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The duration of load effect in tension perpendicular to the grain for Douglas fir McDowall, Bruce J. 1982

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CJ THE DURATION OF LOAD EFFECT IN TENSION PERPENDICULAR TO THE GRAIN FOR DOUGLAS FIR. BY Bruce J . McDowall B.E.(Hons)., The U n i v e r s i t y of Auckland, New Zealand, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1982. © Bruce J . McDowall, 1982. 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 of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the 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 study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of 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 not be allowed without my w r i t t e n p e r m i s s i o n . Department of (CiUll- ^ / 0 6 / A J 5 V " R W 6 The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) i i ABSTRACT The d u r a t i o n of l o a d problem i n wood s u b j e c t e d to s t r e s s e s in t e n s i o n p e r p e n d i c u l a r to the g r a i n i s s t u d i e d both a n a l y t i c a l l y and e x p e r i m e n t a l l y . A n a l y t i c a l l y , v i s c o e l a s t i c models d e r i v e d from the p r i n c i p l e s of f r a c t u r e mechanics are developed and d i s c u s s e d with r e s p e c t to t h e i r a p p l i c a t i o n to the f a i l u r e mechanism of wood i n t e n s i o n and bending. To assess the accuracy of the p r e d i c t i o n s of these models, they are compared a g a i n s t the experimental r e s u l t s from d u r a t i o n of loa d t e s t s c a r r i e d out on Douglas f i r i n t e n s i o n p e r p e n d i c u l a r to the g r a i n . T h i s f a i l u r e mode i s hypothesised as r e p r e s e n t a t i v e of the c r i t i c a l f a i l u r e i n i t i a t i n g mode of commercial m a t e r i a l . A specimen design was developed which proved very s u c c e s s f u l i n keeping the c o e f f i c i e n t of v a r i a t i o n of the short term s t r e n g t h q u i t e low f o r wood, approximately 10%. T h i s low v a r i a b i l i t y enabled the t e s t s to be c a r r i e d out with sample s i z e s much smal l e r than those used i n the d u r a t i o n of loa d t e s t i n g of commercial m a t e r i a l . i i i ACKNOWLEDGEMENTS The author would l i k e t o express h i s g r a t i t u d e to h i s re s e a r c h a d v i s o r P r o f e s s o r Borg Madsen, Dept of C i v i l E n g i n e e r i n g f o r h i s guidance support and i n s p i r a t i o n throughout t h i s r e s e a r c h p r o j e c t . He would a l s o l i k e to thank the Department of C i v i l E n g i n e e r i n g t e c h n i c i a n s f o r t h e i r w i l l i n g a s s i s t a n c e i n the manufacture of the apparatus, and h e l p f u l s u g g e s t i o n s . A l s o the f o l l o w i n g people: John S o l e s , B i l l L i p s e t t , Andy Buchanan, Mike McNab, Wally Kee, Alex A p o s t o l i , A l G r o f f , Dana Soong and Diane S o l e s are thanked f o r t h e i r a s s i s t a n c e i n s e t t i n g up and moni t o r i n g two of the experiments. A l s o Dr Ken Johns, Dean of E n g i n e e r i n g at the U n i v e r s i t y of Sherbrooke, who while on s a b b a t i c a l leave at UBC, gave h e l p f u l comments and c r i t i c i s m s at d i f f e r e n t stages of the work. Dr R.O. F o s c h i , adjunct p r o f e s s o r at the U n i v e r s i t y of B r i t i s h Columbia, i s thanked f o r h i s c r i t i c i s m s on the work and fo r h i s suggestions of improvements. In a d d i t i o n Dr D.L. Anderson, p r o f e s s o r at the U n i v e r s i t y of B r i t i s h Columbia, f o r h i s h e l p i n the i n t e r p r e t a t i o n of the v i s c o e l a s t i c models. F i n a l l y the r e s e a r c h a s s i s t a n t s h i p funded by an NSERC grant and a grant of lumber from Bay F o r e s t Products L t d , Vancouver, B.C., are g r a t e f u l l y acknoweldged. i v TABLE OF CONTENTS. CHAPTER 1 INTRODUCTION 1 1.1 D e f i n i t i o n Of The D u r a t i o n Of Load E f f e c t 1 1.2 Background 1 1 .3 Scope 4 1.3.1 I n t r o d u c t i o n 4 1.3.2 The N i e l s e n Model 4 1.3.3 The Step-wise Model 4 1.3.4 S t r e n g t h E f f e c t 4 1.3.5 Mo i s t u r e Content 5 1.3.6 C y c l i c Loading 5 1.4 Summary Of O b j e c t i v e s 6 CHAPTER 2 THEORY 7 2.1 I n t r o d u c t i o n 7 2.2 Models For Duration Of Load 7 2.3 General Theory Of V i s c o e l a s t i c F r a c t u r e Mechanics ... 10 2.4 The Step-Wise Model For The Duration Of Load E f f e c t . 14 2.5 The N i e l s e n Model For The D u r a t i o n Of Load E f f e c t . . 19 2.6 Comparison Of The N i e l s e n And Step-wise Models 22 2.7 Evidence For The N i e l s e n And Step-wise Models 25 2.7.1 I n t r o d u c t i o n 25 2.7.2 Experimental Evidence 26 2.7.3 Q u a l i t a t i v e Evidence 28 2.8 Summary 29 CHAPTER 3 EXPERIMENT DESIGN 30 3.1 I n t r o d u c t i o n L 30 3.2 Crack O r i e n t a t i o n And Propagation 31 3.3 Specimen Development 34 3.4 Apparatus i* • • r 3 6 3.5 Specimen P r e p a r a t i o n I 37 3.6 Tes t Procedure X . 38 3.7 Data A n a l y s i s .Vr- 39 3.8 Determination Of Ts 42 3.9 E r r o r s 44 3.10 Summary 45 CHAPTER 4 RESULTS 46 4.1 I n t r o d u c t i o n 46 4.2 Experiment D e s c r i p t i o n s 47 4.2.1 Experiment No. 1 , C y c l i c - 1 47 4.2.2 Experiment No.2, Two Crack Lengths 48 4.2.3 Experiment No.3, Three Crack Lengths 50 4.2.4 Experiment No.4, C y c l i c - 2 51 4.2.5 Experiment No.5, Mo i s t u r e Test 52 4.2.6 Experiment No.6, Ts 53 4.3 Summary 55 CHAPTER 5 DISCUSSION 56 5.1 I n t r o d u c t i o n 56 5.2 Experimental Method And A n a l y s i s 56 5.2.1 The Normal D i s t r i b u t i o n 56 V 5.2.2 F r a c t u r e Toughness 57 5.2.3 F i t Of The Creep And L i m i t S t r e n g t h Parameters .. 59 5.2.4 Short Term Str e n g t h B i a s 60 5.2.5 F i n i t e - I n f i n i t e Medium Adjustment 61 5.3 Creep F u n c t i o n s And Parameters 63 5.3.1 I n t r o d u c t i o n 63 5.3.2 Creep F u n c t i o n s 64 5.3.3 Creep Parameters 64 5.4 The Ts Experiment 66 5.5 Moisture Content E f f e c t 68 5.6 Str e n g t h E f f e c t 70 5.7 D e n s i t y E f f e c t 72 5.8 C y c l i c Loading 73 5.9 Confidence L i m i t s 75 5.10 General L i m i t a t i o n s 76 5.10.1 Ts Experiment 76 5.10.2 S t r e n g t h E f f e c t ; 77 CHAPTER 6 CONCLUSIONS 79 6.1 I n t r o d u c t i o n 79 6.2 Summary 80 6.3 F u r t h e r Research And A p p l i c a t i o n s 81 REFERENCES 84 FIGURES 8 7 APPENDICES 1 2 7 Appendix 1 Test Data 127 Appendix 2 Volume E f f e c t 140 Appendix 3 F a i l u r e C r i t e r i o n 141 Appendix 4 Adjustment C a l c u l a t i o n s 142 Appendix 5 A Step-wise Value For P l a s t i c Y i e l d S t r e s s .148 ABSTRACT • i i ACKNOWLEDGMENTS i 1 1 TABLE OF CONTENTS i v LIST OF FIGURES v i v i LIST OF FIGURES 1 S t r e s s F i e l d At The Crack T i p 88 2 T y p i c a l D uration Of Load P l o t For The Step-wise Model .. 89 3 N i e l s e n And Step-wise Model P l o t s 90 4 R e s u l t s Of Schniewind And Centeno( 1973) 91 5 R e s u l t s Of Bach(l975) 92 6 R e s u l t s Of Debaise Et A l . 0 9 6 6 ) 93 7 The Three Modes Of F a i l u r e 94 8 Induced S t r e s s e s In Tension Perp. To The G r a i n 95 9 Unprepared Specimen 96 10 Mounted Specimen 97 11 Specimen With I n s t r o n Tensometer Attached 98 12 Diagrammatical Specimen, And Crack I n i t i a t i o n 99 13 Apparatus (Diagrammatical) 100 14 Apparatus - 101 15 Apparatus 102 16 Ts Determination 103 17 Ts Determination 103 18 D u r a t i o n Of Load P l o t For Experiment 1 104 19 D u r a t i o n Of Load P l o t For Experiment 2 105 20 Duration Of Load P l o t For Experiment 3 106 21 D u r a t i o n Of Load P l o t For Experiment 4 107 22 D u r a t i o n Of Load P l o t For Experiment 5 108 23 D u r a t i o n Of Load P l o t For Experiment 6 109 24 Ts (Expt 6) S u r v i v o r Strengths 110 25 Normal F i t To Short Term Strength Data, Expt 6 111 26 Tada P a r i s I r w i n d 9 7 3 ) 112 27 E f f e c t Of "a" And "b" Upon The Duration Of Load P l o t ..113 28 E f f e c t Of «, « y 0 and Upon The Duration Of Load P l o t .114 29 Creep Parameters, K a s s d 9 6 9 ) 115 30 Creep Parameters, K a s s d 9 6 9 ) 116 31 Creep Parameters, Schniewind And Barrett(1972) 117 32 Creep Parameters, Schniewind And B a r r e t t ( 1972) 1 17 33 Step-Wise F i t To Expt 6 (Ts) 118 34 N i e l s e n F i t To Expt 6 (Ts) 119 35 Step-Wise F i t To Moisture T e s t , Expt 5 120 36 Step-wise F i t To Expt 2 121 37 Step-wise F i t To Expt 3 122 38 N i e l s e n F i t To Expt 3 123 39 Step-wise F i t To Expt 1 124 40 Step-wise F i t To Expt 4 125 41 Confidence L i m i t s On The S t r e s s R a t i o 126 A1 Volume E f f e c t P l o t By Barrett(1974) 140 A2 Step-wise Model Of The Crack T i p 150 1 CHAPTER 1 INTRODUCTION 1.1 D e f i n i t i o n of the Duration of Load E f f e c t The d u r a t i o n of l o a d e f f e c t i n wood can be d e f i n e d as a r e d u c t i o n of s t r e n g t h under the a c t i o n of a p e r s i s t e n t load over a p e r i o d of time. For example, a specimen of wood loaded to a s t r e s s r a t i o of 0.75 ( i . e . , 75% of i t s short term f a i l u r e s t r e s s ) w i l l not f a i l upon a p p l i c a t i o n of the l o a d . However, at some time l a t e r i t may suddenly f a i l due to a weakening of the specimen that has taken p l a c e w i t h i n that time p e r i o d . 1.2 Background. The development of the LSD code in Canada r e q u i r e s that a f r e s h look be taken at the design procedure fo r timber. Many qu e s t i o n s need to be addressed, amongst them the d u r a t i o n of l o a d phenomenon. For many years the s t r e n g t h r e d u c t i o n versus time curve developed by Wood(l95l) has been used i n design c a l c u l a t i o n s i n North America and other p a r t s of the world. The a p p l i c a b i l i t y of t h i s curve to design seems q u e s t i o n a b l e when c o n s i d e r i n g that small dry, s t r a i g h t g r a i n e d , c l e a r specimens of Douglas f i r are being used to represent commercial m a t e r i a l s , 2 o f t e n with major d e f e c t s and s t r e s s r a i s e r s such as knots, c r a c k s , g r a i n d e v i a t i o n s , e t c . The f a i l u r e of the small c l e a r t e s t s conducted i s normally i n i t i a t e d i n compression p a r a l l e l to the g r a i n . T h i s f a i l u r e mode does not represent the most f r e q u e n t l y encountered f a i l u r e mode i n commercial m a t e r i a l , where crack propagation i s observed to occur i n a plane p a r a l l e l with the g r a i n , caused e i t h e r by s t r e s s e s i n t e n s i o n p e r p e n d i c u l a r to the g r a i n induced^ at the knots, or shear s t r e s s e s . To s a t i s f y the need f o r a more a c c u r a t e treatment of the d u r a t i o n of loa d e f f e c t , c u r r e n t r e s e a r c h i s d e v e l o p i n g i n two ways. B a r r e t t and Foschi(1978a,1978b) have p o s t u l a t e d a damage accumulation model based on creep r u p t u r e . The damage rate i s given as the sum of a s t r e s s dependent term and a damage dependent term. The model al l o w s f o r a s t r e s s t h r e s h o l d below which the damage r a t e vanishes.. By c a l i b r a t i o n a g a i n s t experimental data the co n s t a n t s of the r e s u l t i n g e x p r e s s i o n can be o p t i m i s e d , to gi v e a very f l e x i b l e t o o l u s e f u l i n r e l i a b i l i t y s t u d i e s using v a r i o u s l o a d i n g h i s t o r i e s . Gerhards(1979) a l s o proposes a s i m i l a r model. The theory of v i s c o e l a s t i c f r a c t u r e mechanics has been a p p l i e d to the d u r a t i o n of load e f f e c t i n wood. The ba s i c assumption i s that crack propagation i n wood i s of a l i n e a r v i s c o e l a s t i c - p l a s t i c nature. In order to p r e d i c t the time to f a i l u r e , the necessary m a t e r i a l behaviour and parameters are chosen and the e x p r e s s i o n s d e r i v e d . Nielsen(1978,1980) and Kousholt(1980) using one set of parameters have d e r i v e d a model 3 (now r e f e r r e d to as the N i e l s e n model) where the crack i s s a i d to lengthen i n a continuous f a s h i o n . Brincker(1981,1982) d e r i v e s a model with c h a r a c t e r i s t i c s very s i m i l a r to those of the N i e l s e n model. However, one m a t e r i a l parameter i s d i f f e r e n t , and the cra c k i s assumed t o lengthen i n small s p u r t s . T h i s model w i l l be r e f e r r e d to as the step-wise model. Johns and Madsen (1982) have a p p l i e d the theory of the N i e l s e n model to commercial 2"x6" Douglas f i r boards i n bending ( f a i l u r e i n i t i a t e d i n t e n s i o n p e r p e n d i c u l a r to the g r a i n ) under constant s t r e s s , and have found good agreement between experiment and theory. I t i s the N i e l s e n and the step-wise models that t h i s r e s e a r c h i s aimed at i n v e s t i g a t i n g . These models have a f i r m foundation upon the concepts of f r a c t u r e mechanics, and u n l i k e e m p i r i c a l models u t i l i z e m a t e r i a l parameters which are seen to have a p h y s i c a l s i g n i f i c a n c e . These f e a t u r e s g i v e the v i s c o e l a s t i c models p o t e n t i a l to represent the d u r a t i o n of load behaviour i n wood. In order to t e s t the v i s c o e l a s t i c models i n the most r e a l i s t i c manner, i t was decided that c r a c k s propagating l o n g i t u d i n a l l y , s t r e s s e d i n t e n s i o n p e r p e n d i c u l a r to the g r a i n i n the opening mode be used. For reasons e x p l a i n e d i n Chapter 3 t h i s mode of f a i l u r e was expected to be r e p r e s e n t a t i v e of the c r i t i c a l f a i l u r e i n i t i a t i o n mode i n commercial m a t e r i a l and i t i s p o s t u l a t e d that t h i s r e p r e s e n t s a lower bound or worst case f o r d u r a t i o n of l o a d behaviour of commercial m a t e r i a l . 4 1.3 Scope. 1.3.1 I n t r o d u c t i o n . In order to i n v e s t i g a t e the v i s c o e l a s t i c models, experiments were c a r r i e d out and t h e i r r e s u l t s compared with the p r e d i c t i o n s of the v i s c o e l a s t i c models. In p a r t i c u l a r , the experiments were designed to i n v e s t i g a t e the f o l l o w i n g f e a t u r e s . 1.3.2 The N i e l s e n model. To determine e x p e r i m e n t a l l y the a p p l i c a b i l i t y of the N i e l s e n model to wood, by v a r y i n g c r i t i c a l parameters and comparing the p r e d i c t e d behaviour to the experimental r e s u l t . 1.3.3 The Step-wise model. To determine e x p e r i m e n t a l l y the a p p l i c a b i l i t y of the s t e p -wise model to wood, by v a r y i n g c r i t i c a l parameters and comparing the p r e d i c t e d behaviour to the experimental r e s u l t . 1.3.4 Strength E f f e c t . One of the f e a t u r e s of the N i e l s e n and step-wise models i s that they both p r e d i c t a s t r e n g t h e f f e c t . For example, f o r two separate groups of specimens, each group with a d i f f e r e n t average short term s t r e n g t h , loaded to the same s t r e s s r a t i o , 5 the times to f a i l u r e w i l l be d i f f e r e n t . The higher s t r e n g t h specimens should f a i l e a r l i e s t . To t e s t t h i s aspect of the t h e o r i e s , h i g h s t r e n g t h specimens and low s t r e n g t h specimens are t e s t e d at the same s t r e s s r a t i o and t h e i r r e l a t i v e behaviours compared. 1.3.5 Moisture Content. In the past r e s e a r c h has been c a r r i e d out to t e s t the e f f e c t of moisture content on creep ( i n c r e a s e d moisture content tends to g i v e i n c r e a s e d c r e e p ) . However the r e l a t i o n s h i p between moisture content and the d u r a t i o n of lo a d e f f e c t has not been e s t a b l i s h e d . T h i s p a r t of the study attempts to l i n k the i n c r e a s e d creep due to a higher moisture content, to e a r l i e r f a i l u r e times i n the d u r a t i o n of l o a d e f f e c t as p r e d i c t e d by the N i e l s e n and step-wise models. Ther e f o r e specimens of high moisture content are t e s t e d , and t h e i r behaviour compared with the behaviour of somewhat drye r specimens. 1.3.6 C y c l i c Loading. T r a d i t i o n a l l y when d e s i g n i n g a wooden s t r u c t u r e , i t i s necessary to determine the maximum design load, (which f o r r o o f s i n Canada i s o f t e n the snow l o a d ) . T h i s maximum snow loa d i s assumed to a c t f o r a t o t a l of only two months d u r i n g the l i f e of the b u i l d i n g . T h e r e f o r e a d u r a t i o n of lo a d f a c t o r of 1.15 i s used a c c o r d i n g to the Madison Curve developed by Wood(l95l). 6 More r e a l i s t i c a l l y , a s e r i e s of annual snow loads i s a p p l i e d , not a l l of the them the maximum design l o a d . In order that the d u r a t i o n of l o a d behaviour of t h i s l o a d i n g c h a r a c t e r i s t i c be b e t t e r understood, i t was deci d e d that a simple experiment be performed. By means of comparative t e s t s between samples loaded c o n t i n u o u s l y and samples loaded f o r s e v e r a l p e r i o d s of time adding up to the c o n t i n u o u s l y loaded time, the obj e c t was to determine i f a d i f f e r e n c e i n the d u r a t i o n of loa d e f f e c t e x i s t s f o r these d i f f e r e n t l o a d i n g h i s t o r i e s . 1.4 Summary of O b j e c t i v e s . 1. To i n v e s t i g a t e the d u r a t i o n of loa d behaviour i n t e n s i o n p e r p e n d i c u l a r to the g r a i n over time p e r i o d s between one hour and three months, and compare with t h a t p r e d i c t e d by the models based on the theory of v i s c o e l a s t i c f r a c t u r e mechanics as proposed by N i e l s e n and by B r i n c k e r . 2. To i n v e s t i g a t e the e f f e c t of the a p p l i e d s t r e s s l e v e l upon the d u r a t i o n of load e f f e c t . 3. To i n v e s t i g a t e the e f f e c t of a high moisture content upon the d u r a t i o n of load e f f e c t . 4. To i n v e s t i g a t e the e f f e c t of recovery p e r i o d s ( c y c l i c l o a d i n g ) upon the d u r a t i o n of l o a d e f f e c t . 5. To comment upon the o b s e r v a t i o n s reached, t h e i r subsequent a p p l i c a t i o n s to design p r a c t i c e and suggested d i r e c t i o n s f o r f u r t h e r r e s e a r c h . 7 CHAPTER 2 THEORY 2 . 1 I n t r o d u c t i o n . This chapter e x p l a i n s the basic assumptions and theory of the N i e l s e n and step-wise models, along with other models for du r a t i o n of loa d . I t then o u t l i n e s the theory of the Nie l s e n and step-wise models and demonstrates the ready a p p l i c a t i o n of both models to the larg e body of research already a v a i l a b l e i n the f i e l d s of f r a c t u r e mechanics and dur a t i o n of load i n wood. 2 . 2 Models For Duration Of Load. Models for du r a t i o n of load take one of f i v e forms: r e g r e s s i o n to s t r a i g h t l i n e s or curves, regression to a proposed mathematical model, phenomenological approaches, l i n e a r f r a c t u r e mechanics models, or v i s c o e l a s t i c f r a c t u r e mechanics models. T r a d i t i o n a l l y , regressions to a s t r a i g h t l i n e on semi-log paper have been most widely used. In f a c t , f o r an experiment c a r r i e d out w i t h i n a l i m i t e d time domain, these regressions give a good f i t to the trend of the data. They f a i l abysmally however, when the r e s u l t s of one researcher are compared with those of another; no r a t i o n a l e can be provided to l i n k the two 8 because of the inescapable v a r i a t i o n s i n m a t e r i a l p r o p e r t i e s and t e s t c o n d i t i o n s . These r e g r e s s i o n s do however p r o v i d e a good r e c o r d of d u r a t i o n of l o a d experiments performed i n the past, and as such are a very v a l u a b l e source of i n f o r m a t i o n f o r the understanding of the d u r a t i o n of l o a d behaviour of f u l l s c a l e members. Rec e n t l y , more f l e x i b l e mathematical models, as proposed by Gerhards(1979) and B a r r e t t and Foschi(1978a,1978b) are being used. These models i n c o r p o r a t e the concept of accumulating damage and t h e r e f o r e p r o g r e s s i v e f a i l u r e ( b e a r i n g a c l o s e resemblance t o the f a t i g u e equations of m e t a l s ) , thus y i e l d i n g a f l e x i b l e model f o r d u r a t i o n of l o a d . A good f i t to the experimental data can be ob t a i n e d . The damage accumulation model i s h i g h l y developed, t o the p o i n t where the inherent v a r i a b i l i t y of each parameter i s c o n s i d e r e d , so that the model i s very u s e f u l i n c a r r y i n g out r e l i a b i l i t y s t u d i e s f o r v a r y i n g l o a d h i s t i o r i e s and l o a d i n g c o n d i t i o n s . Other f a i l u r e p r e d i c t i n g c r i t e r i a are based upon energy methods. For example, the C r i t i c a l T o t a l Energy method p r e d i c t s f a i l u r e to occur when the t o t a l energy transformed i n the m a t e r i a l reaches a c e r t a i n v a l u e . The C r i t i c a l R e v e r s i b l e Energy method (o f t e n r e f e r r e d to as the Reiner Weisenberg f a i l u r e c r i t e r i o n ) uses K e l v i n c h a i n s and dashpots t o model m a t e r i a l behaviour, and f a i l u r e i s p r e d i c t e d to occur when the e l a s t i c a l l y s t o r e d energy reaches . a c r i t i c a l v a l u e . These methods are c a l l e d phenomenological. They o f f e r a p h y s i c a l i n t e r p r e t a t i o n of behaviour but are not based on p h y s i c a l 9 f a i l u r e mechanisms. Thus i f l i n e a r c o n s t i t u t i v e equations are a p p l i e d , they can not account f o r accumulating damage as does the crack propagation mechanism of f r a c t u r e mechanics. Mindess Nadeau and Barrett(1975,1976), and Mindess B a r r e t t and Spencer(1979) c a r r i e d out t e s t s on Douglas f i r i n t e n s i o n p e r p e n d i c u l a r to the g r a i n u s i n g bending and double t o r s i o n methods. By assuming the p r i n c i p l e s of l i n e a r f r a c t u r e mechanics, the r e s e a r c h e r s developed an experimental and a n a l y t i c a l method whereby long term d u r a t i o n of l o a d behaviour was e x t r a p o l a t e d from the r e s u l t s of r e l a t i v e l y short d u r a t i o n t e s t s . They concluded, however, that the i n v e s t i g a t i o n s were "...not very f r u i t f u l . . . " , and that a more g e n e r a l model a b l e to i n c l u d e v i s c o e l a s t i c and time dependent e f f e c t s was r e q u i r e d . Although f r a c t u r e mechanics i n wood has been s t u d i e d f o r many yea r s , r e s e a r c h has always tended to be segmental i n nature, l a c k i n g a model capable of l i n k i n g together the many f a c e t s of f r a c t u r e i n wood. Creep experiments, d u r a t i o n of load experiments, f r a c t u r e toughness experiments, f l u c t u a t i n g moisture content experiments and many others have been undertaken i n the r e s e a r c h l a b o r a t o r i e s of the world on v a r y i n g s p e c i e s . However, none of these have been combined in order to p r o v i d e a deeper understanding of the d u r a t i o n of load e f f e c t . T h i s e m p i r i c a l evidence r e q u i r e s a t h e o r e t i c a l framework, w i t h i n which the v a r y i n g behaviours observed can be u n i f i e d and used f o r the p r e d i c t i o n of response to c o n d i t i o n s other than those t e s t e d . The N i e l s e n and step-wise models allow t h i s , by l i n k i n g the m a t e r i a l p r o p e r t i e s w i t h i n a g e n e r a l theory where t h e i r 10 r e l a t i v e e f f e c t s upon the behaviour of wood under a s u s t a i n e d l o a d can be e v a l u a t e d , p r o v i d i n g two u s e f u l models f o r d e v e l o p i n g an understanding of the d u r a t i o n of l o a d phenomenon. 