EXPERIMENTAL STUDIES ON FRACTURE OF NOTCHED WHITE SPRUCE BEAMS by WILSON W.S. LAU B.A.Sc U n i v e r s i t y of Windsor, 1984 THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept t h i s t h e s i s as conforming . t o the r e q u i r e d THE standard UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1987 © Wilson W.S. Lau, 1987 In presenting degree at this the thesis in University of partial fulfilment of British Columbia, I agree freely available for reference and study. I further copying of department publication this or of thesis for by his or her representatives. requirements that the for an advanced Library shall make it agree that permission for extensive scholarly purposes may be It is granted by the understood that head of copying my or this thesis for financial gain shall not be allowed without my written permission. Department of Civil Engineering The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date the April, 1987 ABSTRACT The f r a c t u r e a r i s i n g from problem the due sudden to change the of singular geometric stresses properties around c r a c k s and notches was s t u d i e d both a n a l y t i c a l l y and experimentally. The f a i l u r e specimens were models derived mechanics methodology, critical stress conducted t o of the cracked by using linear and the elastic notched fracture which l e d to the d e t e r m i n a t i o n of the intensity determine factors. fracture Experiments toughness were for different modes as w e l l as the e f f e c t of v a r i a t i o n s i n the c r a c k - f r o n t width, specimen size and moisture content. Subsequently, f a i l u r e s u r f a c e s f o r cracks and notches were developed based on the experiments undertaken, d e s c r i b i n g i n t e r a c t i o n between mode I To v e r i f y the r e l i a b i l i t y and mode II f r a c t u r e an application, degrees-cracked beam and design a 90 loading to the r a p i d crack conditions. i i results literature. curves for degrees-notched propagation under a 90 beam are presented. These curves allow the p r e d i c t i o n of the loads due the toughness. of these experiments, the obtained were compared with the p u b l i s h e d As i n each case failure different Table of Contents ABSTRACT i i LIST OF TABLES vi LIST OF FIGURES vii ACKNOWLEDGEMENTS 1. xi INTRODUCTION 1 1.1 The Problem 1 1.2 Previous Research 3 1.3 Objective 7 1. 4 Scope 1.4.1 7 Introduction 7 1.4.2 Mode I F r a c t u r e Toughness 8 1.4.3 8 Mode II F r a c t u r e Toughness 1.4.4 Specimen Size E f f e c t 8 1.4.5 Moisture Content E f f e c t 1.4.6 1.5 2. Interaction Loading Curves for 9 Mixed Mode Summary of O b j e c t i v e s 9 9 THEORY 11 2.1 Introduction 11 2.2 A p p l i c a t i o n of LEFM on wood s t r u c t u r e s 11 2.3 Formulation of 2.4 the S t r e s s Intensity Factors for Cracks 17 Stress I n t e n s i t y F a c t o r s f o r Sharp Notches 26 2.4.1 26 Introduction 2.4.2 Formulation of the problem f o r notches ...27 2.4.3 Implementation in a Finite Element Program 38 2.5 E f f e c t of Factors Specimen Size on S t r e s s Intensity 43 2.5.1 E f f e c t of M a t e r i a l Heterogeneity 2.5.2 E f f e c t of the Change of the S t r e s s State 2.6 Effect of Toughness Moisture Content on 2.7 Mixed Mode F r a c t u r e i n White Spruce 43 Fracture .44 49 51 Experiment Parameters 54 3.1 Introduction 54 3.2 Crack O r i e n t a t i o n and Propagation 54 3.3 Materials 56 3.4 Specimen P r e p a r a t i o n 57 3.5 Experimental Displacement 3.6 Treatment Measurement 4.1 Introduction 4.2 Experiment No.1, 4.4 4.5 4.6 Load and of Data 59 62 EXPERIMENT DESCRIPTIONS AND 4.3 of RESULTS 65 . 65 Mode I F r a c t u r e Toughness 65 4.2.1 Experimental Design and Procedure 65 4.2.2 Results 66 Experiment No.2 C r a c k - f r o n t Width F r a c t u r e Toughness Effect on 75 4.3.1 Experiment Design and Procedure 75 4.3.2 Results 77 Experiment No.3, C r a c k - f r o n t Length V a r i a t i o n ..84 4.4.1 Experiment Design and Procedure 84 4.4.2 Results 84 Experiment No.4 Moisture Content E f f e c t 89 4.5.1 Experiment Design and Procedure 89 4.5.2 Results 91 Experiment No.5, Mode II F r a c t u r e Toughness ....94 iv 4.6.1 Experiment Design and Procedure 94 4.6.2 R e s u l t s 4.7 95 Experiment No.6, Mixed Mode Mid-cracked Beams .103 4.7.1 Experiment Design and Procedure 103 4.7.2 R e s u l t s 4.8 Experiment No.7, DCB under Mixed-mode Loading .114 4.8.1 4.9 105 Experiment Design and Procedure 114 4.8.2 R e s u l t s 119 Experiment No.8, Notched Beam Specimens 122 4.9.1 122 Experiment Design and Procedure 4.9.2 R e s u l t s 5. 6. 125 4.10 Summary 127 DISCUSSION 128 5.1 Introduction 128 5.2 S t r e s s I n t e n s i t y Factor I n t e r a c t i o n Curve Cracks for 5.3 Stress I n t e n s i t y Factor I n t e r a c t i o n Curve Notches for 5.4 Application 128 131 138 CONCLUSIONS AND RECOMMENDATIONS 149 6.1 Conclusions 149 6.2 Recommendations f o r f u t u r e r e s e a r c h 150 BIBLIOGRAPHY 151 APPENDIX I 1 55 APPENDIX II . 1 67 APPENDIX III 170 v LIST OF TABLES TABLE Page 1 E l a s t i c p r o p e r t i e s of s t r u c t u r a l m a t e r i a l s 32 2 E f f e c t of the annual r i n g s angle on 71 3 E f f e c t of c r a c k - f r o n t width on K , ^ f o r compact t e n s i o n specimens of white spruce f o r l o n g i t i i d i n a l propagation 78 4 E f f e c t of c r a c k - f r o n t length on t e n s i o n specimens of white spruce 86 5 The 6 E f f e c t of moisture content l o n g i t u d u n a l propagation 7 E f f e c t of the annual r i n g s angle on 8 Results of the mid-cracked beam specimens 107 9 Results 121 10 Results of the notched beam specimens 11 Stress i n t e n s i t y f a c t o r s problems for s i z e c o e f f i c i e n t f o r CTSs and DCB compact specimens on K ^ ^ , f o r CTSs in vi 92 100 K-J.-J.^ of the two-loads beam t e s t s for various 90 126 sharp crack LIST OF FIGURES FIGURE Page 1 Three b a s i c modes of crack s u r f a c e displacement 13 2 T h r e e - l e t t e r system f o r d e s i g n i n g o r i e n t a t i o n r e l a t i o n s i n wood 15 3 Coordinate system used f o r d e s c r i b i n g the s t r e s s e s ahead of a crack 19 4 F i r s t primary s i n g u l a r s t r e s s f i e l d 5 Second primary s i n g u l a r s t r e s s f i e l d 0° crack f o r a 0° crack...25 for a 25 6 N o t a t i o n f o r r e a l space transformed c o o r d i n a t e s 33 7 N o t a t i o n and coordinates f o r . n o t c h r o o t s 33 8 Determinant o f eigenequation f o r notch i n Douglas fir 36 9 S t r e s s s t a t e i n specimen 4-6 10 A s t e e l crack s t a r t e r with crack t i p r a d i u s l e s s than 0 . 5um LVDT gage f o r c e n t e r l i n e d e f l e c t i o n measurement 58 60 12 M o d i f i e d LVDT gage f o r measuring displacement 60 13 T y p i c a l l o a d vs displacement curves 14 R e l a t i o n s between T _ / 15 C o n f i g u r a t i o n of compact t e n s i o n specimen (CTS) 67 16 C o n f i g u r a t i o n of double c a n t i l e v e r beam specimen (DCB) 67 17 The experimental setup o f the compact t e n s i o n specimen t e s t °8 18 F i n i t e element mesh f o r the CTS 19 F i n i t e element mesh f o r the DCB specimen 20 Mode I f r a c t u r e toughness angle 11 k k a n d i c K u / longitudinal sliding K I I C v a r i a t i o n w i t h annual v i i 61 • 6 4 7 0 70 rings LIST OF FIGURES - Cont'd FIGURE Page 21 Cumulative d i s t r i b u t i o n curve o f the compact specimen 22 Cumulative d i s t r i b u t i o n curve o f the double c a n t i l e v e r beam 74 23 E f f e c t o f s p e c i f i c g r a v i t y on mode I f r a c t u r e toughness 24 E f f e c t of c r a c k - f r o n t width on mode I f r a c t u r e toughness.(Based on the W e i b u l l weakest l i n k model)..80 25 E f f e c t o f c r a c k - f r o n t width on mode I f r a c t u r e toughness . (Based on the s t r e s s - s t a t e model) 26 Specimen c o n f i g u r a t i o n specimen 27 E f f e c t of the c r a c k - f r o n t toughness 28 E f f e c t of the moisture content on mode I f r a c t u r e toughness 93 29 Experimental setup of the mode I I f r a c t u r e specimen 96 30 F i n i t e element mesh f o r the mode I I f r a c t u r e toughness specimen 96 31 Applied 97 32 The apparatus and the experimental setup of the mode II f r a c t u r e toughness t e s t specimen 98 33 Mode I I f r a c t u r e toughness v a r i a t i o n w i t h annual r i n g s angle 34 Cumulative d i s t r i b u t i o n curve o f the e n d - s p l i t beam specimen 102 35 Specimen c o n f i g u r a t i o n of the 45 deg. beam 104 36 Specimen c o n f i g u r a t i o n of the 90 deg. beam 104 37 F i n i t e element mesh f o r the mid-cracked beam specimen 38 o f the c r a c k - f r o n t length tension 76 82 length 85 on mode I f r a c t u r e 88 toughness load versus l o n g i t u d i n a l displacement Experimental and c o r r e c t e d cumulative p r o b a b i l i t y curves of the mode I f r a c t u r e toughness f o r the 90 degrees beam viii '^ 101 J - ° u 109 L I S T OF FIGURES - 39 40 Cont'd E x p e r i m e n t a l and c o r r e c t e d c u m u l a t i v e p r o b a b i l i t y c u r v e s o f t h e mode I I f r a c t u r e t o u g h n e s s f o r t h e 9 0 d e g r e e s beam. 110 Method o f n o r m a l i z i n g t h e c u m u l a t i v e p r o b a b i l i t y c u r v e s o f t h e 4 5 d e g r e e s m i d - c r a c k e d beam Ill 41 K J [ " ^ T I i n t e r a c t i o n d i a g r a m b a s e d on t h e m i d - c r a c k e d beam r e s u l t 113 42 Specimen c o n f i g u r a t i o n specimen of the two-loads beam 115 43 Experiment setup of the two-loads beam s p e c i m e n 116 44 Failure the two-loads beam s p e c i m e n 120 45 Typical load-displacement beam s p e c i m e n envelope of curve of the two-loads 123 46 Specimen c o n f i g u r a t i o n f o r n o t c h e d beam 47 Interaction between K ^ K ^ and K ^ / K ^ 124 f o r c r a c k e d 129 beam specimens 48 49 50 Interaction ^Tl^HC ^ o r c u r v e s d e r i v e d between K ^ / K ^ a n d ^beam s p e c i m e n s c r a c e 130 d I n t e r a c t i o n models between K ^ / K ^ and K ^ / K ^ ^ c r a c k e d beam s p e c i m e n s 77. . . . V Interaction relations between K-j. and f ° r 132 for 134 notches 51 Interaction 52 I n t e r a c t i o n curve for notches with A = 1 . 7 0 D i m e n s i o n l e s s S I F s f o r p u r e moment and p u r e s h e a r l o a d i n g s as a f u n c t i o n o f n o t c h - t o - d e p t h r a t i o f o r 9 0 d e g r e e s c r a c k e d beam 139 54 Several 141 55 D e s i g n ' c u r v e s f o r 9 0 d e g r e e s c r a c k e d beam as f u n c t i o n of n o t c h - t o - d e p t h r a t i o 53 curve f o r notches with A=1.60 136 sharp c r a c k problems ix 137 a 143 LIST OF FIGURES - Cont'd FIGURE 56 57 Page K / P versus notch length for v a r i o u s notch depth of a 2"x8" beam • T 146 C r i t i c a l load versus notch length f o r v a r i o u s notch depth of a 2 "x8 " beam 148 x ACKNOWLEDGEMENTS The author would like to express h i s sincere to Dr. R.O. Foschi and Dr. J.D. Barrett for their and invaluable advice gratitude guidance in the preparation of this thesis. The author would also l i k e to thank Conroy Lum for h i s assistance in using the f i n i t e element program NOTCH, Mr. D. Postgate and Mr. B. Merkli for their helpful suggestions and assistance in the manufacture of the apparatus, and also the Departments Harvesting of C i v i l Engineering and Wood Science and of the University of B r i t i s h Columbia for the use of their laboratories and equipment. xi 1. INTRODUCTION 1 . 1 THE PROBLEM In p r a c t i c a l situations sliced, dried, drilled, wood i s sawn, flaked, and occur processes. In other c a s e s , gaps between "butt-joint"), (which are notches notches o c c u r r i n g at beams lead to a concentrations. A l l analyzed by Manual, as of at ordinary well without these the f o r Engineering the notch a by result the The Design and stress a can stress not Timber Canadian term laminated in which in be Design design code, i n Wood, use a f o r the present of a notch i n a of the intense c o r n e r . Furthermore, stress the code specimens. theoretical understanding s i n g u l a r i t i e s i n wood should be of p r a c t i c a l of these importance. A p p l i c a t i o n of f r a c t u r e mechanics to wood i s with s t r u c t u r a l f a i l u r e these re-entrant geometry notch root the governing does not cover s p l i t e d Obviously, in or ends of boards sawcut, cases consideration concentration a t a of lap joints in change reduced net depth to account beam, by Flaws of each denoted s t r e s s formulae. as CAN3-086-M84: Code usually open butt or sudden s i n g u l a r i t y formed result formed chipped, fastened. d e f e c t s unavoidably laminated timber as a chopped, by c a t a s t r o p h i c crack concerned propagation. In many a p p l i c a t i o n s , f r a c t u r e mechanics techniques a r e used to e l i m i n a t e such f a i l u r e s by s p e c i a l m a t e r i a l c o n t r o l or by d e f i n i n g maximum crack 1 or flaw s i z e s that tests can be 2 t o l e r a t e d i n the s t r u c t u r e . As wood s t r u c t u r e cannot be mentioned above, flaws i n a totally controlled, so fracture mechanics methods must be used t o assess the a l l o w a b l e load on the s t r u c t u r e . Near the v i c i n i t y of a notch root or a crack t i p , the s t r e s s at every point i n t h i s r e g i o n i s s u b j e c t e d t o plane s t r e s s or s t r a i n c o n d i t i o n s which can be expressed i n terms of the stress intensity factors. Therefore, the determination of the a l l o w a b l e l o a d i s the same as o b t a i n i n g the c r i t i c a l values of the s t r e s s i n t e n s i t y Depending on the mode of crack stress intensity factors K I, II, factors. t i p deformation, are d e s i g n a t e d with a I I I f o r the cracked specimen (zero the subscript notch angle cracks) which corresponds to the opening, forward shear and tranverse shear modes For the s u b s c r i p t A, are used B f a c t o r s f o r the primary of deformation. to and d e f i n e the notches, stress the secondary intensity stress fields respect i v e l y . Critical cracked different values f o r the pure mode specimen species information about have been of wood. studied There the c r i t i c a l f a i l u r e of a cracked specimen. is, I and mode II f o r a and published however, a lack v a l u e s f o r the mixed for of mode 3 1.2 PREVIOUS RESEARCH Studies of the (1913) who a cracks or notches were i n i t i a t e d by Inglis made the stress analysis of an e l l i p t i c a l hole in uniformly stressed elastic plate. A crack can be represented by an i n f i n i t e s i m a l l y narrow e l l i p s e . Based Inglis' theory, G r i f f i t h (1921) formulated his well energy c r i t e r i o n for b r i t t l e fracture which was Irwin (1948) and Orowan (1955) on known extended by to apply to metallic solids where p l a s t i c deformation takes place at crack t i p s . Later, Savin (1961), using his photoelastic stress at re-entrant corners. Up method, analyzed to that moment, the the study of cracks and notches was mainly based on the energy method or individual stress experiment results concentration to determine the factors for s p e c i f i c geometries. Linear e l a s t i c fracture mechanics (LEFM) i s based on the e l a s t i c solution of the crack t i p stress f i e l d where the yielding has been highly l o c a l i z e d Williams (1957) has solved the expressing the displacements u — stresses in a crack front. problem of a cracked plate, i j / the at strains terms of i n f i n i t e r and 6, the where r e i j ' ^ ant series of and 6 the singular and regular terms of are polar coordinates, with the crack t i p at the o r i g i n . Later, Irwin (1957) found that the f i r s t singular term always dominate in the stress formulae and expressed the stress and strain equations introducing the Kj stress intensity factor. A comprehensive analysis of crack t i p e l a s t i c stresses, strains and displacements, using the stress intensity factor 4 method, was small of studied by L i u (1965&-1966) . s c a l e y i e l d i n g , SSY, characterizing the displacements within Liu has crack stresses, application of l i n e a r (LEFM), to cracked f a c t o r s has been shown (1973), Broek material isotropic material. body was the for the s t r e s s e s in a f r a c t u r e mechanics. elastic using fracture mechanics stress intensity the e f f e c t i v e method by (1984) extension f i r s t made by S i h et at capable f i e l d zone which forms Hellan The of and to be an (1982), case strains fundamental b a s i s of the l i n e a r e l a s t i c The the shown that K i