n p a p e r more t i m e i s r e q u i r e d t o t e a r t h r o u g h specimens... w i t h h i g h e r num-ber o f p l i e s . T h i s i s shown i n F i g . 12 ( e x c l u d i n g a l l l o w e s t p l y specimens., f o r each p a p e r grade) as a c u r v i l i n e a r r e l a -t i o n s h i p . S i n c e more t o t a l t e a r i n g e n e r g y per p l y i s r e q u i r e d when t e a r i n g o v e r l o n g e r t i m e , or a t a l o w e r r a t e , i t can t h e n be c o n c l u d e d t h a t more t o t a l t e a r i n g e n e rgy p e r p l y i s r e -q u i r e d when t e a r i n g a specimen w i t h h i g h e r number o f p l i e s . The r e s u l t s w i t h bag paper and p a r c e l wrap show t h a t i t i s n e c e s s a r y t o s l i g h t l y m o d i f y t h e c o n c l u s i o n d e s c r i b e d above. 55 T h a t i s , w h i l e t e a r i n g energy i s s t i l l t i m e - d e p e n d e n t , and more t e a r i n g energy i s r e q u i r e d t o t e a r a s i n g l e s h e e t when h i g h e r number o f p l i e s a r e t o r n s i m u l t a n e o u s l y , some i n h e r -ent s h e e t p r o p e r t i e s may have p r o f o u n d e f f e c t s . In f a c t , t e a r i n g e n e rgy c a l c u l a t e d i n t h i s s t u d y i s t h e t o t a l amount o f e n e r g y r e q u i r e d t o t e a r a s p e c i m e n . Any f a c t o r which can a l t e r s pecimen t e a r r e s i s t a n c e a l s o a l t e r s t e a r i n g e n e r g y . The r e s u l t s shown i n F i g . 13 c o n f i r m e d t h i s d i r e c t r e l a t i o n -s h i p . T a k i n g p a r c e l wrap f o r example, t h e i r r e g u l a r f i b r e d i s t r i b u t i o n n o t e d i n t h e paper s h e e t c a u s e d i r r e g u l a r t e a r r e s i s t a n c e , t h e r e f o r e c a u s e d t h e same . t o t a l t e a r i n g energy v a r i a t i o n no m a t t e r what t i m e was r e q u i r e d t o t e a r a g i v e n s p e c i m e n . Bag paper a l s o showed t h e same e f f e c t . F o r un-known r e a s o n s t h e t e a r i n g s t r e n g t h o f bag paper f i r s t de-c r e a s e d t h e n i n c r e a s e d as t h e 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 was i n c r e a s e d ( F i g . 5 ) . The same t e n d e n c y i s shown f o r t e a r i n g e n e rgy w i t h t h e same m a t e r i a l ( F i g . 1 1 c ) . These modi-f i c a t i o n s r e a f f i r m t h e c o n c l u s i o n o f W i n t e r b o t t o m and Minor ( 2 9 ) , t h a t t h e p o s i t i v e r e l a t i o n s h i p between t e a r v a l u e 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 i s not i n h e r e n t i n t h e t e a r t e s t e r , but i n t h e t y p e o f m a t e r i a l b e i n g t e s t e d . The t e a r i n g energy and t e a r d i s t a n c e r e l a t i o n s h i p f o r o n i o n s k i n p a p e r as shown i n F i g . 14 i s l i n e a r i n t h a t t h e i n -c r e a s e o f energy d i s s i p a t e d per p l y i s a l i n e a r f u n c t i o n o f t e a r d i s t a n c e . The l o n g e r t h e t e a r d i s t a n c e , p r o p o r t i o n a l l y , t h e more t e a r i n g energy r e q u i r e d p e r p l y . T h i s t e a r i n g 56 e n e r g y p e r p l y and t e a r d i s t a n c e r e l a t i o n s h i p i s f u r t h e r a f f e c t e d by t h e 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 . The h i g h e r t h e number o f p l i e s , t h e more t e a r i n g e n e rgy p e r p l y i s r e q u i r e d t o t e a r t h r o u g h t h e same d i s t a n c e compared t o fewer p l i e s . T h i s may be b e c a u s e s p e c i m e n s h a v i n g h i g h e r number o f p l i e s were t o r n at s l o w e r r a t e , and as a r e s u l t r e q u i r e d more t e a r i n g e n e r g y . The t e a r i n g e n e rgy and d i s -t a n c e r e l a t i o n s h i p o f t h e I D - p l y o n i o n s k i n specimen i n F i g . 14 f u r t h e r d e m o n s t r a t e s t h a t t h i s s p e c i m e n , b e c a u s e o f i t s low t e a r r e s i s t a n c e , behaved d i f f e r e n t l y from o t h e r specimens c o m p r i s i n g h i g h e r number o f p l i e s . The a c c e l e r a t i o n o f each t e a r i n g v e l o c i t y , 2c ( E q u a t i o n Hl3^] ) , was o b t a i n e d from .the s e c o n d d e r i v a t i v e o f t h e t e a r d i s t a n c e - t i m e , E q u a t i o n Cl l] . When t h e a v e r a g e o f t h r e e r e -p l i c a t i o n s was p l o t t e d v e r s u s 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 , a l i n e a r r e l a t i o n s h i p was o b t a i n e d f o r n e w s p r i n t , bag p a p e r , 5 5 . 5 - l b . wrapper and p a r c e l wrap, r e s p e c t i v e l y ( F i g . 1 5 ) . Onion s k i n m a i n t a i n e d a s t r a i g h t l i n e up t o about 40 p l i e s b e f o r e c h a n g i n g t o a c u r v i l i n e a r f o r m ( F i g . 