2.3 General Theory of V i s c o e l a s t i c F r a c t u r e Mechanics The important f e a t u r e s of the theory of v i s c o e l a s t i c f r a c t u r e mechanics as they apply to t h i s study, are o u t l i n e d below. Wood i s a composite m a t e r i a l c o n s i s t i n g mainly of polymers. I t i s assumed to be a homogeneous continuum and thus concepts such as s t r e s s and s t r a i n can be a p p l i e d . The concept of the very important s t r e s s i n t e n s i t y f a c t o r k i s i n t r o d u c e d and d e f i n e d Cc i s the h a l f crack l e n g t h ) . (2-1 ) k 2 = <r2jrc i I f the a p p l i e d s t r e s s a i s s u f f i c i e n t t h a t k reaches K (the c r i t i c a l s t r e s s i n t e n s i t y f a c t o r otherwise known as the f r a c t u r e toughness) then f a i l u r e occurs immediately. Second, i t must be a p p r e c i a t e d that i f a sharp crack i s assumed, there e x i s t s a s i n g u l a r s t r e s s f i e l d at the crack t i p . For a s t r a i g h t crack i n a s u r f a c e where the s t r e s s i s a p p l i e d p e r p e n d i c u l a r to the crack plane i n an i n f i n i t e sheet (2-2) k = <t (21TX) 1/ 2 ( r e f e r to f i g u r e 1 ) 11 where k = the s t r e s s i n t e n s i t y f a c t o r and x = the d i s t a n c e from the crack t i p . Having developed the i d e a l e l a s t i c s t r e s s c o n d i t i o n at the crack t i p i t i s necessary to i d e n t i f y f u r t h e r f e a t u r e s which w i l l i n f l u e n c e the f a i l u r e c h a r a c t e r i s t i c s . For v i s c o e l a s t i c m a t e r i a l s the r e l a t i o n s h i p between the s t r e s s and the s t r a i n i s a f f e c t e d by the time dependency ( v i s c o e l a s t i c i t y ) of the m a t e r i a l . By a p p l y i n g the extended e l a s t i c / v i s c o e l a s t i c correspondence p r i n c i p l e and u s i n g the Laplace t r a n s f o r m a t i o n , a general e x p r e s s i o n r e l a t i n g the s t r e s s s t r a i n displacement and time v a r i a b l e s can be developed. If the v i s c o e l a s t i c c reep f u n c t i o n J ( t ) and s t r e s s i n t e n s i t y are known, then the time a t which crack propagation occurs can be determined as f o l l o w s . For an a r b i t r a r y plane body with a constant a p p l i e d s t r e s s and a crack opening (not propagating lengthwise) i n a s i n g l e mode, then k ( t , t 0 ) can be d e f i n e d as the s t r e s s i n t e n s i t y at any time t a f t e r time t 0 . As the crack i s not l e n g t h e n i n g k ( t , t 0 ) i s a constant equal to k ( t 0 ) . In c o n t r a s t , the deformation at the crack t i p i s not constant but a f u n c t i o n of time (assuming a s t a t i o n a r y crack) because of the v i s c o e l a s t i c i t y of the m a t e r i a l . Thus a deformation f u n c t i o n can be d e f i n e d as (2-3) d ( t , t 0 ) = P k ( t , t 0 ) J ( t , t 0 ) where p = 1 plane s t r e s s and p = (1-I/ 2) f o r plane s t r a i n . 12 N o t i c e that the u n i t s of the deformation f u n c t i o n are m1/2 and not m as one would expect. T h i s f u n c t i o n c o u l d perhaps b e t t e r be r e f e r r e d to as a creep and s t r e s s f u n c t i o n , so that the u n i t of d i s t a n c e i s not i m p l i e d . As both the s t r e s s i n t e n s i t y f a c t o r and the deformation f u n c t i o n w i l l a f f e c t crack growth, i t i s necessary to r e l a t e t h i s knowledge i n some way, so a f a i l u r e c r i t e r i o n may be e s t a b l i s h e d . T h e r e f o r e a f a i l u r e c r i t e r i o n c a l l e d Fc i s assumed as a t h r e s h o l d v a l u e . I f Fc i s reached, crack propagation ensues. Some f u n c t i o n of k and d can be used t o a t t a i n t h i s c r i t i c a l value i . e . , (2-4) F( k,d ) £ Fc i m p l i e s f a i l u r e . N o t i c e however that a paradox has been in t r o d u c e d due to the assumption of a s i n g u l a r s t r e s s at the crack t i p . I f a crack i s f i r s t c o n s i d e r e d at time t=t-, when i t i s on the verge of moving, and again a f t e r a very small time has passed at time t=t+ then, F( k ( t - ) , d ( t - ) ) = Fc At the moment of f i r s t movement, the new crack t i p parameters are k(t+) = k ( t - ) ( because the crack has not yet lengthened s u f f i c i e n t l y to warrant a change i n the value of k ( t ) ), and d(t+) < d ( t - ) ( because the deformation at time t+ equals the deformation at 13 time t 0 )• T h e r e f o r e F( k(t+),d(t+) ) < Fc and the crack i s no longer a t the f a i l u r e c r i t e r i o n and has e f f e c t i v e l y been unable to propagate. T h i s i s the so c a l l e d c rack propagation paradox; which says that i f the crack propagation c r i t e r i o n i s f u l f i l l e d at time t - f then because of the movement, the crack propagation parameters immediately assume valu e s such that the crack propagation c r i t e r i o n i s no longer f u l f i l l e d at time t+. T h e r e f o r e crack growth i s stopped before i t has e f f e c t i v e l y s t a r t e d . At l e a s t two methods e x i s t whereby the s i n g u l a r i t y at the crack t i p can be removed. One can apply l i n e a r e l a s t i c p l a s t i c i t y at the crack t i p , f i r s t d e r i v e d by Dugdale(1960). Or one can assume the crack t i p to have a c u r v a t u r e g r e a t e r than z e r o . Each method introduce an a d d i t i o n a l m a t e r i a l constant and both can be used s u c c e s s f u l l y , although the c a l c u l a t i o n s i n v o l v e d become h i g h l y complex due to the changing boundary c o n d i t i o n s f o r a propagating c r a c k . Note that the N i e l s e n model u t i l i s e s the f i r s t assumption and the m a t e r i a l constant here r e f e r r e d to i s i n f a c t the p l a s t i c l i m i t s t r e n g t h However t r a d i t i o n a l forms of s o l u t i o n to the s t r e s s a n a l y s i s at the c r a c k t i p assumed the crack t i p to have a c u r v a t u r e g r e a t e r than z e r o . T h i s method of a n a l y s i s has l i m i t e d a p p l i c a b i l i t y to the r e a l f a i l u r e mechanism of wood and has not to the authors knowledge been a p p l i e d to the v i s c o e l a s t i c problem. As o u t l i n e d i n the next s e c t i o n , Brincker(1981) i n t r o d u c e s a t h i r d m a t e r i a l 14 parameter which enables an a d d i t i o n a l s o l u t i o n to the v i s c o e l a s t i c crack propagation problem. 2.4 The Step-Wise Model f o r the Du r a t i o n of Load E f f e c t A new crack propagation model i s obtained by assuming that the s m a l l e s t d i s t a n c e other than zero that a crack can grow equals 6, where 6 i s a m a t e r i a l c o n s t a n t . A p p l y i n g t h i s m a t e r i a l constant to crack growth behaviour, the crack t i p i s now c o n s t r a i n e d to move a d i s t a n c e 6 between the times t - and t+. T h e r e f o r e i n s p i t e of the s i n g u l a r s t r e s s f i e l d , the paradox of no crack growth i s removed. The s i m p l i c i t y of t h i s approach i s u s e f u l . I f a s t a t i o n a r y crack i s c o n s i d e r e d , the time i t takes t o reach a stage where growth i s about to occur can be c a l c u l a t e d from the f a i l u r e c r i t e r i o n (2-4). At t h i s time the crack t i p begins moving at time t - and stops a very s h o r t time l a t e r a t time t+, having t r a v e l l e d a d i s t a n c e 6. Subsequently the time taken to reach the f a i l u r e c r i t e r i o n can again be c a l c u l a t e d using the new values of k and d. By approaching the a n a l y s i s i n such a step-wise manner the time to f a i l u r e can be determined. If the creep f u n c t i o n i s such that even at i n f i n i t e time the f a i l u r e c r i t e r i o n i s not f u l f i l l e d , then the crack w i l l remain s t a t i o n a r y . I f however the f a i l u r e c r i t e r i o n i s s a t i s f i e d the i n s t a n t the lo a d i s a p p l i e d , then the crack w i l l propagate immediately and c a t a s t r o p h i c a l l y , as does an e l a s t i c m a t e r i a l when loaded to the c r i t i c a l s t r e s s i n t e n s i t y . Thus two 15 l i m i t s of crack growth can be de f i n e d . One where the crack remains s t a t i o n a r y , the other where the crack moves at a high v e l o c i t y . Between these extremes e x i s t s u b c r i t i c a l cracks loaded to a s u b c r i t i c a l s t r e s s , but which grow i n a c o n t r o l l e d manner u n t i l a c r i t i c a l crack length i s reached where the f a i l u r e c r i t e r i o n i s i n s t a n t l y s a t i s f i e d , and c a t a s t r o p h i c f a i l u r e ensues. The step-wise model allows for three separate phases of crack growth. Phase 1. I n i t i a l l y , only an opening of an i n t r i n s i c flaw (not lengthening) occurs. Note that t h i s stage of growth i s only a f f e c t e d by two parameters. F i r s t the creep f u n c t i o n i . e . , the greater the rate of creep as defined by the creep f u n c t i o n , the f a s t e r the crack w i l l widen, and the shorter the time f o r the crack to reach the f a i l u r e c r i t e r i o n . Second, the s t r e s s r a t i o 9 , defined as the a p p l i e d s t r e s s d i v i d e d by the short term strength of the specimen. The higher the s t r e s s r a t i o the f a s t e r the crack w i l l open to the stage where the f a i l u r e c r i t e r i o n i s s a t i s f i e d . The end of phase 1 s i g n i f i e s the s t a r t of l o n g i t u d i n a l crack propagation. The time taken to do t h i s i s c a l l e d Ts i . e . , the time to the s t a r t of l o n g i t u d i n a l propagation and the s t a r t of phase 2. Phase 2. I f the length of the i n t r i n s i c flaw i s l e s s than the c r i t i c a l crack length ( f o r a p a r t i c u l a r l e v e l of a p p l i e d s t r e s s ) , then the crack begins to propagate i n a slow step-wise manner. The rate of propagation remains dependent upon the creep f u n c t i o n i n the same way as f o r phase 1. However, the 16 r a t e of crack propagation i s a l s o dependent upon the step l e n g t h 6. I f the step l e n g t h i s low, t h i s i m p l i e s a h i g h value of *, and the r e s i s t a n c e to crack propagation i s i n c r e a s e d because more energy i s r e q u i r e d to f a i l the f i b r e s thus slowing p r o p a g a t i o n . Propagation c o n t i n u e s at a s l o w l y i n c r e a s i n g r a t e u n t i l the c r i t i c a l crack l e n g t h i s achieved. Phase 3 i s then i n i t i a t e d , the crack propagating r a p i d l y i n a c a t a s t r o p h i c manner to f a i l u r e . The time at which c a t a s t r o p h i c f a i l u r e ensues i s r e f e r r e d to as Teat. By s u b s t i t u t i n g s u i t a b l e m a t e r i a l c o n s t a n t s i n t o the s t e p -wise model a p l o t of the l o g a r i t h m of the time to f a i l u r e (Teat) versus s t r e s s r a t i o (0) i s o b t a i n e d . T y p i c a l values f o r wood p r e d i c t the behaviour shown i n f i g u r e 2. N o t i c e that both the Ts and Teat curves have negative c u r v a t u r e . For h i g h s t r e s s r a t i o s , the opening of the crack due to the e l a s t i c s t r a i n s i s very n e a r l y a l l the opening r e q u i r e d to i n i t i a t e f a i l u r e . T h e r e f o r e v i s c o e l a s t i c e f f e c t s are of minimal s i g n i f i c a n c e , and the d u r a t i o n of l o a d e f f e c t i s almost non-e x i s t e n t , y i e l d i n g a d u r a t i o n of load p l o t c l o s e to h o r i z o n t a l , i n d i c a t i n g t h a t the s t r e s s at f a i l u r e i s n e a r l y independent of the r a t e of l o a d i n g . Assume the creep f u n c t i o n (2-5) J ( t ) = (1/E) ( 1 + a t b ) or f o r a u n i t of time, t , , J ( t ) = (1/E) ( 1 + a ( A t / t 1 ) ) 17 and the f a i l u r e c r i t e r i o n ( 2 - 6 ) Fc = k d = k ( t 0 , A t ) d ( t 0 , A t ) = G where At = t - t 0 , At £ 0 G = G r i f f i t h s s u r f a c e energy per u n i t area, and E = Youngs modulus. Refer a l s o to Appendix 3 where the f a i l u r e c r i t e r i o n Fc = kd i s d i s c u s s e d i n more d e t a i l . As a l r e a d y mentioned, k( t 0 , A t ) i s a constant with time, y i e l d i n g (2-7) k( t ' , A t ) = k( f ) S u b s t i t u t i n g (2-3) i n t o ( 2 - 6 ) y i e l d s (2-8) <r 2 i rC/> ( 1 + a ( A t / t , ) b ) = GE S i m i l a r l y f o r the short term s t r e n g t h c0 ( At = 0 ) (2-9) ff02nc0p = GE N o t i c e that (2-9) i s equal to the G r i f f i t h Energy Balance as i n (2-20) D e f i n i n g the s t r e s s r a t i o 9 (2-10) 9 = tfAo 18 and n o r m a l i s i n g we get (2-11) At = t,M ( ( C 0 / C ) ( 1 / 9 2 ) -1 ) , M = (1/a) </b N o t i c e that t h i s equation i s only v a l i d f o r v a l u e s of c up to the c r i t i c a l c r a c k l e n g t h . When c tends t o the c r i t i c a l c rack l e n g t h , At tends to zero and the crack v e l o c i t y tends to i n f i n i t y . The equation does not d e s c r i b e the growth of the cra c k a f t e r the stage where the c r i t i c a l c rack l e n g t h has been a t t a i n e d i . e . , i t only d e s c r i b e s phase 1 and phase 2 behaviour. Phase 3 growth (very r a p i d and c a t a s t r o p h i c ) d e s c r i b e s crack growth a f t e r the c r i t i c a l crack l e n g t h has been achieved. Ts - Teat can be determined by the f o l l o w i n g where the s u b s c r i p t c r i s an a b b r e v i a t i o n f o r c r i t i c a l . T h i s i n t e g r a l can be s o l v e d n u m e r i c a l l y f o r any value of b, however i f b= 1/2, 1/3, or 1/4 e t c . , then the i n t e g r a l can be expressed i n c l o s e d form such that (2-12) (2-13) Ts-Tcat = t,M ( F ( 0 ) / ( 9 2 * 2 ) ) 19 where (2-14) <t>2 = i r / ( 2 c 0 ) ( r e f e r a l s o to s e c t i o n 2.6) and as an example, f o r b=l/3, (2-15) F(9) = l / ( 4 9 a ) - 3/(29 2) + 9 2/2 + 3/4 - 3/2 l n ( 9 2 ) From (2-11) i f c / c 0 = 1, then At = Ts which i m p l i e s Vb (2-16) Ts = t,M ( 1/9 2 - 1 ) S u b s t i t u t i n g t h i s r e s u l t i n t o (2-13) then (2-17) Teat » t,M ( (1/9 2 - D 3 + F ( 9 ) / ( 9 2 * 2 ) ) 2.5 The N i e l s e n Model f o r the Du r a t i o n Of Load E f f e c t . The assumptions and development of the N i e l s e n model are giv e n by Nielsen(1978,1980), Kousholt(1980) and N i e l s e n and Kousholt(1980). Johns and Madsen(l982) p r o v i d e f u r t h e r e x p l a n a t i o n s , and a p p l i e d the model to d u r a t i o n of l o a d t e s t r e s u l t s . The N i e l s e n model i s founded upon the Dugdale B a r e n b l a t t model f o r a crack i n a t h i n p l a t e with an a p p l i e d t e n s i l e s t r e s s <r remote from the cr a c k of l e n g t h 2c. The cr a c k i s bound at each end by a p l a s t i c i s e d zone where the y i e l d s t r e n g t h i s equal 20 to a constant u l t i m a t e value <r, the maximum a t t a i n a b l e s t r e n g t h of the p l a s t i c m a t e r i a l i n the mouth of the crack t i p . By assuming a crack shape, the crack opening displacement can be determined, from which the c r i t i c a l value of energy i n t e n s i t y per u n i t s u r f a c e i s obta i n e d , thus d e f i n i n g the s t r e s s at f a i l u r e , a c c o r d i n g to the f r a c t u r e toughness equation (2-1). The Dugdale B a r e n b l a t t model when a p p l i e d to a l i n e a r e l a s t i c m a t e r i a l y i e l d s the G r i f f i t h s t r e s s i n t e n s i t y r e l a t i o n s h i p . The ra t e of crack d e r f o r m a t i o n and l o n g i t u d i n a l crack propagation are f u n c t i o n s of the v i s c o e l a s t i c c o n s t a n t s of the m a t e r i a l . T h e r e f o r e there a r i s e s the need f o r a creep f u n c t i o n J ( t ) r e l a t i n g s t r e s s e s to the s t r a i n s . By a p p l y i n g an extended v e r s i o n of the e l a s t i c / v i s c o e l a s t i c correspondence p r i n c i p l e as presented by Schapery(1975), the crack opening can be e v a l u a t e d even as the boundary c o n d i t i o n s are changing and moving. Appl y i n g the above theory, a f o r m u l a t i o n i s d e r i v e d such that an estimate of the time to f a i l u r e can be made, i f given: 1. A v i s c o e l a s t i c creep f u n c t i o n J ( t ) f o r the m a t e r i a l , 2. A c r i t i c a l crack l e n g t h or c r i t i c a l s t r e s s l e v e l as d e f i n e d by the c r i t i c a l s t r e s s i n t e n s i t y f a c t o r , 3. A l e v e l of a p p l i e d s t r e s s c, 4. A value of the l i m i t s t r e s s «, i n the y i e l d i n g p l a s t i c r e g i o n of the crack t i p . The f o r m u l a t i o n d e r i v e d i s of the same form as the ex p r e s s i o n d e r i v e d f o r the step-wise model i . e . , (2-17). The same three phases of crack growth e x i s t f o r the N i e l s e n model as they do i n the step-wise model: phase 1 where the 21 i n t r i n s i c flaw opens; phase 2 where a f t e r r e a c h i n g Ts at the end of phase 1, the crack propagates i n a steady continuous manner; fo l l o w e d by phase 3 where the crack f a i l s c a t a s t r o p h i c a l l y because the crack has reached the c r i t i c a l c rack l e n g t h ( f o r the l e v e l of a p p l i e d s t r e s s ) . Note that Nielsen(1978) has developed h i s model to i n c l u d e the cases of ramp l o a d i n g , and combined ramp and constant l o a d i n g . T h i s allows ready comparison of ramp, const a n t , and combined l o a d i n g t e s t d a ta. For specimens ramp loaded t o f a i l u r e , the lack of such a r e l a t i o n s h i p caused Pearson(1972) to assume an e f f e c t i v e loaded time of 20% of the t o t a l ramp time, in order to c o l l a t e t e s t data with v a r y i n g l o a d i n g h i s t o r i e s onto one r e p r e s e n t a t i v e p l o t . For constants r e p r e s e n t a t i v e of wood, the N i e l s e n model y i e l d s the f o l l o w i n g u s e f u l r e l a t i o n s h i p fo r comparing constant and ramp lo a d t e s t s . For two i d e n t i c a l specimens which f a i l at the same s t r e s s r a t i o , one under ramp l o a d i n g , the other under constant l o a d i n g , the ramp loaded specimen w i l l s u r v i v e approximately three times as long as the constant loaded specimen (although t h i s was u n v e r i f i e d by t e s t s ) . T h i s i s a reasonable c o n c l u s i o n c o n s i d e r i n g that f o r h a l f of the time the ramp loaded specimen i s being t e s t e d , the s t r e s s e s are only h a l f of those which w i l l e v e n t u a l l y cause f a i l u r e , and as such are c o n t r i b u t i n g l i t t l e to the creep and or crack propagation of the specimen. As the t h e o r i e s of the N i e l s e n and step-wise models are s i m i l a r , the step-wise model can a l s o be used to i n c l u d e ramp l o a d i n g and combined l o a d i n g (ramp and c o n s t a n t ) . 22 2.6 Comparison of the N i e l s e n and Step-wise Models. As a l r e a d y o u t l i n e d , the assumptions u t i l i s e d i n these two models are q u i t e d i f f e r e n t . However, the models s t i l l e x h i b i t the same ge n e r a l behaviour. In t h i s regard the models are s i m i l a r and t h e r e f o r e e i t h e r c o u l d be used to f i t d u r a t i o n of l o a d data p o i n t s , by i n t r o d u c i n g a p p r o p r i a t e parameters. If one d i d t h i s however, one would f i n d that the two models ( a l l other parameters the same i . e . , "a", nb", and «,) would not y i e l d s i m i l a r p r e d i c t i o n s of the Teat l i n e . In p a r t i c u l a r , the N i e l s e n model p r e d i c t s a l a r g e amount of time spent i n phase 2, p l a c i n g l e s s emphasis on Ts; while the step-wise model p r e d i c t s Ts being much c l o s e r to Teat. The two models y i e l d f u n c t i o n s of the form as given i n (2-17). The d i f f e r e n c e i s i n the value of F ( e ) . For b= 1/3 the N i e l s e n model g i v e s (2-18) F(9) = 2.24 ( 1/(3G 6) - 3/(26") + 3/G2 - 11/6 + l n ( 9 2 ) ) and the step-wise model as s t a t e d i n (2-15). For a=l/3 b=1/3, and *=0 .2 the graph of f i g u r e 3 i s p l o t t e d . Note the d i s t i n c t d i f f e r e n c e i n the p r e d i c t e d times spent in phase 2. The step-wise model i m p l i e s that once the crack has begun to propagate at time Ts, i t soon lengthens and f a i l s at time Teat, whereas f o r the N i e l s e n model, c o n s i d e r a b l e time i s spent i n phase 2. I t was decided that t h i s fundamental d i f f e r e n c e i n behaviour should be i n v e s t i g a t e d , to determine which model ( i f any) i s the more r e a l i s t i c ( r e f e r to Chapter 3 f o r the 23 experimental method adopted). In order to enable a d i r e c t comparison between the N i e l s e n and step-wise models, Brincker(1982) has r e l a t e d the step l e n g t h 6 to the p l a s t i c l i m i t s t r e s s <r, i n the f o l l o w i n g manner. The s i n g u l a r i t y s t r e s s e s at the crack t i p are d e s c r i b e d as i n (2-2). I f K i s the c r i t i c a l s t r e s s i n t e n s i t y f a c t o r and 6 i s the crack step length, then the cohesion t e n s i o n <s, ( e q u i v a l e n t to the p l a s t i c l i m i t s t r e n g t h i n the N i e l s e n model) can be d e f i n e d as the average s t r e s s over 6 ( r e f e r a l s o to Appendix 5) The G r i f f i t h Energy Balance s t a t e s that (2-20) pK2 = GE and f o r short term s t r e n g t h , (2-21 ) p < t 0 2 i t c 0 = GE D e f i n i n g <t> as the s t r e n g t h r a t i o (as i n the N i e l s e n model) (2-22) 0 = c0 / then from (2-19), (2-20), (2-21) and (2-22) (2-19) K (2-23) * 2 = 6 / ( 2 c 0 ) 24 T h i s r e s u l t seems reasonable i n that the stronger the p l a s t i c m a t e r i a l i n the r e g i o n of the crack t i p (a higher value f o r the p l a s t i c l i m i t s t r e s s c^) the lower the s t e p l e n g t h 6. T h i s r e s u l t a l s o i m p l i e s that the higher the short term s t r e n g t h , the l a r g e r the s t e p l e n g t h 6 ( f o r a constant * , ) . I f the s t r e s s r a t i o i s c o n s i d e r e d , i t can be seen that the higher the s h o r t term s t r e n g t h * 0 , the higher a must be to maintain the same s t r e s s r a t i o . T h e r e f o r e f o r two specimens loaded to the same s t r e s s r a t i o (but with d i f f e r e n t values of short term str e n g t h ) the specimen with the high short term s t r e n g t h w i l l a l s o have the l a r g e r s t e p l e n g t h 6 and w i l l f a i l sooner. From (2-19) i t can be seen that 6 i s a f f e c t e d by two opposing i n f l u e n c e s . F i r s t c\, which i f i n c r e a s e d tends to decrease 6. T h i s i s because more energy i s absorbed i n the process of f a i l u r e and t h e r e f o r e the step l e n g t h i s s h o r t e r . Second the f r a c t u r e toughness K which i f i n c r e a s e d tends to i n c r e a s e 6. Because a high K value i n d i c a t e s high l e v e l s of s t o r e d s t r a i n energy i n the m a t e r i a l surrounding the crack, then each crack step must be longer so that f o r a constant value of a high value of energy Fc can be absorbed at each step. 25 2.7 Evidence fo r the N i e l s e n and Step-wise Models. 