s tip the e l a s t i c For but restricted of LEFM (1965) who small region Knott to to orthotropic derived formulae surrounding a crack tip in an o r t h o t r o p i c body. In the application of LEFM to the strength of s t r u c t u r e s , i t i s not only the form of s i n g u l a r i t y but the magnitude of the s t r e s s e s near the root of the also fractured surface what i s needed. T h i s r e q u i r e s the computation of the s o - c a l l e d s t r e s s i n t e n s i t y f a c t o r s f o r the singularity. And boundary c o n d i t i o n s , the except f o r some simple geometry and solutions always (1971,1972,1974) require introduced method to compute the which incorporated field. This compatibility conventional method was numerical method. a calibrated finite s i n g u l a r terms also conditions satisfies at the f i n i t e elements and extended Walsh element s t r e s s i n t e n s i t y f a c t o r s using the method a to apply to i n the the displacement equilibrium interface the m o d i f i e d LEFM, between and the elements. T h i s orthotropic materials and 5 r i g h t angle notches. However, the method f o r c o m p a t i b i l i t y at the elements monotonic A account i n t e r f a c e s , which enhances convergence. method suggested compatible displacement with does not singular by Benzley (1974), using formulation for a f i n i t e 'enrichment' thoroughly. The method terms, solves has been shown a element, this problem to be e f f i c i e n t and reliable. T h i s conformable F o s c h i and B a r r e t t model was extended (1976) f o r the a n i s o t r o p i c case and by G i f f o r d and H i l t o n bodies with displacement (1978) to analyze c r a c k s i n 12-node i s o p a r a m e t r i c elements and by used isotropic a coarser mesh. The a p p l i c a t i o n of LEFM t o the a n a l y s i s of the stress f i e l d at the root of a sharp notch of a r b i t r a r y notch angle was f i r s t proposed stress intensity by L e i c e s t e r factors correspond r e s p e c t i v e l y stress f i e l d of the (1971). f o r notches, to the eigenfields symmetrical symmetry. These element with notches, has The been s t u d i e d the secondary notch t i p axes in notches of elastic the Gross and special Mendelson and with finite i n L e i c e s t e r ' s paper which i s a by the i n connection were i n c o r p o r a t e d V-notch, which fi terms "opening mode" to method to analyze notches Walsh (1982). K , notches only i f the respect factors and A governing the are commonly used sharp cracks are a p p l i c a b l e t o are K primary and s t r e s s d i s t r i b u t i o n . However, the " s l i d i n g mode" which He introduced the with case of (1972) 6 using the boundary c o l l o c a t i o n procedure and a l s o by L i n and Tong (1980) using According the s i n g u l a r to the f r a c t u r e i s assumed factor attains a to be linear to fracture occur when critical i n t e n s i t y f a c t o r . This f i n i t e element method. value mechanics the — intensity the c r i t i c a l factor i s material determined e x p e r i m e n t a l l y stress model, stress dependent and has for different species of wood. Schniewind and Centeno (1971) presented K -, values f o r I( the s i x p r i n c i p a l systems wood; using the e n d - s p l i t beam method, (1971) found the K I I C values (1982) a l s o e s t a b l i s h e d baltic redwood by on orthotropic the a mixed elements due to Douglas-fir B a r r e t t and f o r Hemlock. Hunt and mode t e s t method between mode interaction materials for Foschi Croager the mode II f r a c t u r e toughness f o r i n t e r a t i o n curve e x i s t e d studies of propagation has the f a c t curve been that I and the notch always have a 1/Vr s i n g u l a r i t y . A (1986), which incorporates mode I I . for isotropic restricted d i f f e r e n t types of s i n g u l a r i t i e s , assuming to root The and cracked can whereas the sharp as a have cracks method suggested by the eigenvalue X an Lum failure parameter, has proved t o be a reasonable approach to s p e c i f y the c r i t i c a l describe notch root s t r e s s c o n d i t i o n s interaction curves and may be used to f o r notches as well as f o r cracks. Studies on the c r a c k - f r o n t it influences the p r e d i c t i o n of structures. Barrett (1976) width e f f e c t have shown that the f a i l u r e load f o r found a size effect large due to 7 c r a c k - f r o n t width theory on Kj^. f o r cracks using the Weibull's whereas Ewing (1979), has a l s o shown the e x i s t e n c e of this effect on h i s stress-state (1969,1973) has s t u d i e d the magnitude of notches by based considering a model. size c o e f f i c i e n t Leicester a size effect on implied on the nominal s t r e s s . The moisture Williams content e f f e c t was (1979) on compact tension Madsen (1985) on the shear s t u d i e d by Ewing & specimen and by Dolan & strength. 1.3 OBJECTIVE The o b j e c t i v e of t h i s t h e s i s i s t o i n v e s t i g a t e the behavior of white spruce under fracture the mode I, mode II and the combined mixed mode l o a d i n g c o n d i t i o n s . The e f f e c t s - s u c h the v a r i a t i o n of the moisture content and the as crack-front width w i l l a l s o be s t u d i e d . 1 .4 SCOPE 1.4.1 INTRODUCTION In order notches under to develop a loading, e s t a b l i s h the c r i t i c a l f a c t o r s and r e s u l t s p a r t i c u l a r , the following design experiments values were f o r the were compared with experiments were features. procedure for cracks carried stress published designed to and out t o intensity ones. examine In the 8 1.4.2 MODE I FRACTURE TOUGHNESS To determine e x p e r i m e n t a l l y the c r i t i c a l opening mode s t r e s s i n t e n s i t y f a c t o r a compact tension specimen value of the f o r white spruce and a double specimen. E f f e c t of crack o r i e n t a t i o n with cantilever respect using beam to the g r a i n was a l s o c o n s i d e r e d . 1.4.3 MODE II FRACTURE TOUGHNESS To determine e x p e r i m e n t a l l y the c r i t i c a l value of the s l i d i n g mode s t r e s s i n t e n s i t y f a c t o r f o r White Spruce by the e n d - s p l i t beam method. E f f e c t of the crack o r i e n t a t i o n was a l s o considered f o r cracks propagating along the g r a i n . 1.4.4 SPECIMEN SIZE EFFECT The strength assessed from of complex laboratory tests structures on s c a l e d are frequently models. T h i s is most simply done by use of the assumption that a s c a l e model and a f u l l - s i z e structure stress l e v e l . stress fail However, f o r s t r u c t u r e s stress f i e l d s , this were c a r r i e d will may not be out t o study a t the same nominal containing singular the r e a l case. the dependence of the mode I i n t e n s i t y f a c t o r on the s i z e of the specimen and a l s o to study the a p p l i c a b i l i t y of the c r a c k - f r o n t developed by B a r r e t t the mode I s t r e s s factor. Experiments (1976) on width theory intensity 9 1.4.5 MOISTURE CONTENT EFFECT Many p u b l i s h e d toughness have shown this but design were conducted moisture content effect on intensity factor and a model 1.4.6 the formula are l a c k i n g e f f e c t . Experiments residual that fracture of wood depends on the moisture content as w e l l as the temperature, for papers the mode I to account t o study the critical i s proposed stress i n terms of stresses. INTERACTION CURVES FOR MIXED MODE LOADING Since wood producing both i s often opening i n t e r a c t i o n curves subjected and loading deformation modes, are necessary to p r e d i c t strength of the s t r u c t u r e s . establish sliding t o combined Experiments the i n t e r a c t i o n curves the f a i l u r e were c a r r i e d out t o f o r both cracks and notches. 1.5 SUMMARY OF OBJECTIVES 1. To determine for 2. white 3. stress intensity factor spruce. To determine factor the mode I c r i t i c a l the mode II c r i t i c a l stress intensity f o r white spruce. To study the specimen s i z e e f f e c t on the mode I f r a c t u r e toughness. 4. 5. To study the moisture toughness of the white spruce. To establish content e f f e c t the family of mixed on the mode fracture failure 10 interaction curves for cracks and notches of white spruce. 6. To discuss practice and the results, their suggested directions application for further to design research. 2. THEORY 2.1 INTRODUCTION T h i s chapter e x p l a i n s the b a s i c theory f r a c t u r e mechanics as a p p l i e d to cracked and notched computation wood of specimens. the stress The of l i n e a r (zero angle theory intensity elastic notch) requires factors, the which are a s s o c i a t e d with the s i n g u l a r s t r e s s e s near the crack t i p or notch r o o t . For s i m p l i c i t y , the theory w i l l be introduced starting from a sharp crack i n an i s o t r o p i c and o r t h o t r o p i c specimen. T h i s w i l l be extended to the more g e n e r a l cases of This combined information, determined critical stress p r e d i c t i o n of the u l t i m a t e with intensity the notches. experimentally factors, makes s t r e n g t h of f r a c t u r e the structures possible. The s i z e and the moisture been proven content of the specimen to have pronounced e f f e c t on the s t r e n g t h of the specimen and w i l l be d i s c u s s e d i n the f o l l o w i n g sections. T h i s chapter w i l l end with the theory of i n t e r a c t i o n of had stress intensity curves factor. 2.2 APPLICATION OF LEFM ON WOOD STRUCTURES The concept of linear elastic fracture determining the f r a c t u r e f a i l u r e mode have r e c e i v e d a several years. abundant amount Fracture and s t r e n g t h i n of a t t e n t i o n mechanics 1 1 mechanics i n the i s concerned for wood past with 12 structural failure average s t r e s s by catastrophic crack below the normal f a i l u r e propagation at stress level. In many a p p l i c a t i o n s , f r a c t u r e techniques a r e used to e l i m i n a t e t h i s kind of f a i l u r e by c o n t r o l l i n g the flaw s i z e or i n the cases that these d e f e c t s are unavoidable, by c o n s i d e r i n g the e f f e c t of the flaw on the allowable Complete s t u d i e s of l o a d on the s t r u c t u r e s . f r a c t u r e behavior s t r e s s a n a l y s i s aspects and the cover both r e s i s t a n c e of the the material to the s t r e s s imposed. In t h i s chapter, develop the s i g n i f i c a n t s t r e s s a n a l y s i s d e t a i l s and r e l e v a n t parameters, which will govern the the purpose i s to failure strength of s t r u c t u r e s c o n t a i n i n g cracks or notches. The existence redistribution of a crack method. The g r e a t e s t of stress in a or a notch w i l l body due be analyzed a t t e n t i o n should be to the by the LEFM p a i d to the e l e v a t i o n of s t r e s s e s a t the v i c i n i t y of the crack t i p which w i l l u s u a l l y be accompanied by a t l e a s t some p l a s t i c i t y other non-linear e f f e c t s . The s u r f a c e s dominate the d i s t r i b u t i o n of crack of a crack s t r e s s e s near high and or a notch or around the t i p s i n c e they are s t r e s s - f r e e boundaries of the body. Other remote boundaries and loading f o r c e s a f f e c t only the i n t e n s i t y of the l o c a l s t r e s s f i e l d a t the t i p . The s t r e s s f i e l d s near d i v i d e d i n t o three a crack basic types, or a each a s s o c i a t e d with a l o c a l mode of deformation as i l l u s t r a t e d i n Figure In mode I propagation, the crack to themselves; i n mode II notch t i p can be the s u r f a c e s 1. s u r f a c e s open normal slide tangential to Figure 1 - Three b a s i c modes of crack s u r f a c e displacement CO 1 4 themselves slide longitudinally; tangential to direction superposition of these general in and t h e wood i s a highly material with usually a tearing case anisotropic, varying of planes surface free fields. the principal refers is letters, the f i r s t fracture s u r f a c e , and t h e s e c o n d which the c r a c k plane these (1957). to propagation o f symmetry w i l l a s shown i n identified with refers be six Figure a pair of normal t o the to the d i r e c t i o n in the d i r e c t i o n intensity for placed cracks solutions crack exists the i s o t r o p i c stress solutions are available of planes. propagates. s o l u t i o n s of symmetrically the t h r e e modes o f p r o p a g a t i o n , propagation of and orthotropy, systems of p r o p a g a t i o n the computation different t h e g r a i n . Due t o t h e h i g h F o r each of t h e The in porous propagation necessary. system of and know, crack specifying checks, w i t h i n the o r t h o t r o p i c planes 2. A crack-tip are natural cleavage of the The As we properties directions principal to to describe local heterogeneous mechanical surfaces fashion. longitudinal-radial occurs along system the perpendicular associated stress The specimen a III t h r e e modes i s s u f f i c i e n t longitudinal-tangential For and 3-dimensional deformation directions. i n mode themselves propagation t h e most and sharp crack problem factors. A variety cracks, particularly in isotropic have been require summarized of for m a t e r i a l s , and many by P a r i s and S i h u. t.r. Figure 2 - Three-letter wood, r.l. l.r. r.t. l.t. system f o r d e s i g n i n g a c c o r d i n g to orientation relations A . P . S c h n i e w i n d and R . A . P o z n i a k in 16 Since the formulation on continuum mechanics considered, of t h i s f r a c t u r e theory where m a t e r i a l is based d i f f e r e n c e s are not i t i s expected that f r a c t u r e mechanics should equally a p p l i c a b l e to orthotropic materials Similar of formulation wood. intensity factors in a n i s o t r o p i c m a t e r i a l s has a l s o been proposed by P a r i s and S i h (1965). had orthotropy stress such as be Walsh (1972) i n v e s t i g a t e d the effect on computed s t r e s s i n t e n s i t y f a c t o r s for geometry and sufficient concluded length, o r t h o t r o p i c c l o s e l y . However, symmetrical and on cracked products, that f o r it and several specimen isotropic results would appear only for the Thus, to apply LEFM must aware the dependence of s t r e s s factor on the m a t e r i a l p r o p e r t i e s of the Another assumption that is of agree case skew-symmetrical s e l f - e q u i l i b r a t i n g infinite plates. one rectangular of of loading on wood intensity specimen. necessary to apply LEFM the c o n d i t i o n of small s c a l e y i e l d i n g (SSY) around the is crack t i p . Under t h i s c o n d i t i o n , K can c h a r a c t e r i z e the crack stresses and equivalent to s t r a i n s even w i t h i n the plastic zone smaller that the radius of e l a s t i c the p l a s t i c zone. This radius ^ p^ being r c o n d i t i o n can if valid K £ > c much be s a t i s f i e d , r e in p r i n c i p l e , the enough specimen. The is f i e l d zone ( r ) . S i n c e i s p r o p o r t i o n a l to the specimen s i z e , always tip one uses SSY a large ASTM recommended s i z e requirements for measurements a r e : 2.5 ( y e 2 > 1 ) 17 where a i s the l e n g t h of the c r a c k , L i s the d i s t a n c e ahead of the crack f r o n t , o-y i s the y i e l d i n g s t r e s s or s t r e s s that r e s u l t s i n gross r e » that deformation. The c o n d i t i o n of Tp i s a s u f f i c i e n t but not necessary c o n d i t i o n f o r the validity of restrictive the in LEFM. terms The condition of specimen could size necessary c o n d i t i o n f o r the v a l i d i t y be unduly requirements. The of the LEFM i s that K would be able t o c h a r a c t e r i z e the crack t i p s t r e s s or s t r a i n component at the l o c a t i o n of the d e f i n e d f r a c t u r e p r o c e s s . The zero notch root assumption w i l l be imperative f o r applying LEFM on notch problem. In order t o have SSY at the notch r o o t , the notch other dimensions, r a d i u s should be small compare to so that v a r i a t i o n of the notch r a d i u s does not i n f l u e n c e the surrounding s t r e s s and s t r a i n fields. Ewing and W i l l i a m s (1979) had s t u d i e d the importance of the sharpness of the i n i t i a l notch on the f r a c t u r e toughness of Scots Pine and found toughness that the mode I f r a c t u r e tends to i n c r e a s e as the r a d i u s i n c r e a s e s . 