1 5 ) . I t s h o u l d be n o t e d t h a t a l l f i v e l i n e s i n t e r s e c t e d t h e y - a x i s a t a l m o s t t h e same p o i n t , which r e p r e s e n t e d t h e a c c e l e r a t i o n o f t h e s e c t o r z e r o - s w i n g v e l o c i t y . T h i s p r e s e n t s e v i d e n c e t h a t a l l d a t a were measured w i t h s m a l l r a n g e o f e x p e r i m e n t a l e r r o r . L i n e a r r e l a t i o n s h i p s between t e a r a c c e l e r a t i o n and number o f p l i e s means p r o p o r t i o n a l d e c r e a s e i n t e a r i n g v e l o c i t y w i t h 57 r e s p e c t t o i n c r e a s e i n number o f p l i e s . Onion s k i n was t h e weakest member among t h e f i v e k i n d s o f paper t e s t e d . The change i n r e l a t i o n s h i p from l i n e a r t o c u r v i l i n e a r a t about 40 p l i e s may be b e c a u s e s p e c i m e n s - w i t h t o o many p l i e s had t h e same b u l k e f f e c t as d e m o n s t r a t e d by S w a r t o u t and S e t t e r -holm ( 2 4 ) . In a d d i t i o n , d i s p l a c e m e n t o f o u t e r p l i e s due t o specimen b e n d i n g when t e a r i n g i s e x p e c t e d t o be more s e r i o u s when specimen g r o s s t h i c k n e s s i s l a r g e . T h i s d i s p l a c e m e n t o f o u t e r p l i e s can be e x p e c t e d t o p r o d u c e some s e r i o u s e f f e c t on t h e d i s t r i b u t i o n o f s t r e s s e s , hence slow down t e a r i n g v e -l o c i t y by r e d u c i n g a c c e l e r a t i o n . The n e g a t i v e l i n e a r r e l a t i o n s h i p between t e a r i n g a c c e l e r -a t i o n w i t h r e s p e c t t o number o f p l i e s w i t h i n a c e r t a i n range can be used t o r e a s o n , t h a t , when t h e t o t a l s pecimen t h i c k -ness i s not t o o l a r g e , f r i c t i o n between n e i g h b o r i n g p l i e s , d i s p l a c e m e n t o f o u t e r p l i e s and i n c r e a s e i n b e n d i n g r e s i s t a n c e due t o i n c r e a s e i n t h e number o f p l i e s o r specimen b u l k do not have a s i g n i f i c a n t e f f e c t on t e a r i n g e n e r g y r e q u i r e m e n t s . O t h e r w i s e , t h e r e would be a c u r v i l i n e a r i n s t e a d o f l i n e a r r e -l a t i o n s h i p . Review o f P r e s e n t T e a r T e s t Knowledge (1) Energy d i s s i p a t i o n c o n c e p t The energy d i s s i p a t i o n c o n c e p t has been a d o p t e d f o r c a l i -b r a t i n g t h e E l m e n d o r f t e a r t e s t e r . Members o f t h e I n s t i t u t e 58 o f Paper C h e m i s t r y (16) have a p p l i e d i t t o e x p l a i n some paper t e a r i n g phenomena. The methods and m a t h e m a t i c a l model o f t h i s s t u d y a r e d e r i v e d a c c o r d i n g t o t h e c o n c e p t o f e n e rgy d i s s i p a t i o n . K i n e t i c e n e rgy d i f f e r e n c e between s e c t o r net e n e r g y and r e s i -d u a l energy a r e used t o c a l c u l a t e t e a r i n g e n e r g y . T h i s t e a r -i n g e n e rgy i s t h e n an e x p r e s s i o n o f t h e amount of e n e rgy ex-pended i n t e a r i n g a s h e e t o f p a p e r ; o r i t can be c o n s i d e r e d as t h e amount o f e n e r g y d i s s i p a t e d w i t h i n t h e s h e e t . The p r e v a l e n t t e a r i n g s t r e n g t h i s measured a c c o r d i n g t o t h e energy d i s s i p a t i o n c o n c e p t as w e l l . Hence, t e a r i n g s t r e n g t h s h o u l d r e l a t e p r o p o r t i o n a l l y t o t e a r i n g e n e r g y . T e a r i n g s t r e n g t h r e s u l t s , shown i n F i g . 5 w i t h r e s p e c t t o number o f p l i e s , a r e f u r t h e r p l o t t e d i n F i g . 13 w i t h r e s p e c t t o t h e t o t a l t e a r i n g e n e r g y per p l y o f t h e c o r r e s p o n d i n g number o f p l i e s . A l l l o w e s t p l y s p e c i m e n s f o r each k i n d o f paper have been e x c l u d e d from t h i s f i g u r e b e c a u s e o f t h e i r s p e c i a l b e h a v i o r . P o s i t i v e r e l a t i o n s h i p s were o b t a i n e d f o r a l l f i v e k i n d s o f p a p e r . A l l showed t h a t h i g h e r t e a r i n g s t r e n g t h i s always accompanied by h i g h e r e n ergy r e q u i r e m e n t f o r t e a r i n g a s i n g l e s h e e t . T h i s d i r e c t e v i d e n c e c o n f i r m e d t h a t t h e p r e s e n t p r e v a l e n t t e a r i n g s t r e n g t h i s a measure o f t h e amount o f e n e rgy d i s s i p a t e d i n t e a r i n g a p a r t i c u l a r s h e e t o f p a p e r , hence i t i s a measure o f t h e t e a r r e s i s t a n c e o f a paper a g a i n s t t h e i n c i d e n t t e a r i n g f o r c e . 59 (2) Dynamic p r o p e r t y o f paper In o r d e r t o d e t e r m i n e whether t e a r r e s i s t a n c e , m e a s u r e d i n t h e c o n v e n t i o n a l way, was b a s i c a l l y a dynamic p r o p e r t y o f paper which r e f l e c t e d t h e r e s p o n s e t o a h i g h r a t e o f l o a d i n g , H i g g i n s (15) e s t a b l i s h e d good c o r r e l a t i o n between c o n v e n t i o n -a l t e a r f a c t o r s and maximum l o a d measured d u r i n g v e r y slow t e a r i n g t e s t at a r a t e o f s t r a i n o f 0.64 mm/min. by u s i n g a D.F.P. ( D i v i s i o n o f F o r e s t P r o d u c t s ) r h e o m e t e r ( 2 2 ) . He c o n -c l u d e d t h a t t h e r a t e o f l o a d i n g was not t h e b a s i c f a c t o r which c o n t r i b u t e s t h e s p e c i a l i t y i n t h e c o n v e n t i o n a l t e a r t e s t . As t h i s s t u d y has shown, t h e c o n v e n t i o n a l t e a r t e s t i s b a s i c a l l y a f f e c t e d by r a t e o f t e a r o r r a t e o f l o a d i n g . The d i f f e r e n c e between H i g g i n s c o n c l u s i o n and t h i s s t u d y a r o s e f r o m t h e d i f f e r e n t a p p r o a c h e s u s e d . H i g g i n s used v e r y slow t e a r i n g at c o n s t a n t r a t e o f s t r a i n . T h i s s t u d y used a h i g h r a t e o f l o a d i n g and specimen s t r a i n was not c o n s i d e r e d . The l a t t e r a p p r o a c h i s c o n s i d e r e d t o r e l a t e b e t t e r t o t h e c o n v e n -t i o n a l t e a r t e s t . (3) R ate o f t e a r e f f e c t and l i m i t a t i o n o f s c a l e r e a d i n g I t has been p r o v e d i n t h i s s t u d y t h a t t e a r r a t e has an e f f e c t on t h e f i n a l t e s t v a l u e . F i g u r e s 11a, l i b , and l i d i l l u s t r a t e t h a t t h e v a r i a t i o n o f t e a r i n g e n e rgy i s l e s s p r o -nounced when l o n g e r t i m e i s r e q u i r e d t o t e a r t h r o u g h a s p e c i -men, o r when specimen t e a r r e s i s t a n c e i s l a r g e . I t has a l s o 60 been f o u n d t h a t v a r i a t i o n i n t e a r i n g e n e rgy w i t h t i m e becomes l e s s p r o n o u n c e d when t h e s e c t o r s c a l e r e a d i n g i s around and more t h a n 50. Hence, i t i s p r e f e r a b l e f o r t h e s t a n d a r d t e s t method t o have a specimen which w i l l g i v e a s e c t o r r e a d i n g a r ound or more t h a n 50, r a t h e r t h a n t h e 40 recommended i n TAPPI S t a n d a r d T 414 t s - 6 4 ( 2 6 ) . T h i s i s b e c a u s e a t s c a l e r e a d i n g 40, t e a r i n g e n e rgy v a r i a t i o n i s s t i l l not s t a b l e . Specimens used i n t h i s s t u d y c o v e r e d a wide r a n g e o f t e a r v a l u e s . A l l f i v e k i n d s o f paper showed t h a t t e a r i n g b e h a v i o r at s c a l e r e a d i n g s between 0 t o 15 i s d i f f e r e n t from t h a t w i t h s c a l e r e a d i n g s above 15. Hence, i t i s not adequate t o compare t e s t r e s u l t s which a r e o b t a i n e d from two d i f f e r e n t t e a r i n g b e h a v i o r s . W i n t e r b o t t o m and Minor (29) s u g g e s t e d u s i n g s i n g l e samples as l o n g as t h e s e d i d not g i v e v a l u e s t o o low f o r a c c u r a c y . I f r e s u l t s , and t h e i n t e r p r e t a t i o n o f t h e p r e s e n t s t u d y a r e c o r r e c t , W i n t e r b o t t o m and M i n o r ' s s u g g e s t i o n s h o u l d not be a d o p t e d , s i n c e t e a r i n g energy v a r i a t i o n o v e r t h e low t e a r r e s i s t a n c e r a n g e i s not s t a b l e . CONCLUSIONS The f o l l o w i n g c o n c l u s i o n s can be drawn from r e s u l t s o f t h e s e t e a r i n g e n e rgy s t u d i e s . 1. The e n e r g y r e q u i r e d t o t e a r p a p e r s o f t h e s t u d y was t i m e - d e p e n d e n t . More t e a r i n g e n e r g y was needed p e r u n i t p l y when t h e t e a r r a t e was s l o w e r . V a r i a t i o n i n t o t a l t e a r i n g e n e rgy p e r p l y was a f f e c t e d more when t o t a l t i m e was s h o r t . T h i s e f f e c t was r e d u c e d by p r o -l o n g i n g t e a r t i m e . 2. L o nger t i m e was r e q u i r e d p e r u n i t s h e e t when h i g h e r num-ber o f p l i e s were t o r n s i m u l t a n e o u s l y , and more energy was r e q u i r e d p e r p l y when t e a r i n g a specimen h a v i n g h i g h e r number o f p l i e s . 3. A l t h o u g h t h e energy r e q u i r e d t o t e a r t h r o u g h a specimen was t i m e - d e p e n d e n t , and more energy was r e q u i r e d when h i g h e r number o f p l i e s were t o r n s i m u l t a n e o u s l y , i n h e r e n t m a t e r i a l p r o p e r t i e s , p a r t i c u l a r l y s h e e t d i s c o n t i n u i t i e s , may have p r o f o u n d l y e f f e c t e d some t e a r i n g e n e r g y r e s u l t s . 4. The c o n v e n t i o n a l i n t e r n a l t e a r d i s t a n c e - t i m e r e l a t i o n s h i p was f o u n d t o be a p a r a b o l i c form which can be e x p r e s s e d i n terms o f a se c o n d d e g r e e e q u a t i o n . 5. The l i n e a r i n c r e a s e i n t o t a l t e a r i n g e n e rgy as t e a r d i s -t a n c e i s i n c r e a s e d i s p o s i t i v e . The r a t e o f i n c r e a s e i s f u r t h e r a f f e c t e d by 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 . 62 The l i n e a r r e l a t i o n s h i p between t e a r a c c e l e r a t i o n 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 i m p l i e s t h a t f r i c t i o n between n e i g h b o u r i n g p l i e s , d i s p l a c e m e n t o f o u t e r p l i e s and i n c r e a s e i n b e n d i n g r e s i s t a n c e due t o i n c r e a s e i n number o f p l i e s d i d not s i g n i f i c a n t l y change t e a r i n g e nergy r e q u i r e m e n t s . The p o s i t i v e r e l a t i o n s h i p f o u n d between c o n v e n t i o n a l t e a r i n g s t r e n g t h and t e a r i n g energy p r o v e s t h a t t h e e n e r g y d i s s i p a t i o n c o n c e p t i s a d e q u a t e . From f i n d i n g s o f t h i s s t u d y , i t i s s u g g e s t e d t h a t t h e s t a n d a r d t e s t method s h o u l d t r e a t a specimen g i v i n g a s e c t o r s c a l e r e a d i n g a r ound or more t h a n 50, and t h a t i n no c a s e s h o u l d s e c t o r s c a l e r e a d i n g s l o w e r t h a n 15 be c o n s i d e r e d a c c e p t a b l e . REFERENCES 1. A n d e r s s o n , •. and 0. F a l k . 1966. Spontaneous c r a c k f o r m a t i o n i n p a p e r . Svensk P a p p e r s t i d . 69: 91-99. 2. B a l o d i s , V. 1963. 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 . 14: 284-304. 3. B e r g e a , E. 1942. The c a l i b r a t i o n o f E l m e n d o r f ' s t e a r i n g t e s t e r . Svensk P a p p e r s t i d . 45: 71-73. 