2.7.1 I n t r o d u c t i o n . For many years t e s t s on the creep, s t r e n g t h , and d u r a t i o n of l o a d p r o p e r t i e s of wood have been researched e x t e n s i v e l y , but mostly as separate t o p i c s . Research i n t o f r a c t u r e mechanics and d u r a t i o n of l o a d i n wood, does y i e l d c e r t a i n t r e n d s , i n s p i t e of the inherent v a r i a b i l i t y of wood o f t e n encountered. Moreover the N i e l s e n and step-wise models f i t these trends and serve to e x p l a i n the anomalies other r e s e a r c h e r s chose to d i s c o u n t or e x p l a i n i n vague terms. However one q u a l i f i c a t i o n must be made on t h i s statement. In no way can the N i e l s e n and step-wise models support the t r a d i t i o n a l , s t r a i g h t l i n e d u r a t i o n of load l i n e s as developed by Wood, C l o u s e r , Youngs and H i l b r a n d e t c . , which are c o l l a t e d and summarised by Pearson(1972). These l i n e s e x h i b i t behaviour f a r d i f f e r e n t from that proposed • by the N i e l s e n and step-wise models. For reasons more f u l l y e x p l a i n e d in Chapter 3 these summary r e s u l t s are regarded as n o n r e p r e s e n t a t i v e of the i n i t i a t i n g f a i l u r e mode of commercial lumber i . e . , i n t e n s i o n p e r p e n d i c u l a r to the g r a i n . The bending t e s t r e s u l t s summarised by Pearson i n c l u d e t e s t s on many s p e c i e s , with v a r y i n g moisture contents and t e s t c o n d i t i o n s . A l l of the t e s t s are undertaken using c l e a r m a t e r i a l (most of which have a c r o s s s e c t i o n a l area of 1 cm 2), the f a i l u r e mode of which i s u s u a l l y i n compression of the top f i b r e s . In t h i s case, these r e s u l t s can i n no way represent the d u r a t i o n of load 26 behaviour of commercial m a t e r i a l , because d i f f e r e n t f a i l u r e modes may have separate d u r a t i o n of l o a d c h a r a c t e r i s t i c s . For example, Schniewind and Pozniak(1971) found that no d u r a t i o n of l o a d e f f e c t e x i s t s f o r c r a c k s propagating i n the t a n g e n t i a l r a d i a l plane. 2.7.2 Experimental Evidence. The a p p l i c a b i l i t y of f r a c t u r e mechanics to wood was f i r s t i n v e s t i g a t e d by Atack e t . a l . ( 1 9 6 1 ) . L a t e r , P o r t e r ( l 9 6 4 ) a l s o found that a G r i f f i t h - t y p e r e l a t i o n s h i p e x i s t e d between the crack l e n g t h and the f r a c t u r e s t r e s s . Since then c o n s i d e r a b l e experimental work on f r a c t u r e mechanics i n wood, some concerned with d u r a t i o n of l o a d has been r e p o r t e d and i s o u t l i n e d below. Researchers continue a f t e r Pearson i n p l o t t i n g s t r a i g h t l i n e s through the l o g a r i t h m of time to f a i l u r e versus s t r e s s r a t i o p l o t s . T h i s i s i l l u s t r a t e d i n the paper by Schniewind and Centeno(1973), where both short and long d u r a t i o n t e s t s are c a r r i e d out on Douglas f i r i n t e n s i o n p e r p e n d i c u l a r to the g r a i n . A s i g n i f i c a n t d i f f e r e n c e i s noted between the slope of the s t r a i g h t l i n e s f i t t e d f o r the two d i f f e r e n t time domains. The ramp l o a d t e s t s with f a i l u r e times between 0.5 seconds and 100 minutes y i e l d e d a very f l a t l i n e on the d u r a t i o n of load p l o t . The constant l o a d t e s t s with f a i l u r e times between 45 minutes and 18 days (corresponding s t r e s s r a t i o s of 0.9 to 0.7) y i e l d e d a much steeper l i n e on the d u r a t i o n of l o a d p l o t . Refer to f i g u r e 4 f o r p l o t s of these r e s u l t s . T h i s i n c r e a s i n g slope 27 of the p l o t with time supports the p r e d i c t i o n of the N i e l s e n and step-wise models, as do the f i n d i n g s of the other r e s e a r c h e r s mentioned i n the f o l l o w i n g . In t e s t i n g unnotched Scotch Pine i n t e n s i o n p e r p e n d i c u l a r to the g r a i n Bach's(1975) data ( f i g u r e 5) c o u l d perhaps more a p p r o p r i a t e l y be f i t t e d to a curve of negative c u r v a t u r e , p r o v i d i n g a much b e t t e r f i t at high s t r e s s r a t i o s . Mindess B a r r e t t and Spencer (1979) found with bending t e s t s on Douglas f i r i n t e n s i o n p e r p e n d i c u l a r to the g r a i n that f r a c t u r e s t r e n g t h i n c r e a s e d with displacement r a t e . They noted that t h e i r r e s u l t s underestimated the s e v e r i t y of the d u r a t i o n of l o a d e f f e c t when compared with other d u r a t i o n of load data i n t e n s i o n p e r p e n d i c u l a r to the g r a i n . T h i s i s best e x p l a i n e d by the f a c t t h a t t h e i r times to f a i l u r e were between 0.5 seconds and one day, and they are comparing t h e i r data with that of Bach(l975), whose times to f a i l u r e were e s s e n t i a l l y between one hour and s i x months, y i e l d i n g a steeper l i n e . Mindess Nadeau and Barrett(1975,1976) found when t e s t i n g Douglas f i r in t e n s i o n p e r p e n d i c u l a r to the g r a i n that "At long times the slopes (of the d u r a t i o n of l o a d p l o t s ) seem to i n c r e a s e c o n s i d e r a b l y . . . " . Recent rese a r c h at the U n i v e r s i t y of B r i t i s h Columbia by Johns and Madsen(l982) has a l s o r e v e a l e d a negative c u r v a t u r e f o r d u r a t i o n of l o a d t e s t s i n bending on commercial m a t e r i a l . C o n t i n u i n g r e s e a r c h by F o s c h i and Barrett(1982) i s a l s o y i e l d i n g s i m i l a r behaviour i . e . , negative c u r v a t u r e . Thus the p r e d i c t i o n s of the N i e l s e n and step-wise models 28 are w e l l supported by the experimental evidence i n the p r o f e s s i o n a l l i t e r a t u r e . 2.7.3 Q u a l i t a t i v e Evidence. The e x p l a n a t i o n s of the behaviour of wood under d u r a t i o n of l o a d as t o l d by other r e s e a r c h e r s i n the f i e l d of crack propagation, propose the same g e n e r a l mechanisms and l o g i c as i s a p p l i e d i n the theory and development of the v i s c o e l a s t i c f r a c t u r e mechanics models. Kollman(1963) d e s c r i b e s f a i l u r e as a process of i n i t i a l submicroscopic y i e l d i n g , f o l l o w e d by growth of m i c r o s c o p i c deformation and c r a c k s , f i n a l l y l e a d i n g to macroscopic f a i l u r e . T h i s i s i l l u s t r a t e d by t e s t s at high s t r e s s e s , where i r r e g u l a r i t i e s i n creep curves are i n t e r p r e t e d as sudden s t r u c t u r a l changes. Debaise et a l . ( l 9 6 6 ) made s i m i l a r o b s e r v a t i o n s . By monitoring a c o u s t i c emissions of wood as i t f r a c t u r e d , they hypothesised three main phases of f r a c t u r e : flaw n u c l e a t i o n , flaw growth, and unstable f r a c t u r e . They proposed d i f f e r e n t crack growth mechanisms depending on the crack c o n d i t i o n s , as shown i n f i g u r e 6. As an o b s e r v a t i o n they noted emissions o c c u r r i n g f o r very low l e v e l s of s t r e s s (20% of the f a i l u r e s t r e s s ) . T h i s f i n d i n g supports the v i s c o e l a s t i c models which p r e d i c t some amount of damage (be i t ever so small) to occur, even at very low s t r e s s r a t i o s . 29 2.8 Summary. As presented i n t h i s Chapter, the assumptions and theory of the N i e l s e n and step-wise models are based upon the theory of v i s c o e l a s t i c f r a c t u r e mechanics. Moreover, the p r e d i c t i o n s of the models are supported q u a l i t a t i v e l y and q u a n t i t a t i v e l y by the contemporary r e s e a r c h i n t o d u r a t i o n of l o a d . As such, the N i e l s e n and step-wise models are a u s e f u l advance in p r o v i d i n g a framework w i t h i n which the d u r a t i o n of l o a d phenomenon can be understood and q u a n t i f i e d . 30 CHAPTER 3 EXPERIMENT DESIGN 3.1 I n t r o d u c t i o n . The o b j e c t of the experiments i s to study and attempt to understand the propagation of cra c k s i n wood. I t i s necessary to model the cra c k s e x i s t i n g i n commercial m a t e r i a l as c l o s e l y as p o s s i b l e , and observe t h e i r behaviour, so c o n c l u s i o n s can be obtained which remain r e l e v a n t to the r e a l design s i t u a t i o n s . T h i s chapter o u t l i n e s the important c o n s i d e r a t i o n s ( i n the o p i n i o n of the author) with regard to the s e l e c t i o n of a s u i t a b l e crack geometry, mode of f a i l u r e , specimen design and e x p e r i p e n t a l method. 31 3.2 Crack O r i e n t a t i o n and Propagation. Due to the a n i s o t r o p i c nature of wood, many d i s t i n c t planes of f a i l u r e are p o s s i b l e . Even w i t h i n a plane, a crack may propagate i n one or both of the t a n g e n t i a l d i r e c t i o n s . I t may a l s o f a i l i n one of three modes. Because of the l a r g e number of p o s s i b i l i t i e s , a r a t i o n a l e i s developed here which shows that the f a i l u r e mode i n t h i s experiment i s such that i t i s the most r e l e v a n t r e p r e s e n t a t i o n of the c r i t i c a l f a i l u r e mode i n commercial m a t e r i a l . Three modes of f a i l u r e ( f i g u r e 7) are p o s s i b l e . I t i s suggested here that the c r i t i c a l mode of f a i l u r e f o r commercial lumber i s i n t e n s i o n p e r p e n d i c u l a r to the g r a i n i n the opening mode. Bending f a i l u r e i n commercial lumber i s g e n e r a l l y i n i t i a t e d at the knots. A crack i n i t i a t e d at the knot, curves around the knot, and i s then observed to propagate l o n g i t u d i n a l l y . As can be seen from f i g u r e 8 the slope of g r a i n at knots undergoes c o n s i d e r a b l e change. Upon the a p p l i c a t i o n of a t e n s i l e s t r e s s , the l o n g i t u d i n a l f i b r e s of the wood become s t r e s s e d i n t e n s i o n . Because df the curva t u r e of these f i b r e s , l a t e r a l s t r e s s e s are i n t r o d u c e d . These s t r e s s e s are i n t e n s i o n p e r p e n d i c u l a r to the g r a i n , the weakest plane of f a i l u r e f o r wood. Thus a small crack i n the plane of the f i b r e s c l o s e to the knot w i l l be s t r e s s e d i n the opening mode. T h i s small crack c o u l d be i n i t i a t e d at knots or by a d e f e c t i n the wood f i b r e s , or by a n i s o t r o p i c shrinkage s t r e s s e s . Schniewind and others have pro v i d e d comparisons of the f r a c t u r e toughness i n the v a r i o u s planes and d i r e c t i o n s , as w e l l 32 as some d u r a t i o n of l o a d c h a r a c t e r i s t i c s . The f r a c t u r e toughness r e s u l t s of Schniewind and Centeno(1973) can be separated i n t o two c a t e g o r i e s , each with separate values of f r a c t u r e toughness. T e s t s where f r a c t u r e o c c u r r e d i n a plane p a r a l l e l to the l o n g i t u d i n a l d i r e c t i o n y i e l d e d f r a c t u r e toughness v a l u e s of approximately one seventh the values where propagation was a c r o s s the g r a i n . T h i s supports the concept that f r a c t u r e would n a t u r a l l y occur i n the weakest plane i . e . , i n the plane of the g r a i n as a l r e a d y e x p l a i n e d . In order f o r a crack i n a three dimensional medium to propagate i n one d i r e c t i o n , i t must a l s o propagate i n the p e r p e n d i c u l a r d i r e c t i o n i . e . , a penny shaped crack propagates i n two d i r e c t i o n s w i t h i n the one plane. Within the t a n g e n t i a l l o n g i t u d i n a l plane Schniewind(1977) found the l o n g i t u d i n a l d i r e c t i o n of propagation e x h i b i t s a much more s e r i o u s d u r a t i o n of l o a d e f f e c t than the r a d i a l d i r e c t i o n , due to l e s s " b l o c k i n g or blunting"- by the c e l l w a l l s f o r the l o n g i t u d i n a l case. T h i s i n d i c a t e s that t e s t s should be c a r r i e d out with the crack propagating l o n g i t u d i n a l l y . Wood i n bending can f a i l due to the formation of compression w r i n k l e s i n the compression zone. T h i s f e a t u r e however i s u s u a l l y c o n f i n e d to the behaviour of the high s t r e n g t h m a t e r i a l , because the t e n s i o n zone i s f r e e of d e f e c t s . The low s t r e n g t h m a t e r i a l , which i s the most important when c o n s i d e r i n g safe design s t r e s s e s , u s u a l l y f a i l s by l o n g i t u d i n a l c rack propagation i n the t e n s i o n zone. These f a i l u r e s are u s u a l l y i n i t i a t e d at knots and d e f e c t s where s t r e s s e s i n t e n s i o n 33 p e r p e n d i c u l a r to the g r a i n are induced. In order to o b t a i n the best specimen d e s i g n , the c o r r e c t , c h o i c e between two a l t e r n a t i v e f a i l u r e planes needs to be made. The c r a c k s c o u l d propagate i n e i t h e r the t a n g e n t i a l l o n g i t u d i n a l plane or the r a d i a l l o n g i t u d i n a l p l a n e . On a macroscopic l e v e l , many commercial boards i n bending are observed to f a i l i n the r a d i a l l o n g i t u d i n a l plane. However, on a c l o s e r examination of the i n i t i a l f a i l u r e s u r f a c e at the knot, i t i s observed that the f a i l u r e has been i n i t i a t e d i n the t a n g e n t i a l l o n g i t u d i n a l plane. T h e r e f o r e i t was decided to t e s t c r a c k s propagating l o n g i t u d i n a l l y i n the t a n g e n t i a l l o n g i t u d i n a l plane i n the opening mode. I t i s p o s t u l a t e d that, t h i s mode of f a i l u r e would be r e p r e s e n t a t i v e of the i n i t i a t i o n of f a i l u r e i n commercial m a t e r i a l . As o u t l i n e d i n S e c t i o n 5 . 3 . 3 , the t a n g e n t i a l l o n g i t u d i n a l plane y i e l d s the h i g h e s t creep r a t e of any of the creep planes, i n d i c a t i n g ( a c c o r d i n g to the p r e d i c t i o n of the N i e l s e n and step-wise models) that the d u r a t i o n of load behaviour w i l l be most c r i t i c a l i n t h i s p l a n e . Therefore by t e s t i n g t h i s mode of f a i l u r e , the most c r i t i c a l f a i l u r e mode f o r wood i s ach i e v e d . N o t i c e that the crack c o n d i t i o n s f o r the t e s t specimen are not f a r removed from those f o r , a crack i n an i n f i n i t e sheet. However f o r commercial m a t e r i a l , the d i r e c t i o n of propagation of the crack may change as the crack lengthens, thus changing the s t r e s s f i e l d as the crack t i p . Because commercial specimens have a l o t of inherent redundancy (cracks propagating i n the reg i o n s of highest s t r e s s w i l l propagate and f i n d themselves i n 34 re g i o n s of lower s t r e s s ) and t h e r e f o r e w i l l take a c o n s i d e r a b l y longer time to f a i l than do the sma l l t e s t specimens used i n t h i s study which have no redundancy at a l l . Once the crack i n the s m a l l t e s t specimens begins to propagate there i s no chance t h a t the s t r e s s f i e l d w i l l d i m i n i s h , i t only i n t e n s i f i e s a c c o r d i n g t o (5-1). I t i s t h e r e f o r e proposed that the t e s t specimen used does not r e p l i c a t e the d u r a t i o n of l o a d behaviour of commercial m a t e r i a l . However, because the c r i t i c a l f a i l u r e i n i t i t a t i n g modes of both are s i m i l a r , i t i s proposed that the t e s t specimen can be a p p l i e d as a model, always p r o v i d i n g a lower bound to the d u r a t i o n of l o a d behaviour of commercial m a t e r i a l , and that the d u r a t i o n of l o a d c h a r a c t e r i s t i c s of the t e s t specimens be r e f l e c t e d i n the d u r a t i o n of l o a d behaviour of commercial m a t e r i a l . 3.3 Specimen Development. There are many s u i t a b l e specimen geometries to choose from. T e s t s have been c a r r i e d out i n double t o r s i o n by Mindess Nadeau and Barrett(1975,1976). T h i s geometry however r e q u i r e s s o p h i s t i c a t e d apparatus f o r each specimen and i s t h e r e f o r e i n a p p r o p r i a t e to t e s t i n g l a r g e sample s i z e s , e s p e c i a l l y over a long d u r a t i o n . Schniewind(1977) i n v e s t i g a t e d the t e n s i l e s t r e n g t h and d u r a t i o n of l o a d e f f e c t s i n t e n s i o n p e r p e n d i c u l a r t o the g r a i n , u s i n g notched beams i n bending. The disadvantage of t h i s method l i e s i n the f a c t that the s t r e s s i n t e n s i t y i n c r e a s e s markedly as the crack propagates, making a n a l y s i s more 35 d i f f i c u l t . A l s o , the specimen i s not r e p r e s e n t a t i v e of a r e a l flaw i n wood as i t only c o n s i d e r s h a l f a c r a c k , and does not a l l o w f o r the time dependent e f f e c t of the m a t e r i a l surrounding the other h a l f . S i n g l e edge notched specimens encounter the same problem. However, i f the d u r a t i o n of l o a d behaviour of c r a c k s of the nature as modelled by the s i n g l e edge notch specimen was to be determined, then the use of t h i s specimen geometry would be most a p p r o p r i a t e . Even f o r t e s t specimens of wood which are prepared from the same board, i f one wants co n f i d e n c e i n the r e s u l t s many specimens are r e q u i r e d , and the v a r i a b i l i t y of s t r e n g t h w i t h i n the p o p u l a t i o n of each t e s t needs to be as low as p o s s i b l e . Time, space, apparatus and labour requirements, r e q u i r e the specimen to be of r e l a t i v e l y small dimensions. P r e l i m i n a r y t e s t s were conducted by c u t t i n g a 2"x6" board i n t o 6" l e n g t h s , and s t r e s s i n g them i n t e n s i o n p e r p e n d i c u l a r to the g r a i n a f t e r i n s e r t i n g a c e n t r a l l y p l a c e d 1" l o n g i t u d i n a l through crack i n the specimen. However because the loads r e q u i r e d to f a i l such specimens were c o n s i d e r e d to be too l a r g e and the number of specimens taken from each board too few, a much smaller specimen and crack l e n g t h was f i n a l l y chosen. Subseqently a specimen design which avoided the l i m i t a t i o n s of other specimen d e s i g n s was developed. The specimens are s m a l l , e a s i l y manufactured and loaded, with crack c o n d i t i o n s s i m u l a t i n g the c r i t i c a l c r a c k s of commercial m a t e r i a l . In a d d i t i o n t h i s specimen design proved very s u c c e s s f u l i n reducing the c o e f f i c i e n t of v a r i a t i o n to q u i t e low v a l u e s , f o r wood. A 36 r e c t a n g u l a r p r i s m a t i c specimen, with the t e n s i l e l o a d i n g p l a t e s on the small ends, and a through crack i n i t i a t e d c e n t r a l l y i n the l o n g i t u d i n a l t a n g e n t i a l plane was used. Refer to f i g u r e s 9, 10, 11 and 12 f o r a v i s u a l p r e s e n t a t i o n . 3.4 Apparatus. In order to t e s t l a r g e numbers of specimens simultaneously the apparatus used must be compact i n nature. The m a t e r i a l s used were inexpensive and u n s o p h i s t i c a t e d . Maximum use was made of the m a t e r i a l s a l r e a d y a v a i l a b l e at the Department of C i v i l E n g i n e e r i n g S t r u c t u r e s Laboratory ( l e a d weights, s t e e l , Glulam beams e t c . ) Three separate racks were b u i l t , a l l o w i n g a t o t a l of 110 specimens to be t e s t e d s i m u l t a n e o u s l y . Each specimen was suspended from a mounting a t t a c h e d to the top of the support beam. The f r e e lower end of the specimen was a t t a c h e d to ( t h e l o a d p o i n t of a l e v e r arm whose fulcrum was underneath the support beam. A l e a d weight of approximately 20kg was f i x e d to the a p p r o p r i a t e l o c a t i o n on the l e v e r arm to give a load u n d e r e s t i m a t i n g the f i n a l l o a d on the specimen. For a c l e a r e r p i c t o r i a l p r e s e n t a t i o n r e f e r to f i g u r e s 13, 14 and 15. In order to apply the exact amount of l o a d to the specimen a small weighed bag of sand was suspended from the end of the l e v e r arm. For the short term t e s t s the Department of C i v i l E n g i n e e r i n g Satec t e s t i n g machine NO.20GBN 1002 with the 2000 l b l o a d c e l l was used on speed c o n t r o l (10% medium) y i e l d i n g a ramp 37 l o a d f a i l u r e time of approximately 45 seconds. To c a l i b r a t e the l e v e r arms, a d i a l guage p r o v i n g r i n g c a l i b r a t e d a g a i n s t the Satec l o a d c e l l was used. In determining moisture c o n t e n t s , the oven and s c a l e s i n the Department of C i v i l E n g i n e e r i n g S o i l s Laboratory were u t i l i s e d . The c r o s s s e c t i o n a l area of the specimens was measured at the crack c r o s s s e c t i o n by v e r n i e r c a l i p e r s . 