2.3 FORMULATION OF THE STRESS INTENSITY FACTORS FOR CRACKS The formulation displacement of the solution analyzed the stress and f i e l d s a s s o c i a t e d with each mode using the LEFM methodology f o l l o w s i n the manner the method of of Westergaard as (1939). two-dimensional plane symmetric and skew-symmetric s t r e s s the crack p l a n e . of Irwin (1957) based Mode I I I can Mode I and II problems can with a pure be the f i e l d s with respect be t r e a t e d as on to shear 18 problem. From the the resulting cracks handbook by stress isotropic case are and given Hiroshi Tada displacement as follows fields with the (1973), for the notation referred to Figure 3. Mode I : a = - x ( o = n r ) 1 cos £ / [1 - s i n £ s i n |£] + a 2 2 J xo 2 — cos £ (2*r)l>2 y = y + 0(rl'2) [ l + s i n £ s i n |£] + 0 ( r l / 2 ) 2 2 2 T X 2 — s i n i cos i (2Trr)l/2 1 1 cos—+0(rl/2) • 2 (2.2a) (2.2b) (2.2c) and for plane strain (with higher order terms omitted) o T z xy The = v(a = T yz x + a ) y (2.2d) = 0 (2.2e) corresponding u = ^ [r/(27r)] v = ^ [r/(2n)]l/ 1 / 2 2 displacements are: cos J - [ l - 2v + s i n |£] (2.2f) sin f (2.2g) [2 - 2v - cos |£] Figure 3 - C o o r d i n a t e system used f o r d e s c r i b i n g the o f a c r a c k . From P . C . P a r i s and G . C . S i n stresses ahead 20 = 0 w (2.2h) Mode II : — /o (2*1- a x v _ a _ = X II (2Trr)l/2 y + a + 0(rl'2) xo . 9 9 39 , i/o s i n TT cos -r- cos + 0(r ) 1 / z s (2^)1/2 y s i n £ [2 + cos | - cos 2 2 2 2 2 2 2 2 2 and for plane strain (with higher order terms omitted) (2.3d) T xz =T yz = 0 (2.3e) with dispalcements: u v = — i - [r/(2Tr)]l/2 s i n 1 [2 - 2v + cos 29 2 [-1 + 2v + s i n Y~] (2.3f) (2.3g) 21 f w =0 (2.3h) Mode III : T XZ T ' a u J (2Tvr)l/ x z Q + 0(rl/2) ( 2 . 4 a ) 2 cos i + 0(rl/2) y z x w e sinf + T hn ( 2 > 4 b ) (2Ttr)l/2 =ff=a=t =0 y z xy -IiI[(2r)/,r]l/2 v s i n |. (2.4c) w ( 2 > 4 d ) =v = 0 (2.4e) Equations (2.2) and (2.3) have been written for the case of plane s t r a i n , but can be changed to plane stress taking a = 0 and replace v by As seen from by v/(]+v). equations (2.2) and (2.3), the formulae include higher order terms such as uniform stresses p a r a l l e l to cracks, o XQ and r X Z O f ^ ) . But a n ^ terms of root of r, 0 ( r 1 / / as the value of r approaches square these terms can be neglected t i p ) , the singular term 1/Vr the equations. the order of 0, ( i . e . close since to the crack becomes the governing term in 22 The parameters c a l l e d the i- I I f stress intensity respectively, as parameters Kj , K shown in are factors F i g 1. these equations n for the It i s coordinate-independent, are three modes found that these so they can be thought of as the magnitude of the stress f i e l d s surrounding the crack other t i p . The parameters, K, boundaries conditions Consequently, formulae for are determined and the by imposed their evaluation the loads. come from complete stress analysis for the specimen configuration a and loading. A crack stress f i e l d for is represented by a unique intensity factors. Since certain loading and geometry combination of the three they are stress correlated parameters , the f a i l u r e c r i t e r i o n w i l l depends on a l l three. From the Equations (2.2), (2.3) that the stress intensity (ForceJxtLength) ^/ . - 2 Since and factors they are (2.4), we have observe units linear factors of in a linear e l a s t i c stress solution, the stress intensity factors are l i n e a r l y related to the applied loads. Sih et a l (1965) derived formulae for the stresses in a small region surrounding a crack t i p of orthotropic material using a complex variable formulation Their results for a crack and coincident with the negative for the LEFM method. p a r a l l e l to a material x-axis (Figure 3) are : axis Symmetric (about x-axis) plane loading, -6,6, "1 2 Kj 6 "2 P n : H K, /2nr K I 1 3 •6 6 r /2Trr 2 1 2 ^ ~ 1 2 B B B l ,1 S 1 •xy and Skew-symmetric (about x-axis) plane loading, c = *II " r 1 , ? r 1 r B B l /2iri X K II ? "^ 0 K D Re[ 1 1 *V*? *7 II 1 ( B l B M 2 where B. = /cos 9 J . sin 9 J + ie i = /^T Re = real part i= 1,2 24 B? =• e B. -E 2 [« + (,c -l)l/2] 2 2 2 -1 (2.6g) [< - ( < - l ) ! / ] 2 2 -1 (2.6h) > f o r plane s t r e s s (2.61) xy And d i r e c t l y ahead of the crack the stresses are : II xy /2TTX where the K's values conditions, highly (2.7) • 2 TTX are dependent on dependent remote on geometry, boundaries and s l i g h t l y dependent on orthotropic parameters (for f i n i t e bodies). Some t y p i c a l symmetric and skew-symmetric are shown in Figure co-ordinates are stress f i e l d s 4 and Figure 5. The stresses in polar obtained by doing the transformation as follows : a a o r = a cos 9 + o s i n 6 + 2a cos8 sin9 x y xy (2.8a) o = a x (2.8b) 2 2 sin 9 + a y 2 cos 8 - 2a cos9 sln9 xy 2 - ( o - a ) cos8 sin9 + o ( c o s 9 - s i n 9 ) 2 r 6 y x The symmetric only exist i f the 2 (2.8c) x y and skew-symmetric crack stress fields i s p a r a l l e l to an axis of material symmetry. Then, the stresses will elastic around the crack t i p 25 so* Primary stress field E : E . . = 20.00 lao* i T 270* Figure 4 - F i r s t primary s i n g u l a r 0° c r a c k stress field for a Primary stress E : E = 20.00 E :G X y = field 0.90 = 0.02 Positive Negative Figure 5 - Second primary s i n g u l a r 0° c r a c k stress field for a 26 w i l l be the superposition of some linear combination of these two f i e l d s . 2.4 STRESS INTENSITY FACTORS FOR SHARP NOTCHES 2.4.1 INTRODUCTION The stress characterize the intensity stress factors fields plane. However, the l i t e r a t u r e is mostly concerned have surrounding shown the to fracture on stress intensity with sharp cracks. few papers on notches, the been factors Although there method was based on the are nominal fracture stress combined with limited experimental results. Leicester(1971) has presented a new method for the analysis of the stress f i e l d s at notch of the root of a mathematically arbitrary notch method indicates that angle. in general s i n g u l a r i t i e s of stress at the orthotropic structural The application there are sharp of two this stress roots of notches in t y p i c a l materials. The magnitude of stress f i e l d s are noted as the stress intensity factors these for notch angle. He used the indices of A and B referring to the two stress f i e l d s and these two stress f i e l d s are d i f f e r e n t from the symmetry and skew-symmetry stress f i e l d s in most cases. The significance of the stress intensity factors in the fact that a c r i t e r i o n for crack propagation from notch root may be formulated as follows: lies the 27 where K A C and Kg^, the c r i t i c a l factors, and the interaction equation stress intensity (2.9) are determined by direct measurement. For the s p e c i f i c case of a zero angle the notation Kj , Kj£, K J J and ^HQ notch, i . e . , a crack, w i l l replace the notation K A , K ^, Kg and Kg^. A 2.4.2 FORMULATION OF THE PROBLEM FOR NOTCHES The formulation of the .problem follows the method proposed by Lum (1986). The equations of equilibrium under zero body forces, in cartesian co-ordinates, are: 3a 3a _xy_ = 0 3y 3x 3a 3a +- S L 3x 3? = 0 (2.10a) (2.10b) The strain-displacement kinematic relationships are: X 3u 3x (2.11a) 3y_ (2.11b) 3y xy 3u 3y t 3v 3x (2.11c) 28 and the stress-strain equations for an orthotropic material with plane stress condition are : o x E = x F T x u E _ — yx ° y y (2.12a) J a u y xy E ~ E x y x y G a G (2.12b) a xy G = xy (2.12c) xy From strain-energy considerations i t i s known that : u E xy _ u yx E x (2.13) y Equations (2.11a,b,c) may be shown to result in the following equation of compatibility : 3e , , 3e 2 8 x 3 3 e 2 x 2 + y 3y y 2 £ 3x Equations (2.14) 2 (2.l0a,b) are s a t i s f i e d when the stress components are expressed by Airy's stress function 0 through 3<b 2 3x 2 (2.15a) 7 29 (2.15b) 3y 2 A | % - = -a 3x3y xy (2.15c) ^ ' Substitude (2.l5a,b,c) into (2.12a,b,c) we get the expressions for strains in term of Airy's stress functions : £ E x - ^ - ( f ) X 3y 2 J 3x z E " ^ ( ^ ) x E y y 3x x (2.16a) 2 3y z xy " " f x V ^ ( 2 Substituding compatibility (2.l6a,b,c) into the equation ' 1 6 C ) of (2.14), one f i n a l l y obtains : (2.17a) 3x^ e 2 3x 3y 2 2 3y * I 30 where E E e y < - I ^ y ^ 2 t ( 1 / G xy) - < W " <W- In order to solve equation ' <2 17b) (2.17a), i t may also be written as : t T I TK -+« ^-]* 1 3x + a 1 2 1 i i 3y 3x 2 I I 2 1 dy 1 =0 (2.18a) 2 where 1 a I ll o I/ , 2 1 + (< -l) ' = —2 [< 1 e a r ? 2 J ] (2.18b) J (2.18c) 2 L N Since ^ ~ x ' equation (2.18a) was derived from the equilibrium, stress-strain and compatibility equations, the solution of this equation will satisfy a l l the governing equations. Also, the parameters that govern the solution of the notch root stress f i e l d s defined as (2.17a,b). w i l l be the constants e and K These two parameters are material dependent and have values between 0.1 and 10. For isotropic materials, e=«=1.0 and for White Spruce, e=2.13 K=1.60 with axes in the longitudinal-tangential plane, and e=1.87 K=1.98 31 with axes in the longitudinal-radial plane. The representative range for the e l a s t i c properties of structural materials published by Leicester (1971) i s shown in Table 1. The solution of the f i e l d equation and II indicates that with subscripts I two further transformed co-ordinates may be defined. These are x =x =x y "V l = a (2.19a) Il II (2.19b) y and shown in Figure 6. The field equation (2.18), written in the new co-ordinates, i s : [jl.+ _!!_][_>!_ • _ » ! _ ] » . 0 V V »V 3 For the biharmonic V (v ($)) 2 2 case (2.20) '-II* a i~ n> a equation (2.20) reduces to a equation, and in terms of polar co-ordinates : = 0 where the operator V (2.21) 2 is Table Elastic Properties of 1 Structural Materials Type o f M a t e r i a l Isotropic solid Typical Wood ( LR & L T p l a n e s Typical plywood o f Typical fibre-reinforced plastic From R . H . L e i c e s t e r balanced (1971) ) condition e K 1.0 1.0 2.0 2.0 1.0 4.0 2.0 4.0 33 Figure 6 Figure 7 N o t a t i o n f o r r e a l space and transformed c o o r d i n a t e s . (a) C o o r d i n a t e s o f r e a l space, (b) C o o r d i n a t e s s y s t e m I . ( c ) C o o r d i n a t e s y s t e m I I . F r o m R.H. L e i c e s t e r (1971) N o t a t i o n and c o o r d i n a t e s f o r n o t c h F r o m R.H. L e i c e s t e r (1971) roots. 34 In general, for the orthotropic materials a^ta^^, and the solution may be written as where <f>j and are harmonic functions of the co-ordinates systems I and II respectively. The solution to equation (2.22) may be sought in the product form : i = r i f ( V " 1 > J 1 (2.24) 1 where X i s a constant. A suitable pair of harmonic functions for the solutions is the following : $ = A ^ * c o s ( X 6 ) + A r ^ sin(X6 .) I 'n = V I I c o s where A C 1 x e n 2 ) Vii + (2.25a) ] s i n ( x e n> (2.25b) to A^ are arbitrary constants. Substituting (2.25) into (2.l5a,b,c) leads to the following : a = A(A-l)r^" (l/a2)[-A cos(X0 -29 ) - A sin(X6^26 )j 2 x 1 I 1 2 + X(X-l)r . (l/ 2 )[-A cos(X9 -2e ) X 2 I a I 3 I I i I r - A^slnUe^-Ze )] (2.26a) 35 o = X(X-l)r^ y 1 + X(X"l)r^ o [ A c o s ( X 8 - 2 6 ) + A sin(Xe .-29 )] 2 I [A cos(Xe 2 3 2 I ] ; I ] i - 2 e ) + A sin(Xe .-26 )] i I lt i] = X ( X - l ) r ^ ~ ( l / a ) [ A , s i n ( X 9 - 2 9 ) - A,cos(X9 I I 1 I I 2 I 2 xy T1 T (2.26b) II -29 )] I T J + X(X-l)r^~ (l/a )[A sin(X9 .-29 ) - A ^ c o s U g ^ g ^ ] (2.26c) 2 3 I I The constants A 1 I ] I I to A^ are obtained by substitution eqnations (2.26a,b,c) in the boundary conditions OQ= ° of R Q = Q and 6=9^, as shown in Figure at the notch edges along 0=0 A 7. These four conditions lead to a matrix equation : a 2i a. 31 . a m a !2 3 22 !3 a 23 Zk a a_ _ 32 \ l a 33 • -3 a a = 0 ^ For any particular notch, s a t i s f i e d when a determinant | a | m n (2.27) a the eigenequation (2.27) i s value of X has been found i s zero. Figure 8 is a such that the plot of this determinant for a notch in Douglas f i r . It can be shown that the eigenvalues of a eigenfield are limited to the range singular 1 < X < 2. Within this -3-0 I 1-0 I I 1-5 2-0 I I I I I 2-5 3-0 3-5 4-0 Parameter A Figure 8 - Determinant of eigenequation f o r notch F r o m R.H. L e i c e s t e r ( 1 9 7 1 ) i n Douglas fir. CO cn 37 range i t i s found that there i s always at least one eigenvalue, and i f the notch angle i s small there w i l l be two eigenvalues. Denote these two eigenvalues, X and X A The eigenfields associated with these defined and as 'primary' respectively. In general, B with X < X . A g eigenvalues w i l l 'secondary' the primary be eigenfields eigenfield will dominate at the notch root as r approaches 0 except in pure shear mode where the secondary eigenfield w i l l Since equation govern. (2.27) i s a homogeneous additional constraint must be applied equation, an besides the four boundary conditions. The magnitude of the stress f i e l d s fulfil this requirement. The stress intensity factors corresponding to the primary and the secondary stress w i l l be denoted by K of K and K A B A and K B will fields respectively. The d e f i n i t i o n s are quite arbitrary but in most cases of orthotropic material, they are defined as follows : o (e=n) e = K. /(2-jrr) c (e=fi.) r9 K /(2*r) 2-A, (2.28a) 2-X, (2.28b) B where fi i s the angle of crack propagation and is usually the grain angle in the wood. For K TT the special case of a sharp crack, K A = Kj and Kg = . However, the terms "opening mode" and "sliding mode", 38 which are usually associated applicable to notches with a sharp only i f the notches are crack, are symmetrical with respect to the axes of e l a s t i c symmetry. The failure criterion propagation of a characterizing crack from a sharp the onset of notch corresponds to these two stress intensity factors, K and K , reaching some A 6 c r i t i c a l values K and K AC B C respectively. From the equations of stress f i e l d s , we can derive the equations of strain and displacement stress intensity factors. The f i e l d in terms of the method for obtaining these factors i s discussed in the next section. 2.4.3 IMPLEMENTATION IN A FINITE ELEMENT PROGRAM A singular compute the finite element method stress intensity mesh consists of three t r a n s i t i o n a l elements factors. The regions — t h e notch t i p , the elements remote i s manipulated finite to element elements around the from the notch t i p , and the in between to assure the compatibility of the elements. Around the notch t i p , the quadratic displacement field is enriched by the singular terms as follows : u i = a i + a 2 c + a 3 n+ a k^ 2 + a 5 C + a n + a c n + a ?n 2 6 2 ? n 2 8 + K ^ U . n ) + K g.(c,n) B (2.29) where $ and TJ are natural co-ordinates. Substituting the nodal displacements (2.29), we get : {u.} = [M]{a} + K {f.} + K {g.} A B or {a} = [M]-l[{u - K { f i} A i} - K {g.}] B Substituting back into equation (2.29), u i(?.n) = [N(c,n)]{u.} + K A [^(Cn) " + Kg [ g ^ . - l ) - [N( ,Ti)]{f.}] C [N(c,n)]{g }] i Using the v i r t u a l work p r i n c i p l e results in / « {<5}[B] [[D][B]{6} + {K} [D]{J2}]dV = « {6}{R} V T T T for a v i r t u a l change in {6}, and T 40 / 6 V (K}{n} [[D][B]{6> + {K} [D}{fi}]dV = T T T T 6 {K}{R } (2.34) k for a v i r t u a l change in {K}. Combining equations (2.33) and (2.34), we get : L [S] {Pj} {V* [C {P } 2 T [c n 1 2 {P } " 2 ] [c ] 21 {6} {R} (2.35) ] [c ] 22 where [S] = / [B] V [D][B]dV (2.36) {PJ = / [B] V [D]{Q }dV (2.37) [C J ] = / {Q.} [D]{n.}dV v (2.38) 41 The element s t i f f n e s s matrix can be assembled to form the global stiffness the global load vector. The elements far from the notch w i l l be ordinary matrix, and the element load quadratic, regular vector, tip elements, whereas the t r a n s i t i o n a l elements are introduced in a t r a n s i t i o n zone between f u l l y singular and regular elements. The displacement field of the t r a n s i t i o n a l elements w i l l be : u.(?,n) = [NU,r,)]{u.} + K where at B + R(c,n){K [f (c,n) A i [N( ,n){f.}] c [g.