4. B r e c h t , W. and 0. Imset. 1933/1934. The i n i t i a l t e a r and t h e t h r o u g h - t e a r r e s i s t a n c e o f p a p e r s . Z e l l s t o f f &. P a p i e r 13: 564-567; 14: 14-16. (Not s e e n , from r e f e r e n c e 1 6 ) . 5. C a r s o n , F.T. and L.W. S n y d e r . 1928. I n c r e a s i n g t h e c a p a -c i t y o f t h e E l m e n d o r f 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 s e e n , from r e f e r e n c e 8 ) . 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. ed. 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 York, 2113 pp. 7. C l a r k , J . D'A. 1932. C a l i b r a t i o n o f E l m e n d o r f t e a r i n g t e s t e r . Paper T r a d e J . 9 4 ( 1 ) : 33-34. 8. Cohen, W.E. and A . J . Watson. 1949. The measurement o f 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 . P r o c . APPITA 3: 212-244. 9. C o t t r a l l , L.G. 1932. T e a r i n g r e s i s t a n c e of p a p e r . Paper Maker B r i t i s h Paper T r a d e J . 8 3 ( 3 ) : T5127-132. (Not s e e n , from B.I.P.C. 2 ( 9 ) : 246. 10. D i n w o o d i e , J.M. 1965. The r e l a t i o n s h i p between f i b r e mor-p h o l o g y and paper p r o p e r t i e s : A r e v i e w o f l i t e r -a t u r e . T a p p i 48: 440-447. 11. E l m e n d o r f , A. 1920. T e s t i n g t h e t e a r i n g s t r e n g t h o f p a -p e r . Paper T r a d e J . 7 0 ( 1 6 ) : 213, 215, 217. (Not s e en, from r e f e r e n c e 1 6 ) . 12. Fanselow, J.R. and J . L . F a n s e l o w . I960. The use of t h e p r o d u c t o f b u r s t and t e a r v a l u e s as 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: 205-215A. 63 R e f e r e n c e s ( c o n t ' d ) 64 13. G i e r t z , H.W. and T. H e l l e . I 9 6 0 . On t h e t e a r s t r e n g t h o f p a p e r . Norsk S k o g i n d . 14: 455-469. (Not se e n , from B.I.P.C. 3 1 ( 7 ) : 1006.) 14. H a r d a c k e r , K.W. and J.A. Van den A k k e r . 1950. I n s t r u -m e n t a t i o n s t u d i e s L I X . 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 . T a p p i 3 3 ( 1 1 ) : 109-11BA. 15. H i g g i n s , H.G. 1958. The s t r u c t u r e and p r o p e r t i e s o f pa-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 12: 1-24. 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. T e a r i n g s t r e n g t h o f p a p e r . Paper T r a d e J . 1 1 8 ( 5 ) : 13-16. 17. J o n e s , H.W.H. and W. G a l l a y . 1952. T e a r i n g s t r e n g t h and i t s r e l a t i o n s h i p t o b a s i s w e i g h t . P u l p Paper Mag. Can. 53: 116-120. 18. L i , J.C.R. 1964. 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 , pp. 176-181. 19. M a l l e t t , E . and R. Marx. 1923/1924. C a l i b r a t i o n o f t h e E l m e n d o r f t e a r i n g t e s t e r . Paper Maker A s s o c . T e c h . S e c . P r o c . 4: 212-222. (Not se e n , from r e f e r e n c e 8 ) . 20. Ranee, H.F. 1949. 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 o f p a p e r . W o r l d ' s Paper T r a d e Rev. 131 ( 3 ) : T e c h . Sup. 1-7; ( 7 ) : T e c h . Sup. 9-13. 21. Rydholm, 5.A. 1965. P u l p i n g P r o c e s s e s . 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 York. 1269 pp. 22. S a n t e r , L., H.G. H i g g i n s and J.W.P. N i c h o l l s . 1955. The 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. Rheometer. A u s t . J . A p p l . S c i . 6: 197-207. 23. S e a r s , F.W. and M.W. Zemansky. 1952. C o l l e g e P h y s i c s . A d d i s o n - W e s l e y P u b l i s h i n g Co., Cambridge, pp. 22-23. 24. S w a r t o u t , J.T. and V.C. S e t t e r h o l m . 1963. E f f e c t o f num-ber 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. Res. Note No. FPL-05. 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 Paper I n d u s t r y . 1949. C o n d i t i o n i n g 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 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 Paper I n d u s t r y , New Y o rk. 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 Paper I n d u s t r y . 1964. 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 t s - 6 4 , 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 o f P u l p and Paper I n d u s t r y , New Y o r k . 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 . P a r t I I . The i n f l u e n c e o f s t i f f n e s s . Svensk P a p p e r s t i d , 56: 173-177. 28. Wink, W.A. and R.H. Van E p e r e n . 1963. Does t h e Elmen-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 ? T a p p i 46: 323-325. 29. W i n t e r b o t t o m , M. and J . E . M i n o r . 1937. The t e s t f o r t e a r i n g r e s i s t a n c e . Paper In d . 1 8 ( 1 1 ) : 928-930. (Not s een, from B.I.P.C. 7 ( 7 ) : 2 3 4 . ) . 66 T a b l e 1 P a r a m e t e r s o f t h e f i v e p a p e r g r a d e s used i n t h e s t u d y * ** *** PAPER CALIPER BASIS WEIGHT TEARING GRADE m i l g/m 2 STRENGTH q/ s h e e t Onion s k i n 1.70 28.5 14.1 N e w s p r i n t 3.41 52.1 42.7 Bag pa p e r 2.88 48.8 57.0 5 5 . 5 - l b . wrapper 5.90 90.3 128.0 P a r c e l wrap 6.65 97.6 156.0 # D a t a were measured a c c o r d i n g t o TAPPI S t a n d a r d T411 m-44. B a s i s w e i g h t i s d e f i n e d as t h e we i g h t o f paper i n grams p e r s q u a r e meter. Data were measured a c c o r d i n g t o TAPPI Stan d a r d . T410 bs-61. *** C r o s s m a c h i n e - d i r e c t i o n t e a r i n g s t r e n g t h s measured 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 . 