3.5 Specimen P r e p a r a t i o n In order to maintain s i m i l a r samples, the specimens were numbered a c c o r d i n g to t h e i r l i n e a l p o s i t i o n i n the board. By s e l e c t i n g specimens i n numerical order, the samples f o r t e s t i n g were e q u a l l y r e p r e s e n t a t i v e of the l e n g t h of the board. For example, f o r a simple t e s t with only one d u r a t i o n of l o a d t e s t and one short term c o n t r o l t e s t , the p o p u l a t i o n would be d i v i d e d i n t o two i d e n t i c a l groups of specimens; one to be t e s t e d immediately to give the short term s t r e s s d i s t r i b u t i o n of the p o p u l a t i o n and the other to be t e s t e d under constant s t r e s s f o r a long d u r a t i o n of time, over which the f a i l u r e times are recorded. I n d i v i d u a l specimen p r e p a r a t i o n c o n s i s t s of the f o l l o w i n g s t e p s . 1. Squaring the edges of the board i n the p l a n e r . 2. C l e a v i n g the 2x6 i n t o two 1x6's using a t a b l e saw. 3. P l a n i n g both halves to equal t h i c k n e s s . 4. C u t t i n g and numbering specimens at 1.5 inch i n t e r v a l s along 38 the two boards. 5. Marking the c e n t r a l hole p o s i t i o n and the waist or necked down re g i o n with a template and c u t t i n g out the necked region on a small band saw ( f i g u r e 12). 6 . D r i l l i n g the c e n t r a l and end ho l e s i n the specimen. 7. F i x i n g the s t e e l end p l a t e s i n p o s i t i o n by i n s e r t i n g the screws u s i n g an e l e c t r i c d r i l l . 8. I n i t i a t i n g the crack by hammering a small symmetrical spear of known width c o n s t r u c t e d from a brass rod and a razor blade, through the c e n t r a l hole ( f i g u r e 12). 9. Measuring and r e c o r d i n g the c r o s s s e c t i o n a l area at the crack plane. 10. The l o a d i n g arrangement and the t e s t procedure i s d e s c r i b e d i n the f o l l o w i n g s e c t i o n . 3.6 Test Procedure For the constant l o a d t e s t s , the l e v e l of a p p l i e d s t r e s s i s c r i t i c a l because i t c o n t r o l s the r a t e at which f a i l u r e s occur. Depending on the s t r e s s l e v e l , the t e s t w i l l vary i n i t s d u r a t i o n . Based on past experience and d e s i r e d t e s t performance a d e c i s i o n i s made as to the most s u i t a b l e s t r e s s l e v e l to app l y . The time r e q u i r e d f o r the completion of the t e s t i s very s e n s i t i v e to the l e v e l of a p p l i e d s t r e s s , t h e r e f o r e great care i s r e q u i r e d i n determining the optimum s t r e s s l e v e l . In order to lo a d the specimens to the chosen s t r e s s l e v e l , the c r o s s s e c t i o n i s f i r s t measured and the r e q u i r e d a x i a l l o a d 39 c a l c u l a t e d . The l e a d weight i s f i x e d i n the a s s i g n e d p o s i t i o n on the l e v e r arm, which i s then c a l i b r a t e d u s ing the proving r i n g . The specimen i s p o s i t i o n e d on the apparatus and g e n t l y loaded, with the balance l o a d (a weighed bag of sand) t i e d to the end of the l e v e r arm. The l o a d i n g time i s noted, and subsequently the f a i l u r e time (or otherwise) i s a l s o recorded. In the e a r l y stages of the t e s t s , p e r s o n a l monitoring of the t e s t and r e c o r d i n g of the f a i l u r e times was continuous. However, as the experiment progressed, the t e s t was l e f t unattended f o r i n c r e a s i n g d u r a t i o n s as the frequency of f a i l u r e s d e c l i n e d . I f one or more specimens f a i l e d w i t h i n one of these i n t e r v a l s , the f a i l u r e times were i n t e r p o l a t e d i n a l i n e a r manner. T h i s approach worked w e l l because the rate of f a i l u r e s i s approximately p r o p o r t i o n a l t o the l o g a r i t h m of time. A f t e r a t e s t had progressed f o r s e v e r a l weeks i t was only necessary to r e c o r d the f a i l u r e s every two days. 3.7 Data A n a l y s i s . The short term f a i l u r e s t r e s s e s f o r a p a r t i c u l a r crack c o n d i t i o n showed some spread, however much l e s s than f o r uncracked specimens with c o e f f i c i e n t s of v a r i a t i o n as low as 0.06. Therefore i t was necessary to propose a model r e p r e s e n t a t i v e of the d i s t r i b u t i o n of short term s t r e n g t h s . The Gaussian Normal D i s t r i b u t i o n was chosen, as i t seemed reasonable that the v a r i a t i o n i n short term s t r e n g t h was l a r g e l y due to random e c c e n t r i c i t i e s i n the p o s i t i o n s of the end p l a t e s and i n 40 the p o s i t i o n of the necked down r e g i o n , thus superimposing random bending moments over top of the a x i a l l o a d . The mean value and the standard d e v i a t i o n of the short term s t r e s s e s were c a l c u l a t e d . The W e i b u l l model i s o f t e n used to model the s t r e n g t h d i s t r i b u t i o n of commercial m a t e r i a l . T h i s model i s based upon the "weakest l i n k " p r i n c i p l e e.g., a long chain with many l i n k s has a higher p r o b a b i l i t y of f a i l u r e than does a short chain with fewer l i n k s . In t h i s experiment, the numerous very small c r a c k s i n the wood are not i n i t i a t i n g f a i l u r e . Rather, only one of the " l i n k s " i s being t e s t e d , because the crack l e n g t h which i n i t i a t e s f a i l u r e i s c o n t r o l l e d . T h e r e f o r e the a p p l i c a t i o n of the W e i b u l l model i s c o n s i d e r e d i n a p p r o p r i a t e i n t h i s study. A f t e r a specimen has f a i l e d under constant l o a d , i t i s necessary to determine i t s s t r e s s r a t i o so that the f a i l u r e can be recorded on the s t r e s s r a t i o versus time to f a i l u r e p l o t . Before the s t r e s s r a t i o can be c a l c u l a t e d the o r i g i n a l short term s t r e n g t h of the specimen needs to be estimated. T h i s i s determined by the method of ranking. Because the specimens i n a d u r a t i o n of l o a d t e s t f a i l i n a sequence, the f a i l u r e times of the specimens are ranked. Using the d u r a t i o n of load t e s t sample s i z e , ranking i s a l s o a p p l i e d to the Normal d i s t r i b u t i o n of short term s t r e n g t h s a l r e a d y determined. From t h i s a ranked l i s t of short term s t r e n g t h s i s ob t a i n e d . Each ranked f a i l u r e time i s then a s s i g n e d the short term s t r e n g t h of equal rank. For example, the f i r s t specimen to f a i l i n the d u r a t i o n of loa d t e s t i s as s i g n e d a short term s t r e n g t h equal to the lowest 41 s t r e n g t h specimen taken from the Normal d i s t r i b u t i o n of short term s t r e n g t h . By d i v i d i n g the value of the s h o r t term s t r e n g t h by the l e v e l of a p p l i e d s t r e s s , an estimate of the s t r e s s r a t i o i s determined f o r each specimen. The p l o t of s t r e s s r a t i o versus the l o g a r i t h m of time to f a i l u r e i s used e x t e n s i v e l y , as seen i n Chapter 2 where the r e s u l t s of other r e s e a r c h e r s are quoted. In subsequent chapters the r e s u l t s of t h i s study are a l s o presented i n t h i s manner. In most t e s t s no specimens broke on l o a d i n g . However, s e v e r a l specimens d i d f a i l as soon as the l o a d was a p p l i e d . These f a i l u r e s were recorded on the d u r a t i o n of l o a d p l o t s but were as s i g n e d a short time to f a i l u r e and p l o t t e d with a d i f f e r e n t symbol. At f i r s t i t may seem alarming t h a t t h a t even towards the end of the experiments, there does not seem to be any slowing i n the r a t e of f a i l u r e s . However i t must be r e a l i s e d that the time s c a l e i s a l o g a r i t h m i c one, and i f the r e s u l t s were p l o t t e d with a r e a l time a x i s , the c u r v a t u r e would not be n e g a t i v e , but p o s i t i v e , with very l i t t l e change i n s t r e n g t h at long l o a d i n g times. In c a l c u l a t i n g the s t r e s s r a t i o , a l t e r n a t i v e approaches c o u l d have been adopted. By u t i l i s i n g t h e . s h o r t term s t r e n g t h s of the s u r v i v o r s of the d u r a t i o n of l o a d t e s t s , the short term s t r e n g t h s of the specimens which f a i l e d d u r i n g the t e s t s can be c a l c u l a t e d . For example, i f the t e s t i s terminated at a time where only 50% of the specimens have f a i l e d , then 50% of the specimens w i l l be s u r v i v o r s and w i l l be ramp loaded to f a i l u r e . 42 The s t r e n g t h of these s u r v i v o r s may i n d i c a t e that the short term c o n t r o l e s t a b l i s h e d from the sho r t term c o n t r o l sample i s not the best one to represent the short term s t r e n g t h of the d u r a t i o n of l o a d t e s t sample. The a p p r o p r i a t e adjustment can then be made. In t h i s study t h i s approach was not adopted f o r two reasons. F i r s t , many of the t e s t s d i d not have a s i g n i f i c a n t number of s u r v i v o r s . Second, because some of the s u r v i v o r s may be damaged they can no longer be used to represent the short term s t r e n g t h d i s t r i b u t i o n of the o r i g i n a l sample. 3.8 Determination of Ts. Ts i s (as a l r e a d y mentioned) the time at which the crack begins to propagate l o n g i t u d i n a l l y , and s i g n i f i e s the beginning of phase 2. D i f f e r e n t approaches c o u l d have been taken in order to determine the l e n g t h of time spent in phase 2. Rather than monitoring l o n g i t u d i n a l crack growth v i s u a l l y , a d i f f i c u l t technique even with the a i d of the most s o p h i s t i c a t e d equipment, a d i f f e r e n t method was adopted. I t was decided to t r y and " c a t c h " specimens at the time when the c r a c k s are lengthening i . e . , the crack i s undergoing phase 2. By s t o p p i n g a d u r a t i o n of l o a d t e s t while there are numerous s u r v i v o r s , some of the s u r v i v o r s w i l l be "caught" i n phase 2, while the remainder w i l l s t i l l be i n phase 1. The s u r v i v o r s undergoing phase 2 w i l l have accumulated some damage i . e . , t h e i r c r a c k s w i l l have lengthened. By f a i l i n g a l l s u r v i v o r s i n a ramp loa d t e s t , the number of damaged s u r v i v o r s (and the extent of damage) can be determined, 43 because the damaged specimens w i l l f a i l at lower s t r e s s e s as p r e d i c t e d by the s t r e s s i n t e n s i t y equation (2-1). For example, a sample of 20 specimens i s shown i n f i g u r e 16. I f the constant s t r e s s i s removed a f t e r 8 specimens have f a i l e d , then i t can be seen that specimens 9 and 10 w i l l by t h i s stage have accumulated some damage, i f the Ts l i n e i s i n p o s i t i o n ( 1 ) . These two specimens w i l l t h e r e f o r e f a i l at reduced s t r e s s e s as shown i n f i g u r e 17. I f the Ts l i n e i s at p o s i t i o n (2) however, specimens 11 12 and 13 w i l l a l s o be expected to accrue some damage. Th e r e f o r e , the number of damaged specimens f i x e s the p o s i t i o n of one p o i n t on the Ts l i n e , i n r e l a t i o n to the Teat l i n e . The number of damaged specimens i s d e f i n i n g the amount of time being spent i n phase 2, t h e r e f o r e determining the r e l a t i v e p o s i t i o n s of the Ts and Teat l i n e s . From (2-17), the d i s t a n c e between the Ts and Teat l i n e s i s c o n t r o l l e d by the F (e) / ( 0 2 * 2 ) term. For a p a r t i c u l a r s t r e s s r a t i o 9 and value of F (e) d e f i n e d by the model and the creep parameters chosen, the only unknown parameter i s <t>. By f i t t i n g the experimental r e s u l t s to (2-17), an estimate of <t> can be determined, which in turn y i e l d s an estimate of a,. When r e f e r r i n g to t h i s experiment as the Ts experiment, i t should be remembered that the i m p l i c a t i o n s go f u r t h e r than j u s t d etermining Ts. In e s t i m a t i n g and commenting upon i t s v a l i d i t y , the d e t e r m i n a t i o n of Ts becomes very important i n p r o v i d i n g a f u r t h e r t e s t of the p r e d i c t i o n s of the v i s c o e l a s t i c f r a c t u r e mechanics models, as they p e r t a i n to the time dependant 44 f a i l u r e of wood. 3.9 E r r o r s The random e r r o r s i n t h i s experiment are s m a l l . The e r r o r in measuring the c r o s s s e c t i o n i s very small and l e s s than 1%. Small random e r r o r s c o u l d occur i n the reading of the Satec and in the reading of the d i a l gauge when c a l i b r a t i n g . I t i s estimated that the maximum e r r o r p o s s i b l e i n the l e v e l of a p p l i e d s t r e s s i s i n the order of l e s s than 2%. However due to the e c c e n t r i c i t i e s of end p l a t e s and of the specimens themselves, v a r i a b i l i t y o c c u r r e d i n the l e v e l of maximum s t r e s s i n the specimens. As d i s c u s s e d f u r t h e r i n S e c t i o n 5.2.2 the maximum s t r e s s s at the c r i t i c a l c r o s s s e c t i o n was 10% higher (on the average) than the nominal a p p l i e d s t r e s s . Confidence l i m i t s on the r e s u l t s w i l l be d e a l t with i n the d i s c u s s i o n . 45 3.10 Summary To simulate the most c r i t i c a l f a i l u r e mechanism o c c u r r i n g i n the low p e r c e n t i l e s of commercial m a t e r i a l , the f o l l o w i n g specimen geometry was deci d e d upon. The specimen c o n s i s t e d of a u n i a x i a l t e n s i l e specimen with a c e n t r a l l y p l a c e d through crack, p r o v i d i n g a low c o e f f i c i e n t of v a r i a t i o n f o r the short term s t r e n g t h . The plane of the crack was the t a n g e n t i a l l o n g i t u d i n a l plane and the d i r e c t i o n of propagation was l o n g i t u d i n a l . The a p p l i e d s t r e s s was i n t e n s i o n p e r p e n d i c u l a r to the g r a i n , s t r e s s i n g the crack i n the opening mode. The method of ranking was assumed, so that the short term s t r e n g t h s of a c o n t r o l t e s t (approximated to the Normal D i s t r i b u t i o n ) c o u l d be a s s i g n e d to the specimens of the constant l o a d t e s t which were ranked a c c o r d i n g t o t h e i r f a i l u r e times. By d i v i d i n g the short term s t r e n g t h by the l e v e l of a p p l i e d s t r e s s , the s t r e s s r a t i o f o r each specimen was c a l c u l a t e d . The s t r e s s r a t i o of each- specimen was p l o t t e d a g a i n s t the lo g a r i t h m of the time t o f a i l u r e i n the development of the d u r a t i o n of load p l o t s . The d u r a t i o n of l o a d behaviour of t h i s s m a l l t e s t specimen i s only expected to p r o v i d e a lower bound (worst case) to the behaviour of commercial m a t e r i a l , which because of the redundancy o c c u r r i n g a r o u n d the knots, tends to demonstrate an improved d u r a t i o n of l o a d performance. However, the d u r a t i o n of lo a d trends determined from the small t e s t specimens would s t i l l be expected t o r e f l e c t themselves i n the behaviour of commercial m a t e r i a l . 46 CHAPTER 4 RESULTS 4.1 I n t r o d u c t i o n . T h i s chapter g i v e s a b r i e f d e s c r i p t i o n of the purpose of the experiments and t h e i r r e s u l t s f o r experiment Nos 1 through 6 (as o u t l i n e d i n Appendix 1). A d e t a i l e d d i s c u s s i o n of the i m p l i c a t i o n s i s presented i n Chapter 5. At the commencement of t h i s t h e s i s , i t was planned that the comparison of specimens loaded i n a st e p f u n c t i o n manner ( r e f e r r e d to as c y c l i c l o a d i n g ) with specimens loaded c o n s t a n t l y , be the major f e a t u r e of the r e s e a r c h . However as the study progressed, t h i s emphasis changed markedly as an i n t e r e s t i n the v i s c o e l a s t i c f r a c t u r e mechanics models developed. T h i s change i n emphasis i s r e f l e c t e d i n the sequence of the experiments which are d e s c r i b e d i n the order i n which they were c a r r i e d out. 47 4.2 Experiment D e s c r i p t i o n s . 4.2.1 Experiment No.l, C y c l i c - 1 . Experiment No.1 served the purpose of comparing c y c l i c l o a d i n g (or i n t e r m i t t e n t l o a d i n g ) with constant l o a d i n g , i n order to determine i f the assumption of l i n e a r l y adding loaded times i s v a l i d . In p r a c t i c e , d e s i g n e r s only c o n s i d e r a l i n e a r a d d i t i o n of the loaded times as c o n t r i b u t i n g to the design d u r a t i o n of l o a d thus i g n o r i n g the time f o r which a member i s unloaded. The step f u n c t i o n c y c l e of 3 hours under l o a d followed by 3 hours unloaded was chosen f o r the f o l l o w i n g reasons. As t h i s was the f i r s t experiment, an i n i t i a l estimate of the d u r a t i o n of lo a d behaviour was d e s i r e d . T h e r e f o r e the t o t a l t e s t d u r a t i o n d i d not need to be very l o n g . In order to perform a s i g n i f i c a n t number of c y c l e s w i t h i n t h i s time a 3 hour c y c l e time was most a p p r o p r i a t e . T h i s c y c l e was performed 3 times each day, and at the end of the t h i r d l o a d i n g p e r i o d a 9 hour recovery p e r i o d was allowed i n order f o r the author to o b t a i n some s l e e p . A t o t a l of 20 c y c l e s was performed i n t h i s manner. The recovery p e r i o d of 3 hours was c o n s i d e r e d adequate, because at l e a s t 90% of the t o t a l creep recovery w i l l have occ u r r e d by t h i s time, a c c o r d i n g to Kass(l969) who c a r r i e d out creep and recovery t e s t s i n t e n s i o n p e r p e n d i c u l a r to the g r a i n . At t h i s e a r l y stage i n the experiments, the method of i n s e r t i n g the c e n t r a l crack had not been p e r f e c t e d . The razor 48 blade used i n t h i s experiment only protruded from one si d e of the brass rod to which i t was at t a c h e d . In order to i n i t i a t e the crack, i t was d r i v e n through the c e n t r a l hole twice, each time c u t t i n g out one h a l f of the cr a c k . Because of t h i s procedure, c o n t r o l over the crack l e n g t h was not as good as i n the subsequent experiments where double edged blades were manufactured and used. As can be seen i n f i g u r e 18, no marked d i f f e r e n c e i n behaviour was observed. Due to the small sample s i z e , experiment No.4 was c a r r i e d out at a l a t e r date i n order to improve the confidence of the r e s u l t . 4.2.2 Experiment No.2, Two Crack Lengths. Experiment No.2 was designed to t e s t the s t r e n g t h e f f e c t , as p o s t u l a t e d by the N i e l s e n and step-wise models i . e . , f o r two specimens loaded to the same s t r e s s r a t i o , but whose short term s t r e n g t h s are d i f f e r e n t , the times to f a i l u r e w i l l be d i f f e r e n t . The high s t r e n g t h specimens w i l l f a i l sooner than the low s t r e n g t h specimens. The p r e d i c t e d d i f f e r e n c e in behaviour becomes more pronounced at longer times. In order to ob t a i n two groups of specimens with d i f f e r e n t s h o r t term s t r e n g t h s , two samples were prepared from the same board. One sample had a crack of t o t a l l e n g t h 6.22mm and the other a t o t a l crack l e n g t h of 10.8mm. Accord i n g to the f r a c t u r e toughness equation (2-1) the small crack l e n g t h specimens w i l l f a i l at a higher s t r e s s . In order that the same s t r e s s r a t i o be 49 a p p l i e d to both samples, a higher l e v e l of s t r e s s i s a p p l i e d to the s h o r t crack l e n g t h specimens. In t h i s way, the behaviour of specimens with d i f f e r e n t s t r e n g t h s but loaded to the same s t r e s s r a t i o s , can be s t u d i e d i n order to t e s t the p r e d i c t i o n of the s t r e n g t h e f f e c t . In a n a l y s i n g experiment Nos 2 and 3, a method was employed which served to combine a l l of the short term s t r e n g t h r e s u l t s i n t o one p o p u l a t i o n . For each experiment, the d i f f e r e n t specimen groups had d i f f e r e n t crack l e n g t h s and s l i g h t l y d i f f e r e n t crack geometries. They t h e r e f o r e had s l i g h t l y d i f f e r e n t F(c/b) f a c t o r s . F(c/b) i s the f a c t o r which i s a p p l i e d to the s t r e s s i n t e n s i t y to compensate f o r the lack of an i n f i n i t e medium, as c a l c u l a t e d by Tada, P a r i s and Irwin(l973) ( f i g u r e 26). Refer a l s o to S e c t i o n 5.2.5 where the F(c/b) f a c t o r i s d i s c u s s e d f u r t h e r . When the short term t e s t s were c a r r i e d out, the a c t u a l f r a c t u r e toughness ( i n c o r p o r a t i n g F(c/b) ) was c a l c u l a t e d f o r each specimen and the Normal curve f i t t e d to the e n t i r e p o p u l a t i o n . The s t r e s s r a t i o was c a l c u l a t e d by d i v i d i n g the a p p l i e d s t r e s s i n t e n s i t y f a c t o r by the a c t u a l f r a c t u r e toughness, r a t h e r than by d i v i d i n g the a p p l i e d s t r e s s by the short term s t r e n g t h . By u t i l i s i n g t h i s method, the data p o i n t s changed l i t t l e , but the method of a n a l y s i s was more c o n s i s t e n t . T h i s experiment ( f i g u r e 19) showed only l i m i t e d s e p a r a t i o n f o r the data p o i n t s of the two crack s i z e s , f o r two reasons. F i r s t , the s t r e n g t h e f f e c t as p r e d i c t e d by the Madsen and Johns(!982) f i t of the N i e l s e n model was l a t e r found 50 e x p e r i m e n t a l l y to be l e s s s i g n i f i c a n t than expected. Work by Madsen and Johns i n d i c a t e d a more s i g n i f i c a n t s t r e n g t h e f f e c t because of a high a^/Y value of approximately 82MPa f i t t e d to the N i e l s e n model (Y was an unknown f u n c t i o n of random v a r i a b l e s d e s c r i b i n g the c h a r a c t e r i s t i c s of the crack f o r any p a r t i c u l a r p i e c e of lumber). Second, the l e v e l of a p p l i e d s t r e s s was too h i g h , c ausing the experiment to terminate i n a time too short f o r the s t r e n g t h e f f e c t to become very apparent. T h i s was caused by the f a c t that the specimens from the board used i n experiment No.2 f a i l e d unexpectedly sooner than those taken from the board f o r experiment No.1. In s p i t e of these l i m i t a t i o n s the t r e n d of the data p o i n t s at long times i s i n agreement with the t r e n d as p r e d i c t e d by the N i e l s e n and step-wise models i . e . , the long crack l e n g t h specimens (low strength) s u r v i v e d the l o n g e s t when loaded to the same s t r e s s r a t i o . 