(c.,n) - [N(c,n)]{g }]} (2.39) i i s bilinear R($,TJ) singular-singular singular-regular t r a n s i t i o n function with boundary boundary. and i s The equal use of R=1 to 0 at the bilinear t r a n s i t i o n a l elements ensures conformity between elements. After forming the s t i f f n e s s matrix, the solution method w i l l be the same as for an ordinary f i n i t e element From the calculated magnitude of K and Kg values, A the eigenfields, problem. which indicate the we can get the Kj and KJJ stress intensity factors by assigning the amounts of opening and shear-sliding in each eigenfields. Thus, for a direction of crack propagation ^, — K = /2TT 1 2 " lim (r r*0 A A A Q 9 — = /2TT o (r,9 = VA A l i m ( K f ( 0 = *) + Kg r r+0 A A f ( 6 - i | 0 + ... B (2.40) 42 - K n 2 _ A T (r,e= ( ) X = /2TT lim ( r r+0 = /2TT lim (K g (9=*) + K r ° r+0 rQ A 1 A l B g (9=*) + ... (2.41) B then I K l K = = K / 2 L F / 2 AV 9 = [ A A 7 K F * ) f 0 r X < ( 9 = * ) + K B f ( B H ) l f o r A = X A B (2.42a) B (2.42b) X A and K I I K II = V 2 i r K A A "* 8 (e for 1 ) For A R notches, w (2.43b) A > the (2.43a) B for A, = = /2ir [K g (9=*) + K g (6-*)] B°B A <X primary eigenfield will always dominate the stress and strain f i e l d s around the notch root except loading in pure shear-sliding mode. Lum (1986) utilized comparison (1972). this with has method the developed and the has collocation program shown method NOTCH, good used which results by in Gandhi 43 From equation (2.28) we can observe that the 2-X dimension A of K which depends on the material e l a s t i c properties and grain angle. Therefore, for A and Kg is (Force)x(Length) different angle of notches, we have d i f f e r e n t dimensions of stress intensity factors. Thus, in order to derive a f a i l u r e c r i t e r i o n for sharp notches, we must include the parameter X in our c r i t e r i o n . 2.5 EFFECT OF SPECIMEN SIZE ON STRESS INTENSITY FACTORS In predicting the strength of large structures, experiments were usually done by using the scale models. This employs an assumption that a scale model and a f u l l - s i z e structure w i l l f a i l at the same nominal structures at a may stress. However, f a i l u r e of lower nominal stress large indicates that there be a specimen size effect. Most of the s t a t i s t i c a l model previous based on literature the weakest has applied a link p r i n c i p l e to account for the strength of reduction. Such a theory assumes that f a i l u r e of a single volume element leads to the failure of the whole specimen. This has an obvious analogy with the strength of the a chain in which weakest link govern strength. 2.5.1 EFFECT OF MATERIAL HETEROGENEITY Barrett (1976) has applied this method to compare I fracture toughness d i f f e r i n g thickness as data obtained multiple from specimens crack-fronts. Using Mode of the 44 Weibull analogy, Barrett defined the cumulative d i s t r i b u t i o n function for c r i t i c a l stress crack-front width B intensity factor assuming all cracks have with total the same nominal stress intensity factor : F(K I C ) = 1 - exp[-(K where k and I C / ) m m k B] ( are the shape and scale 2 . 4 4 ) parameters respectively. An expression relating the fracture toughness of two specimens with the crack-front widths, B and B*,is : IC K* K = c ,B* B 1/k (2.45) ; Obviously, i f we plot Kj^, log-log scale, the against specimen width on slope of the regression line should CHANGE OF THE STRESS STATE a be -1/k. 2.5.2 EFFECT OF THE Another model to explain the crack-front width effect is proposed by Ewing (1979) based on the stress state in the specimen, and the drying in particular the residual stresses induced by which increases the stresses which cause a decrease. toughness and constraint 45 A useful model for describing plane stress-plane strain effects has been used in other u t i l i t y here. Figure 9 materials shows a cross and has some section diagram of a specimen, the region near the surface within the area H i s in a plane stress condition. The region in the middle i s in the plane strain stress condition. Let denote the c r i t i c a l plane fracture toughness stress condition respectively. r K' c 2HK K C1 + (D-2H) K C2 + 2H D (K C2 Usually K C1 Then the for D>2H w i l l be : (2.46a) " C1 < K and average c l K be for the plane strain c r i t i c a l stress intensity factor K' DK' c and (2.46b) } C 2 so that we'll have a positive slope i f we plot K' versus inverse thickness and an intercept of c K at D = 0 . The depth of -1 C1 the value H of the plane stress region can be computed from : H (2.47) where i s the y i e l d stress or the stress which results in gross deformation. The plane strain condition can only exists as long as there i s enough thickness to the constraint and in general : provide Figure 9 - Stress state in specimen 47 D > 4H (2.48) In general cases, i f D<2H, plane stress condition and K' c strain condition w i l l then we'll have a complete = K . For D>2H, then plane c2 result in the middle that decrease the average stress intensity factor. For the notched effect on the c r i t i c a l specimen, there is a similar size stress intensity factor proposed by Leicester (1973). From equations (2.26a,b,c), we can rewrite i t for the plane stress at f a i l u r e due to the primary stress singularity f i e l d as follows : s = A f (2.49) where a^j i s the stress component around the notch root and Af i s a constant that depends on the notch root angle, material and d e t a i l of the notch root. From the dimensional analysis, we get : (2.50) where a i s the normal stress, 48 L i s a characteristic length denoting the size of the element, B i s a dimensionless constant depends on the geometry and loading. Hence for a structural element of s p e c i f i c shape and loading, we can write : CT of " V S >0 L S where a £ i s Q (2.51) the value of o at f a i l u r e Q and A 1 is a constant. S i m i l a r i l y for the secondary °of L q = A 2 where A stress f i e l d : q > 0 2 i s another (2.52) constant. Since s>q i t follow that (2.51) always predominates provided the structural element i s s u f f i c i e n t l y large. We can also write (2.51) for two similar specimen with different size : 0.. L„ — °f2 - (r-f 2 (2.53) L where 1 and a£ a 2 r e t n e nominal stress at f a i l u r e and Lj and 1>2 are the corresponding respectively. characteristic lengths 49 There are also two conditions required to apply the size c o e f f i c i e n t factor on two d i f f e r e n t specimens. F i r s t l y , the d e t a i l of the notch roots must be i d e n t i c a l (not scaled) and secondly, the predicted fracture stress must be less than about 70% of the nominal proportional l i m i t stress for unnotched members. Based on the experimental results, the values of s follow a trend with the variation of the notch angle. 2.6 EFFECT OF MOISTURE CONTENT ON FRACTURE TOUGHNESS Fracture toughness values kiln dried state, but show thickness the toughness variation in the also varies with moisture content. In green timber, water i s contained within the c e l l lumens and c e l l walls, and drying to the fibre saturation point (approximately 27%) removes free water from the c e l l lumens. Below this value, water i s removed from the c e l l walls, resulting in shrinkage and s i g n i f i c a n t changes. It was expected that decreasing moisture content property strength would increase below 27%. Since with the drying process increases strength, then i t i s reasonable to propose that the residual stresses w i l l increase K'^ providing no cupping and consequential cracking occurs. Debaise, Porter, and Pentoney (1966) found that g of Western white pine varied according to : G ITLC = 1 , 2 8 + °' 1 1 2 5 ( e " 2 4 3 >^ " ~ e M / 6 ) (2.54) ITLC 50 where 6 = temperature (0°K) M = moisture content (% ovendry weight) G ITLC = c r ^ t i c a 1 strain energy release rate for the TL system We can relate G r = ITLC - K a.. vl J 1 I ' I T L C to K IC for TL system by : a. ~ i z 1 — * ( 2 . 5 5 ) where • _ J • " 3 1- and 22 a77 2a,„+a, "12^66^/2 2 a J + (2-56) n a ,a 2f tc., e 1 1 2 are the usual anisotropic compliances. Substituting into (2.54) for c r i t i c a l value : K I T L = 2 C[1.28 + 0.1125(9-243)(l-e~ M / 6 )] (2.57) c where C = / 1/ ' a.. " a2 2 _ 1 1 (2.58) 51 A similar equation can be derived for the critical fracture toughness of white spruce along the TL system. 2 . 7 MIXED MODE FRACTURE IN WHITE SPRUCE In recent years, attention has been fracture of wood in the opening shown that fracture independent material to the mode (mode I ) . It has been toughness, property dedicated K , is JC of wood a geometry characterizing the stress f i e l d around cracks, which govern the i n i t i a t i o n of a crack propagation. The fracture f a i l u r e of wood for mode, mode II, has also (1977). Foschi subjected been investigated However,wood sliding by Barrett structural members and are often to complex loading condition that result in mixed mode fracture. Available information is limited, and w i l l be discussed under this mixed mode b r i e f l y in the following. The mixed mode fracture in wood was f i r s t studied by Wu ( 1 9 6 7 ) . He carried out mixed wood and proposed the f i r s t mode loading tests on balsa interation curve for the mixed mode fracture in the form: K I K II (ir^) 2 = (2.59) 1 Similar experiments ( 1 9 7 4 ) on pine and an had been conducted interaction curve stress intensity factors i s in the form : by Leicester for Kj and K J J 52 I II v—+ K IC 1 y K (2.60) IIC Although most of the investigators favor the idea an interaction relation exists, based on their experiments there i s no effect s l i d i n g mode Williams and Birch on Scots (1976), pine, concluded on the opening mode caused by that that f a i l u r e due to the the shear stress. Their proposed failure criterion i s : l K^-l K ( K II > 0 <' ) ) 2 Recently, Woo and Chow (1979) 61 investigated the mixed mode fracture in Kapur and Gagil using the single-edge notch and center-crack there i s some mode II stress specimen. Their results has shown interaction relation between intensity factors under that the mode I and combined loading conditions. More recently, Mall and Murphy (1983) have studied the mixed mode fracture f a i l u r e of eastern red spruce by means of single-edge notch and center-crack specimen with crack i n c l i n a t i o n s in the TL system. Their various results, have shown that the c r i t e r i o n for the mixed mode fracture f a i l u r e of red spruce i s same as equation (2.59). A more general form for the interaction curve for mixed mode fracture may be proposed : 53 K T K IC K TT (2.62) nc where constants a i s determined experimentally. The interaction phenomenon w i l l also be v a l i d for mixed mode loading of sharp notches. As we know, the dimension the c r i t i c a l stress intensity depends on the parameter dependent. In order notches, we need method i s to X, factor which i s material and to generate the to have for sharp the same generate a family of notches geometry interaction curve for dimensions. A rational of interaction curves each with a different X, where X has a range of 1.5 to 2.0. Then each curve w i l l apply to a l l the mixed mode cases in the same dimension. For the special case of sharp crack, X=1.5. To describe mixed mode fracture of each species of wood for both cracks and notches, we need to generate the interaction curve for intermediate values, 1.7, 1.8, 1.9 and 2.0. for example, For s p e c i f i c X =1.5, geometry and 1.6, loading condition, the value of X i s determined and the appropriate curve to estimate the K T and K TT values at f a i l u r e . 3. EXPERIMENT PARAMETERS 3.1 INTRODUCTION The aim of carrying out experiments a p p l i c a b i l i t y of the theories to i s to verify real p r a c t i c e . Different experiments were designed to study the v a l i d i t y of stress intensity f a i l u r e mode and factors methodology strength of critical in predicting the real the the structures. This chapter outlines the important considerations with regard to the selection of a fracture geometry, mode of failure, specimen preparation, and some guidelines on the experiment procedure and measurements. 3.2 CRACK ORIENTATION AND PROPAGATION Since wood i s a typical orthotropic material, one must specify the mode of crack propagation so as to describe the i n i t i a l crack surface plane, the crack may in a combination orientation (see sec.2.2). Within propagate in d i f f e r e n t directions of the three d i f f e r e n t fundamental a and modes (see Figure 1). Therefore, in preparing the specimen for the experiments, attention must be paid to orientation. Also, due to the large number of of fracture orientation, a set of appropiate the crack possibilities experiments must be designed in order to simulate the fracture behavior in real structures. F i r s t of a l l , the determination of mode I fracture toughness may lead to six different sets of experiments for 54 55 each d i f f e r e n t crack TR, where orientation, namely TL,RL,LT,LR,RT or the f i r s t letter perpendicular to the crack refers to the surface and refers to the crack propagation direction the second direction. However, the fracture toughness results of Schniewind and Centeno have shown that the fracture longitudinal direction the values where strength has values (1973) for the plane in approximately propagation i s across letter one-tenth the grain. This implies that the fracture would always occur in the weakest natural cleavage plane, i . e . , either the tangential-radial plane or the radial-longitudinal plane. In addition, in order to obtain the best design of the experiment, the correct choice between two alternative f a i l u r e planes need to be made. On a macroscopic l e v e l , most of the commercial boards under bending are observed to fail in the r a d i a l longitudinal plane. However, a closer look at the crack i n i t i a t i o n w i l l reveal that the f a i l u r e has been i n i t i a t e d in the tangential-longitudinal plane. Therefore i t was decided to conduct the experiments category and it i s postulated representative of the i n i t i a t i o n of with specimen in this that it would be f a i l u r e in commercial material. In addition, some of the specimens with different orientation such as RL system were tested in order to study the effect of orthotropy on the opening mode and the s l i d i n g mode fracture toughness. 56 3.3 MATERIALS Thirty two k i l n - d r i e d nominal 2-in by 8-in (Picea glauca) boards were purchased studies of between tension specimens, board K j.c study the KJQ, one-metre mid-crack compact From another eleven board, H C an< ^ beam specimens mixed specimens mode were were used loading. cut sixty and another twenty-four for the study of notched beams. Twenty tested for the one-metre long between to study modulus stressing them under three-point displacement. Due to the supports, axis — board of loading cut variations long specimen elasticity were (MOE) obtained was by the stresses induced were tested about their approximately 9000 MPa. at weak on the f l a t -- to eliminate the bearing e f f e c t . average MOE of loading and measuring the high bearing the specimens the specimens were of the one-metre average to Sixty-six mid-cracked beams under the condition of mixed mode and used For 125mm by 120mm by 38mm, were cut randomly long end-split K Spruce from one sawmill. variation of from 10 different boards. two one-metre White The Since obtaining l o c a l modulus of e l a s t i c i t y and Poisson ratios is d i f f i c u l t , published values of these constants were used in the analysis. (E =10163 L G =663 MPa, G =700 LT U TL~ LR 0.0194 ). These regression equations MPa, MPa, E =494 MPa, T ^ =0.337, *> =0.40, LR LT E =830 R v =0.0275, RL e l a s t i c parameters were computed as proposed by Bodig MPa, and from Goodman (1973). A l l the specimens had been previously kiln-dried and were stored for a period of at least one month in a 57 temperature and humidity controlled room of 70°C maintained a nominal 9% (±1%) equilibrium moisture which content. These boards were straight-grained with the dominant system of propagation being TL. 3.4 SPECIMEN PREPARATION In order to maintain similar samples, the specimens were cut in a mass production pattern to ensure uniform dimensions of the testing specimens. bandsaw which produced a The cracks were kerf of about first 1/8" cut by so as to have enough room to accomodate the crack starter (see Figure and the smooth plastic fracture toughness testing. Then just prior to testing, the bandsawn with the the mode 10) II notch was sharpened plate for a crack starter to extend the sawn notch approximately 0.1 i n . An importance consideration of the specimens prepared is the sharpness of the i n i t i a l crack. As we assumed a notch root hypothesis, we need crack sharpness. to have some control of the strength of d r i l l e d notches i s less than 10% for notch roots below 5 mm in diameter. an root Thus (1974) diameter on a single the point has shown the that influence of Leicester zero fracture crack starter with included angle of approximately 10° and a t i p radius of less than 0.5Mm was used. It i s postulated that this would the same result as the cracks made by a razor blade. give 58 Figure 10 A s t e e l crack s t a r t e r with crack t i p r a d i u s less than 0.5um 5 9 3 . 