67 T a b l e 2 Avera g e t e a r d i s t a n c e , t i m e , v e l o c i t y , e n e r g y and s t r e n g t h v a l u e s f o r f i v e paper g r a d e s AVERAGE PAPER NO. TEAR TEAR ZERO-SWING SECTOR NET TEARING* OF DISTANCE TIME VELOCITY ENERGY STRENGTH GRADE PLIES (cm) ( s e c ) (cm/sec) ( e r g ) ( g / s h e e t ) z e r o -swing 0.2 0.078 15.13 114.43 0.5 0.098 1.0 0.127 19.84 196.73 1.5 0.150 2.0 0.171 24.19 292,53 2.5 0.190 3.0 0.209 27.83 387.20 3.5 0.226 4.0 0.243 31.20 486.68 4.3 0.254 32.20 518.53 0.00 AlTERTGT PAPER NO. TEAR TEAR TEARING TEARING TEARING OF DISTANCE TIME VELOCITY ENERGY STRENGTH GRADE PLIES (cm) ( s e c ) (cm/sec) ( e r g ) ( g / s h e e t ) o n i o n 0.2 0.084 14.48 0.96 0.5 0.104 1.0 0.133 18.92 1.77 1.5 0.157 2.0 0.179 23.21 2.32 2.5 0.200 3.0 0.220 26.91 2.51 3.5 0.239 4.0 0.255 30.16 3.20 4.3 0.264 31.01 3.95 These a r e c r o s s - m a c h i n g d i r e c t i o n t e a r i n g s t r e n g t h T a b l e 2 c o n t ' d 68 AVERAGE PAPER GRADE NO. OF PLIES TEAR DISTANCE TEAR TIME TEARING VELOCITY TEARING ENERGY (cm) ( s e c ) (cm/sec) ( e r g ) 0.2 0.087 14.60 0.39 0.5 0.108 1.0 0.137 18.78 1.02 1.5 0.161 2.0 0.184 22.76 1.68 2.5 0.204 3.0 0.225 26.23 2.16 3.5 0.245 4.0 0.262 29.36 2.79 4.3 0.270 30.06 3.33 0.2 0.094 13.57 0.56 0.5 0.114 1.0 0.146 17.19 1.23 1.5 0.174 2.0 0.198 20.76 1.93 2.5 0.221 3.0 0.243 23.85 2.57 3.5 0.263 4.0 0.284 26.62 3.31 4.3 0.294 27.33 3.63 0.2 0.096 12.56 0.59 0.5 0.120 1.0 0.157 15.43 1.30 1.5 0.186 2.0 0.214 18.28 2.09 2.5 0.240 3.0 0.263 20.67 2.89 3.5 0.288 4.0 0.312 23.05 3.68 4.3 0.323 23.59 4.01 0.2 0.097 12.45 0.53 0.5 0.122 1.0 0.160 14.64 1.28 1.5 0.191 2.0 0.223 16.84 2.15 2.5 0.250 3.0 0.276 18.68 3.04 3.5 0.302 4.0 0.330 20.57 3.93 4.3 0.344 21.05 4.24 TEARING STRENGTH ( g / s h e e t ) o n i o n s k i n 20 4 0 60 70* 11.80 13.00 14.00 14.30 * r e s u l t o f a s i n g l e measurement w i t h o u t r e p l i c a t i o n 69 T a b l e 2 c o n t ' d AVERAGE PAPER NO. TEAR TEAR TEARING TEARING TEARING OF DISTANCE TIME VELOCITY ENERGY STRENGTH GRADE PLIES (cm) ( s e c ) (cm/sec) ( e r g ) ( g / s h e e t ) o n i o n s k i n 80 0.2 0.097 11.53 0.60 0.5 0.124 1.0 0.167 13.02 1.40 1.5 0.204 2.0 0.238 14.54 2.34 2.5 0.270 3.0 0.300 15.86 3.27 3.5 0.331 4.0 0.365 17.25 4.22 4.3 0.382 17.60 4.55 15.30 news- 5 0.2 0.084 14.31 2.40 p r i n t 0.5 0.105 1.0 0.134 18.73 4.26 1.5 1.158 2.0 0.181 22.91 6.01 2.5 0.202 3.0 0.221 26.42 7.66 3.5 0.240 4.0 0.258 29.66 9.39 4.3 0.267 30.51 10.62 33.60 10 0.2 0.086 14.35 1.15 0.5 0.106 1.0 0.138 18.07 3.36 1.5 0.164 2.0 0.187 21.51 6.13 2.5 0.208 3.0 0.229 24.54 8.60 3.5 0.250 4.0 0.270 27.43 11.05 4.3 0.279 28.09 12.39 41.60 15 0.2 0.093 13.46 1.59 0.5 0.114 1.0 0.147 16.64 3.89 1.5 0.176 2.0 0.200 19.75 6.50 2.5 0.225 3.0 0.247 22.49 8.95 3.5 0.270 4.0 0.292 25.08 11.48 4.3 0.302 25.66 12.62 42.70 70 T a b l e 2 c o n t ' d AVERAGE PAPER GRADE NO. OF PLIES TEAR DISTANCE TEAR TIME TEARING VELOCITY TEARING TEARING ENERGY STRENGTH (cm) ( s e c ) (cm/sec) ( e r g ) 0.2 0.094 12.98 1.51 0.5 0.118 1.0 0.153 15.29 3.99 1.5 0.185 2.0 0.213 17.69 6.80 2.5 0.240 3.0 0.264 19.72 9.64 3.5 0.290 4.0 0.315 21.75 12.51 4.3 0.328 22.28 13.52 0.2 0.102 11.74 1.82 0.5 0.130 1.0 0.168 13.51 4.22 1.5 0.204 2.0 0.237 15.34 7.00 2.5 0.268 3.0 0.297 16.95 9.74 3.5 0.326 4.0 0.356 18.51 12.62 4.3 0.372 18.93 13.57 0.2 0.083 14.86 2.04 0.5 0.102 1.0 0.130 19.00 8.14 1.5 0.155 2.0 0.177 23.11 12.70 2.5 0.198 3.0 0.217 26.59 16.90 3.5 0.236 4.0 0.254 29.76 21.91 4.3 0.263 30.51 26.50 0.2 0.087 14.65 1.18 0.5 0.107 1.0 0.138 18.81 4.54 1.5 0.163 2.0 0.186 21.94 8.65 2.5 0.208 3.0 0.227 24.98 12.52 3.5 0.248 4.0 0.267 27.91 16.19 4.3 0.277 28.67 17.93 news-p r i n t 20 25 bag p a p e r 45.10 46.60 60.00 58.20 T a b l e 2 c o n t ' d 71 AVERAGE PAPER NO. TEAR TEAR TEARING TEARING TEARING OF DISTANCE TIME VELOCITY ENERGY STRENGTH GRADE PLIES (cm) ( s e c ) (cm/sec) ( e r g ) ( g / s h e e t ) bag pape r 10 14 I B 22 ) e c ) r g ) 0.2 0.089 14.23 1.32 0.5 0.110 1.0 0.142 17.35 4.61 1.5 0.169 2.0 0.194 20.46 8.32 2.5 0.217 3.0 0.238 23.08 12.09 3.5 0.260 4.0 0.281 25.62 15.84 4.3 0.293 26.32 17.22 0.2 0.091 13.53 1.64 0.5 0.113 1.0 0.147 16.15 4.73 1.5 0.177 2.0 0.203 18.78 8.29 2.5 0.229 3.0 0.253 21.10 11.75 3.5 0.275 4.0 0.298 23.22 15.51 4.3 0.312 23.87 16.69 0.2 0.095 12.63 1.93 0.5 0.119 1.0 0.155 14.62 4.99 1.5 0.190 2.0 0.220 16.78 8.43 2.5 0.246 3.0 0.273 18.53 11.97 3.5 0.301 4.0 0.326 20.29 15.60 4.3 0.343 20.86 16.72 0.2 0.096 12.24 1.80 0.5 0.123 1.0 0.163 13.20 4.98 1.5 0.201 2.0 0.236 14.25 8.68 2.5 0.268 3.0 0.300 15,17 12.37 3.5 0.333 „ 4.0 0.365 16.11 16.22 4.3 0.387 16.44 17.43 57.00 57.00 57.60 59.40 T a b l e 2 c o n t ' d 72 AVERAGE PAPER NO. TEAR TEAR TEARING TEARING TEARING OF DISTANCE TIME VELOCITY ENERGY STRENGTH GRADE PLIES (cm) ( s e c ) (cm/sec) ( e r g ) ( g / s h e e t ) 5 5 . 5 - l b . 