4.2.3 Experiment No.3, Three Crack Lengths. Because of the l i m i t a t i o n s of experiment No.2 a l r e a d y mentioned, experiment No.3 was a l s o aimed at t e s t i n g the s t r e n g t h e f f e c t as given by the v i s c o e l a s t i c f r a c t u r e mechanics models. By t e s t i n g three samples of specimens, each taken from the same p o p u l a t i o n , each with a d i f f e r e n t crack s i z e (and t h e r e f o r e s t r e n g t h ) , the s t r e n g t h e f f e c t was i n v e s t i g a t e d . The three crack lengths ( t o t a l ) were 10.06, 6.22, and 3.86 mm. T h i s experiment ( f i g u r e 20) was a success i n a p p l y i n g the c o r r e c t l e v e l of a p p l i e d s t r e s s , which y i e l d e d an average time 51 to f a i l u r e of approximately 10 days. At the end of the t e s t where the s t r e n g t h e f f e c t i s most pronounced, the data p o i n t s of the three crack s i z e s separated out i n manner p r e d i c t e d by the N i e l s e n and step-wise models, i . e . , f o r the same s t r e s s r a t i o , the h i g h s t r e n g t h (small crack) specimens f a i l e d sooner than the medium crack specimens, which i n tu r n f a i l e d at s h o r t e r times than the l a r g e crack (low stre n g t h ) specimens. Because the use of a very small crack was employed, the short term t e n s i l e s t r e s s at f a i l u r e became q u i t e high, averaging 2.57MPa. T h i s caused some of the specimens ( i n a p r e l i m i n a r y t e s t ) t o f a i l at the screw conne c t i o n a t the end p l a t e . At t h i s stage of the r e s e a r c h the p r a c t i c e of w a i s t i n g or necking down of the specimens over t h e i r middle p o r t i o n s was i n t r o d u c e d f o r the more h i g h l y s t r e s s e d specimens, i n order t o a v o i d end f a i l u r e s . A l l f u r t h e r t e s t s used necked specimens. 4.2.4 Experiment No.4, C y c l i c - 2 . As a l r e a d y mentioned, experiment No.4 ( f i g u r e 21) was a repeat of experiment No.1, but using a d i f f e r e n t board and an i n c r e a s e d sample s i z e , i n order that i n c r e a s e d c o n f i d e n c e c o u l d be expressed i n the pre v i o u s r e s u l t . A c y c l e of 2 hours on and 4 hours o f f was made p o s s i b l e by the a s s i s t a n c e of h e l p e r s , each r e s p o n s i b l e f o r l o a d i n g and unloading 2 hours l a t e r . T h i s c y c l e was maintained c o n t i n u o u s l y , day and n i g h t . Some small v a r i a t i o n s occurred i n the d u r a t i o n of the recovery p e r i o d , but the d u r a t i o n of the 52 loaded time was w e l l monitored. T h i s experiment s u c c e s s f u l l y confirmed the p r e v i o u s r e s u l t of experiment 1, by demonstrating that specimens loaded c y c l i c a l l y g i v e the same d u r a t i o n of l o a d p l o t as specimens loaded c o n s t a n t l y , assuming only the t o t a l loaded time i s p l o t t e d . 4.2.5 Experiment No.5, Moisture T e s t . Experiment No.5 was aimed at i n v e s t i g a t i n g the i n f l u e n c e of moisture content on the d u r a t i o n of l o a d e f f e c t , and t e s t i n g the r e s u l t a g a i n s t that p r e d i c t e d by the v i s c o e l a s t i c f r a c t u r e mechanics models. In order to do t h i s , two d u r a t i o n of load experiments were c a r r i e d out, one wet and one dry, each with t h e i r own short term s t r e n g t h c o n t r o l . Both samples were subsets of a l a r g e r sample. The dry t e s t was conducted w i t h i n the atmosphere of the t e s t room, sub j e c t to s m a l l f l u c t u a t i o n s i n humidity and temperature. The wet specimens were s a t u r a t e d in water f o r one week, and i n order to maintain t h e i r high moisture content f o r the d u r a t i o n of l o a d t e s t , were wrapped i n f i l t e r paper and enclosed i n p l a s t i c bags with f r e e water i n the bottom. In t h i s manner the high moisture content of the specimens was s u c c e s s f u l l y maintained f o r the d u r a t i o n of the experiment. As can be seen i n f i g u r e 22, the wet specimens f a i l e d much sooner than the dry. The r e s u l t s i n d i c a t e t h a t the times to f a i l u r e f o r the wet specimens are approximately one order of magnitude sooner than f o r the dry specimens. 53 As d i s c u s s e d f u r t h e r i n S e c t i o n 5.2.2, the wet specimens became curved upon immersion i n water. The c u r v a t u r e was about an a x i s i n the l o n g i t u d i n a l t a n g e n t i a l plane, p o i n t i n g i n the l o n g i t u d i n a l d i r e c t i o n . The e c c e n t r i c i t i e s induced by t h i s c u r v a t u r e reduced the short term s t r e n g t h s of the wet specimens c o n s i d e r a b l y . However the e s t i m a t i o n of the s t r e s s r a t i o remains v a l i d because the short term s t r e n g t h c o n t r o l s were a l s o wet and curved and t h i s reduced value was used i n the c a l c u l a t i o n s . 4.2.6 Experiment No.6, Ts. The aim of experiment No.6 was to determine the p o s i t i o n of the Ts l i n e and determine T h i s aim has been more f u l l y d e s c r i b e d i n S e c t i o n 3.8. The t e s t c o n s i s t e d of 34 short term c o n t r o l specimens and 89 l o a d d u r a t i o n specimens, a l l with t o t a l crack lengths of 6.22mm. In order that the p o s i t i o n of the Ts p o i n t be determined most a c c u r a t e l y , the l e v e l of a p p l i e d s t r e s s i s very important. In p a r a l l e l with the s t r e n g t h e f f e c t , the s e p a r a t i o n of the Ts l i n e and the Teat l i n e become more s i g n i f i c a n t at longer times. T h e r e f o r e i n order that the specimens be loaded to the c o r r e c t l e v e l of a p p l i e d s t r e s s f o r an a p p r o p r i a t e t e s t d u r a t i o n (2 weeks), 10 of the constant l o a d specimens were p r e t e s t e d f o r 3 days before the complement of the specimens were loaded. The d e t e r m i n a t i o n of the t e r m i n a t i o n time of the constant l o a d experiment or o f f - l o a d time i s a l s o important i n the design 54 of a sound Ts experiment. The o f f - l o a d time was not determined at the commencement of the constant l o a d t e s t , but r a t h e r during the t e s t , where the best d e c i s i o n c o u l d be made i n l i g h t of the d u r a t i o n of l o a d c h a r a c t e r i s t i c s of the t e s t so f a r . E s s e n t i a l l y , the d e c i s i o n when to o f f - l o a d was made upon the b a s i s of s t r e s s r a t i o . On the one hand, the longer the d u r a t i o n of the experiment, the more d i s t i n c t becomes the d i f f e r e n c e between the Ts p o i n t and the Teat l i n e . On the other hand, i f o f f - l o a d i s delayed too long, the c o n f i d e n c e i n the s t r e s s r a t i o of the s u r v i v o r s begins to d i m i n i s h . In a d d i t i o n , i f the t e s t i s l e f t f o r too long, a l l of the s u r v i v i n g specimens may demonstrate damage i n which case the Ts p o i n t f a i l s to be d e f i n e d , because the t o t a l number of damaged s u r v i v o r s can not been determined. The r e s u l t s of t h i s experiment are i l l u s t r a t e d in f i g u r e s 23 and 24. As p r e d i c t e d by the N i e l s e n and step-wise models, many s u r v i v o r s from the d u r a t i o n of l o a d t e s t demonstrated c o n s i d e r a b l e damage, with two short term f a i l u r e s t r e s s e s as low as l.90MPa. Note that the weakest specimen i n the short term c o n t r o l t e s t of sample s i z e 34 ( f i g u r e 25) had a short term s t r e n g t h of 2.l8MPa, c o n s i d e r a b l y higher than the minimum value found i n the s u r v i v o r s . The d e t e r m i n a t i o n of the Ts p o i n t as given i n f i g u r e 23 was based upon the o b s e r v a t i o n that 20 s u r v i v i n g specimens i n c u r r e d damage ( f i g u r e 24), the 21st s u r v i v o r being p l a c e d to the r i g h t of the c o n t r o l l i n e . The s t r e s s r a t i o of the l a s t specimen (the 59th) to f a i l i n the d u r a t i o n of l o a d t e s t was 0.730. The 55 s t r e s s r a t i o of the 79th (59th p l u s 20) specimen was 0.704. 4.3 Summary T h i s chapter has served to pr o v i d e an o u t l i n e of the day to day c o n s i d e r a t i o n s , and r e s u l t s of the experiments performed. Each experiment c o n t r i b u t e d to a b e t t e r understanding of how best to design the next one. Thus the experimental method became more e f f i c i e n t , and b e t t t e r c o n t r o l was e x e r c i s e d over the experiments, so that they t e s t e d the a p p r o p r i a t e f e a t u r e s more p r e c i s e l y . 56 CHAPTER 5 DISCUSSION 5.1 I n t r o d u c t i o n T h i s chapter f i r s t e x p l a i n s the methods of a n a l y s i s a p p l i e d i n the p r e s e n t a t i o n of the r e s u l t s , and then d i s c u s s e s the creep f u n c t i o n used, and other reseach i n t o the phenomenon of v i s c o e l a s t i c creep. F o l l o w i n g i s a d e t a i l e d d i s c u s s i o n of a l l of the experiments performed i n t h i s study, i n c l u d i n g comments on the r e s u l t i n g i m p l i c a t i o n s . F i n a l l y , mention i s made of the c o n f i d e n c e which can be expressed i n the r e s u l t s , and the g e n e r a l l i m i t a t i o n s of the experiment. i 5.2 Experimental Method and A n a l y s i s . 5.2.1 The Normal D i s t r i b u t i o n In the experimental a n a l y s i s , a Normal curve was f i t t e d to the s h o r t term s t r e n g t h d a t a . By a p p l y i n g the assumption of ranking ( i . e . , the weakest specimen f a i l s f i r s t ) , t h i s model was used to determine the short term s t r e n g t h s of the constant load specimens. T h i s curve p r o v i d e d a good f i t to the mid-range 57 data, however the f u l l extent of the t a i l s of the curve are not r e a l i s e d by the experimental data ( f i g u r e 25). Thus when the s t r e s s r a t i o i s c a l c u l a t e d , the f i r s t one or two p o i n t s s l i g h t l y o v erestimate the s t r e s s r a t i o , while the l a s t few p o i n t s underestimate i t . T h i s was the only l i m i t a t i o n of using the Normal model. 5.2.2 F r a c t u r e Toughness The specimen design used i n t h i s experiment was an o r i g i n a l one. However a comparison of the f r a c t u r e toughness values obtained here can s t i l l be made with those of Schniewind and Centeno(1973) who used a d i f f e r e n t specimen. In t e s t i n g a notched bending specimen (D. F i r ) i n the t a n g e n t i a l l o n g i t u d i n a l plane with propagation i n the l o n g i t u d i n a l d i r e c t i o n , they obtained a f r a c t u r e toughness value of 410 kPa m1'2. F o l l o w i n g i s a l i s t of the f r a c t u r e toughness values obtained i n t h i s study from the specimens with a c e n t r a l l y p o s i t i o n e d c r a c k . 58 Expt No, T i t l e F(c/b) « 0(MPa) 2c(nun) Frac.Toug.(kPa ml/2) 1 C y c l i c - 1 1.05 1.923 11.0 270 2 Two Crack 1.05 1.908 10.8 260 2 Two Crack 1.02 2.578 6.22 260 3 Three Crack 1.05 2.043 10.06 257 3 Three Crack ' 1.02 2.470 6.22 244 3 Three Crack 1.05 3.260 3.86 254 4 C y c l i c - 2 1.04 2.472 6.22 250 5 M o i s t u r e , Wet 1.04 1.742 6.22 180 5 M o i s t u r e , Dry 1.04 2.417 6.22 250 6 Ts 1.04 2.465 6.22 250 N o t i c e that experiment No.2 y i e l d s c o n s i s t e n t f r a c t u r e toughness v a l u e s . Experiment No.3 shows however, a higher v a r i a b i l i t y i n the f r a c t u r e toughness v a l u e s . The reason that the wet specimens have such a low value, i s not due to a low s t r e n g t h , but rather to the f a c t that the specimens became curved upon immersion i n water. In order to determine the e f f e c t of t h i s c u r v a t u r e , a tensometer was a t t a c h e d to an uncracked specimen ( a l t e r n a t e l y to oppposing f a c e s ) and the specimen p u l l e d t o a s t r a i n e q u i v a l e n t to the breaking s t r a i n . From t h i s measurement i t was estimated that the e c c e n t r i c i t y i n the wet specimens was induc i n g a s t r e s s d i s t r i b u t i o n at the c r i t i c a l c r o s s s e c t i o n where the maximum s t r e s s was 150% of the nominal. A s i m i l a r t e s t on dry specimens i n d i c a t e d an i n c r e a s e over the nominal s t r e s s of 10% was reasonable. In determining the e f f e c t s of e c c e n t r i c i t y and specimen c u r v a t u r e , t h i s t e s t i n g was only approximate. Because 59 the specimens had v a r y i n g e c c e n t r i c i t i e s and c u r v a t u r e s , the va l u e s of 150% and 110% were estimates of the order of magnitude of the e f f e c t s o n l y . By a p p l y i n g these adjustments, the wet and dry values of f r a c t u r e toughness become almost i d e n t i c a l . T h i s r e s u l t i s i n keeping with the assumption that the f a i l u r e mechanism i n commercial m a t e r i a l i s i n t e n s i o n p e r p e n d i c u l a r to the g r a i n . Madsen(1973,1975) a l s o found no d i f f e r e n c e i n the s t r e n g t h of wet and dry m a t e r i a l at the lower q u a n t i l e s , where knots i n i t i a t e f a i l u r e i n t e n s i o n p e r p e n d i c u l a r to the g r a i n . 5.2.3 F i t of the Creep and L i m i t Strength Parameters The Clouser creep f u n c t i o n parameters "a" and "b" (2-5) can be used to f i t the models to the d u r a t i o n of load data , p o i n t s . In a p p l y i n g (2-17), these parameters have an e f f e c t upon the slope of the d u r a t i o n of lo a d p l o t , and a l s o upon the p o s i t i o n along the time a x i s , ( f i g u r e 27). For a low "b" value the d u r a t i o n of lo a d l i n e w i l l be f l a t t e r than f o r a high "b" va l u e . The "a" parameter c o n t r o l s the p o s i t i o n along the d u r a t i o n of loa d curve on the time a x i s . A high "a" value tends to d i s p l a c e the d u r a t i o n of lo a d l i n e to the l e f t . The «, c,, and <t0 parameters c o n t r o l the d i s t a n c e between the Ts and Teat l i n e s ( f i g u r e 28). For a constant value of s t r e s s r a t i o 9 (<f/a0), the d i s t a n c e between the Ts and Teat l i n e s can be i n c r e a s e d i f the s t r e n g t h r a t i o # ( * o / * i ) r i s decreased by i n c r e a s i n g *,. The s t r e n g t h r a t i o can a l s o be 60 decreased by de c r e a s i n g * 0 and h o l d i n g a, c o n s t a n t . But t h i s changes the s t r e s s r a t i o . However the s t r e s s r a t i o can s t i l l be h e l d constant i f a lower s t r e s s a i s a l s o a p p l i e d . In t h i s case ( f o r a constant s t r e s s r a t i o ) , * has been decreased by i n c r e a s i n g * 0 and « si m u l t a n e o u s l y . T h i s e f f e c t i s known as the s t r e n g t h e f f e c t and i s demonstrated i n S e c t i o n 5.5, where the r e s u l t s of the experiment are compared with the p r e d i c t i o n s of the models. 5.2.4 Short Term Strength Bias When f i t t i n g the N i e l s e n and step-wise model curves to the data, a b i a s i s i n t r o d u c e d i f i t i s assumed that the ramp loa d s t r e n g t h based upon a 45 second f a i l u r e time, i s e q u i v a l e n t to the short term s t r e n g t h where the f a i l u r e time i s i n f i n i t e s i m a l . In the N i e l s e n model f o r example i f b=l/3, and #=0.3, then a c c o r d i n g to Nielsen(1978) f o r a ramp loa d to f a i l u r e t e s t of 45 seconds the e q u i v a l e n t time to f a i l u r e under constant l o a d i s approximately equal to 15 seconds. For Teat equals 15 seconds under constant l o a d , the s t r e s s r a t i o 9 equals 0.98. Thus i n t h i s case a b i a s due to an e r r o r of 2% i n the e s t i m a t i o n of the r e a l short term s t r e n g t h i s i n t r o d u c e d . As t h i s b i a s i s small and w i t h i n the bounds of experimental e r r o r , i t was ignored i n the experimental a n a l y s i s . For the Wet Moisture t e s t , however, the b i a s i s more s i g n i f i c a n t because of the much s h o r t e r times to f a i l u r e f o r the constant load t e s t s . In a d d i t i o n , the compliance of the wet specimens was found to be higher by a 61 f a c t o r of two, thus i n c r e a s i n g the ramp load f a i l u r e time. Even so, t h i s was not i n c o r p o r a r e d , because the c o r r e c t n e s s of any creep f u n c t i o n used i s very d i f f i c u l t to a s c e r t a i n over the very sho r t times i n q u e s t i o n . 5.2.5 F i n i t e - I n f i n i t e Medium Adjustment. A f u r t h e r s i m p l i f y i n g assumption has been made i n the c a l c u l a t i o n s . A l l of the equations where the theory of v i s c o e l a s t i c f r a c t u r e mechanics are d e r i v e d , assume an i n f i n i t e medium. However i n t h i s experiment, the crack i s propagating i n a f i n i t e medium and t h e r e f o r e , as i t propagates, F(c/b) i s not constant but g r a d u a l l y i n c r e a s i n g . F(c/b) i s the f a c t o r which i s a p p l i e d to the s t r e s s i n t e n s i t y to allow f o r a f i n i t e medium, as c a l c u l a t e d by Tada, P a r i s and I r w i n ( l 9 7 3 ) , ( f i g u r e 26). Ther e f o r e the s t r e s s i n t e n s i t y equation (2-1) becomes (5-1 ) k 2 = ( F ( c / b ) ) 2 tf2 i r e In the Ts experiment, f o r example, F(c/b) equals 1.04 at the s t a r t of the experiment. Due to the subsequent i n c r e a s e i n h a l f crack l e n g t h (from 3.11mm to a c r i t i c a l crack length of 5.7mm over the d u r a t i o n of the experiment), specimens about to f a i l have an F(c/b) value of 1.11. In order t o c a l c u l a t e both the c r i t i c a l crack l e n g t h and the F(c/b) value of 1.11 at the a p p l i e d l e v e l of s t r e s s , the crack l e n g t h must be i t e r a t e d using (5-1) u n t i l the c r i t i c a l s t r e s s i n t e n s i t y f a c t o r i s obtained. 62 The i n c r e a s i n g F(c/b) f a c t o r tends to a c c e l e r a t e crack growth at a f a s t e r r a t e than f o r the i n f i n i t e medium case where F(c/b) remains constant at a value of 1.0. In order to q u a n t i f y the r e s u l t i n g d i f f e r e n c e i n the times of propagation, the incremental c a l c u l a t i o n s shown i n Appendix 4 were performed. A changing F(c/b) value was a p p l i e d to the step-wise model f o r the case of b=1/5, so the c a l c u l a t i o n s were a p p l i c a b l e to the Ts experiment. These showed that the e x i s t e n c e of an i n f i n i t e medium would tend to i n c r e a s e the time spent i n phase two by approximately 30%. A l l o w i n g f o r the e f f e c t of Ts, the i n c r e a s e i n the t o t a l time to f a i l u r e became only 19%. F u r t h e r c a l c u l a t i o n s showed that the s h i f t of 19% was p r a c t i c a l l y independent of the s t r e s s r a t i o , thus implying that an adjustment i n the "a" parameter c o u l d be used to allow f o r the e f f e c t of the f i n i t e specimen s i z e . In g e n e r a l these adjustments were ignored when f i t t i n g parameters to the data because the a p p r o p r i a t e adjustment can be made at a l a t e r date, and i n any case, the value of "a" i s not very s e n s i t i v e to a small change i n the p o s i t i o n of the d u r a t i o n of l o a d curve along the time a x i s . However when comparing the N i e l s e n and step-wise models, the adjustment needs to be made before v a l i d c o n c l u s i o n s can be made. T h i s i s p a r t i c u l a r l y important i n the Ts experiment where the a p p r o p r i a t e adjustment w i l l a f f e c t the d i f f e r e n c e between the Ts and Teat l i n e s , an important f e a t u r e of the Ts a n a l y s i s . The method a p p l i e d i n the case of the Ts experiment was to leave the experimental data alone, and f i t the v i s c o e l a s t i c 63 model curve to a p o s i t i o n 19% to the r i g h t of the experimental data as o u t l i n e d i n s e c t i o n 5.4 where the Ts experiment i s d i s c u s s e d . Note that the 19% i n c r e a s e i n the time to f a i l u r e i s a r e a l time value, and that on the l o g a r i t h m i c time p l o t the in c r e a s e does not appear q u i t e so s i g n i f i c a n t because the lo g a r i t h m of 1.19 i s only 0.076. 5.3 Creep Functions and Parameters. 5.3.1 I n t r o d u c t i o n . An attempt was made to measure the creep of the specimens under constant l o a d c o n d i t i o n s u s i n g the I n s t r o n extensometer shown i n f i g u r e 1 1 . However, f l u c t u a t i o n s i n the s t r a i n caused by small humidity and temperature changes made a r e a l i s t i c a n a l y s i s of the r e s u l t i n g data i m p o s s i b l e , and the t e s t i n g was d i s c o n t i n u e d because a room with humidity c o n t r o l was not ava i l a b l e . In t h i s study, the Clouser creep f u n c t i o n (2-5) has been assumed. In order to f i t the experimental d u r a t i o n of load data, the a p p r o p r i a t e parameters were a p p l i e d to the v i s c o e l a s t i c f r a c t u r e mechanics models. T h i s s e c t i o n endeavours to p r o v i d e r e l e v a n t i n f o r m a t i o n so t h a t , i n the context of past r e s e a r c h , the creep f u n c t i o n can be b e t t e r understood, and an a p p r e c i a t i o n of r e a l i s t i c creep parameters ac h i e v e d . 