5 EXPERIMENTAL MEASUREMENT OF LOAD AND DISPLACEMENT In the experiments, centerline deflection was measured using a linear variable d i f f e r e n t i a l transducer cases (see Figure 1 1 ) , 12) Figure was displacement while a modified used (COD) to measure and the (LVDT) in a l l LVDT (as shown the in crack . opening relative longitudinal displacement of the crack in a cracked beam. In obtaining some criterion the was critical employed stress to intensity determine factors, the critical f a i l u r e load during the testings. Load-deflection plots were generated at the time of testing on an X-Y recorder. these curves were similar to fracture tests, a those encountered in the similar method was in the ASTM test, but approximately 5 % of i n i t i a l test. Three experiments (as shown in growth i s crack length different curves 13), and growth to valid K during the the IC load P Q corresponding to a 5 % offset from the i n i t i a l slope was to compute the c r i t i c a l the limited in a were encountered Figure ASTM adopted to check v a l i d i t y of the results. Small amounts of slow crack are allowed Since used stress intensity factors. The compact tension specimens were done in the humidity room with provided an Instron autographic displacement. Cross head which produced A l l the universal recordings speed of 0 . 5 f a i l u r e s in about beam testing tests (approx. 2 0 ° C ) with a Satec were of machine the which load mm/min was and selected 1 minute. done at room temperature testing machine which provided 60 F i g u r e 11 - LVDT gage f o r c e n t e r l i n e measurement deflection c o n n e c t t o X-Y p l o t t e r F i g u r e 12 - M o d i f i e d LVDT gage f o r m e a s u r i n g l o n g i t u d i n a l s l i d i n g displacement P =P Q max P„<<P Q max P„<P Q max Q < O DISPLACEMENT (COD o r M i d s p a n Deflection) Figure 13 - T y p i c a l Load (b) 1 . 2 P * p Q m a x Vs Displacement Curves: (acceptable);(c) (a)P =P 1.2P <P Q Q max (acceptable); TunSgeeptable ) 62 plotting of the load and displacement. Cross head speed was controlled so as to produce f a i l u r e s at about 2-4 minutes. The moisture content at the time of testing and the specific gravity were determined for a l l specimens tested. 3.6 TREATMENT OF DATA An average failure load as well calculated for each experiment. as the variance were Moisture content data were also collected to assure that a l l the specimens were under uniform conditions and provide tested a basis for adjusting data as required. The values of the deflection these experiments except that used for determining the i s not significant the load deflection curve i s failure loads. As mentioned e a r l i e r , the stress intensity factor i s l i n e a r l y related the applied f a i l u r e loads load in a linear were used in elastic to compute material, the c r i t i c a l to so the stress intensity factors by the following equation : K ic = F x K 1 = 1.2 (3.1) where P Q i s the f a i l u r e load; P * i s the arbitrary entered into the f i n i t e element computer program to compute the corresponding stress the load critical stress intensity factor Kj*, intensity specimen geometry and loading. factor and K j ^ i s for the specific 63 Calculated used to stress intensity factors were collected generate the interaction curves for cracks notches. The curves were expected to have a form similar a quadrant of an e l l i p s e as shown in Figure derive a new as proposed by many 14. In order to empirical formula, need to spreaded f a i l u r e points on the curves. Different and to investigators v e r i f y the equation we and have or evenly experiments were designed to generate the curve.s as well as to study the moisture constant effect and the specimen size e f f e c t . The d e t a i l s of each experiment as well as the w i l l be described in the next chapter. results 64 4. EXPERIMENT DESCRIPTIONS AND RESULTS 4.1 INTRODUCTION This chapter gives a f u l l description of the procedures results of the experiments. The eight experiments and conducted w i l l be discussed in separate sections, namely No.1 to No.8. Emphasis was put on studies intensity factors which tend f a i l u r e . Effects of crack-front of mode I c r i t i c a l to dominates in wood fracture variation of the moisture content and width on the mode I fracture toughness was studied. At the beginning of this that the c r i t i c a l constant stress for stress thesis, i t was intensity specific geometry, also expected factors would be a material and crack orientation. However, as the experiments progressed, i t was found that K JC may also depends on the actual size of the specimen. Experiments were conducted to study the effect the variation of the uncracked length of in front of the crack on the mode I fracture toughness. 4.2 EXPERIMENT N0.1, MODE I FRACTURE TOUGHNESS 4.2.1 EXPERIMENTAL DESIGN AND PROCEDURE Experiment No.1 was designed to determine fracture toughness for white spruce. Effects the mode I of the grain orientation and specific gravity were also investigated. Two type of specimens were used to study the fracture --the ASTM compact tension specimen 65 toughness (CTS) and the double 66 cantilever beam specimen The length specimen was to as (DCB) as shown in Figure 15 and 16. width ratio recommended thickness of 38 mm. of by A t o t a l of the the compact ASTM tension Standard 75 specimens were with machined from two 20ft boards with the dominant system of propagation being TL. The CTS i s also designed to the grain orientation well as the between on the mode board study the effect of I fracture toughness as variation of K . IC The crack opening displacement (COD) was measured by the modified LVDT and a l l the testing was done with an Instron Testing Machine in a humidity room at 70°F and 50 per cent relative humidity. Crosshead speed was maintained at 0.5mm/min produced f a i l u r e s in about 1 minute. The experiment which setup i s shown in Figure 17. For the double cantilever specimens were cut from were previously beam test, three 20ft conditioned to a total of kiln-dried boards approximately 10 per 18 and cent moisture content. The tests were carried out at room temperature with the Satec Testing Machine at a rate of 0.2mm/min which caused f a i l u r e in about 2 minutes. . 4.2.2 RESULTS In order to determine the mode I fracture toughness, we need to compute the geometrical correction factors for the CTS as well as the DCB specimens. This was with the finite element program NOTCH, both accomplished using quadratic 67 100 N K K = 0.06901 II= ° D = 1 in. Figure 15 C o n f i g u r a t i o n of Compact Tension Specimen P = 100 (CTS) N K•••= 0 : 0 4 0 5 MPa/ra x 50 mm —* Figure 16 C o n f i g u r a t i o n of Double C a n t i l e v e r Beam specimen (DCB) MPa^ 68 Figure 17 The experimental tension specimen. setup of the compact 69 isoparametric, singular-enriched elements. An arbitrary load of 100 N was selected in obtaining the stress factors and the c r i t i c a l fracture toughness intensity were easily obtained by eqn. (3.1). The f i n i t e elements mesh for the CTS and the DCB specimens are shown respectively in Figure 18 and Figure 19. Table 2 summarizes results for both cases for d i f f e r e n t grain orientations. A trend of decreasing mode I fracture toughness from TL system to RL system i s apparent. A plot of Kj^ versus annual rings angle i s shown in Figure 20 and a line was f i t t e d by least-squares to the data. Thus, K I C = 0.3933 - 2.1323 x 10"3 e where K j ^ i s the mode (4.1) I fracture toughness, MPai/m ; 8 is the angle between the crack and the growth ring angle as defined in Figure 20. Figure 21 and Figure distribution functions of 22 shows the the K J Q for the CTS cumulative and DCB specimens. They were b e s t - f i t t e d by the Weibull d i s t r i b u t i o n curves which are often used to model the strength distribution of wood products. From Table 2, i t i s obvious that the K j ^ values for the CTS i s lower than the DCB specimens. This implies there i s some relationship between the KJQ and the specimen size, and this was investigated in the later phase of the experiments. 70 steel elements F i g u r e 18 singular elements F i n i t e E l e m e n t Mesh f o r t h e CTS P \V singular elements P F i g u r e 19 F i n i t e Element Mesh f o r the DCB specimens Table Effect Specimen Type of the A n g l e to RL(deg) Annual sample no. 2 Rings Angle on K M.C. (7„) S.G. P P m x / p ( p /m) M a ^IC (psi/Tn) 0 - 10 6 9 .54 0 . 349 1. 02 0.3872 352.31 11 - 20 4 9 .40 0. 354 1.04 0.3650 332.17 21 - 30 1 10 . 0 0 0 . 336 1.06 0.3666 333.60 31 - 40 5 8 .44 0. 374 1.05 0.3153 286.96 41 - 50 12 8 .49 0 . 386 1.03 0.2972 270.44 51 - 60 8 8 .96 0. 380 1.03 0.2822 256.80 61 - 70 1 9 .00 0. 400 1.05 0.3061 278.55 71 - 80 3 9 .00 0. 393 1.03 0.2245 204.30 81 - 90 33 9 .33 0. 388 1.05 0.2051 228.41 17 9 .43 0 . 370 1.05 0.3879 352.99 90 0.50- S 6 a, 20 30 ANNUAL Figure 20 40 50 RINGS A N G L E 60 70 ( DEG. ) t Mode I F r a c t u r e Toughness V a r i a t i o n w i t h Annual Rings Angle MODE I CUMULATIVE DISTRIBUTION Compact T e n s i o n S p e c i m e n K DATA (SOLID L I N E ) : N=33 DATA ST.DV. WE I BULL 0.138 5 2 I L E : 0. 133 0.010 0.196 5CUILE: 0.192 0.010 0.205 MEAN: 0 . 2 0 5 0.051 9 5 * I L E : 0.315 0.038 0.301 OF DISP: 0.250 0.249 ic-° F = 5 o.o 0.15 F i g u r e 21 INTERVAL CHI-SQUARE F I T : 1.39 3-PAR WE I BULL (DASHED): SHAPE = 1 .6554 SCALE = 0.0920 LOCA. = 0.1227 1 1 1 0.2 0.25 0.3 K I C > ST.DV. 0.007 0.011 0 .051 0.027 1 0.35 1 0.4 1 1 0.^5 0.5 MPa/m Cumulative D i s t r i b u t i o n Curve o f the Compact T e n s i o n Specimen MODE I CUMULATIVE DISTRIBUTION D o u b l e C a n t i l e v e r Beam DATA (SOLID L I N E ) : N=17 DATA ST.DV. WE 1 BULL ST .DV 0 .037 5 X I L E : 0.291 0.020 0.285 0.016 5 0 * I L E : 0.391 0.010 0.392 0 .056 MEAN: 0.387 0.051 0.386 0 .020 9 5 J I L E : 0.502 0.035 0.468 OF DISP: 0.132 0.144 8.2095 F = 1 0.4099 INTERVAL CHI-SQUARE F I T : 1.52 2-PAR WE I BULL (DASHED): SHAPE = 8.2095 SCALE = 0.4099 oo 0.2 i— 0.3 —1— 0.4 K F i g u r e 22 IC' 0.5 0.6 0.7 0.8 0.9 I .0 M P a /in Cumulative D i s t r i b u t i o n Curve of t h e D o u b l e C a n t i l e v e r Beam. 75 The c r i t i c a l mode I stress intensity factor, K , , I( the CTS of white spruce in the TL and propagation were 0.205 MPa/m RL system of and 0.387 MPa/m of wood has not been tested for mechanics fracture present values of K white pine i s about 0.190 parameters. However, for the TL mode of JC western MPa/m and for western red cedar i s MPa/m obtained by Johnson (1973). The mode I fracture toughness for the DCB the TL system previously are comparable with their counterparts IC obtained for other species; K about 0.185 crack respectively, (see Table 2) This species this for of crack propagation spscimens is approximately in 0.387 MPa/m . Average moisture content recorded specimens i s 9.1% and average s p e c i f i c gravity based on the ovendry weight-ovendry volume. the dependence of K 23 for the CTS JC and for all NO.2 is 0.38, An example of on s p e c i f i c gravity i s shown in Figure it is apparent that there corelation between the s p e c i f i c gravity and the K 4.3 EXPERIMENT tested CRACK-FRONT WIDTH EFFECT IC ON is no . FRACTURE TOUGHNESS 4.3.1 EXPERIMENT DESIGN AND PROCEDURE Experiment variation of the No.2 was designed to study the effect of the crack-front width on toughness. A total of thirty-eight the mode I fracture CTSs were cut into different widths, nominally B = 7,15,21,29,38 mm and five mostly 0.30 COMPACT B = TENSION 38mm 0.25 H A A A (TJ a, A 0.20 H A A 0.15 o.io-i 0.32 A^ A r 0.34 0.36 0.38 SPECIFIC Figure 23 0.40 0.42 0.44 0.46 GRAVITY Effect of Specific Gravity on Mode I Fracture Toughness. cn 77 in the TL s y s t e m . The s p e c i m e n p r o c e d u r e was same expected that the as used stress geometry in Experiment intensity p r o p o r t i o n a l t o t h e w i d t h of and factor the loading No.1. It is is inversely t h e s p e c i m e n , so i f we double the w i d t h of the specimen, t h e s t r e s s i n t e n s i t y f a c t o r be h a l f of b e f o r e . A l l the temperature and h u m i d i t y Testing s p e c i m e n s were c o n t r o l l e d room tested with will i n the the Instron Machine. 4.3.2 RESULTS The c r i t i c a l width are given critical values, intensity f a c t o r s , K -. s i z e as w e l l parameters f o r a v a r i a t i o n of the Weibull m values of coefficient of d e v i a t i o n t o average a s t h e shape 2-parameters k and , I( (COV) i . e . , r a t i o o f s t a n d a r d sample f o r each i n T a b l e 3. Shown a r e t h e a v e r a g e v a l u e s stress variation stress i n t e n s i t y f a c t o r s obtained ( k ) and s c a l e model. c a n be The explained (m) large by the l i m i t e d t e s t s f o r each c a s e . The influence of t o u g h n e s s c a n be d e r i v e d The c r a c k - f r o n t w i d t h , crack-front width by t h e W e i b u l l B ,mm, on weakest the fracture link model. i s also incorporated i n the d e r i v a t i o n and t h e r e l a t i o n s h i p o b t a i n e d i s : F ( K ) = 1 - e p[-(K /m) B] (4.2) k K X where k respectively. IC and m are the shape and scale paramters Table Effect of Width,B of white Sample crack-front width spruce for Specific on for longitudinal P . gravity ? PQ 3 size specimens propagation Average K a mm compact-tension MPa/m I C C . V . Shape „ parameter % k Scale parameter m 7 5 0.335 1.08 0.2570 2 .15 47.98 0.2598 15 7 0.378 1.05 0.2282 9 .33 10.95 0.2384 21 6 0.395 1.03 0.2435 6, . 1 8 22.31 0.2501 29 7 0.375 1.04 0.1805 1 0 .. 8 7 12.93 0.1887 38 10 0.389 1.05 0.1768 2 7 ., 1 5 4.23 0.1940 79 Considering two different width, B and B*, the effect of variation of B on median values of Kj^. can be derived by evaluating equation (4.2) at F = 0.5. Then, we get : IC If we plotted Kj^ values against specimen width B on a log-log plot, the slope of the regression l i n e w i l l be -1/k. The least-squares method was used to f i t the regression l i n e to the data. The relationship between K -. and crack-front I( width obtained i s : log K I C = -0.3795 - 1/4.423 l o g where B i s the crack B R front 2 = 0.735 width,mm; (4.4) KJQ i s the c r i t i c a l stress intensity factor, MPa/m . A plot of this relationship results obtained by Barrett (1976) i s shown in Figure 24 and for Douglas f i r i s also presented in the graph. The parameters k and m obtained Barrett and (1976) are 7.41 discrepany between the two curves 0.41 respectively. i s due to the by The difference in the e l a s t i c properties of two species of wood. From equation (4.2) we get the cumulative for K IC : distribution 1 l o g K . = -0.379 • '° i n s 1 r , c 4.423 log B 1 0 1 0 R =0.735 o Legend A WHITE SPRUCE CTS O DOUGLAS FIR CTS 0.14 F i g u r e 24 i i i i i i 111 10 B m m i i 111— 100 I I I I I I I I 1000 E f f e c t o f Crack-front Width on Mode I F r a c t u r e Toughness (Based on the W e i b u l l Weakest L i n k Model) CO o 81 F(K ) = 1 - exp[-(K IC IC /0.384) 4 - 4 2 3 B] (4.5) which i s comparable to the results of Douglas-fir. The results of these tests relationship exists between the that a fracture toughness and the crack-front width. I t has been shown the crack-front width confirmed that K increases which can deceases as be explained by J C the weakest-link p r i n c i p l e . Another consistent model of the crack-front effect i s based on the stress state width in the specimen. From equations (2.46b) and (2.47) we have : K C H - C1 K + 2 H / D ( K C2 " C 1 K } 1_ 2TT where OQ i s the y i e l d stress or that stress which results in gross deformation. A graph of Kj^ versus inverse thickness i s shown in Figure 25 and a regression line i s plotted based on the four data points. From the intercept (2.46b) and (2.47), we get : at 1/D = 0 and equations F i g u r e 25 E f f e c t of Crack-front Width on Mode I F r a c t u r e Toughness (Based on the S t r e s s - s t a t e Model) 83 K = 0.1414 K_» = 0.435 C2 H MPa/m MPa/m = 6.95 mm The transition thickness from strain w i l l be D = 2H which 13.90 mm, a plane stress condition > 13.90 mm, to plane i s equal to 13.90 mm. When D < the basic plane stress value, For D plane stress w i l l be maintained, so w i l l be achieved. the plane strain condition will result in a decline in Kj£ . As seen from Figure 24 and Figure 25, both the Weibull theory model and the with the data less than stress-state model points. However, the transition f i t t e d very i f a specimen has a thickness, according well width to the stress-state theory, i t i s postulated that the width has no effect on the fracture toughness. For the Weibull model, i t is proposed that the effect w i l l exist specimen thickness i s . Since usually less than 10 mm, no matter what the the transition thickness i s the deviation between the Weibull model and the stress-state model i s not significant at all and both models behaved well in the experiments conducted. 