2 0.2 0.080 13.32 12.85 wrapper 0.5 0.101 1.0 0.132 17.88 18.42 1.5 0.158 2.0 0.182 22.35 21.34 2.5 0.202 3.0 0.223 25.92 25.68 3.5 0.241 4.0 0.259 29.12 31.30 4.3 0.270 30.06 33.31 116.00 4 0.2 0.084 13.39 6.21 0.5 0.108 1.0 0.140 17.16 12.38 1.5 0.167 2.0 0.192 20.68 19.67 2.5 0.214 3.0 0.235 23.60 27.18 3.5 0.255 4.0 0.277 26.41 34.50 4.3 0.289 27.23 36.94 122.00 6 0.2 0.092 13.17 4.62 0.5 0.115 1.0 0.149 15.98 11.52 1.5 0.180 2.0 0.207 18.77 19.40 2.5 0.231 3.0 0.254 21.09 27.46 3.5 0.278 4.0 0.300 23.33 35.74 4.3 0.315 24.06 38.17 127.10 8 0.2 0.097 12.26 4.91 0.5 0.123 1.0 0.161 14.30 11.82 1.5 0.195 2.0 0.226 16.39 19.78 2.5 0.254 3.0 0.281 18.16 27.80 3.5 0.308 4.0 0.334 19.87 36.15 4.3 0.352 20.46 38.65 133.10 T a b l e 2 c o n t ' d 73 AVERAGE PAPER NO. TEAR TEAR TEARING TEARING TEARING OF DISTANCE TIME VELOCITY ENERGY STRENGTH GRADE PLIES (cm) ( s e c ) (crn/sec) ( e r g ) ( g / s h e e t ) p a r c e l wrap ) e c ) m/se r g ) 0.2 0.083 14.21 13.48 0.5 0.104 1.0 0.134 18.67 22.37 1.5 0.158 2.0 0.181 22.79 32.94 2.5 0.201 3.0 0.221 26.34 40.18 3.5 0.240 4.0 0.258 29.52 51.05 4.3 0.268 30.39 56.72 0.2 0.090 14.22 4.44 0.5 0.111 1.0 0.143 17.67 13.54 1.5 0.170 2.0 0.193 20.95 24.36 2.5 0.215 3.0 0.236 23.75 35.04 3.5 0.258 4.0 0.277 26.42 45.89 4.3 0.290 27.25 49.11 0.2 0.094 12.63 6.95 0.5 0.121 1.0 0.156 15.82 14.31 1.5 0.184 2.0 0.211 18.73 23.44 2.5 0.236 3.0 0.261 21.31 32.02 3.5 0.283 4.0 0.305 23.58 41.71 4.3 0.320 24.40 44.16 0.2 0.103 11.94 6.16 0.5 0.131 1.0 0.170 13.73 14.65 1.5 0.205 2.0 0.237 15.51 24.61 2.5 0.267 3.0 0.296 17.07 34.50 3.5 0.325 4.0 0.354 18.62 44.77 4.3 0.372 19.09 48.03 160.00 164.00 157.90 163.80 74 T a b l e 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 c o n d u c t i v e p l i e s u sed i n t h e s t u d y METHOD CONDUCTIVE PREPAR- SHARPNESS TEARING MATERIAL ATION OF STEPS STRENGTH TIME ( g / s h e e t ) 4 p e r f o r a t e d c o n d u c - medium wors t 55.3 t i v e p a p e r 5 g r a p h i t e l i n e s longe.st medium 55.5 6 s i l v e r p a i n t l i n e s s h o r t e s t b e s t 54.4 C o n t r o l no c o n d u c t i v e ma- 55.5 t e r i a l added T h e s e a r e c r o s s m a c h i n e - d i r e c t i o n t e a r i n g s t r e n g t h s (n = 5 ) . 75 T a b l e 4 T e s t o f s i g n i f i c a n c e f o r p a r t i a l r e g r e s s i o n c o e f f i c i e n t f o r t h e n e w s p r i n t t e n - p l y specimen SOURCE SUM OF SQUARES D.F. MEAN SQUARE R e g r e s s i o n due t o t , t 2 , and t 3 1,890.2742 (1) Due t o t and t 2 , i g n o r i n g 1,890.0254 t 3 (2) A d d i t i o n due t o t 3 R e s i d u a l T o t a l 0.2488 0.2259 1,890.5000 3 630.0914 2 945.0127 1 0.2488 6 0.0376 9 16,739.9** 25,106.6** 6.6* s i g n i f i c a n t a t 5% l e v e l s i g n i f i c a n t a t 1% l e v e l 76 T a b l e 5 T e a r d i s t a n c e - t i m e r e l a t i o n s h i p s f o r f i v e 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 r e p l i c a t i o n s PAPER GRADE NO. OF PLIES REPLI-CATION REGRESSION EQUATIONS n = 10 a b (T) c C T ) R* S E E z e r o 0 1 -6.3734 64.8480 524.552 0.99988 0.2504 swing 0 2 -7.1554 77.4525 476.396 0.99987 0.2600 0 3 -7.6369 82.6602 460.313 0.99983 0.3030 o n i o n 10 1 -6.6601 65.1748 464.815 0.99989 0.2435 s k i n 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 news- 5 1 -6.8220 69.4849 440.333 0.99991 0.2233 p r i n t 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 v a l u e s a r e h i g h l y s i g n i f i c a n t a t 1,% l e v e l , a l l e x c e e d i n g 0.999. T a b l e 5 c o n t ' d 77 PAPER GRADE news p r i n t bag p a p e r NO. RE P L I - REGRESSION EQUATIONS R S E C OF n = 10 t PLIES CATION a b (T) c C 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 T a b l e 5 c o n t ' d 78 PAPER NO. REP L I - REGRESSION EQUATIONS R S E F OF n = 10 t GRADE PLIES CATION a b (T) c (Td) 5 5 . 5 - 2 1 -5.5841 63.2721 441.483 0.99976 0.3579 l b . wrapper 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 p a r c e l 1 1 -6.7607 59.2432 462.346 0.99996 0.1434 wrap 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 F i g . 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 p r i n c i p l e . Angles are defined by the v e r t i c a l l i n e and the l i n connecting the se c t o r center of mass and axis of ro ta t i o n . 0, = i n i t i a l angle before swinging or t e a r i n g a specimen," 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 energy, and 0^ = angular displacement of s e c t o r center of mass re p r e s e n t i n g s e c t o r r e s i d u a l energy 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 s t r e s s - s t r a i n - t i m e r e l a t i o n s h i p s f o r p o l y m e r i c m a t e r i a l s . springs \ dashpots elastic visco-elastic I time dependent flow load 81 F i g . 3. C o n v e r s i o n o f e n e r g i e s i n t h e . b a l l i s t i c -t y p e t e a r i n g p r o c e s s . begin finish tearing tearing top SECTOR VERTICAL POSITION bottom 82 F i g . 4. S c h e m a t i c d i a g r a m s i l l u s t r a t i n g t h e c o n v e r s i o n o f energies i n , a, s e c t o r swing w i t h o u t s p e c i m e n , b. s e c t o r swing when t e a r i n g a s p e c i m e n . mgy mgy mgy mgy •Tot "Tot m g y ^ l = mgy 1 2 ( = + m g y 1 + j ( v 1 , t 1 ) "Tot = mgy 1 2 = 2-IT!V/2 + + T . E . = t o t a l e n e r gy of a s y s t e m , = s e c t o r mass, = g r a v i t a t i o n a l a c c e l e r a t i o n , = v e r t i c a l p o s i t i o n o f t h e s e c t o r c e n t e r o f mass b e f o r e s w i n g , = v e r t i c a l d i s p l a c e m e n t o f t h e s e c t o r c e n t e r o f mass, = t a n g e n t i a l v e l o c i t y o f t i m e t-^, at z e r o - s w i n g , = t a n g e n t i a l v e l o c i t y a t time t y , a t t e 3 r i n g - s w i n g , and | ( v ^ , t ^ ) &. j(v2»t2) = f r i c t i o n terms i n c a s e a and b., ' r e s p e c t i v e l y ^ 83 F i g . 5. R e l a t i o n s h i p s between c r o s s m a c h i n e - d i r e c t i o n t e a r i n g s t r e n g t h 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 p a p e r g r a d e s . 160 A onion skin -. • . : newsprint ' " * • '•' -• - . ..V . -e bag paper M O - 55-5-lb wrapper '- -0- parcel wrap •f-hee hee 1 20 10 V . 