64 5.3.2 Creep F u n c t i o n s . The C l o u s e r creep f u n c t i o n used i n t h i s study i s w e l l supported by other r e s e a r c h . Clouser(1959) used i t to model creep behaviour i n bending over r e l a t i v e l y long time spans. I t has a l s o been used to d e s c r i b e creep i n compression and t e n s i o n p e r p e n d i c u l a r to the g r a i n by Youngs(1957), Schniewind and B a r r e t t ( 1 9 7 2 ) , and Schniewind(1967). Other f u n c t i o n s can a l s o be a p p l i e d to model creep behaviour. Of p a r t i c u l a r prominence are the t r a d i t i o n a l mechanical models c o n s i s t i n g of K e l v i n c h a i n s . Schniewind(1968) p r o v i d e s a very adequate summary of the many creep models developed, both l i n e a r and n o n - l i n e a r . Although the above mechanical models o f f e r more f l e x i b i l i t y and can allow f o r non-l i n e a r e f f e c t s , they can become q u i t e complicated. As such they lend themselves to t h e o r e t i c a l a n a l y s i s , but on a more a p p l i e d l e v e l , the very simple power f u n c t i o n employed by Clouser i s more p r a c t i c a l . 5.3.3 Creep Parameters. T h i s s u b s e c t i o n c o n c e n t r a t e s on d i s c u s s i n g the creep parameters obtained i n other r e s e a r c h . Only r e s e a r c h on t e s t s i n t e n s i o n p e r p e n d i c u l a r t o the g r a i n i s d i s c u s s e d . Kass(l969) researched E a s t e r n White Pine with an average s p e c i f i c g r a v i t y of 0.39, s t r e s s e d r a d i a l l y i n t e n s i o n p e r p e n d i c u l a r to the g r a i n at 12% moisture content. He t e s t e d f o r times up to 1833 hours, monitoring the creep and subsequent 65 recovery f o r d i f f e r e n t l e v e l s of a p p l i e d s t r e s s with a specimen c r o s s s e c t i o n of 12mm x 10mm. By p l o t t i n g h i s r e s u l t s ( f i g u r e s 29 and 30), valu e s of a=0.12 and b=0.29 were obt a i n e d . Creep appeared to remain l i n e a r up to s t r e s s e s of 60% (2.28MPa) of the average short term s t r e n g t h (3.80MPa). In a d d i t i o n , Kass's recovery t e s t s showed that f o r loaded times of 30 hours or l e s s , 90% of the t o t a l recovery o c c u r r e d w i t h i n a recovery time equal to the o r i g i n a l loaded time. Schniewind and Barrett(1972) performed t e s t s on Douglas F i r s t r e s s e d at v a r y i n g angles to the g r a i n . For a moisture content of 10%, with the specimen s t r e s s e d i n the t a n g e n t i a l l o n g i t u d i n a l plane, t h e i r r e s u l t s are r e p r i n t e d i n f i g u r e 31 and r e p l o t t e d i n f i g u r e 32. By drawing a s t r a i g h t l i n e through the second and f i n a l p o i n t s of the p l o t , approximate parameters of a=0.l3 and b=0.26 are obt a i n e d . By s i m i l a r l y p l o t t i n g the r e s u l t s of specimens s t r e s s e d i n the r a d i a l l o n g i t u d i n a l plane, s i g n i f i c a n t l y l e s s creep o c c u r r e d at longer times, y i e l d i n g approximate values of a=0.11 and b=0.21. Notice that the r e s u l t s obtained are not p e r f e c t l y l i n e a r . When the l o g a r i t h m of f r a c t i o n a l creep i s p l o t t e d a g a i n s t the l o g a r i t h m of time, the t r e n d of the data r e v e a l s d i m i n i s h i n g creep at longer times. T h i s would mean that i f t h i s data were f i t t e d i n t o the v i s c o e l a s t i c f r a c t u r e models, the time to f a i l u r e curves would begin to l e v e l out as a r e s u l t . T h i s r e s u l t c a s t s doubt upon the continued use of the simple creep f u n c t i o n a l r e a d y assumed. However i t should be noted that C l o u s e r ' s f u n c t i o n g i v e s a c o n s e r v a t i v e r e s u l t , and as such maintains i t s u s e f u l n e s s . T h i s 66 i s p a r t i c u l a r l y t rue when i t i s remembered that the experiments c a r r i e d out i n t h i s study d i d not have a c o n t r o l l e d environment. In c o n t r a s t , other r e s e a r c h e r s endeavoured to maintain a constant environment as t h e i r experiment proceeded. By having an u n c o n t r o l l e d environment, i t i s to be expected that the creep not only be more e r r a t i c but a l s o , i n g e n e r a l , more severe. In summary, i t can be seen that f o r the two softwoods d i s c u s s e d , the creep parameters are q u i t e c o n s i s t e n t i . e . , a=0.12 and 0.13, with b=0.29 and 0.26. 5.4 The Ts Experiment I f , as seems reasonable, damage i s a c c r u i n g i n the form of crack growth then the N i e l s e n and step-wise models can be f i t t e d to the data. The parameters "a", "b" and <r, can be used to f i t the models to the d u r a t i o n of load data p o i n t s , as o u t l i n e d i n S e c t i o n 5.2.3. The c h o i c e e x i s t s between continuous growth (the N i e l s e n model) and step-wise growth (the step-wise model). A f t e r f i t t i n g the a p p r o p r i a t e parameters, the Ts experiment can be used to compare the r e l a t i v e m e r i t s of the two models. The step-wise model was f i t t e d to the experiment shown in f i g u r e 33. Note that the Ts l i n e i n t e r s e c t s the Ts p o i n t determined by the Ts experiment, and that the slope of the l i n e i s made p a r a l l e l to t hat of the data. In order to s i m u l t a n e o u s l y s a t i s f y these two requirements, a=0.343 and b=1/5 were chosen. A l l o w i n g f o r the adjustment of Teat r e f e r r e d to i n s e c t i o n 5.2.5, tf,=16Mpa i s 67 found to be the best v a l u e . A p p l y i n g the same value s of "a" and "b" and the 19% adjustment to the f i t of the N i e l s e n model, as were used i n the f i t of the step-wise model, a value of tf^S.SMPa i s obtained ( f i g u r e 34). T h i s d i f f e r e n c e i n the value s of <s % (16MPa compared wi t h 5.5MPa) i s because the F(©) v a l u e s as given i n (2-17), are higher f o r the case of continuous growth (the N i e l s e n model) than f o r the step-wise model. In order that the term F(e) / ( 6 2 * 2 ) remains constant (the d i s t a n c e between the Ts and Teat l i n e s i s f i x e d i n the Ts experiment), <t> must i n c r e a s e . T h i s i s achieved by de c r e a s i n g to 5.5Mpa. Thus i t becomes unclear as to which model one should use, the step-wise model with tf,=16 MPa or the N i e l s e n model with 5.5MPa, because both provide an adequate f i t to the experimental d a t a . In order to e s t a b l i s h an estimate of a lower bound f o r in a t e s t independent of the Ts experiment, a new experiment was performed. The idea was to t e s t high q u a l i t y m a t e r i a l without i n i t i a l c r a c k s and of very small c r o s s s e c t i o n a l area i n t e n s i o n p e r p e n d i c u l a r to the g r a i n . In t h i s way, the f a i l u r e s u r f a c e ( i n the l i m i t as the specimen becomes very small) c o u l d be assumed t o be t o t a l l y p l a s t i c , with the f a i l u r e s t r e s s approaching the p l a s t i c l i m i t s t r e n g t h As small flaws w i l l always e x i s t , however, t h i s method can only p r o v i d e a lower bound to the value of <s\. Nine very small specimens were prepared and f a i l e d i n u n i a x i a l t e n s i o n . The specimens were cut on a small band saw, sandpapered to a c r o s s s e c t i o n a l dimension 6 8 of approximately 6mmx5mm, the ends glued i n t o short s e c t i o n s of tube which were pinned to the c h a i n s a l r e a d y a t t a c h e d to the crossheads of the Satec t e s t i n g machine. (For a f u r t h e r d i s c u s s i o n of t h i s experiment r e f e r to appendix 2 ) . With a time to f a i l u r e of approximately 10 seconds, the maximum s t r e s s achieved was 14MPa (Mean=9.2MPa, Std Dev=2.7MPa). T h i s s t r e s s r u l e s out the a p p l i c a b i l i t y of the N i e l s e n model to the data of the Ts experiment. However i t c o i n c i d e s w e l l with the p r e d i c t e d p l a s t i c l i m i t s t r e s s of *,=16MPa as p r e d i c t e d by the step-wise model. Because of the small flaws i n the small t e n s i l e specimens, the maximum s t r e s s reached by them cannot be expected to a t t a i n the e , value of 16MPa obtained from the Ts experiment. 5.5 Moisture Content E f f e c t . The m o t i v a t i o n behind experiment No.5 (the moisture t e s t ) was t w o f o l d . F i r s t , there was the d e s i r e to determine the e f f e c t of an i n c r e a s e i n moisture content on the d u r a t i o n of load e f f e c t . Second, an i n c r e a s e i n moisture content i s accompanied by an i n c r e a s e i n the amount and r a t e of creep. T h e r e f o r e the p r e d i c t i o n of the v i s c o e l a s t i c models (more s e r i o u s d u r a t i o n of l o a d e f f e c t f o r high creep) c o u l d be t e s t e d . Experiment No.5 achieved both of these aims. The i n c r e a s e d moisture content was found to i n t e n s i f y the d u r a t i o n of load e f f e c t , as p r e d i c t e d by the v i s c o e l a s t i c models. In f i t t i n g the step-wise model to the data, tf,=16MPa was assumed. A value of b=0.20 was f i t t e d ( f i g u r e 35) to the dry 69 c o n t r o l data p o i n t s with a=0.343. For the wet data p o i n t s the l i n e was much f l a t t e r , and a "b" value of 0.15 was f i t t e d . Note however that the "a" value was much higher at 0.81. T h i s f i t of the parameters does not agree with the trend i n the parameters as found by Youngs(1957). He found "b" to i n c r e a s e , and "a" to remain approximately c o n s t a n t . Youngs(l957) researched Northern Red Oak with an average s p e c i f i c g r a v i t y of 0.55, s t r e s s e s at 3 angles (0, 45 and 90 degrees) to the growth r i n g s , sometimes v a r y i n g the temperature, and other times v a r y i n g the moisture content. For a moisture content of 12% at s t r e s s e s l e s s than 70% of f a i l u r e , t e s t i n g i n the t a n g e n t i a l l o n g i t u d i n a l plane, he ob t a i n e d values (on the average) of a=0.64, and b=0.l6. I n c r e a s i n g the moisture content to approximately 75% y i e l d e d a=0.67 and b=0.26. T h i s lack of agreement between the creep parameters of Youngs and the creep parameters s u b s t i t u t e d i n t o the step-wise model c o u l d be due to many t h i n g s ! F i r s t , Youngs was t e s t i n g Oak, a hardwood not comparable i n c e l l s t r u c t u r e to that of the Douglas F i r t e s t e d here. Second, the change i n the moisture content f o r t h i s experiment was much more severe than f o r the t e s t s on the Oak. T h i r d , the c u r v a t u r e of the wet specimens c o u l d have changed the creep c h a r a c t e r i s t i c s of the wet moisture t e s t . For these reasons, experiment No.5 proves u s e f u l i n a l l o w i n g a gross estimate of the e f f e c t of high moisture content on d u r a t i o n of loa d behaviour. From the r e s u l t s obtained, the times to f a i l u r e of the wet specimens were reduced by 70 approximately one order of magnitude and may present a t e s t method f o r a c c e l e r a t i n g d u r a t i o n of l o a d t e s t s . 5.6 Strength E f f e c t . Experiment No.2 d i d not show c l e a r l y the s t r e n g t h e f f e c t as p r e d i c t e d by the v i s c o e l a s t i c models. T h i s was because the average s t r e s s r a t i o a p p l i e d was too h i g h , causing the experiment to terminate i n a r e l a t i v e l y short time. As the s t r e n g t h e f f e c t only becomes more obvious at longer times, o n l y a small s e p a r a t i o n of the data p o i n t s ( f i g u r e 36) was evident where the step-wise model was f i t t e d to the data assuming tfl=16MPa, (a=0.29,b=l/3). Experiment No.3 more c l e a r l y demonstrated the s t r e n g t h e f f e c t . For the purposes of comparison of the r e s u l t s of the three crack s i z e s , i t was decided to again employ the F(c/b) f a c t o r . Thus the a c t u a l l e v e l of a p p l i e d s t r e s s f o r an i n f i n i t e medium becomes 1.015 x 2.572 = 2.61 MPa f o r the case of the small crack specimen. S i m i l a r l y , the medium and l a r g e crack specimens become 1.86, and 1.64MPa r e s p e c t i v e l y . Applying these v a l u e s of c to the step-wise model y i e l d s the l i n e s as shown i n f i g u r e 37 where a=0.l82 and b=l/3. S i m i l a r l y the N i e l s e n model i s f i t t e d as i n f i g u r e 38. From the above f i g u r e s i t appears that the models are underestimating the s t r e n g t h e f f e c t . However the l a r g e s h i f t of the data p o i n t s f o r the small crack s i z e c o u l d be due to a change i n the creep f u n c t i o n as e x p l a i n e d below. 71 I t should be r e a l i s e d t h a t the most d i f f i c u l t t h i n g to do when t r y i n g to determine the extent of the s t r e n g t h e f f e c t i s to be sure t h a t i t r e a l l y i s the s t r e n g t h e f f e c t that i s being observed. The author f e e l s t h a t the s e p a r a t i o n of the data p o i n t s f o r the small crack s i z e , from the data p o i n t s of the medium and l a r g e crack s i z e specimens, i s due i n p a r t to a n o n l i n e a r change ( i n c r e a s e ) i n the creep, as o u t l i n e d i n S e c t i o n 5.3.3. Creep i s r e p o r t e d by Kass(l969) to become n o n - l i n e a r at s t r e s s l e v e l s of approximately 60% of the short term s t r e n g t h , or about 2.2MPa. By o b s e r v i n g the c l o s e n e s s of the f i t t e d l i n e s f o r the medium and l a r g e s i z e s , i t i s understandable that these l i n e s d i d not become d i s t i n c t l y separate f o r short times. By the end of the experiment, however, the s e p a r a t i o n of the data p o i n t s was i n keeping with the s t r e n g t h e f f e c t behaviour as p r e d i c t e d by the step-wise model. The l a r g e s e p a r a t i o n of the data p o i n t s f o r the small crack s i z e i s i n t e r p r e t e d - a s being not e n t i r e l y due to the s t r e n g t h e f f e c t , but a l s o due to an i n c r e a s e i n creep. T h e r e f o r e the <r=2.6lMPa l i n e used, does not f i t the experimental p o i n t s but overestimates the time to f a i l u r e . Other d u r a t i o n of l o a d r e s e a r c h e r s have found d i f f e r i n g behaviours f o r d i f f e r e n t l e v e l s of a p p l i e d s t r e s s . Madsen and Johns(l982) f i t t e d the N i e l s e n model ( i n c l u d i n g the s t r e n g t h e f f e c t ) to t h e i r data with <r1/Y=82MPa. T h i s high value was used to f i t the spread of the data p o i n t s f o r the d i f f e r e n t s t r e n g t h l e v e l s i . e . , the spread of the data was wholly a t t r i b u t e d to the s t r e n g t h e f f e c t , with the a p p r o p r i a t e value of <t>, assuming the 72 creep f u n c t i o n to remain c o n s t a n t . I t seems however that part of the spread of the data f o r d i f f e r e n t s t r e n g t h s c o u l d be a t t r i b u t e d to the e f f e c t of n o n l i n e a r creep at the higher l e v e l s of s t r e s s . Creep t e s t s by Fouquet(1979) on boards i n bending where the s t r e s s e s are p a r a l l e l to the g r a i n , "...suggest that the r e l a t i o n between the creep deformation and a p p l i e d s t r e s s i s not n e c e s s a r i l y l i n e a r f o r s t r e s s l e v e l s higher than 4500psi (31.0MPa)." N o t i c e that the maximum s t r e s s a p p l i e d i n the t e s t s by Madsen and Johns i s 3830psi (26.4MPa). F o s c h i and Barrett(1982) r e s u l t s , with t e s t s on 2"x6" boards i n bending, a l s o demonstrate a s e p a r a t i o n of the d u r a t i o n of l o a d curves f o r the d i f f e r e n t l e v e l s of a p p l i e d l o a d . 5.7 D e n s i t y E f f e c t . Although only one s p e c i e s was t e s t e d , c e r t a i n c h a r a c t e r i s t i c s of d u r a t i o n of load behaviour were observed. Notably, a d i s t i n c t i o n e x i s t e d between the d u r a t i o n of load behaviour f o r specimens s i m i l a r except f o r t h e i r d e n s i t i e s . In p a r t i c u l a r , the boards f o r experiment No.1 and experiment No.2 were of markedly d i f f e r e n t d e n s i t i e s , 600 and 400 kg/m3 r e s p e c t i v e l y . The g r a i n o r i e n t a t i o n s were the same, but the d u r a t i o n of l o a d behaviours q u i t e d i s t i n c t . The high d e n s i t y m a t e r i a l s u r v i v e d f o r a much longer time than the low d e n s i t y m a t e r i a l . The f i t of the parameters can be seen as i n f i g u r e s 36 and 39, where a=0.29 i s f i t t e d f o r the high d e n s i t y m a t e r i a l , and a=0.56 f o r the lower d e n s i t y m a t e r i a l . The 73 e s s e n t i a l d i f f e r e n c e f o r these two experiments i s assumed to be in the creep f u n c t i o n , however the high d e n s i t y of the m a t e r i a l c o u l d a l s o i n d i c a t e a higher value of cy should be a p p l i e d . T h i s would a l s o tend to i n c r e a s e the time to f a i l u r e as r e q u i r e d . Because n e i t h e r <r1f "a" or "b" values are known, i t i s d i f f i c u l t to determine the exact cause of the d i f f e r e n c e i n the d u r a t i o n of loa d behaviours f o r the experiments Nos 1 and 2. Because of t h i s , the c o n c l u s i o n s to be drawn from t h i s r e s u l t are only t e n t a t i v e , but do i n d i c a t e that m a t e r i a l c o u l d be expected to s u r v i v e longer i f the d e n s i t y of the wood i s i n c r e a s e d . 5.8 C y c l i c Loading. Experiments Nos 1 and 4 both demonstrated that the d u r a t i o n of l o a d behaviour under c y c l i c step f u n c t i o n l o a d i n g i s e s s e n t i a l l y the same as f o r constant l o a d i n g , i f only the t o t a l loaded time i s c o n s i d e r e d . For a p l o t of experiments 1 and 4, r e f e r to f i g u r e s 39 and 40. Assuming <j, = 16MPa, the step-wise model i s f i t t e d with parameters a=0.47 and b=l/4, to experiment 4 i n f i g u r e 40. Before t h i s experiment was completed, s p e c u l a t i o n was made as to the p o s s i b l e outcomes. The c y c l i c experiment c o u l d y i e l d behaviour s i m i l a r or d i s s i m i l a r to the constant l o a d experiment. I f s i m i l a r behaviour r e s u l t e d , then o b v i o u s l y the creep recovery phase has no e f f e c t . I f d i s s i m i l a r , the c y c l i c r e s u l t c o u l d be 74 l e s s , or more severe than the constant l o a d r e s u l t . In the case where the c y c l i c r e s u l t might be l e s s c r i t i c a l , i t c o u l d be argued that the creep recovery i s having a b e n e f i c i a l e f f e c t , because p a r t of the time i n a newly loaded c y c l e i s spent i n reachin g the creep s t r a i n a l r e a d y reached at the end of the pre v i o u s c y c l e . For the case where the c y c l i c t e s t might be found more c r i t i c a l than the constant t e s t , i t was hypothesized that d u r i n g the recovery, a wedging a c t i o n c o u l d occur i n the r e g i o n of the crack t i p . As the p l a s t i c m a t e r i a l r e f u s e s to be completely compressed, i t would tend to wedge the t i p of the crack open, and i n so doing c o u l d cause h i g h s t r e s s e s to remain (or even i n t e n s i f y ) w i t h i n the recovery p e r i o d . In t h i s way, more severe d u r a t i o n of l o a d behaviour c o u l d be i n t e r p r e t e d . However, n e i t h e r of these p o s s i b i l i t i e s was r e a l i s e d , as the specimens loaded c y c l i c a l l y and the specimens loaded c o n s t a n t l y d i d not show d i s s i m i l a r d u r a t i o n of load behaviour. 75 5.9 Confidence L i m i t s . An e f f o r t was made to e s t a b l i s h the l e v e l of confidence i n the d u r a t i o n of loa d r e s u l t s p l o t t e d . When a ranked specimen f a i l s , we are r e l a t i v e l y sure of the c o r r e c t time to f a i l u r e . However, we are not so sure of i t s o r i g i n a l s h o r t term s t r e n g t h , as s i m i l a r samples have v a r y i n g d i s t r i b u t i o n s of short term s t r e n g t h . The v a r i a b i l i t y i n s t r e n g t h of the q u a n t i l e s can be c a l c u l a t e d a c c o r d i n g to the equation as given by Bu r y ( l 9 7 5 ) , (5-2) Var(q) = q ( l - q ) / (n f ( x ) 2 ) where Var(q) = the v a r i a n c e of the q u a n t i l e q n = the sample s i z e and f ( x ) = the frequency of the sample at the q u a n t i l e q. T h i s formula i s not exact, but a good approximation f o r any p r o b a b i l i t y d e n s i t y f u n c t i o n , f o r l a r g e n. In f i g u r e 25, the 95% confi d e n c e l i m i t s on the short term s t r e n g t h f o r the Ts experiment are i l l u s t r a t e d . By d i v i d i n g the p o i n t s on the confidence l i n e s by the l e v e l of a p p l i e d s t r e s s , the 95% con f i d e n c e l i m i t s on the s t r e s s r a t i o a re obtained as i l l u s t r a t e d i n f i g u r e 41. 76 5.10 General L i m i t a t i o n s 5.10.1 Ts Experiment The r e s u l t s of the Ts experiment d i s c u s s e d i n S e c t i o n 5.4 are s t a t e d very p r e c i s e l y . By r e f e r r i n g to the conf i d e n c e l i m i t s shown i n f i g u r e 24, i t can be seen that with 95% con f i d e n c e we can say that at l e a s t 18 specimens were damaged. The r e f o r e i t seems that the Ts p o i n t i s w e l l d e f i n e d . However, of those 18 or 20 damaged s u r v i v o r s , i t c o u l d be argued that the assumption of ranking has not h e l d completely t r u e . T h e r e f o r e of those 20 s u r v i v o r s , some may have had a short term s t r e n g t h a s s i g n e d them a c c o r d i n g to the assumption of ranking ( f i g u r e 24). However, i t c o u l d be argued that some of the s u r v i v o r s are merely specimens with low short term s t r e n g t h s which, f o r some reason not understood w i t h i n the l i m i t s of v i s c o e l a s t i c f r a c t u r e mechanics, d i d not f a i l at the time as p r e d i c t e d w i t h i n the assumption of ranking that the weaker specimen w i l l always f a i l e a r l i e r . Rather, they s u r v i v e d to the end of the experiment, and are now mistakenly r e f e r r e d to as s u r v i v o r s with reduced s t r e n g t h . In t h i s case we would expect the s t r e s s r a t i o of the Ts p o i n t to be a higher value, thus reducing the time spent i n phase 2, which r e q u i r e s a higher <t> v a l u e , which i n t u r n i m p l i e s a lower v a l u e . Conversely, i t i s p o s s i b l e that many of the f a i l e d specimens were of a higher short term s t r e n g t h than assumed by ranking, thereby y i e l d i n g a higher <s, v a l u e . 77 Desp i t e the p o s s i b i l i t y that the assumption of ranking may l e a d to an i n c o r r e c t estimate of * 1 f t h i s i s u n l i k e l y and ff,=16MPa remains as the best estimate a v a i l a b l e . 