84 4.4 EXPERIMENT NO.3, CRACK-FRONT LENGTH VARIATION 4.4.1 EXPERIMENT DESIGN AND PROCEDURE Because of the differences between the mode I fracture toughness values obtained from the CTS and DCB test results, Experiment No.3 was conducted. It aimed at investigating the influence of the crack-front length on the mode I fracture toughness. It was expected that the fracture toughness would follow a trend as the crack-front length i . e . , the length in front of the crack increases. Experiments were carried out with the CTS with different crack-front length, nominally = 50,75,100,125 mm, where L crack i s maintained at i s defined in Figure 26. the same length as in L The experiment no.1. Five specimens were prepared for each case and a t o t a l of twenty experiments specimens were were conducted cut in from room four boards. temperature The with a •average moisture content of 9%. The load was plotted against the COD recorded by the modified LVDT on a X-Y p l o t t e r . design f a i l u r e load were determined in the same way as Standard E399 and the time to f a i l u r e were about 1 The ASTM minute for a l l cases. 4.4.2 RESULTS Table 4 summarize designed to determine crack-front length, moisture content and results of within-board the relation L. Average between values of experiments K , and the KJQ , I( specific gravity for each length C.V., are Figure 26 Specimen C o n f i g u r a t i o n o f the C r a c k - f r o n t Length Specimen. oo Table Effect of crack-front tension Board No. specimens Length L , mm length of white Sample size 4 o n K^^ f o r compact spruce Average M.,C. Average S.G. % Average K I C , MPa/m C.V. / o A 75 100 125 2 1 2 8..43 8..49 8,.68 0 .413 0 .420 0 .414 0..2304 0..2275 0..2222 2., 24 0..00 9..36 B 50 75 100 125 2 2 1 1 8..81 9..36 8..96 8..96 0 0 0 0 .412 .419 .410 .408 0..2817 0..2918 0..2762 0,.2569 4.,63 5., 28 0.,00 0..00 C 50 75 100 125 2 1 2 1 9..69 9..68 9..75 9..54 0 0 0 0 .335 .334 .322 .342 0..2076 0..1894 0..2193 0.. 1 8 0 5 3.,86 0.,00 7 .4 1 0.,00 D 50 100 125 1 1 1 8..99 8,.98 8,.93 0 .416 0 .413 0 .405 0..2640 0..2518 0,.3145 0.,00 0.,00 0.,00 87 also shown and the results are plotted in results do not seem too helpful as i t i s Figure 27. The not obvious any trend exists for a l l cases except for board A, for which the values of K , decrease as the length increases. The lack I( of consistent trends can be attributed to the limited number of specimens tested. toughness However, comparing between surprisingly, the the DCB CTSs the average and specimens the specimens, fracture toughness the CTSs. This may have different causes. F i r s t l y , the CTSs were prepared from many different values twice as great as gave DCB fracture boards which implies a higher degree of v a r i a b i l i t y , whereas the DCB specimens were prepared from only three boards. This can be observed by examining the c o e f f i c i e n t of variations for both cases, as shown in F i g . 21 and 22. The difference of the fracture toughness between and DCB specimens may also be explained by the condition in the specimens. As mentioned stress-state before, fracture i n i t i a t i o n constant, K J Q , i s thickness The amount of material also be small. To which yields at ensure this, CTSs the dependent. the crack t i p must specimens must be s u f f i c i e n t thickness so that a t r i a x i a l state of stress exist at the flaw t i p . The lower l i m i t occurs as the width approaches i n f i n i t y and i t i s c a l l e d the fracture considered to be can It i s postulated that Kj^ decreases as the crack-front width increases. strain of toughness. a geometric The Kj C at this invariant material point plane is property. The size requirements recommended by L i u (1983) for this SSY Legend 50 1 1 1 1 60 70 80 90 L F i g u r e 27 1 100 1 110 1 120 A BOARD A O BOARD B © BOARDC O BOARD D 1 130 (mm) E f f e c t of C r a c k - f r o n t Length on Mode I F r a c t u r e Toughness 89 and plane strain condition are : (4.6a) or (4.6b) where t i s the specimen thickness, m; a , l are defined in Fig. 26. This means that i f the left-hand term i s less that one, then a plane stress condition w i l l e x i s t ; otherwise,' the plane strain condition w i l l result in a decline in K Table 5 shows the data for the cases specimens, the left-hand J C . of CTSs and DCB term for the CTSs i s larger than the value for the DCB specimens. This indicates that the DCB specimens have a condition closer to the plane stress condition which implies a higher value of K j ^ , which agrees with the results obtained. Further analysis of this difference i s included in Appendix I I . 4.5 EXPERIMENT NO.4 MOISTURE CONTENT EFFECT 4.5.1 EXPERIMENT DESIGN AND PROCEDURE This experiment fracture toughness uses Kj^, to examine in white spruce with the variation of the p r a c t i c a l l y Table 5 The s i z e c o e f f i c i e n t Specimen Sample a L for t CTSs a n d DCB s p e c i m e n s ^ 5 (—!£.) Q - ^ t ^ t/^ Oy m MPa a C Y) K-j.^, type size mm mm mm L CTS 33 50 50 38 0.038 2.80 0.01288 2.9503 DCB 17 250 700 38 0.1064 2.80 0.04800 2.2164 91 important parameter — moisture content. Thirty-two compact tension specimens were prepared; most of them were in the TL system of crack propagation. The specimens were randomly divided into two groups and each group was conditioned in two different environments. were placed in a dry a i r oven at 105°C Sixteen to reduce the moisture content to zero; the remainder were conditioned in a humidity room to achieve a moisture content of 6%. When combined with the results from the 9% M.C. CTS of Experiment No.1, the three sets of specimens allows evaluation of K j ^ at different moisture content. Specimen size experimental procedures were exactly the same as No.1. The tests were conducted and Experiment in the same humidity room. 4.5.2 RESULTS Table 6 summarized the results from each set of experiments and a plot i s given in Figure 28. As can be seen in the figure, the ovendried specimens f a i l e d at lower loads than the wet. The results indicate moisture content on K - shows I( that a decrease the effect of as the moisture content decreases from 9% to 0%. The average values for the kiln-dried condition,with moisture content approximately 9%, is 0.2051 MPa/m , and an average value of 0.1752 MPa/m for the oven-dried specimens. This shows a decrease of 15% which is consistent with Technical Research the results Centre of the Newsletter of Finland According to the results obtained (1986) of for spruce. by P.D.Ewing (1979), the Table Effect Moisture content of moisture Sample s ± z & content on K T r 6 f o r CTS i n l o n g i t u d i n a l A v e r a g e K-j-^ MPa/m C.V. propagation Specific % gravity /o 0.00 16 0.1752 21.46 0.383 6.34 16 0.1957 17.32 0.347 9.10 33 0.2051 24.67 0.380 0.22 COMPACT TENSION i C SPECIMEN 0.20 a, 0.18 A K 0.16 2 I C = 0.03051 + 0.01362 ( 1 - e H 2 4 -r 6 6 MOISTURE CONTENT F i g u r e 28 E f f e c t of Moisture M / 6 ) 8 10 ( % ) Content on Mode I F r a c t u r e Toughness CO 94 effect of moisture content on K J C generally increase as the moisture content decreases from maximum at the K j ^ values decrease. around This 10% and agrees then with the results shows an 20% to a tend obtained to from Experiment No. 4. An exponential using the curve was least-squares fitted technique to the data and the points following relationship i s obtained for M^10 % : K R 2 2 = 0.0305144 + 0.013619 (1 - e~ ) (4.7) M/6 C = 0.9688 where K J Q i s the mode I fracture toughness, MPay/m ; M i s the moisture content, %. This equation shows experimental data points, this equation toughness w i l l shows a good agreement with the as shown in Figure 28. However, limitation increase as long in that the fracture as the moisture content increases. According to Ewing's theory, this equation should only be v a l i d in the range between 0% and 10%. 4.6 EXPERIMENT NO.5, MODE II FRACTURE TOUGHNESS 4.6.1 EXPERIMENT DESIGN AND PROCEDURE Although mode I is usually dominant in crack propagation, for certain loading situations mode II can also 95 be of significance. In order to generate the interaction curve between mode I and mode II at f a i l u r e ; or to study the wood structure under the mode II f a i l u r e condition, mode II fracture toughness values are required. The end-split beam specimen suggested by Barrett, and Foschi (1977) was adopted here to study the mode II fracture toughness. The test method i s shown in Figure 29 and the mesh for the f i n i t e element program i s shown in Figure 30. During the experiments, the r e l a t i v e displacement of the crack surfaces at points A and B shown in Figure 31 was recorded by a modified LVDT and plotted against the applied loads. F r i c t i o n induced by the crack closure was avoided by placing a smooth as p l a s t i c plate between the sawn notch shown in Figure 29. Twenty-four specimens were prepared the bandsawn notches were extended by a steel knife starter just before the testing. The specimens were and crack tested in the Satec Testing Machine under room temperature and the crack orientation was recorded for each specimen. The experimental setup i s shown in Figure 32. 4.6.2 RESULTS A t y p i c a l load versus crack displacement curve i s also shown in Figure 31 which indicates prior to the crack starting were determined using some slow crack growth to propagate. The f a i l u r e the 5% offset l i n e load as suggested ASTM Standard E399. The experiment results also indicated dependence of mode II fracture toughness on in a the crack 96 Applied Load i 300 mm -500 184 92 mm 1 38 mm smooth p l a s t i c p l a t e 900 mm 50 mm F i g u r e 29 F i g u r e 30 E x p e r i m e n t Setup o f t h e Mode I I F r a c t u r e Toughness Specimen. F i n i t e Element Mesh f o r t h e Mode I I F r a c t u r e Toughness Specimen. 97 smooth plastic plate Figure 31 A p p l i e d Load versus L o n g i t u d i n a l Displacement. Figure 32 The apparatus and the experimental setup of the mode II fracture toughness test specimen. vD co 99 orientation as shown in Table 7. The K j angle, 6 I C values were plotted against the annual in F i g 33. The b e s t - f i t t e d rings polynomial curve calculated by the least-squares method i s also shown in the figure, and the following relationship was derived : K = 1.8798 + 2.5114 x 1 0 9 - 2.4828 x K T ^ e _2 (4.8) 2 where KJJ^ i s the mode I I fracture toughness, MPa/m ; 0 i s the annual rings angle, The curve shows a maximum degree. at a rings angle of 50 degrees which indicates a mode of cross-grain plane f a i l u r e . This i s reasonable as the cross-grain f a i l u r e always require a higher failure load. The average toughness values obtained for white mode I I fracture spruce were 2.16 MPa/m and 2.05 MPa/m for the TL and RL systems respectively. These values are comparable with the results obtained by (1981) for white spruce -- K n c Barrett for the TL mode i s 1.89 MPa/m which i s in the same order of magnitude. Figure 34 shows for K I I C based only the cumulative d i s t r i b u t i o n on a l l the specimens system. The c o e f f i c i e n t funmction tested in the TL of varience i s 16% with the percentile being 1.925. MPa/m . fifth Table Effect Specimen type " End-split Beam o f the annual Angle to , degree Sample g i z e 7 rings angle M.C. on K S.G. I I C K n c MPa/m % K IIC psi/Tn 0 -- 10 2 13. 38 0. 3 8 4 2.0180 1 8 3 6 . 33 11 -• 20 2 13. 75 0. 369 2.1000 1 9 1 1 . 00 21 -- 30 0 31 -- 40 3 12. 83 0. 4 1 0 2.2597 2 0 5 6 . 30 4 1 -- 50 4 13. 13 0. 4 1 8 2.6935 2 4 5 1 . 09 51 -- 60 2 13. 78 0. 3 7 1 2.3942 2 1 7 8 . 72 61 -- 70 1 14. 50 0. 4 2 3 3.0098 2 7 3 8 . 90 71 -- 80 1 12. 75 0. 3 2 4 1.9359 1 7 6 1 . 70 81 -- 90 8 12. 9 1 0. 345 2.1561 1962. 01 - - - 4- system 90 deg re cs Is equ Iv elent to T s y s t e m 0 degre e is e q u i v o lent to RL • 3- • a. • • • • • • *~] • <> 1- 10 20 30 ANNUAL Figure 33 40 50 RINGS A N G L E 60 70 80 90 ( DEG. ) Mode I I F r a c t u r e Toughness V a r i a t i o n w i t h Annual Rings Angle MODE II CUMULATIVE DISTRIBUTION E n d - s p l i t Beam K F = 1 -( IIC 15.229 5JILE 50ZILE MEAN 952ILE OF DI5P 2.2329 DATA (SOLID L I N E ) : N=8 DflTR ST.UV. WEIBULL 5T.DV I .925 0 . 0 1 4 1 .837 0 . 190 2.163 0.151 2.179 0.073 2.156 0 . 1 7 8 2.157 0.173 2.428 0.047 2.399 0.081 0.082 0.080 INTERVAL CHI-SQUARE F I T : 0.00 2-PAR WE I BULL (DASHED) : SHAPE = 15.2286 SCALE = 2.2329 .0 0.5 1 .0 1 .5 ~l 2.5 1 2.0 K Figure 34 IIC I— 3.0 3.5 MPa/m Cumulative D i s t r i b u t i o n Curve o f the E n d - s p l i t 4.0 -1.5 Beam Specimens. 1 03 4.7 EXPERIMENT NO.6, MIXED MODE MID-CRACKED BEAMS 4.7.1 EXPERIMENT DESIGN AND PROCEDURE Since wood structural complex loading members are often subjected conditions fracture, a f a i l u r e c r i t e r i o n of combinations or that result in mixed mode should be expressed in interations of the mode I to terms and mode II f a i l u r e modes. Experiment investigate the fracture strength of mid-cracked beams under the mixed mode loading No.6 condition. was designed Difficulties designing the specimen geometry to were encountered in in order to produce the desired ratio of Kj to K j j . This had and error, and f i n a l l y the to be done by 45 degrees and trial the 90 degrees p r i n c i p a l phases. In the mid-cracked beams were adopted for testing. This investigation had two f i r s t phase, the 45 degrees mid-cracked beams were loaded at the centerpoint as well as at a distance of 200 mm from support as shown cut in Figure 35. Twenty-four specimens and prepared for each case with mostly in the TL the were system of crack propagation. The second phase consists of the 90 degrees mid-cracked beams loaded at the centerpoint Sixteen specimens were prepared the tests were as 36. for the TL propagation and performed in the Satec room temperature. shown in Figure Testing Machine at 104 P = 100 N K. = 7.55xl0~ MPa/ni K__ = -0.0187 MPa/m Type A 100 N Kj= 6.97x " P = K„ II A Figure 35 = MPa^ 0.0102 MPa/m Type B Specimen Configuration o f the 45 deg. Beam (a) Loading at the centerpoint. (b) Loading at 200mm from the l e f t support. P = 100 N -3 MPa/m K = 6.309x10 x K II 0.0134 MPa>/m cr A Type C Figure 36 Specimen C o n f i g u r a t i o n o f the 90 deg. Beam 105 Prior to testing, the band-sawn sharpened with the crack starter notches were also in the direction of the sawn crack. In fact, the cracks propagated along the grain at maximum load. The Kj and K J J values corresponding to a load of 1 0 0 N were calculated with the program NOTCH using the elements meshes shown in Figure 37. The predicted of propagation was values for both determined by comparing directions from direction the Kj and K J J the computer higher value of Kj and K J J indicates finite output. a lower f a i l u r e A load during the testing. The predicted f a i l u r e direction i s also shown in Figure 3 5 and Figure 3 6 . Because of the d i f f i c u l t i e s of measuring the COD values in the experiments or the opening at the lowest midpoint the i n i t i a l crack, the mid-span deflections were of measured instead. 4.7.2 RESULTS Table 8 experiment summarizes the results with the average K As can be seen I C ,K from Figures 3 5 , 3 6 I I C beams. Since cracks phase of values shown as well. and Table 8 , there are two potential directions of propagation mid-cracked for each for the 9 0 started at degrees the weaker direction of the two, i t implies a double chance of f a i l u r e . The experimental curves following formulae : were corrected based on the 106 Figure 3 7 F i n i t e Element Mesh f o r t h e M i d - c r a c k e d Beam Specimen. (a) 45 degrees M i d - c r a c k e d Beam. (b) 90 degrees M i d - c r a c k e d Beam. Table 8 Results o f the mid-cracked beam specimens Specimen Sample Average size K „ I Average T type MPa/m C K I C.V. T T O I C„ MPa/m '* , C . V . Predicted prop. /. _. K J J K K V A/ C . V . I I C R " /O 7 * ° Mean Erection K L 'Mean I C Type A 20 0.3013 0.7470 19.0 0 0.7765 9.0 0.3433 12.5 100 Type B 18 0.3539 0.5183 18.7 180 0.9073 6.4 0.2341 14.9 100 Type C 16 0.3084 0.6571 17.4 0.8496 3.8 0.3160 10.7 100 •k _R 0 , 180 i s the percentage o f the specimens f a i l e d i n the p r e d i c t e d direction. 108 F F - 1 - [1 - F ] 2 £ £ (4.9) - 1~ / l-F c (4.10) E where F E i s the experimental cumulative p r o b a b i l i t y . F c i s the corrected cumulative p r o b a b i l i t y . These cumulative Figure 3 8 and Figure distribution curves 39 with together are shown in the b e s t - f i t t e d Weibull curves and equations. It can be seen that the curves are rotated to the right i . e . , higher values of K I C and K I I C correspond to the same percentile. For the mode I stress degrees beams, the K the experimental J C intensity factors of the 9 0 has a mean value of 0 . 