100 -X 1-(D Z LU 80 -or t— 60 - ^ o - a - — — g _ — a - -e o z or < 40 -LU 20 0 -A & • i i . A — & • i • — A A 1 0 . 5 10 15 20 25 newsprint, bag paper, 55-5-lb wropper, parcel wrap ' : I | I ' I I 10 20 30 40 50 60 70 80 o n i o n s k i n NUMBER OF PLIES TORN SIMULTANEOUSLY 84 F i g . 6. 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 m a t e r i a l s . a. P e r f o r a t e d e l e c t r i c a l c o n d u c t i v i t y paper as c e n t e r p l y . b. G r a p h i t e c o n d u c t i v e l i n e s a p p l i e d t o t h e c e n t e r p l y . c. S i l v e r c o n d u c t i v e l i n e s a p p l i e d t o t h e c e n t e r p l y . d. 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 t o t h e c e n t e r p l y . (see F i g . 7 ) . 86 F i g . 7. 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 f o r 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 c o n d u c t i v e m a t e r i a l s , r e s p e c t i v e l y . F i g . 8. E l e c t r i c a l c i r c u i t used f o r m e a s u r i n g t h e t e a r d i s t a n c e - t i m e r e l a t i o n s h i p . 88 F i g . 9. S et-up used f o r t h e s t u d y 1. T h w i n g - A l b e r t I n s t r u m e n t Co. No. 60-100 E l m e n d o r f t e a r i n g t e s t e r , 2. T e k t r o n i x , I n . , t y p e 564 s t o r a g e o s c i l l o s c o p e e q u i p p e d w i t h 3B4 t i m e base and 3A3 d u a l t r a c e d i f -f e r e n t i a l a m p l i f i e r p l u g - i n u n i t s . 3. Type 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 t o t h e o s c i l l o s c o p e . 4. B a t t e r y e l i m i n a t o r . 5. 6 - v o l t b a t t e r y . 6. 6-decade r e s i s t a n c e k i t . 7. V o l t m e t e r . 8. P i l o t box. T E A R DISTANCE, ( Cm) I-1-in CD cr — i Ci 0) CD n U3 Qi zr l-i TJ n O a c TJ H-ro 05 < h rt-ro • 01 • n ro c 1 3 c+ 5" H-CO 3 H CD O H -n CD I—1 TJ DJ ru c+ TJ H-ro O H • CD TJ : r H-H-TJ CO ia U5 -tj H-o CO 3 fl) a i-i I • \ CD CT a. • o Z3 P° 3 68 90 F i g . 11a. T e a r i n g e n e r g y , d i s t a n c e and time r e l a t i o n s h i p s f o r u n g l a z e d o n i o n s k i n . Broken and 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 d i s t a n c e s , r e s p e c t i v e l y . LU 5 4 CO CO < 2 or o i-o LU CO e> or LU z LU O z or < LU 3 h J I I ; I I I 2 4 6 8 10 12 14 RELATIVE T E A R TIME, ( I0"2 Sec) 91 F i g . l i b . 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 n e w s p r i n t . 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 d i s t a n c e s , r e s p e c t i v e l y . - 1 4 5 a. | 2 CO to < S I 0 | CC o I-GO O 6 tr UJ z UJ o 4 z or < H 2 2 4 RELATIVE 8 10 12 14^  TEAR TIME, ( I0" 2 Sec) 92 F i g . 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 bag p a p e r . Broken and 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 d i s -t a n c e s , r e s p e c t i v e l y . O z or < 2 - 0 C M ' / / / / l-Ocm 0 2 4 6 8 10 12 14 RELATIVE T E A R TIME, ( 10"2 Sec) 93 F i g . l i d . T e a r i n g e n e r g y , d i s t a n c e 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. wrapper. Broken and 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 d i s -t a n c e s , r e s p e c t i v e l y . 4-3 cm w UJ m CO CO < S or o i-u UJ (O >• o or UJ z UJ or < UJ — 4 0 c m 10 1 0 cm 4 6 8 10 12 RELATIVE TEAR TIME, I I0"2 Sec) 14 94 F i g . l i e . T e a r i n g e n e r g y , d i s t a n c e and time r e l a t i o n s h i p s 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 paper p l i e s and t e a r d i s t a n c e s , r e s p e c t i v e l y . 6 0 0 I t 1 I 1 ; —I _ 1 1 0 2 4 6 8 10 12 14 RELATIVE TEAR TIME, ( 10' 2 Sec) 95 F i g . 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 t i m e p e r u n i t p l y 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 . * u V) lO 16 I 4 12 u 10 2 or UJ i -UJ > r-<. _J UJ or 8 / r / • — onion skin newsprint --S — — X — bog paper — . 55-5-lb wrapper parcel wrap D B— — 0 5 10 15 20 25 newsprint, bag paper, 55-5-lb wrapper, parcel wrap 10 20 30 40 onion 50 skin 60 70 80 N U M B E R 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 96 F i g . 13. R e l a t i o n s h i p s between t e a r i n g s t r e n g t h and t e a r i n g e n e rgy f o r f i v e paper g r a d e s . I65r 164 ^ 163 x i-z LU or o z or < LU «M62 o * 161 u w o a. 160 159 158 16 15 14 13 c o c o • 12 A- onion skin newsprint - -£J bog paper — • 55-5-lb wrapper ©. porcel wrap • 7 11 35 / I 3 4 u> onion skin 12 ,I3+ n e w s p r i n t 14 16 17 bag p a p e r 18 - 4 7 - 60 - 4 6 • -59 -- 4 5 +-c a. -44*1- s s : a. u> 5 a* o» - 4 3 c O 57 -- 4 2 -41 - 56 --.134 -132 130 128 jf u .O 126 7 IN M IN 124 122 120 36 37 38 55-5-lb wrapper 39 43 49 45 47 p a r c e l w r a p TEARING ENERGY / SECTOR MASS / PLY, ( E r g ) 97 F i g . 14. T e a r i n g e n e rgy 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 u n g l a z e d o n i o n s k i n . F i g . 15. R e l a t i o n s h i p s between t e a r i n g a c c e l e r a t i o n 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 p a p e r g r a d e s . 1 0 0 0 ~ 8 0 0 -6 0 0 -4 0 0 -2 0 0 2 4 6 n e w s p r i n t , b a g 10 12 p a p e r , 14 16 18 2 0 5 5 - 5 - l b w r a p p e r , 2 2 2 4 2 6 p a r c e l w r a p 10 2 0 N U M B E R 3 0 o n i o n O F P L I E S 4 0 s k i n T O R N 5 0 6 0 7 0 S I M U L T A N E O U S L Y 8 0 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 Study and V o l t a g e V a r i a t i o n i n t h e T e a r i n g P r o c e s s . a. C i r c u i t f o r m e a s u r i n g t h e t e a r d i s t a n c e - t i m e r e l a t i o n s h i p . 