5.10.2 Strength E f f e c t I t should be emphasised that the a n a l y s i s of the s t r e n g t h e f f e c t experiments i s based upon the assumption that the parameters "a" "b" and <*, are c o n s t a n t . As there w i l l always be some change i n these from specimen to specimen, the f i t of the N i e l s e n and step-wise models i s based upon an average value of "a" "b" and I t c o u l d be argued that the l e v e l of u n c e r t a i n t y i n these parameters from specimen to specimen c o u l d tend t o i n v a l i d a t e attempts t o l i n k the s e p a r a t i o n of the data p o i n t s i n the s t r e n g t h e f f e c t p l o t s to the s t r e n g t h e f f e c t , because the s e p a r a t i o n i s only due to a change i n one or more of the parameters "a" "b" or a,. T h i s l i m i t a t i o n of the s t r e n g t h e f f e c t experiments i s r e a l i s e d by the author and i s r e f e r r e d to in f i t t i n g the step-wise model to experiment 3. In t h i s case the f i t of the models overestimate the time to f a i l u r e f o r the h i g h s t r e n g t h specimens. T h i s i s a t t r i b u t e d to a n o n l i n e a r i n c r e a s e i n the creep parameters because of the p a r t i c u l a r l y high s t r e s s l e v e l . I f however average v a l u e s f o r the parameters are assumed, then the s t r e n g t h e f f e c t experiments are s t i l l v a l i d , i n s p i t e of the p o s s i b l e u n c e r t a i n t i e s which have been minimised by using neighbouring specimens of s i m i l a r geometry and crack l e n g t h . 78 N o t i c e that the c o e f f i c i e n t of v a r i a t i o n f o r the short term s t r e n g t h s of the Ts experiment was q u i t e low f o r wooden specimens at 0.06. T h i s can be l a r g e l y a t t r i b u t e d to the e c c e n t r i c i t i e s of the specimens and t h e i r end p l a t e s . 79 CHAPTER 6 CONCLUSIONS 6.1 I n t r o d u c t i o n The c o n c l u s i o n s reached are based upon the t e s t i n g of 180 Douglas f i r specimens i n the short term, and 320 specimens i n the long term. By c o n t r o l l i n g the crack l e n g t h i n a specimen e s p e c i a l l y designed f o r the t e s t i n g of l a r g e sample s i z e s , good co n f i d e n c e was achieved i n the r e s u l t s with some t e s t s c o n t i n u i n g f o r d u r a t i o n s of up to 2 months. The specimens were t e s t e d i n t e n s i o n p e r p e n d i c u l a r to the g r a i n , u s u a l l y accepted as the c r i t i c a l f a i l u r e i n i t i a t i n g mode of commercial m a t e r i a l . By t e s t i n g i n the t a n g e n t i a l l o n g i t u d i n a l plane, high creep r a t e s were achieved and t h e r e f o r e the d u r a t i o n of loa d experiments were conducted i n what was thought t o be the most c r i t i c a l p l a n e . I t was a l s o f e l t t h a t t h i s plane of f a i l u r e was i n f a c t the plane i n which the f a i l u r e i s i n i t i a t e d , e s p e c i a l l y around knots i n commercial m a t e r i a l . By t e s t i n g i n t h i s way, i t was hoped that a lower bound be achi e v e d on the d u r a t i o n of l o a d behaviour of commercial m a t e r i a l . Due to the g r a i n p a t t e r n s of commercial m a t e r i a l , i t was f e l t that c r a c k s propagating i n the re g i o n s of hi g h e s t s t r e s s w i l l propagate i n t o r e g i o n s of lower s t r e s s , t h e r e f o r e 80 slowing down, i n d i c a t i n g a c e r t a i n amount of redundancy f o r crack propagation i n commercial m a t e r i a l . As expected the f a i l u r e times f o r the t e s t specimens used were q u i t e short when compared with the f a i l u r e times f o r commercial m a t e r i a l s . In f a c t , q u i t e a l a r g e d e t e r i o r a t i o n i n s t r e n g t h was observed even over r e l a t i v e l y s h o r t times i . e . , specimens with s t r e s s r a t i o s of 0.65 f a i l e d w i t h i n 2 months. 6.2 Summary 1. Step f u n c t i o n c y c l i c l o a d t e s t s y i e l d e d d u r a t i o n of loa d responses s i m i l a r to the d u r a t i o n of loa d response of constant l o a d t e s t s . The p e r i o d of the loa d c y c l e s was of the order of s e v e r a l hours. 2. The d u r a t i o n of loa d e f f e c t i s h i g h l y v a r i a b l e from sample to sample depending upon the m a t e r i a l p r o p e r t i e s and the s t r e s s c o n d i t i o n . 3. The concepts and the p r e d i c t i o n s of the v i s c o e l a s t i c f r a c t u r e mechanics models were v e r i f i e d i n the experiments. The models s t u d i e d were the step-wise model and the N i e l s e n model. The f o l l o w i n g c o n c l u s i o n s were made: (a) the step-wise model was found to be the most a c c e p t a b l e of the two models i n v e s t i g a t e d , the N i e l s e n model being found i n c o n s i s t e n t with the r e s u l t of the Ts experiment; (b) the creep parameters f i t t e d ("a" and "b" from the Clouser c reep f u c t i o n ) , were r e p r e s e n t a t i v e of those found i n other r e s e a r c h ; 81 (c) the l i m i t s t r e n g t h « y as determined by experiment was gr e a t e r than the maximum y i e l d s t r e s s of smal l specimens, su p p o r t i n g the p r e d i c t i o n s of the step-wise model. (c) the t r e n d of the d u r a t i o n of load data at long times was in keeping with the s t r e n g t h e f f e c t as p r e d i c t e d by the v i s c o e l a s t i c f r a c t u r e mechanics models. (e) the wet specimens f a i l e d at much s h o r t e r times (one order of magnitude sooner) than the dry specimens (at the same s t r e s s r a t i o ) , s u p p o r t i n g the p r e d i c t i o n of the v i s c o e l a s t i c models, namely that high creep r a t e s w i l l cause r a p i d f a i l u r e . 6.3 Furth e r Research and A p p l i c a t i o n s . Many v a r i a b l e s e x i s t i n the study of the d u r a t i o n of loa d problem. For example, moisture content e f f e c t s p e c i e s e f f e c t and d e n s i t y e f f e c t a l l c o n t r i b u t e to v a r i a t i o n s i n the d u r a t i o n of l o a d behaviour. For a l l of these to be best understood, i t i s i m p r a c t i c a l to t e s t a l l the p o s s i b l e combinations of the v a r y i n g s i z e s and grades of commercial m a t e r i a l . In the o p i n i o n of the author the procedure used i n t h i s study c o u l d (should) c o i n c i d e with f u r t h e r d u r a t i o n of load t e s t s on f u l l s i z e m a t e r i a l , i n order that c o n f i d e n c e can be developed i n the c o r r e l a t i o n of the e m p i r i c a l f u l l s i z e t e s t s , to t e s t s where the crack l e n g t h i s c o n t r o l l e d . With the development of an e m p i r i c a l understanding of the r e l a t i o n s h i p between the two t e s t methods, combined with an understanding of the p r i n c i p l e s of v i s c o e l a s t i c f r a c t u r e mechanics which c o n t r o l the d u r a t i o n of 82 l o a d e f f e c t , the behaviour of commercial m a t e r i a l under v a r y i n g c o n d i t i o n s c o u l d be p r e d i c t e d by the behaviour of t e s t s conducted on small specimens with c o n t r o l l e d crack l e n g t h t e s t e d under the same v a r y i n g c o n d i t i o n s . There are many p o s s i b i l i t i e s f o r developing the v i s c o e l a s t i c f r a c t u r e mechanics models f u r t h e r . For example, one c o u l d develop a f i n i t e element g r i d around a knot which can allow f o r the slope of g r a i n , i n order to c a l c u l a t e the s t r e s s f i e l d . By imposing a small t h e o r e t i c a l crack i n a region of high s t r e s s ( i n t e n s i o n p e r p e n d i c u l a r to the g r a i n ) a procedure c o u l d be developed whereby the s t r e s s c o u l d be c a l c u l a t e d i n the region of the crack, and t h e r e f o r e the crack v e l o c i t y c o u l d be found by the theory of f r a c t u r e mechanics. By a p p l y i n g a time step a n a l y s i s , the time to f a i l u r e f o r t h i s case c o u l d be p r e d i c t e d . The p r e d i c t i o n s of the s o l u t i o n c o u l d be t e s t e d a g a i n s t the experimental o b s e r v a t i o n and r e s u l t ( j u s t as they have been i n t h i s study ). However, because of the need to remain p r a c t i c a l i n the a n a l y s i s of the behaviour of commercial m a t e r i a l , a more e m p i r i c a l approach becomes necessary. The author f e e l s that the v i s c o e l a s t i c f r a c t u r e mechanics models are u s e f u l i n a r r i v i n g at a b e t t e r understanding of the d u r a t i o n of l o a d phenomena. None of the parameters used ( a, b and ) cannot ever be t o t a l l y v e r i f i e d as the only c o n t r o l l i n g f e a t u r e s of the d u r a t i o n of l o a d phenomenon. I t may be that has a f u n c t i o n a l dependance upon the creep parameters or v i c e v e r s a , or there may be another c o n t r i b u t i n g parameter which has not been c o n s i d e r e d i n the d e r i v a t i o n s of the N i e l s e n 83 and step-wise models. Independant of the a b s o l u t e c o r r e c t n e s s of the f u n c t i o n a l form i m p l i e d by the a p p l i c a t i o n of the v i s c o e l a s t i c f r a c t u r e mechanics models, the p r e d i c t e d d u r a t i o n of l o a d behaviour of the wood specimens i n t h i s study was born out by the experiments (the step-wise model p r o v i n g to be the most c o n s i s t e n t ) . 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Pearson,R.G. 1972 "The e f f e c t of d u r a t i o n of load on the bending s t r e n g t h of wood", Holzforschung ( v o l . 2 6 ( 4 ) , pp.153,158). 86 Porter,A.W. 1964. "On the mechanics of f r a c t u r e i n wood", F o r e s t Products J o u r n a l ( v o l . 1 4 ( 8 ) , pp.325,331). Schapery,R.A. 1975. "A theory of crack i n i t i a t i o n and growth i n v i s c o e l a s t i c media, I. T h e o r e t i c a l development", I n t e r n a t i o n a l J o u r n a l of F r a c t u r e ( v o l . 1 1 ( 1 ) , p.141). Schniewind,A.P. 1967. "Creep rupture l i f e of D o u g l a s - f i r under c y c l i c environmental c o n d i t i o n s " , Wood Science and Technology ( v o l . 1 , pp.278,288). Schniewind,A.P. 1968. "Recent progress i n the study of the rheology of wood", Wood Science and Technology ( v o l . 2 , pp.188-206). Schniewind,A.P. 1977. " F r a c t u r e toughness and d u r a t i o n of loa d f a c t o r I I . Duration f a c t o r f o r c r a c k s propagating p e r p e n d i c u l a r to the g r a i n " , Wood and F i b r e ( v o l . 9 ( 3 ) , pp.216,226). Schniewind,A.P. and B a r r e t t , J . D . 1972. "Wood as a l i n e a r o r t h o t r o p i c m a t e r i a l " , Wood Science and Technology ( v o l . 6 , pp.43-57). Schniewind,A.P. and Centeno,J.C. 1973. " F r a c t u r e toughness and d u r a t i o n of load f a c t o r - I " , Wood and F i b r e ( v o l . 5 ( 2 ) , pp.152,159). Schniewind,A.P. and Pozniak,R.A. 1971. "On the f r a c t u r e toughness of Douglas f i r wood", E n g i n e e r i n g F r a c t u r e Mechanics ( v o l . 2 ( 3 ) , pp.223,233). Tada,H. P a r i s , P . And Irwin,G. 1973. "The s t r e s s a n a l y s i s of c r a c k s handbook", Del Research Corp., H e l l e r t o n , P e n n s y l v a n i a . Wood,L.W. 1951. " R e l a t i o n of s t r e n g t h of wood to d u r a t i o n of l o a d " , U.S.D.A. F o r e s t Products Laboratory Report No. R-1916, Madison, Wisconsin.. Youngs,R.L. 1957. "The p e r p e n d i c u l a r to g r a i n mechanical p r o p e r t i e s of red oak as r e l a t e d to temperature, moisture content and time" U.S.D.A. F o r e s t Products Laboratory Report No. 2079, Madison, Wisconsin. 87 FIGURES F i g . 1 S t r e s s f i e l d at the crack t i p . 89 F i g . 2 T y p i c a l d u r a t i o n of l o a d p l o t f o r the Step-wise Model 90 F i g . 3 N i e l s e n and Step-wise model p l o t s . 91 K I C • 2 7 6 I - » 2 l e g t 10 10 10 10 10 10 DURATION OF LOADING (min) FIG. 4. Critical stress intensity factor as a function of time to failure for ranrp loading in the TL system. M E A N TIME TO F A I L U R E (min) Fic. 9. Load level in relation to time to failure in the TL system under constant conditions. F i g . 4 R e s u l t s of Schniewind and Centeno(1973). 92 kg/cm 70 60 Figure 2. - Norminal sus-tained maximum bending stress and time of loading to failure perpendicular-to-the-grain. The data marked with an "x" represents 11 spec-imens that survived 6 months of loading. 50 40 30 20 10 NOMINAL SUSTAINED MAXIMUM BENDING STRESS -I L tr < 2 3 4 5 6 TIME OF LOADING TO FAILURE cr < LU, g | ,LOG I O (MIN.1| F i g . 5 R e s u l t s of Bach(l975) Average shor t term s t r e n g t h = 54.7 kg/m2 93 Flexure » 1 <-/ r / > * J / • i • i r c i / • ri i y U o' oi • 140 120 100 80 60 40 20 0 0.2 0.4 0.6 0.8 01.0 Deflection , inche6 Fig. 3—Accumulative acoustic •million! and load versus de-flection for flexure. O o o o in c o tn jn E UJ 1.0 .8 s 2 Creep in Flexure a 10000 1000 CO 100 § o o 10 10 100 Time, Minutes Emissions at one minute represent the observed number after the start of loading. M a x i m u m load was reached in less tlia:i one minute. fig. 4—Accumulative acoustic emissions and deflection versus time for creep in flexure. F i g . 6 R e s u l t s of Debaise et a l . d 9 6 6 ) 802/Ptiysical Metallurgy Principles Oisplocemsnt of crock iurtocts Fig. 19.50 The three basic fracture modes. in these considerations is the thickness of the plate. In a thick plate, the large depth of metal parallel to the crack front tends to restrict plastic flow parallel to the crack. On the other hand, a crack in a thin plate does not feel this restriction. As a consequence, a crack that passes through a thin plate can draw in the plate The Three Modes of F a i l u r e . F i g . 8 Induced S t r e s s e s i n t e n s i o n p e r p e d i c u l a r to the g r a i n 96 F i g . 10 Mounted Specimen. 99 No 6 37mm Screws Steel Endplate Wood Specimen Necked Region Central Hole Through Crack Crack Initiator F i g . 12 Diagrammatical Specimen, and Crack I n i t i a t i o n , 100 Chain Specimen Beam Pivot Lever Arm <± 190 mm Lead Sandbag F i g . 13 Apparatus (Diagrammatical). 101 102 F i g . 15 Apparatus. 103 in KEY • Constant Load Falure 1 o Expected Failure Test Terminated—•>! Ts(2) Ts(1) *TCat Log^ Time to Failure F i g . 16 Ts Determination. c o fit c a> E o 8. to KEY • Constant Load Falure « Ramp Load Failure Short Term Stress Failure Stress F i g . 17 Ts Determination. 104 1.0 0.9 0.8 o *< or 0.7 to to LU £0.6 to 0.5 0.4 i e x 0 X ox — ex • x ex O- X ©- X O- X-— o- * ©- X-O- X-X-O- X-O- X-o> X-o - x-— O- X-O- X-KEY: _ x Constant Loading L » Cyclic Loading _ Failed on Loading L »-,». Survivor _ Crack = n mm -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 L0G l 0 TIME TO FAILURE - (hours) F i g . 18 Dur a t i o n of lo a d p l o t f o r experiment 1. 105 A 1.0 0.9 0.8 o or 0.7 h t n t o LU £0.6 0.5 0.41 0 0 X ***** »x * X 0 X X X KEY' L • Crack = 6.22mm * Crack = 10.80 mm • i i • • • • i • • « • t • • • • » • • • • i i • • • • i • • • • i • • • • i • •••<•••• i > -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 LOG I O TIME TO FAILURE - (hours) F i g . 19 Duration of l o a d p l o t f o r experiment 2. 1 0 6 i 1.0 0.9 0.5 X X X X • x 0 • X • X S 0.8 b I lo X ac \ 'I. O i 0.7 to t KEY crr\c '• • Crack= 3.84mm i—U.o ~ 1 / 1 o Crack=6.22mm 0 ^ " X x Crack=10.03 mm -«.-o,* Failed on Loading V,*-,*- Survivor 0.4 Y , r . l i . . . . I . . . . i . . . . I . . . . i . . . . I . . . . i . . . . I . . . . i . . . . I . . . . i . . . . I -2.0 -1.0 0.0 1.0 20 3.0 4.0 L0G l 0 TIME TO FAILURE - (hours) F i g . 20 Duration of load p l o t f o r experiment 3. 107 i 1.0 0 X 0.9 — ft A 0.8 o % cr 0.7 -X \ x * °. tO to LU £ 0 . 6 - KEY: x • X • — • Constant Loading to x Cyclic Loading 0.5 — Failed on Loading +.». Survivor 0.4 Crack= 6.22 mm -2.0 -1.0 0.0 1.0 20 3.0 4.0 LOG l o TIME TO FAILURE - (hours) F i g . 21 D u r a t i o n of l o a d p l o t f o r experiment 4. 108 1.0 0.9 -0 0.8 o -0 •* 0 0 0 0 0 0 „ or 0.7 to to LU £0.6 to X X KEY = • Dry 0 0 ° 0 0-* o-X P-\ X X-X-0.5 — « Wet -v« Failed on Loading 0.4 •-,*- Survivor Crocks 6.22 mm 1. . . . 1 w -2.0 -1.0 0.0 1.0 20 3.0 4.0 LOG I O TIME TO FAILURE - (hours) F i g . 22 Dur a t i o n of l o a d p l o t f o r experiment 5. 109 { 1.0 0.9 — • • 0.8 • • • • o or 0.7 '-• L O C O U J £0.6 KEY-to • Failed During Test 0.5 — • Ts Point Crack = 6.22mm 0.4 -2.0 -1.0 0.0 1.0 20 3.0 4.0 LOG | 0 TIME TO FAILURE - (hours) F i g . 23 Du r a t i o n of l o a d p l o t f o r experiment 6. 110 F i g . 24 Ts (Expt 6) S u r v i v o r s t r e n g t h s . SHORT TERM STRESS MPa F i g . 25 Normal f i t to short term s t r e n g t h data, Expt 6. 1 12 T H E C E N T E R C R A C K E D T E S T S P E C I M E N  A. Stress I n t e n s i t y Factor Kj =<rVia F(°/b) Numerical Values at F(a/b) ( I s l d a 1962, 1965 a, b, 1973) Islda's 36 term power s e r i e s of (a / b ) 2 (Laurent s e r i e s expansion of complex st r e s s p o t e n t i a l , 1973) gives p r a c t i c a l l y exact values of F(a/b) up to a/b * 0.9. Numerical values of F(a/b) are shown i n the table. 1.0, • 1 0.8 06 0.2 1 1 1 1 1 1 . M o d e l M o d e l * < M o d e H 1 • 1 i i 0 -4 0.6 0.8 F i g . 26 Tada P a r i s I r w i n d 9 7 3 ) . ^637 1.0 a/b F (a /b ) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .0000 1 .0060 1 .0246 1 .0577 1 .1094 1 .1867 1 .3033. 1 .4882 1.8160 2.5776 1 113 F i g . 27 E f f e c t of "a" and "b" upon the d u r a t i o n of load p l o t . 1 14 F i g . 28 E f f e c t of «, b=l/3 a=0.15. and t r , upon the d u r a t i o n of load p l o t , 115 Figure t O - l 4 . « I l l u s t r a t l n s , the d i rec t p ropor t iona l i ty between t o t a l creep s t r a i n and stress leve l ' o r l i v e c r e e ? durat ions. The heavy s o l i d l ines are the regressions given In Ar^endix A-Z for e ( C O . ) - S,c, f i t t e d for gL - 10(10)60 percent. The dashed Hoes are extensions of the regressions. F i g . 29 Creep parameters, K a s s ( l 9 6 9 ) , O r i g i n a l Data. 116 30 Creep parameters, Kass(!969), R e p l o t t e d Data, a=0.12, 1 17 h — MOOO KB . . . ' i 30 *5 60 Sroin onglf 6 Fig. 10. Average relative creep in the radial-loQgitudinal plane an a function of grain an-gle. Kach point is an average of six specimens 7St*grm90 MOOO i . — ! ! l IS 30 4* 60 75t*gre«9u brain ongip O Fig. 11. Average relative oreep in the tangen-tial-longitudinal plane as a function of grain an-gle. Each point is an average of six specimens F i g . 31 Creep parameters, Schniewind O r i g i n a l Data. o. w w tt U i •J * •< z o and B a r r e t t ( 1 9 7 2 ) , U < tt 8 -2 -1 e U> LOG 1 0 LOADED TIME (HOURS) x o T a n g e n t i a l L o n g i t u d i n a l Plane R a d i a l L o n g i t u d i n a l Plane F i g . 32 Creep parameters, Schniewind and Barrett(1972) , R e p l o t t e d Data, a=0.13, b=0.26. 118 1-01-Tcat • Failed During Test 0.5 [- * T s P o i n t [ Crack = 6.22mm 0.4 --2.0 -1.0 0.0 1.0 2.0 3.0 4.0 LOG I O TIME TO FAILURE - (hours) F i g . 33 Step-Wise f i t to Expt 6 ( T s ) , ff,=l6MPa, a=0.343, b=!/5. 119 1.0 0.9 0.8 < o r 0.7 to to u £0 .6 to 0.5 0.4 Teat KEY* Failed During Test Ts Point Crack = 6.22mm 1 -2.0 -1.0 0.0 1.0 20 3.0 4.0 LOG I O TIME TO FAILURE - (hours) F i g . 34 N i e l s e n f i t to Expt 6 (Ts), «1=55MPa, a=0.343, b=!/5 120 t 1.0 0.9 - •< 0.8 o OC 0.7 -0 0 * ' X LO LO LU £0.6 LO e KEY: Dry v ft— 0.5 X Wet -V" t Failed on Loading 0.4 o-,x- Survivor Crack=6.22mm -2.0 -1.0 0.0 1.0 L0G l 0 TIME TO FAILURE -2.0 3.0 4.0 (hours) F i g . 35 Step-Wise F i t to Moisture t e s t , Expt 5, tf,=l6MPa, Dry; a = 0 .343, b=0.20. Wet; a=0.8l, b=0 . 15. 121 i.ol 0.9 0.8 < cr 0.7 t o t o LU o r r— LO 0.6 0.5 0.4 1 KEY» _ • Crack = 6.22 mm L * Crack = 10.80 mm "• • • • i • • • • ' • • • • i • • • • * • • • • i • • • • * • • • • i • • • • 1 • • • • i • » • • 1 • • * • i * • • • 1 * • • * i -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 LOG | 0 TIME TO FAILURE - (hours) F i g . 36 Step-wise f i t to Expt 2, *,=16MPa, a=0.29, b=l/3. 122 1.0 0 . 9 0 . 8 o i 0 . 7 to to LU £ 0 . 6 t o 0 5 0 . 4 K E Y ' • C r o c k = 3 . 8 4 m m o C r o c k = 6 . 2 2 m m « C r o c k = 1 0 . 0 3 m m ••,-0,* F a i l e d o n L o a d i n g • - , - . » • S u r v i v o r T e a t cr= 261,186.U54 MPa I I I I I 1 I . . . . I . . . . I . . . . I I I II l III II I » - 2 . 0 - U > 0 . 0 1.0 2J0 3 .0 L 0 G l 0 T I M E T O F A I L U R E - ( h o u r s ) 4 . 0 F i g . 37 Step-wise f i t to Expt 3, o-^ieMPa, a = 0.l82, b=l/3. 123 i 1.0 0.9 0.8 o i 0.7 LO LO LU £0 .6 to 05 0.4 KEY: • Crack= 3.84 mm o Crocks 6.22mm « Crocks 10.03mm >.-•,•" Failed on Loading Survivor Teat O-=2.61,1B6,1J64 MPa . U d U L I . . . . I . . . . I . . . . I f . . . I . . . . i . . . . I . . . . I . . . . I . . . . I . . . . I -2X) -1.0 0.0 1.0 2J0 3.0 4.0 L0G, o TIME TO FAILURE - (hours) F i g . 38 N i e l s e n f i t to Expt 3, cr, = l6MPa, a=0.l82, b=1/3. 124 I 1.0 X ^ - ^ 0 X 0.9 — 0 X 0 X . Teat O - X \ 0.8 o t t o- \ x <>• \ > o - X X o- x- \ *>• » O - X-u . 0.7 to to £0.6 to • KEY: x Constant Loading o Cyclic Loading ° - X-» • X-o » ° - X-0.5 ~°.«. Failed on Loading Survivor 0.4 Crack = 11 mm -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 L0G | O TIME TO FAILURE - (hours) F i g . 39 Step-wise f i t to Expt 1, tf,=16MPa, a=0.58, b=l/3. 125 1.0 0.9 0.8 o cr 0.7 to to UJ £ 0 . 6 to 0.5 0.4 — O o ^ X ^ ^ ^ 9 5 X Limits : KEY: \ \ \ l ^ - " T c a t \ \*\ 0 '- • Constant Loading \ v \ L x Cyclic Loading T s - ^ " ^ Failed on Loading - •-,», Survivor L Cracks 6.22 mm -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 L 0 G | 0 TIME TO FAILURE - (hours) F i g . 40 Step-wise f i t t o Expt 4, o-, = 16MPa, a = 0.47, b=1/4. 126 1.0 0.9 '• • ^95% CONFIDENCE LIMITS (about the tr e n d l i n e ) 0.8 o > LU 0.7 - • TREND LINE LO LO Ld £ 0 . 6 -K E Y -LO • Failed During Test 0.5 — • Ts Point 0.4 -2.0 -1.0 0.0 1.0 Z0 3.0 4.0 L O G I O TIME TO FAILURE - ( h o u r s ) F i g . 41 Confidence L i m i t s on the S t r e s s R a t i o . 127 APPENDICES Appendix 1 Test Data T h i s appendix l i s t s the t e s t data of experiments Nos 1 through 6. I t a l s o g i v e s b r i e f comments on the i n d i v i d u a l f e a t u r e s of each experiment. 128 Test Number 1 S t a r t i n g Date Nov 8th, 1981 Test T i t l e , S u b t i t l e C y c l i c - 1 , C y c l i c Crack Length, Hole Diameter (mm) 11, 4.12 Specimen Length (mm), F(c/b) 37.8, 1.05 Board Number 1 De n s i t y (kg/m 3) 590 Moisture Content % (Wet Volume) . . . . . . . 7.3 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 17 Mean, and Standard D e v i a t i o n (Mpa) . . . 1.923, 0.214 Load D u r a t i o n sample s i z e 21 Load D u r a t i o n S t r e s s L e v e l (MPa) 1.550 F a i l u r e s , S u r v i v o r s 6, 15 Comments: The s u r v i v o r f a i l u r e s t r e s s e s (MPa) are as f o l l o w s ; 2.32, 2.22, 1 .62, 1 .93,, 2.07, 2. 