3 0 8 4 MPa/m for cumulative d i s t r i b u t i o n and a value of 0 . 3 5 5 6 MPa/m for the corrected cumulative d i s t r i b u t i o n . In order to determine the normalized interaction between mode I and mode II fracture K J J values critical for mixed mode I mode f a i l u r e and mode curve toughness, the are normalized II stress intensity and by the factors respectively. If we just divide the Kj and K J J values by the average K I C and K J I C respectively, we are comparing the values at d i f f e r e n t percentiles. One method to tackle this problem i s to normalize the Kj and K J J values at f a i l u r e by the K I C and K I I C values which correspond to the same percentile. This method i s shown diagramatically in Figure 4 0 . The s o l i d curve and the dashed curve correspond to the pure mode Kj ( MPaVm) F i g u r e 38 Experimental and Corrected Cumulative P r o b a b i l i t y Curves of the Mode I F r a c t u r e Toughness f o r the 90 degrees beam. o K J J ( MPaVm) F i g u r e 39 Experimental and Corrected Cumulative P r o b a b i l i t y Curves of the Mode I I F r a c t u r e Toughness f o r the 90 degrees beam. o Figure 40 Method of Normalizing the Cumulative P r o b a b i l i t y f o r the 45 degrees mid-cracked beam. Curves 112 I and mode II fracture respectively. Each data K I IC X of cumulative curves the value probability. to Figure 40, we have : l X Weibull point i s divided by which refers to the same level Refering toughness l c K II X 2 K IIC X 2 C where Kj i s the mode I stress intensity factor at f a i l u r e . K J J i s the mode 11 stress intensity factor at failure. Kj^ i s the pure mode I fracture toughness. • K IIC * ^ s fc e P X ,X^,X2,X2Q 1 u r e m °de 1 1 fracture toughness. are as shown in the figure. Both the 45 degree and the 90 degree mid-cracked data points are treated by this method. For the 90 beams data, the corrected cumulative d i s t r i b u t i o n beam degrees functions were used to calculate the required r a t i o s . The average values for the Kj / K I C and K J J / K J I C t h i s experiment are l i s t e d in Table 8 and are plotted Figure 41. for in 1.2 Legend IA A COMPACT TENSION O DOUBLE a 45 DEG B E A M O 9 0 DEG B E A M CANTILEVER 0.8- 0.6 0.4- 0.2- -Q0.2 0.4 0.6 K „ / K Figure 41 0.8 1 1.2 ric K^-K^ I n t e r a c t i o n Diagram based on the M i d - c r a c k e d Beams R e s u l t . co 1 14 4.8 EXPERIMENT NO.7, DCB UNDER MIXED-MODE LOADING 4.8.1 EXPERIMENT DESIGN AND PROCEDURE Experiment no.7 served the purpose of generating more points on the interaction curve and to check the consistency of the interaction curve for describing the mixed mode fracture f a i l u r e for different geometries. The main d i f f i c u l t y in generating a interaction of Kj / K J J (or ( Kj / occurs because of the limited range K IC K * I I / IIC ^ K n fc ^ e s P e c that for most beam geometries, factor often dominates i m e curve n design. It was the mode I stress the fracture failure. found intensity This is equivalent to the f a i l u r e points on the upper region of the interaction curve or a high value of 9 as shown in Figure 41 . The p r a c t i c a l d i f f i c u l t y was separate loads on the controlled to produce fields around the DCB overcome by applying specimens different mode I crack which could two be and mode II stress t i p . This i s the same as superimposing the mode I and mode II DCB loading as shown in Figure 16 and 29. Although the two stress modes are not absolutely independent with respect to the two loads, i t i s possible to control the ratio of Kj to K J J at fracture. The test method i s shown in Figure 42. The midspan load was produced by the Satec Testing Machine while the end-support load was achieved by a hydraulic jack with a c a l i b r a t e d load c e l l . The experimental setup i s shown in Figure 43. Figure 42 Specimen C o n f i g u r a t i o n o f the Two-loads Beam Specimen. 160 ran ' 3/4 I n . p l a c e 1 'xz.zr.n- r—i o r i *Bearing - Load ZJ. -Metal ball cell plate Applied Load - Hydraulic Jack • 3/8 l n . p l a c e CL To Hand pump 3/4 I n . , • 3/« l n . plate Front View Figure 43 Experiment Setup of the Two-loads Beam Specimen. 1 17 The relationship between the stress intensity factors and the applied loads are : Kj K - 4.0488 x lo" * - 9.894 x io"5 P,, 1 II - ' = 1 5 0 6 0 x 1 0 ? 1 ~* (4.12a) (4.12b) p 2 Using the results from before for the TL system : K • ' LV = 0.388 MPa/m K = 2.13 MPa/m 110 For pure mode I fracture : (P2=0) = ^388 = 9 5 8 3 N 4.0488 x 10 + _1 For pure mode II fracture P = : (P^O) = 14143.5 N 1.5060 x I O - 4 Introducing the condition Kj >0 i . e . , that the crack not close, from Equation (4.12a) we have : 4.0488 x 1 0 " P, - 9.894 x 1 0 ~ P 4 5 = 0 does or P < 4.1 2 ? i Assuming the interaction curve to be : Case I : K L e t r I / K K l K ic IC —71? = K II / K IIC l K l nc K = t a n 9 we have, P P = (SrZ+T - r)[1234.93r K 2 + 811.29 K_ ] = 3320.062 (/?2+4 - r)[K ,] II( Case II : ( irK ic ) 2 • + K 1 nc we have, P P 1 = 1//1+P" [2469.9 K 2 = 6640.12 /l+ 2 r IC • r + 1622.6 K ] 1 19 Since case I is more conservative than case II, in designing the fracture c r i t e r i o n , case I i s adopted. The interaction f a i l u r e envelope for the P ~?2 m 1 i- ed x mode loading is shown in Figure 44. Outside the curve i s the f a i l u r e region, where as inside, the specimen i s intact. To obtain the desired ratio of fracture toughness, following procedure was followed : 1. A ratio r was selected. 2. From equations (4.14a) and (4.14b), f a i l u r e loads P 3. 2 the and were obtained. The corresponding Kj K J J values and from equations (4.12a) and (4.12b) were calculated. 4. From Figure 44, a loading path was designed within safe region to achieve the P Experiment 1 and P no.7 was carried out and K J J value at f a i l u r e 2 the calculated. in determining the for two different ratios as Kj shown in Figure 44 together with the loading path. Ten specimens were under room prepared and tested temperature with the dominant for each system of case crack propagation being TL. The applied mid-span load and the were recorded by a X-Y 4.8.2 COD plotter. RESULTS The experiment results Figure 44, with the f a i l u r e the ratio of Kj to corresponding d values K Tl are summarized in Table 9 load shown as well. This values of 74° of 0.64 and 52°. and and gives 0.229, with These values were 15000 n K II ~ I I C K 10000 H 5000 STAGE 1 Legend • PREDICTED FAILURE ENVELOPE A EXPERIMENTAL RESULTS 1 0-<f 1500 2000 2500 3000 3500 P, ( N ) Figure 4 4 Failure Envelope of the Two-loads Beam Specimen O Table 9 R e s u l t s of S p e c i m e n Sample type size P, / P / 9 t h e t w o - l o a d s beam t e s t s Average C.V. Average C . V . . : % % K ] MPa/m MPa/m K * C.V. % x i L K " C.V. J T Average % i i L 5-steps 10 1925.0 3985.4 0.3851 9.6 0.6002 9.6 0.998 5.3 0.282 9.3 3.525 9-steps 9 2800.0 8488.3 0.2938 1 6 . 7 1.2783 16.7 0.754 5.4 0.600 5.9 1.260 1 22 used to generate the interaction curve l a t e r . Theoretically, i t i s possible to attain any intensity factors ratio even for d close to zero, stress however, in practice i t may require the specimen loaded close to the mode II fracture toughness, f a i l e d in mode II which may instead of cause the the mixed mode specimen condition wanted. The onset of the crack propagation were determined by a change of the slope of f i n a l phase as shown the load verses COD curve in the in Figure 45. Some slow crack was observed during the tests. A 5% growth offset method i s used here to determine the f a i l u r e load. 4.9 EXPERIMENT NO.8, NOTCHED BEAM SPECIMENS 4.9.1 EXPERIMENT. DESIGN AND PROCEDURE The f i n a l phase of the experiment concerned the notched beam specimens under the mixed mode loading condition. mentioned before, the stress intensity 2-X has a unit of (Load)x(Length) As factor for notches , which depends on the eigenvalue X. Therefore, the interaction curve for notched specimens include three variables, i . e . , Kj / K »KT.T/ HC K JC and X. Experiment no.9 was designed to produce data points for notches with X = 1.6 and 1.7. Three types of specimens used in these experiments as shown in Figure 46. were 123 \ STAGE 2 STAGE 1 COD Figure 45 Typical Load-Displacement curve of Two-loads Beam Specimen. 124 Applied Load (a) 1 92mm 50mm TYPE A 7S~ 130' \ •450mm Figure 46 Specimen Configurations for Notched Beam (a) Type A specimen; X=1.60 (b) Type B specimen; X=1.60 (c) Type C specimen; X=1.70 125 Eight specimens tests were c a r r i e d were prepared out with for each the Satec room temperature. The i n i t i a l notches the crack starter case and the Testing Machine at were not extended and the loads were by plotted against the midspan deflection for a l l cases. The pure mode for these Xs I and mode were II fracture toughness not done obtaining feasible specimen For example, to obtain the K to test a 77° notch due to the d i f f i c u l t y dimensions and loading and K J C tests I I C of method. for X=1.6, we need in the compact tension specimen, and a 13° notch in shear. 4.9.2 RESULTS The results specimens neutral were summarized are symmetrical axis, initiation at there both experiment data were (4.9) and (4.10). in Table 10. Since the with is a respect equal notches. to the midspan probability Similar preformed according The average corrected of treatment of crack the to the equations Kj and K J J values are given in Table 10 as well. The f a i l u r e of specimens type A and type C was very b r i t t l e and the rate of crack propagation was extremely fast while the f a i l u r e of type B specimens were more ductile due to the slow crack growth. The average values of the stress intensity factors at failure for notches were showed to be less than the values for cracks. It i s expected that the interaction curves will Table 10 R e s u l t s o f the notched beam specimens Specimen Sample M.C. type size % S.G. Average Average MPam MPam 2_X C.V. 2_A Type A 7 8.81 0.361 0.2877 0.4997 17.94 1.60 Type B 8 8.59 0.347 0.2370 1.2044 12.11 1.60 Type C 8 8.00 0.315 0.2122 0.7326 24.54 1.70 127 move towards the origin as the value of X increases. 4.10 SUMMARY This chapter has served to provide the f u l l descriptions the experiment design, procedures and results. of Each experiment contributed to a better understanding of how wood specimens performed under conditions. The different geometries and loading experiments also provide the data points which are necessary in generating the interaction curve for the mode I and mode II stress intensity factor. 5. DISCUSSION 5. 1 INTRODUCTION T h i s chapter first explains the method of generating i n t e r a c t i o n curve f o r a cracked beam, and then presents r e s u l t s obtained. The method i s specimen, to e s t a b l i s h the then extended f a m i l y of o b t a i n the curves, the to a notched interaction curves. 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 the treatment in order to of data i n c l u d i n g assumptions that had been made during the process and the l i m i t a t i o n s of experiments. and notched Finally, an r u l e s f o r the design of cracked the beams beams are suggested. 5.2 STRESS INTENSITY FACTOR INTERACTION CURVE FOR CRACKS Using the r e s u l t s of a l l the cracked specimens t e s t e d specimens, two-loads specimens, mid-crack is possible to cracked beams. establish F i g u r e 47 beam specimens, i t the i n t e r a c t i o n shows a l l curve f o r the the experimental p o i n t s obtained which had been normalized with the mode I and mode II stress intensity f a c t o r s . An (2.65) and data p o i n t s , and i s shown i n F i g u r e in Figure 48 with data critical interation curve has been d e r i v e d by the l e a s t - s q u a r e s technique on equation DCB based 47 with a l l the t h e i r average values. T h i s gives the formula : K I — IC . , I I .2.5587 K + (j7 ) IIC - 1 (5.1) 128 1.5 Legend If- A TWO-LOADS SPECIMENS o 45 DEG BEAMS a 90 DEG BEAMS <£ A P" ° 9> a 8* A.A" V^A 0.5 -i 2.5587 (J<L ) + (IAL ) K ic K = 1 nc -/<\A — A - • A A 0.2 0.4 0.6 K „ / K , F i g u r e 47 0.8 I C I n t e r a c t i o n between K ^ / K ^ and specimens. 1.2 K _/ j_xc K I T f o r c r a c k e d beam F i g u r e 48 I n t e r a c t i o n curve d e r i v e d between f o r cracked beam specimens. KJ/ TC K A N ^ ^n/^xiC — CO o 131 This i s close to the model proposed by Wu (1967),i.e.: ic Sic <-> K 52 This formula i s other suggested replotted in Figure models. The (1974) seems to be too Williams and 49 together linear r e l a t i o n of with Leicester conservative. The model proposed Birch (1976) s l i d i n g mode has no effect that the shear stress causing on the mixed mode f a i l u r e , does not follow the data trend. The present experiments on spruce corroborate the findings of question might be raised is applicable to Wu white (1976) on balsa. whether t h i s interaction other species by of wood, but The relation i t cannot be answered u n t i l further results are obtained. 5.3 STRESS INTENSITY FACTOR INTERACTION CURVE FOR NOTCHES As mentioned before, the stress intensity notches have dimensions of stress*length factors for , where X i s a non-linear function of the material and notch orientation in orthotropic materials. Consequently, X, there may be an infinite for any given value number of compatible of notch geometries and material combinations. In a general parameters are f i e l d s at case of essential the notch a notch, a to uniquely root : indicate the strength of each Kj , minimum specify K J J and of the X. Kj three stress and K J J of the stress f i e l d s while X X = 1.50 c r a c k e d beam s p e c i m e n s . 133 shows the rate at which the stresses change when the distance to the notch root varies. To establish the family of interaction curves, i t i s necessary to obtain interaction curves for X values from 1.5 to 2.0. This i s rather tedious as compared to sharp for which the primary eigenvalues. In order and to secondary obtain data modes cracks, have points equal on the interaction curve, different notch geometries are required. The experiments contributed two and one data data points for X=1.6 point for X=1.7. These are the more general cases since a larger value of X corresponds to a large notch angle. As X approaches the value of 2.0, the notch open to 180 degrees to produce a f l a t surface. Due to assumption the limited number that the shape of notches would be the same as of the data the interaction points, the curves for for the case of cracks had to be made. A plot of the family of the interaction curves obtained is shown in Figure 50. The curves are b e s t - f i t t e d based on the following interaction relationship: A K x + B K^' A/B = 17.84 5 5 8 7 = i ( 5 > 3 ) (5.4) 0.4 X=1.50 A K, + B K,, 2 o X 0.3 H X=1.60 • 5587 =1 B 1.5 2.5573 0.1445 1.6 3.2101 0.1799 1.7 4.2108 0.2360 •A-1.70 ctf cu 0.2 H o.H A Legend TWO-LOADS SPECIMENS .O 45 DEG BEAM B 90 DEG BEAM • LAMBDA=I.6 LAMBDA=1.7 o.o H 0.5 1 2.5 1.5 K ( MPam ) 2_A n F i g u r e 50 I n t e r a c t i o n r e l a t i o n s between K-j. and K-^ f o r notches 135 where , K J J a r e t h e mode I a n d mode I I f r a c t u r e Kj toughness 2 X _ respectively, MPam ; A, B a r e some c o n s t a n t s This give values MPam * , 0.2375 M P a m 0 K I I C 4 of of 0 , 3 2.183 MPa/m , X=1.5, 1.6, equation a s shown K I derived of 0.3880 f o r X=1.5, for 0 , 4 2 SS87 1Z , 0.3115 respectively; , 1.758 MPam for 0 , 3 A more general interaction white spruce using l e a s t - s q u a r e s method on t h e e x p e r i m e n t a l Kj + 0.05605 K MPa/m 1.6, 1.7 1.955 M P a m 1.7 r e s p e c t i v e l y . was C i n F i g u r e 50. the results : = 1-8355 - 1.1525 X + 0.125 \2 (.) 5 5 where , K j J a r e t h e mode I a n d mode I I f r a c t u r e Kj toughness 2-X respectively, MPam ; The i n d i v i d u a l i n t e r a c t i o n c u r v e X=1.6 a n d 1.7 a r e a l s o f o r n o t c h e d beams f o r showned i n F i g u r e 51 a n d F i g u r e 52 respectively. After establishing a b l e t o compute specimen of orientations curves the strength different needs t h e s t r e s s of of c u r v e s , we a r e now any notched geometries, under any k i n d f o r beam d e s i g n these curves the interaction materials loading. seems t o be to obtain cracked and But u s i n g of grain these t o o t r o u b l e s o m e a s one i n t e n s i t y f a c t o r s . However, i n order or a design we c a n alter method f o r t h e F i g u r e 51 I n t e r a c t i o n curve f o r notches w i t h A=1.60 0.8 H 0.4 F i g u r e 52 I n t e r a c t i o n curve f o r notches w i t h X=1.70 138 cracks and notches. 