1 V v W l ! r* I V x X - Y J ['conductive ply "j i 1 Re b. T r a c e o f t h e v o l t a g e v a r i a t i o n on t h e o s c i l l o s c o p e . TEAR DISTANCE, (Cm) 0-2 10 2 0 3 0 4-0 43 I ; ! i — • ' 1 i 0 I 2 3 4 5 T E A R TIME, ( ICT1 Sec) Appendix 1 c o n t ' d 100 c. V o l t a g e v a r i a t i o n i n t h e c o n d u c t i v e p l y d u r i n g t h e t e a r i n g p r o c e s s . The v o l t a g e d i f f e r e n c e (V ) a c r o s s t h e c o n d u c t i v e p l y i s : v = iR r i 9 ~ i x e L_ _ l where: i = c u r r e n t (Amp) R^ = e q u i v a l e n t r e s i s t a n c e , t h e c o m b i n a t i o n o f r e s i s t o r s r ^ t o r ^ g i n p a r a l l e l , o f t h e specimen c o n d u c t i v e p i y . 1 = 1 + 1 + .... + 1 R e r l r 2 r 1 0 A 6 v o l t b a t t e r y i s used i n t h e c i r c u i t ( Appendix l a ) , hence V = 6 = i (R + R ) r 2 0 ^ | o a e 1_ _ l where: R g = r e s i s t a n c e chosen t o be e q u a l t o t h e e q u i v a l e n t r e s i s t a n c e o f t h e c o n d u c t i v e p l y . T h e r e f o r e , v = 6 i R e _ __ 6 Re = 3 v o l t * i ( R a + R B ) R a + R e T h i s i s t h e v o l t a g e d i f f e r e n c e b e f o r e t e a r i n g s t a r t s . The e q u i v a l e n t r e s i s t a n c e , R , i n c r e a s e s as t e a r i n g i s i n p r o g r e s s . Hence, V^ i n c r e a s e s as t e a r i n g p r o g r e s s e s , u n t i l t h e c o n d u c t i v e p l y i s c o m p l e t e l y t o r n . Then, V^ r e a c h e s i t s maximum o f 6 v o l t , as shown i n Appendix l b . 101 APPENDIX 2 I n t e r - r e l a t i o n s h i p Between T e a r D i s t a n c e , Time, V e l o c i t y and Energy i n t h e B a l l i s t i c - t y p e T e a r i n g P r o c e s s . The r e l a -t i o n s h i p between z e r o - s w i n g d i s t a n c e ( L ^ ) and t i m e ( t ^ ) i s : L l " a l + b l * l + c l t l 2 The r e l a t i o n s h i p between t e a r d i s t a n c e ( l _ 2 ) an,d t i m e {t^) i s : L 2 = a 2 + b 2 t 2 + c 2 t 2 2 c 2 ^ 0 C u b l ] Both E q u a t i o n s Q 1 1 a J a n d J ^ l l b J show t h a t t h e - t e a r d i s t a n c e -t i m e r e l a t i o n s h i p i s c u r v i l i n e a r . The r e l a t i o n s h i p between z e r o - s w i n g v e l o c i t y ( v^) and t i m e ( t ^ ) i s : v x = + 2 c 1 t 1 c ^ O L12a~3 Th e r e l a t i o n s h i p between t e a r i n g v e l o c i t y (v,,) and t i m e ( t 2 ) i s : v 2 = b 2 + 2 c 2 t 2 c 2 / ° C 1 2 b J B o t h E q u a t i o n s £ l 2 a ^ | and £ l 2 b 2 | show t h a t t h e v e l o c i t y - t i m e r e l a t i o n s h i p i s l i n e a r . The r e l a t i o n s h i p between z e r o - s w i n g v e l o c i t y (v^) and d i s t a n c e ( L ^ ) can be o b t a i n e d from E q u a t i o n s £lla~ J and QL2a,~J where: v l = b l + 2 ^ 1 t 1 2 0 becomes: A p p e n d i x 2 c o n t ' d 102 S u b s t i t u t i n g E q u a t i o n £ 2 l J i n t o E q u a t i o n £lla2j g i v e s : L, = a, + b. ( V l - b l ) + c. ( V l " b M 2 1 1 1 V 2 C l 1 1 ^ 2c± / 2 2 2 = a + 2 b l v l " 2 b l + c V J L ~ 2 b i v i + bj,  1 4 c 1 1 4 c 1 ^ u 2 2 1 4cj^ 4 c 1 T r a n s p o s i t i o n o f E q u a t i o n [^ 222] g i v e s : v 1 2 = 4 c 1 L 1 - 4 a 1 c 1 + b ^ C 2 3 a H hence: v x = ± J - 4 a l C l + b x 2 C 2 3 b D By t h e same a p p r o a c h , i t can be shown t h a t t h e r e l a t i o n -s h i p between t e a r i n g v e l o c i t y ( v 2 ) and d i s t a n c e ( L 2 ) i s : v 2 2 = 4 c 2 l _ 2 - 4 a 2 c 2 + b 2 2 L 7 2 4 a U v 2 = 4 c 2 L 2 - 4 a 2 c 2 + b 2 2 L~ 2 4 b 3 Hence, t h e 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 be e x p r e s s e d as b r a n c h s (where: v ^ > 0 and v 2 " > 0 ) o f t h e h y p e r b o l a s , E q u a t i o n s Q 2 3 a 3 and C 2 4 a ] . The s e c t o r net e n e r g y i s : E n e t = = ¥ U C 1 L 1 " 4 § 1 C 1 + b l 2 ) ^ The s e c t o r r e s i d u a l energy i s : E . , . = Imv 2 = km (4c_L- - 4a-c_ + b 2 ) \~26~] r e s x d u a l 2 2 2 2 2 2 2 2 1 — —I The d i f f e r e n c e between s e c t o r net and r e s i d u a l e n e r g i e s i s t e a r i n g e n e r g y ( T . E . ) . Appendix 2 c o n t ' d 103 T r , i _ 2 _ i 2 1 / 2 2^  2 1 2 2 2 1 2 _ km | 4 ( c 1 L 1 - c 2 l _ 2 ) + 4 ( a 2 c 2 - a - ^ ) + (b " l 2 " * 2 2>] 1*1 Hence, t e a r i n g e n e rgy r e l a t e s t o t e a r d i s t a n c e l i n e a r l y . The s e c t o r net energy and t i m e ( t ^ ) r e l a t i o n s h i p can be o b t a i n e d by s u b s t i t u t i n g E q u a t i o n L7i2a~J i n t o t h e net energy 1 2 e q u a t i o n ( E n e ^ . = ^ m v j _ )• E . = Imv 2 = km (b. + 2 c.t ) 2 n e t 2 1 2 1 1 1 = km ( b x 2 + 4 b J _ c 1 t 1 +• 4 c 1 2 t 1 2 ) C 2 8 3 The s e c t o r r e s i d u a l e n e r g y ' a n d t e a r t i m e ( t 2 ) r e l a t i o n s h i p can be o b t a i n e d by s u b s t i t u t i n g E q u a t i o n Ql2bH| i n t o t h e r e s i -2 d u a l energy e q u a t i o n ( ^ p e g ^ j y a i = ~msJ2 ^ 2 2 E . , , = imv„ = km (b„ + 2 c~t_) r e s i d u a l 2 2 2 2 2 2 = km ( b 2 2 + 4 b 2 c 2 t 2 + 4 c 2 2 t 2 2 ) C 2 9 3 Hence, t h e ^ t e a r i n g e n e r g y - t i m e r e l a t i o n s h i p i s : T.E. = E . - E . . . net r e s i d u a l = km j T b x 2 - b 2 2 ) + 4 ( b ^ t , - b 0 c 0 t 0 ) 1 1 1 2 W 2*2' + 4 (Cl\2 - c 2 2 t 2 2 3 M Which shows t h a t t e a r i n g e n e rgy r e l a t e s t o t e a r t i m e c u r v i -l i n e a r l y . 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0104518/manifest

Comment

Related Items