14, 2.25, 1 .80, 2.23, 2.35, 2.19, 2.27, 2.09, 2.19. T h i s t e s t was loaded to a constant s t r e s s l e v e l f o r p e r i o d s of three hours, and then allowed to recover f o r p e r i o d s of three hours before the next l o a d c y c l e was a p p l i e d . T h i s p a t t e r n was maintained f o r a t o t a l of twenty complete c y c l e s , (60 hours t o t a l loaded t i m e ) . The r a d i a l - t a n g e n t i a l plane was p a r a l l e l t o , but i n c l i n e d at 35 degrees to the plane of the end p l a t e s . 129 Test Number 1 S t a r t i n g Date Nov 8th, 1981 Test T i t l e , S u b t i t l e C y c l i c - 1 , Constant Crack Length, Hole Diameter (mm) 11, 4.12 Specimen Length (mm), F(c/b) 37.8, 1.05 Board Number 1 Den s i t y (kg/m 3) 590 Moisture Content % (Wet Volume) 7.3 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 17 Mean, and Standard D e v i a t i o n (Mpa) . . . 1.923, 0.214 Duration Load sample s i z e 21 Load Duration S t r e s s L e v e l (MPa) . . . j . . 1.582 F a i l u r e s , S u r v i v o r s 9, 12 Comments: The s u r v i v o r f a i l u r e s t r e s s e s (MPa) are as f o l l o w s ; 1.87, 2.13, 2.12, 1.93, 2.12, 2.12, 2.22, 2.43, 2.13, 2.06, 2.13, 2.26. The r a d i a l - t a n g e n t i a l plane was p a r a l l e l to, but i n c l i n e d at 35 degrees to the plane of the end p l a t e s . Note that t h i s t e s t was terminated at time equals 1580 hours. 130 Test Number 2 S t a r t i n g Date Nov 20th, 1981 Test T i t l e , S u b t i t l e Two S i z e , Large Crack Crack Length, Hole Diameter (mm) 10.8, 4.12 Specimen Length (mm), F(c/b) 37.4, 1.05 Board Number 42 Den s i t y (kg/m 3) 410 Mo i s t u r e Content % (Wet Volume) 10.5 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 11 Mean, and Standard D e v i a t i o n (Mpa) . . . 1.908, 0.093 Du r a t i o n Load sample s i z e 15 Load Duration S t r e s s L e v e l (MPa) 1.639 F a i l u r e s , S u r v i v o r s 15, 0 Comments: The r a d i a l - t a n g e n t i a l plane was p a r a l l e l t o , but i n c l i n e d at 40 degrees to the plane of the end p l a t e s . 131 Test Number 2 S t a r t i n g Date Nov 20th, 1981 Test T i t l e , S u b t i t l e Two S i z e , Small Crack Crack Length, Hole Diameter (mm) 6.22, 3.81 Specimen Length (mm), F(c/b) 37.4, 1.02 Board Number 42 De n s i t y (kg/m 3) 410 Mo i s t u r e Content % (Wet Volume) 10.5 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 11 Mean, and Standard D e v i a t i o n (Mpa) . . . 2.578, 0.161 Du r a t i o n Load sample s i z e 15 Load D u r a t i o n S t r e s s L e v e l (MPa) 2.338 F a i l u r e s , S u r v i v o r s 15, 0 Comments: The r a d i a l - t a n g e n t i a l plane was p a r a l l e l t o , but i n c l i n e d at 40 degrees to the plane of the end p l a t e s . 132 Test Number 3 S t a r t i n g Date Dec 20th, 1981 Test T i t l e , S u b t i t l e . . . . Three s i z e , Large crack Crack Length, Hole Diameter (mm) . . . . 10.06, 5.08 Specimen Length (mm), F(c/b) 37.5, 1.05 Board Numbers 1 02, 1 03, 1 04, 1 05 Den s i t y (kg/m 3) resp 460, 470, 560, 440 Moisture Content % (Wet Volume) 6.0 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 22 Mean, and Standard D e v i a t i o n (Mpa) . . . 2.043, 0.248 Dura t i o n Load sample s i z e 21 Load D u r a t i o n S t r e s s L e v e l (MPa) 1.559 F a i l u r e s , S u r v i v o r s 20, 1 Comments: The s u r v i v o r f a i l u r e s t r e s s was 3.124 MPa. The r a d i a l - t a n g e n t i a l plane was p a r a l l e l t o , but i n c l i n e d at 35 degrees t o the plane of the end p l a t e s f o r boards 103 and 104. For boards 102 and 105 the plane of the g r a i n was p a r a l l e l to the end p l a t e s . 133 Test Number 3 S t a r t i n g Date Dec 20th, 1981 Test T i t l e , S u b t i t l e . . . . Three s i z e , Medium crack Crack Length, Hole Diameter (mm) . . . . 6.22, 3.81 Specimen Length (mm), F(c/b) 32.0, 1.02 Board Numbers 1 02, 1 03, 1 04, 1 05 D e n s i t y (kg/m 3) resp 460, 470, 560, 440 Moisture Content % (Wet Volume) 6.0 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 22 Mean, and Standard D e v i a t i o n (Mpa) . . . 2.473, 0.303 Dura t i o n Load sample s i z e 22 Load D u r a t i o n S t r e s s L e v e l (MPa) 1.823 F a i l u r e s , S u r v i v o r s 18, 3 Comments: The s u r v i v o r f a i l u r e s t r e s s e s were 2.710, 3.079, and 2.641 MPa. The r a d i a l - t a n g e n t i a l plane was p a r a l l e l t o , but i n c l i n e d at 35 spacing of t e x t i s 2 degrees to the plane of the end p l a t e s f o r boards 103 and 104. For boards 102 and 105 the plane of the g r a i n was p a r a l l e l to the end p l a t e s . 134 Test Number 3 S t a r t i n g Date Dec 20th, 1981 Test T i t l e , S u b t i t l e . . . . Three s i z e , Small crack Crack Length, Hole Diameter (mm) . . . . 3.86, 1.91 Specimen Length (mm), F(c/b) 25.0, 1.015 Board Numbers 1 02, 1 03, 104, 1 05 Density (kg/m 3) resp 460, 470, 560, 440 Moisture Content % (Wet Volume) 6.0 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 11 Mean, and Standard D e v i a t i o n (Mpa) . . . 3.260, 0.308 Du r a t i o n Load sample s i z e 22 Load D u r a t i o n S t r e s s L e v e l (MPa) 2.572 F a i l u r e s , S u r v i v o r s 22, 0 Comments: The r a d i a l - t a n g e n t i a l plane was p a r a l l e l t o , but i n c l i n e d at 35 degrees to the plane of the end p l a t e s f o r boards 103 and 104. For boards 102 and 105 the plane of the g r a i n was p a r a l l e l to the end p l a t e s . 135 Test Number 4 S t a r t i n g Date Feb 26th, 1982 Test T i t l e , S u b t i t l e C y c l i c - 2 , C y c l i c Crack Length, Hole Diameter (mm) . . . . 6.22, 2.03 Specimen Length (mm), F(c/b) . . . . 25.0, 1.04 Board Number 102 Density (kg/m 3) resp 440 Moisture Content % (Wet Volume) 6.0 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 17 Mean, and Standard D e v i a t i o n (Mpa) . . . 2.472, 0.203 Duration Load sample s i z e 28 Load D u r a t i o n S t r e s s L e v e l (MPa) 1.90 F a i l u r e s , S u r v i v o r s 28, 0 Comments: T h i s t e s t was loaded to a constant s t r e s s l e v e l f o r p e r i o d s of two hours, and then allowed to recover f o r p e r i o d s of four hours f o r a t o t a l of 23 c y c l e s . Then f o r a f u r t h e r 3 c y c l e s at the c o n c l u s i o n of the t e s t , the p e r i o d was i n c r e a s e d such that the loaded time was 8 hours and the recovery time 16 hours. 136 Test Number 4 S t a r t i n g Date Feb 26th, 1982 Test T i t l e , S u b t i t l e C y c l i c - 2 , Constant Crack Length, Hole Diameter (mm) . . . . 6.22, 2.03 Specimen Length (mm), F(c/b) . . . . 25.0, 1.04 Board Number 102 Den s i t y (kg/m 3) resp 440 Moi s t u r e Content % (Wet Volume) 6.0 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 17 Mean, and Standard D e v i a t i o n (Mpa) . . . 2.472, 0.203 Du r a t i o n Load sample s i z e 27 Load D u r a t i o n S t r e s s L e v e l (MPa) 1.90 F a i l u r e s , S u r v i v o r s 27, 0 Comments: T h i s t e s t served as a c o n t r o l t e s t f o r the C y c l i c , C y c l i c - 2 t e s t . 137 Test Number 5 S t a r t i n g Date March 4th, 1982 Test T i t l e , S u b t i t l e M o i s t u r e , Wet Crack Length, Hole Diameter (mm) . . . . 6.22, 2.03 Specimen Length (mm), F(c/b) . . . . 25.0, 1.04 Board Number 102 Density (kg/m 3) resp 460 Moisture Content % (Wet Volume) 60.0 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 16 Mean, and Standard D e v i a t i o n (Mpa) . . . 1.742, 0.151 Dura t i o n Load sample s i z e 17 Load Du r a t i o n S t r e s s L e v e l (MPa) 1.13 F a i l u r e s , S u r v i v o r s 15, 2 Comments: The ramp f a i l u r e s t r e s s e s of the s u r v i v o r s were 1.697 and 1.719 MPa. 138 Test Number 5 S t a r t i n g Date March 4th, 1982 Test T i t l e , S u b t i t l e M o isture, Dry Crack Length, Hole Diameter (mm) . . . . 6.22, 2.03 Specimen Length (mm), F(c/b) . . . . 25.0, 1.04 Board Number 102 D e n s i t y (kg/m 3) resp 460 Moisture Content % (Wet Volume) 6.0 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 17 Mean, and Standard D e v i a t i o n (Mpa) . . . 2.417, 0.201 D u r a t i o n Load sample s i z e 18 Load D u r a t i o n S t r e s s L e v e l (MPa) 1.838 F a i l u r e s , S u r v i v o r s 13, 5 Comments: The ramp f a i l u r e s t r e s s e s of the s u r v i v o r s were 2.80, 2.91, 2.67, 2.34, and 2.80 MPa. 139 Test Number 6 S t a r t i n g Date May 1st, 1982 Te s t T i t l e . Ts Crack Length, Hole Diameter (mm) . . . . 6.22, 2.03 Specimen Length (mm), F(c/b) . . . . 25.0, 1.04 Board Number 200 Den s i t y (kg/m 3) resp 510 Mo i s t u r e Content % (Wet Volume) 8.1 Temperature ( C e l c i u s ) 22 Short term t e s t sample s i z e 34 Mean, and Standard D e v i a t i o n (Mpa) . . . 2.465, 0.150 Du r a t i o n Load sample s i z e . 89 Load D u r a t i o n S t r e s s L e v e l (MPa) 1.645 F a i l u r e s , S u r v i v o r s . 54, 35 Comments: The ranked f a i l u r e s t r e s s e s (MPa) of the s u r v i v o r s are as f o l l o w s : 1.90, 1.90, 2.05, 2.21, 2.22, 2.22, 2.23, 2.29, 2.31, 2.33, 2.35, 2.36, 2.38, 2.39, 2.40, 2.43, 2.46, 2.53, 2.61, 2.64, 2.68, 2.71, 2.71, 2.73. 140 Appendix 2 Volume E f f e c t As r e f e r r e d to i n S e c t i o n 5.3, small c l e a r specimens were t e s t e d i n a x i a l t e n s i o n i n the t a n g e n t i a l l o n g i t u d i n a l plane. In the f i g u r e A1 below, the r e s u l t s of t h i s t e s t are p l o t t e d on the f i g u r e a l r e a d y p l o t t e d by B a r r e t t ( 1 9 7 4 ) . The volume of 6mmx5mmxl0mm = 300mm3 = 0.005in 3, i s p l o t t e d a g a i n s t the average short term s t r e n g t h found of 9.2MPar or 1330psi. New Data P o i n t / t r h-_ 10 X P R E S E N T A L L O W A B L E WORKING S T R E S S (dry s e r v i c e condi t ion , normal load d u r a t i o n . ' t r a n s f o r m e d v a l u e s 'open s y m b o l : c l e a r m a t e r i a l s o l i d symbol : c o m m e r c i a l m a t e r i a l half sol id: c l e a r gl ued- la minated CSA-086) • *Thut,(1970) a Fox, (1974-1) w F o x . ( 1 9 7 4 - 2 ) A A M a d s e n , (1 972 ) O (Schn iewind and Lyon,(1973) », Pet e r s o n , (1 9 7 3 ) ' -2 10 V -r-10 17 10 V O L U M E (IN.3) FIG. 3. Linear n-erossion equations relating strength to volume for uniformly loaded blocks of commercial and clem 1 >ouglas-fir. F i g . A1 Volume E f f e c t P l o t by Barrett(1974) 141 Appendix 3 F a i l u r e C r i t e r i o n . The f a i l u r e c r i t e r i o n assumed i n (2-6) i s not the only c r i t e r i o n which c o u l d be assumed. Other c r i t e r i a such as, Fc = k d 2 or Fc = k 2 d or o t h e r s , c o u l d be a p p l i e d , however these would no longer s a t i s f y the G r i f f i t h theory of crack p r o p a g a t i o n . The f a i l u r e c r i t e r i o n assumed (2-6) i n t h i s study was Fc = k d = k ( t 0 , A t ) d ( t 0 , A t ) By s u b s t i t u t i n g t h i s c r i t e r i o n , the Ts times of the N i e l s e n and step-wise models c o i n c i d e . T h i s f e a t u r e a l l o w s ready comparison of the two models when f i t t i n g them to the same experimental data. The f a i l u r e c r i t e r i o n chosen by Brincker(1982) was as f o l l o w s , Fc = k d , but where d i s such that Fc = k ( t 0 , A t ) ( 2 d ( t 0 , A t ) - d(t | 0,2At) ) T h i s c r i t e r i o n g i v e s a lower Ts value than the simpler c r i t e r i o n as assumed i n (2-6). In p r i v a t e correspondence with Mr B r i n c k e r , he i n d i c a t e d that the reason he chose t h i s c r i t e r i o n was because i t was based upon "thermodynamic p r i n c i p l e s " . In s p i t e of t h i s , the use of the simpler c r i t e r i o n was maintained, as i t i s i n agreement with the estimate of Ts as determined by N i e l s e n i n h i s development of the N i e l s e n model. 142 Appendix 4 Adjustment C a l c u l a t i o n s . Ths s t r e s s r a t i o i s d e f i n e d as «/<f0. Where the crack i s propagating i n an i n f i n i t e medium, t h i s r a t i o remains c o n s t a n t . However, f o r a f i n i t e medium the s t r e s s a i n c r e a s e s i n the same way as the s t r e s s i n t e n s i t y f a c t o r i n c r e a s e s , a c c o r d i n g to the F(c/b) f a c t o r s as d e f i n e d by Tada P a r i s Irwin(l973) ( f i g u r e 26). T h e r e f o r e f o r a f i n i t e medium the value of 9 as given i n (2-12) i s no longer a c o n s t a n t . In order to a l l o w f o r t h i s , the i n t e g r a t i o n can be c a r r i e d out i n c r e m e n t a l l y , each increment r e p r e s e n t i n g a stage of crack growth between an i n i t i a l and subsequent crack l e n g t h where 8 i s assumed to be a constant over the increment. In the method used below, the F(c/b) f a c t o r a p p l i e d i s the average value f o r the crack increment concerned. 6 w i l l now be d e f i n e d as ©F [. F't i s the r a t i o by which e has i n c r e a s e d because of the f i n i t e medium. For example, i f the i n i t i a l h a l f crack l e n g t h c 0=3.11 mm, and at the end of the increment the h a l f crack l e n g t h has i n c r e a s e d to 3.76 mm, then the F(c/b) values are 1.04 and 1.06 r e s p e c t i v e l y . The average of these values i s 1.05. T h e r e f o r e , F i = 1.05/1.04 = 1.01 If the next increment i s between the crack lengths of 3.76 and 4.41 mm, then the new F(c/b) value equals 1.08. The average value of 1.08 and 1.06 i s 1.07, and F t = 1 .07/1 .04 = 1 .03. By monitoring the change i n 6 as the crack lengthens, a more r e a l i s t i c estimate of the time spent i n phase 2 i s o b t a i n e d . For example, the r e s u l t s of the Ts experiment are 143 analysed i n t h i s manner, as given below. A p p l y i n g (2-12) to in c l u d e a changing y i e l d s A ivU. (A-1 ) Ts-Tcat = 2_ "T~ (F, ef %. where n i s s u f f i c i e n t t h a t the c r i t i c a l crack l e n g t h i s achieved. Using 4 incremental s t e p s , b=l/5 and other f e a t u r e s of the Ts experiment, an example of how (A-1) i s a p p l i e d i s worked below. The i n t e g r a t i o n of (A-1) y i e l d s the f o l l o w i n g e x p r e s s i o n . (A-2) Ts-Tcat= 2. ~T~ I Co —J_ S I Cc r to ^ e f C 4 3 (p.©)* CP,e)' The c r i t i c a l h a l f crack l e n g t h of the Ts experiment i s 5.7mm, with an F(c/b) f a c t o r of 1.16 as d e f i n e d by the f r a c t u r e toughness equation, s u b s t i t u t i n g K=250kPa nW 2 and o-=1.90MPa. The i n i t i a l h a l f crack l e n g t h i s 3.11mm, with an F(c/b) f a c t o r of 1.04. By using 4 increments of crack l e n g t h from 3.11mm to 5.70mm, 4 average val u e s of F(c/b) and 4 F j_ values can be c a l c u l a t e d . These v a l u e s can then be s u b s t i t u t e d i n t o (A-2) and the amount of time spent i n phase 2 i s determined. N o t i c e that by r e p l a c i n g the F\ value s with the value 1.0, the i n f i n i t e medium s o l u t i o n i s obta i n e d . The r e s u l t s of the c a l c u l a t i o n s are shown i n the f o l l o w i n g t a b l e . 144 (UIUI) changing o • • II E Crack Length E E in CN u X) Xi \ XI \ u verage F(c/b) F i ro U E-i (A E-••» to o E-1 in E-Hal: u < 4J X \ o *J \ o 3.11 0.25 1 .04 3.76 0.30 1 .06 1 .05 1.01 0.283 0.348 4.41 0.35 1 .08 1 .07 1 .03 0.020 0.043 5.06 0.40 1 . 1 1 1.10 1 .06 0.002 0.006 5.70 0.46 1 .16 1.14 1.10 0.000 0.000 0.305 0.397 N o t i c e that the time spent i n phase 2 i s 30% higher ( 0.397/0.305 = 1.30 ) f o r the i n f i n i t e medium case where F^ i s assumed to remain constant at a value of 1.0. N o t i c e a l s o that f o r the i n f i n i t e medium case, 93% of the time spent in phase 2 i s w i t h i n the f i r s t increment of crack p r o p a g a t i o n . In order t o be a b l e t o determine the e f f e c t phase 2 has on the time to f a i l u r e , we need to determine the magnitude of Ts with r e s p e c t t o the magnitude of T s - T c a t . C o n t i n u i n g i n the use of the Ts experiment, s u b s t i t u t i n g a=0.343 and b=1/5, f i t s the Ts l i n e through the p o s i t i o n of the Ts p o i n t , and the Teat l i n e i s p a r a l l e l with the slope of the d u r a t i o n of lo a d data p o i n t s . 145 For a s t r e s s r a t i o of 0.7 t h i s y i e l d s from (2-16) Ts = 1.22(M). Assuming #=0.165 ( t h i s assumption can be checked a f t e r the e v a l u a t i o n of has been made, and c o r r e c t e d i f necessary) i m p l i e s 6=0.14 from (2-24), and a l s o i m p l i e s <r, = 2.63/0. 165 = l6.4MPa ( r e f e r (2-23)), where 2.63 i s the short term s t r e n g t h at s t r e s s r a t i o 0.7. Ts-Tcat ( i n f i n i t e medium) = ( M/6 ) 0.397 = 2.48 M Th e r e f o r e , Teat ( i n f i n i t e medium) = M ( 1.22 + 2.48 ) = 3.70 M S i m i l a r l y f o r the f i n i t e medium case, Teat ( f i n i t e medium) = M ( 1.22 + 1.90 ) = 3.12 M A comparison can now be made between the cases of the f i n i t e and i n f i n i t e mediums i . e . , T c a t ( i n f i n i t e ) / T c a t ( f i n i t e ) = 3.70/3.12 = 1.19 In a d j u s t i n g the data p o i n t s of the experiment by 19% to the r i g h t , i t i s found that f o r *=1.90MPa and tf!=16MPa, the step-wise model f i t s the a d j u s t e d p o s i t i o n . T h i s means that the i n i t i a l assumption of #=0.165 remains v a l i d , and that the above c a l c u l a t i o n s do not need to be i t e r a t e d again using an a d j u s t e d value of <t> i n order to get the c o r r e c t f i t . For the case of f i t t i n g the N i e l s e n model, the same procedure i s a p p l i e d . The onl y d i f f e r e n c e i s i n the form of the f u n c t i o n used i n the e s t i m a t i o n of the d u r a t i o n of phase 2. For 1 46 the N i e l s e n model, (Ts-Tcat) i s l a r g e r than the corresponding value i n the step-wise model because F(e) i s l a r g e r , given that "a", "b" and <t> are the same. In order that the N i e l s e n model f i t the same data as the step-wise model, the s t r e n g t h r a t i o <t> must be i n c r e a s e d , thereby d e c r e a s i n g ( T s - T c a t ) . By adopting the same procedure as f o r the step-wise model, a value of 0=0.45 was s u b s t i t u t e d , which gave a value of «r,=5.5MPa. T h i s value a p p l i e d to the N i e l s e n model, f i t s the data p o i n t s d i s p l a c e d 19% to the r i g h t . T h i s appendix has served two f u n c t i o n s . F i r s t , the c a l c u l a t i o n s f o r the adjustments as a p p l i e d i n Chapter 5 are shown. Second, the appendix served t o demonstrate a f u r t h e r a p p l i c a t i o n to which the v i s c o e l a s t i c f r a c t u r e mechanics models can be put. In f r a c t u r e problems i n wood, i t would be uncommon to encounter a s i t u a t i o n where the boundaries of the region c o u l d be assumed as i n f i n i t e . P a r t i c u l a r l y important c o u l d be the case where the crack i s c l o s e t o , or has a l r e a d y impinged upon the boundary. F r a c t u r e mechanics can provide an estimate of the f a i l u r e l o a d i f the value of the f r a c t u r e toughness i s known. V i s c o e l a s t i c f r a c t u r e mechanics can provide an estimate of the d u r a t i o n of l o a d c h a r a c t e r i s t i c s of the c r a c k . Depending on the nature of the m a t e r i a l surrounding the c r a c k , the F(c/b) f a c t o r s may tend t o a c c e l l e r a t e or d e c e l l e r a t e crack growth. By a d j u s t i n g the F(c/b) f a c t o r s i n the same manner as has been a p p l i e d i n t h i s appendix, an estimate of the d u r a t i o n of load behaviour c o u l d be made. I t should be noted that the term 147 F(c/b) i s r e a l l y an o v e r s i m p l i f i c a t i o n f o r cases other than the c e n t r a l through crack used i n the experiments of t h i s study. For other more complex c r a c k geometries, many more f a c t o r s other than the crack l e n g t h as r e l a t e d to the specimen l e n g t h are i n v o l v e d . However, the same p r i n c i p l e s w i l l a p p l y . 148 Appendix 5 A Step-wise Value f o r P l a s t i c Y i e l d S t r e s s Equation (2-19) r e q u i r e s f u r t h e r e x p l a n a t i o n . T h i s equation l i n k s the y i e l d s t r e s s « , to the ste p l e n g t h 6. I f the value of 6 i s determined by assuming an e l a s t i c medium as given by (2-2), then i t i s not p o s s i b l e to assume that a value of (a p l a s t i c y i e l d s t r e s s ) can a l s o be determined from the same model. T h i s i n a b i l i t y to r e l a t e 6 to *, because they each d e r i v e from d i f f e r e n t m a t e r i a l behaviours (one step-wise e l a s t i c and the other continuous e l a s t i c p l a s t i c ) would appear t o i n v a l i d a t e (2 -19). In a d d i t i o n , the concept of a d i s c o n t i n u o u s crack movement, w i t h i n a theory based on the c o n t i n u i t y of s t r e s s e s and displacements, a l s o needs f u r t h e r e x p l a n a t i o n . In a p p l y i n g the f o l l o w i n g r a t i o n a l e to the mechanics of the step-wise model, a r e l a t i o n s h i p between c , and 6 i s developed below. It i s necessary to take the m a t e r i a l i n the mouth of the crack t i p and d i s c r e t i z e i t i n t o segments of len g t h 6. By assuming that these elements each have a uniform s t r e s s and assuming a l s o that they continue to s a t i s f y e q u i l i b r i u m by f o l l o w i n g c l o s e l y the e l a s t i c s t r e s s d i s t r i b u t i o n , then the f o l l o w i n g s t r e s s d i s t r i b u t i o n i s proposed as i n f i g u r e A2 below. A l s o , the d i s c o n t i n u o u s crack movement of the step-wise model i s developed f u r t h e r . Element 1 w i l l f a i l f i r s t . I t w i l l f a i l at a s t r e s s of (the average e l a s t i c s t r e s s over the l e n g t h 6), then have lengthened by a d i s t a n c e 6. Because wood does not demonstrate p l a s t i c i t y but ra t h e r tends to be b r i t t l e at f r a c t u r e , elements 1 2 and 3 are assumed to conform to the e l a s t i c d i s t r i b u t i o n of 149 s t r e s s as shown. Herein l i e s the c e n t r a l d i f f e r e n c e between the Dugdale B a r e n b l a t t model and t h i s one. Dugdale assumes that elements 1 2 and 3 are continuous and that the m a t e r i a l i n t h i s r e g i on i s deforming p l a s t i c a l l y at a constant s t r e s s The model proposed i n f i g u r e A2 assumes t h a t the m a t e r i a l i n the mouth of the crack t i p remains e l a s t i c ( v i s c o e l a s t i c i n t h i s case) u n t i l f a i l u r e , and that f a i l u r e occurs segment by segment in a d i s c r e t e step-wise manner. T h i s approach shows that by averaging the e l a s t i c s t r e s s d i s t r i b u t i o n over a d i s t a n c e 6 from the crack t i p , that an estimate of can be determined a c c o r d i n g to (2-19). 150 F i g . A2 Step-wise model of the crack t i p . 

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