5.4 APPLICATION The application of linear e l a s t i c fracture mechanics methods for beam design allows estimates of the strength of beams. Murphy (1978) had shown that the recent notched code underestimated the effect of the presence of a notch on more the net-section treating the notches provides rules theory. A rational based method and cracks i s presented herein for the strength design has of which of notched beams. These rules were based on t h e o r e t i c a l studies combined with the test data. The design method includes two essential features - the 90 degrees cracked beam and the 90 degrees notched beam. The essential feature of the 90 under investigation i s shown degree cracked beam in Figure 53. The problem geometrical dimensions a, b, d are indicated therein as well as the sign convention of bending moment M and shear force V acting the beam at the cross-section containing the crack on root. Figure 53 also shows the individual contribution of the applied moment and shear to the mode I and mode II stress intensity factors for varying notch-depth ratios. The curves were obtained by using the program "transformed stress NOTCH and followed intensity factors method" proposed the by Murphy (1978). In application, one can obtain the stress intensity factors for specific configuration by evaluating the moment 0.0 I 0.1 I I 0.2 0.3 I 0.4 I 0.5 a/d F i g u r e 53 Dimensionless S t r e s s I n t e n s i t y F a c t o r s f o r pure moment and pure shear loadings as a f u n c t i o n o f notch-to-depth r a t i o f o r 90 degrees cracked beam. ^ VD 1 40 and shear at the notch root and entering the figures. Since the stress intensity factors are also affected by the nominal accuracy of stress f i e l d the surrounding the proposed curves intensity factors depends on how for eigenfield, computing stress similar the nominal f i e l d of the beam being analyzed the stress i s to the stress f i e l d of the beam shown in Figure 53. To v e r i f y the v a l i d i t y of these had been made between the and the the structural comparison results obtained by these program NOTCH. configurations curves, a Five under cases, with different investigated. The notch-to-depth r a t i o i s curves two simple loadings were 0.5 and the five cases are shown in Figure 54. The results are presented in Table 11. As can be seen, the difference between the two methods i s larger in cases and 5, which might be explained between the nominal stress by the high 3 discrepancy f i e l d of the off-center cracked beam and the nominal stress f i e l d of the beam from which the curves were derived. The difference of the beams under uniform loading can be explained in the same manner. Interaction relation shear force can be curves the for between the established stress ultimate moment by using intensity factors the and interaction obtained and substituting the Kj and K J J values by the applied moment and shear. The interaction relation between the moment and shear for 90 degrees crack for various notch-depth ratio i s in Figure 55. The curves show a linear relationship shown between P=100 N Case 1 Case 2 Case 3 b = 38 mm = 38 mm b = 38 mm P = 100 N p = 1 N/mm H 1I I I I i ) a = 92 d = 184 mm mm t 800 mm Case 4 P = 100 mm b = 38 mm d = 184 mm Case 5 b = 38 mm p = 1 N/mm T T i n i n .1 •600 mm —L = 800 F i g u r e 54 1 d = 184 mm Aa = 92 mm mm t Several Sharp Crack Problems. M Table Stress intensity Case factors for various From c u r v e s I MPa/m 11 From NOTCH II MPa/m MP a/iii II l-IPa'/m K K sharp c r a c k K problems Accuracy* of Accuracy o f K. o i a o II o 1 7 ., 2 1 0 x l 0 " 3 0.0154 7 ,005xl0~ 3 0.0149 2,,9 2..7 2 6 .606xl0" 3 0.0107 6 ,372xl0~ 3 0.0101 3..7 5..7 3 0 .0288 0.0614 0,.0317 0.0672 8.,9 8..6 4 9 .606xl0" 0.0137 9,. 2 8 0 x l 0 ~ 0.0133 3..5 3..3 5 0,.0216 0.0461 0,.0241 0.0502 1 0 ., 1 8..3 3 * A c c u r a c y i s b a s e d on t h e r e s u l t s 3 obtained f r o m t h e p r o g r a m NOTCH. F i g u r e 55 Design curves for of notch-to-depth 90 d e g r e e s c r a c k e d beam as a ratio. function 144 moment and s h e a r different with d i f f e r e n t notch-depth nominal total This For af nominal can be e x p r e s s e d maximum b e n d i n g maximum s h e a r beam and s l o p e s for ratio. These r e l a t i o n s h i p terms of the intercepts stress in stress f =3V/2bd formulae fk=6M/bd calculated s 2 in and f o r the depth. gives, 90 d e g r e e s c r a c k e d beam ; + f - J ^ f < 1 (5.6) where a, 0 a r e c o n s t a n t s l i s t e d d i s the t o t a l beam d e p t h , f^ i s t h e maximum b e n d i n g f i s t h e maximum s h e a r s a/d m; stress, MPa; B(/m/MPa) stress, MPa. 0.5 0.0644 0.4 0.0797 0.1815 0.3 0.0787 0.2374 0.2 0.0913 0.3532 0.1 0.1004 0.8544 Equation notches a(dimensionless) below; (5.6) i s a n a l o g o u s from t h e S.A.A. 0.1271 t o the d e s i g n e q u a t i o n s Australian Timber Engineering for Code 145 CA65-1972 (S.A.A. 1972) : 0.3f. + f ——^-< 1 (5.7) S C, F , 3 sj where . s 6M lie2 • min b B dQ C 1.5V Bd . min F • = shear block strength for the species of interest; C3 = constant for s p e c i f i c notch degrees notched beam, the stress intensity show a consistent relationship shear. This means eigenfield of angle. For the 90 with the applied moment the stress a notch factors do not i s very distribution around the sensitive and varies with different loadings and geometries. Consequently, intensity factors for notches have singular f i n i t e element program, and the stress to be computed by and general design using curves for different loadings can not be obtained. One particular end-notched white case spruce of 2x8's a one-metre beam under rectangular centerpoint loading (P) was analyzed with various notch-to-depth The ratio Kj /P were plotted against various notch-to-depth to K J J the notch length for r a t i o , (see Figure ratio i s constant for a given ratio. 56) Since the Kj notch geometry and 147 material, K J J can be obtained from the r e l a t i o n K^j = 1 . 0 9 6 Kj after obtaining the Kj value equation ( 5 . 5 ) and substituting from the figure. K J J by Using 1 . 0 9 6 Kj , the c r i t i c a l mode I and mode II stress intensity factors can be computed. The values obtained are 0 . 3 8 2 3 MPam ° * 4 5 2 1 for the c r i t i c a l 'mode I stress intensity factor and 0 . 3 4 8 7 0 4521 . . MPam ' for the c r i t i c a l mode II stress intensity factor. Using these values and F i g . 5 6 , a applied load against the notch plot of the c r i t i c a l length for various notch depths can be obtained as shown in Figure 5 7 . It should be noted that for notches close to the end support, the stress intensity factors might be affected by the bearing stress, which causes the d i s t o r t i o n of the stress f i e l d around the notch. Therefore the stress intensity factors cannot be represented in the same plot. Although the r e l a t i o n intensity factors complicated the c r i t i c a l and the applied load for notches, configurations between design curves stress seems to be for other and loadings can be established in a way based on the notch length and the notch-to-depth very beam similar ratio. F i g u r e 57 C r i t i c a l Load v e r s u s Notch L e n g t h o f a 2 " x 8 " beam. for various notch depth 6. CONCLUSIONS AND RECOMMENDATIONS 6.1 CONCLUSIONS The linear e l a s t i c fracture mechanics method has been to be appropiate and to apply well to the found characterization of the fracture behavior of wood. The fracture toughness for white spruce has been shown the annual ring orientation in the specimens. It has also been shown that there i s a dependence K IC ' o n t * width of ie to vary with of the mode the crack I fracture toughness, front and the moisture content in the specimens. The mode II c r i t i c a l stress intensity has been shown factor, K J J ^ , to govern the fracture of end-cracked wood beams and i t has also shown a dependence on the annual ring orientation. The mixed the mode I mode interaction and mode curves, which II fracture incorporate toughness, have been presented and applied successfully to the fracture behavior of white fracture spruce. The c r i t e r i o n that toughness i s independent of the mode forward shear I effect has not been observed here while an interaction relationship between Kj and K J J in the mixed more obvious. mode fracture of This relation applied notched specimens, and a family of predicting the onset of established rapid crack for white spruce. 149 white spruce i s equally well to the interaction curves for propagation has been 1 50 Design methods has been provided herein for 90 cracked beam based on the applied moment degrees and shear. Interaction formulae between nominal maximum bending stress and nominal maximum bending stress has been presented which is analogous to the rules outlined by the Australian Design Code. Design also been curves for 90 presented here for a Timber degree notched beam particular case has of a simple-supported mid-span loaded beam. 6.2 RECOMMENDATIONS FOR FUTURE RESEARCH The crack-front width theory has been found to apply well to the experiments conducted and an extension of applying to notches should be studied. The moisture content effect should be investigated with wider range of moisture content variation as well as effect on the mode I p o s s i b i l i t y of applying on notches. The pursuit of a valid size fracture toughness has not been completed successfully and a consistent method to characterize the effect of the specimen size on the mode I fracture toughness should be developed. This work has also given a s i m p l i f i e d picture of mode f a i l u r e for notches which has not been studied mixed before. A proposed design method has also been presented here which can be adopted by the timber design code. Further research should include performing interaction relation experiments for other and other species of wood. to study geometrical this configurations BIBLIOGRAPHY ASTM-E399-B3, "Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials," ASTM Vol. 03.01, Sec.3, 547-582, (1985). Barrett, J.D., "Effect of size on mode I stress factors for Douglas-fir," IUFRO wood meeting, Delft, Holland, (1976). intensity engineering Barrett, J.D., "Mode II stress-intensity factors for cracked wood beams," Engineering Fracture Mechanics, 9, 371-378, (1977). Benzley, S.E., "Representation of singularities with Isoparametric F i n i t e Elements," Int. J. Numerical Methods in Engineering, 8, 537-545, ( 1 974). Bodig, J . , Goodman, J.R., "Prediction of E l a s t i c Parameters for Wood," Wood Science, Vol. 5, No.4, 249-264, (1973). Broek, David, Elementary Engineering Fracture Mechanics, 3rd ed., The Hague: Martinius Nijhoff Publishers, (1982). CAN3-086-M84, Engineering Design in Wood (Working Design), Canadian Standards Association, (1984). Stress Debaise, G.R., Porter, A.W., Pentoney, R.E., Morphology Mechanics of Wood Fracture, Materials standards, 6(10), 493-499, (1966). research and and Dolan, J.D., Madsen, B., "Experimental determination of the shear strength of lumber," Thesis Sec. (Civil) University of British Columbia, (1985). Ewing, P.D., Williams, J.G., "Thickness and moisture content effect in the fracture toughness of Scots Pine," /. Materials Science, 14, 2959-2966, (1979). Foschi, R.O., Barrett, J.D., "Stress intensity factors in anisotropic plates using singular isoparameter elements," Int. J. Numerical Methods in Engi neeri ng, 10, 1281-1287, (1976). Gandhi, L.R., "Analysis of an Inclined Crack Centrally Placed in an Orthotropic Rectangular Plate," Journal of Strain Analysis, 3(3), 157-162, (1972) . G r i f f i t h , A.A., "The phenomena s o l i d s , " Philosophical 221, 163-198, (1921). Gross, B., Mendelson, A., Trans. of R. rupture Soc. and (London) flow in Series A, "Plane E l a s t i o s t a t i c Analysis of 151 152 Plates," Int. Introduction to V-Notched J. Fracture Mechanics, 8, 267-276, (1972). Hellan, K., McGraw-Hill, (1984). Fracture Mechanics, Montreal: Hilton, P.D., Gifford J r . , L.N., "Stress Intensity by Enriched F i n i t e Elements," Engineering Mechanics, 10, 485-496, (1978). Factors Fracture Hunt, D.G., Croager, W.P., "Mode II fracture toughness of wood measured by a mixed mode test method," J. Materials Science Letters, 1, 77-79, (1982). Inglis, C.E., "Stress in a cracked plate due to the presence of cracks and sharp corners," Transact i on of Naval Architects (London), 60, 213, (1913). Irwin, G.R., "fracture dynamics," Fracturing of Metals, 147-166, Am. Society for Metals, Cleveland, (1948). Irwin, G.R., "Analysis of stresses and strains near the end of a crack traversing a plate," /. Appl. Mech., 24(3), ( 1957). Johnson, J.A., "Crack I n i t i a t i o n in wood Science, 6(2), 151-158, (1973). Knott, J.F., Fundamentals of Fracture London: Butterworth, (1973). Leicester, R.H., Australian "The size Conf. 4.1-4.20, (1969). Mech. Dlates," Wood 1st ed., Mechanics, effect of notches," Proc. of Structures 2nd Materials, and Leicester, R.H., "Some Aspects of stress f i e l d s at sharp notches in Orthotropic Materials," CSIRO Aust . Div. Forest Prod. Technol. Paper No. 57, Melbourne, (1971). Leicester, R.H., "Effect of size on the strength of structures," CSIRO Aust. Div. of Building Research Technol. Paper No.71, Melbourne, (1973). Leicester, R.H., "Application of Linear Fracture in the Design of Timber Structures," Conference Australian Fractured Australia, 156-164, (1974). Group 23, Mechanics Proceedings, Melbourne, Leicester, R.H., Walsh, P.F., "Numerical Analysis for Notches or Arbitrary Notch Angle," Proceedings of the International Applied to Conf. Material on Fracture Mechanics Evaluation and Structure Technol. Design, Melbourne, Australia, (1982). Liu, H.W., "Discussion to 'A C r i t i c a l Appraisal of Fracture 153 Mechanics,'" STP 381, ASTM-NASA, 23-26, (1965). Liu, H.W., "Fracture Liu, H.W., "On the fundamental basis of fracture mechanics," fracture mechanics, 17(5), 425-438, (1983). Int. J. Fracture c r i t e r i a of cracked metallic Mech., plate," 2, 393-399, (1966). Engineering Lin, K.Y., Tong, P., "Singular F i n i t e Elements Fracture Analysis of V-Notched Plate," Int. J. Methods in Engi neri ng, 15, 1343-1354, (1980). Lum, C , "Stress Intensity Orthotropic plates using Thesis Sec. ( 1986) . (Civil) for the Numerical Factors for V-Notches in singular f i n i t e elements," University of British Columbia, Mall, S., Murphy, J.F., Schottafer, J.E., "Criterion for the mixed mode fracture in wood," ASCE J. Eng. Mech., 109(3), 680-690, (1983). Orowan, E., "Energy Suppl. c r i t e r i a of , 20, 1575, (1955). fracture," Welding Res. Paris, P.C., "The Mechanics of Fracture Propagation and Solutions to Fracture Arrestor Problems," Document No. D2-2195, Boeing Co., (1957). S.A.A., AS 1720-1979, Timber Engineering Code, Standards Association of Australia, (1975). Sydney: Saven, S.G., Stress Concent r at i on Around Holes, Trans. from Russian by E. Gros., New York: Pergamon Press, (1961). Schniewind, A.P., R.A. Pozniak, "On the Fracture Toughness of Douglas F i r Wood," Engineering Fracture Mechanics, 2, 223-233, (1971). Sih, G.C., Paris, P.C., Irwin, G.R., "On crack in rectilinearly anisotropic bodies," Int. J. Fracture Mechanics, 1, 189-203, (1965). Tada, H., The Stress Analysis of Research Corp., Penn., (1973). Cracks Handbook, Del Technical Research Centre of Finland, Laboratory of Structural Engineering, "Application of Fracture Mechanics: Fracture Toughness of Finnish wood," Research Newsletter, (1986). Walsh, P.F., "The computation of stress intensity factors by a Special F i n i t e Element Technique," Int. J. Solids and Structures, 7(10), 1333-1342, (1971). Walsh, P.F., "Linear fracture mechanics in orthotropic 154 materials," Engineering ( 1972) . Fracture Walsh, P.F., "Linear fracture and right angle notches," Research Technical Melbourne, Westergaard, H.M., Transactions, Am Mechanics, Paper (1974). (1939). 4, Mechanics, 533-541, mechanics solutions for CSIRO Aust. Div. of (Second series) No. 2, "Bearing Pressures. and Soc. Mechanical Engrs., /. zero Building 1-16, Crack," Applied Williams, J.G., Birch, M.W., "Mixed Mode Fracture in Anisotropic Media," Cracks and Fracture STP 601, ASTM, 125-137, (1976). Williams, M.L., "On the stress d i s t r i b u t i o n at the base of a stationary crack," /. Appl. Mech., 24(1), (1957). Woo, C.W., Chow, C.L., "Mixed Media," Proceedings, Appl i cat i on, India, Mode Fracture Fracture Mechanics 387-396, (1979). in in Orthotropic Engineering Wu, E.M., "Application of fracture mechanicxs to anisotropic plates," /. Appli ed Mechani cs , E34(4), 967-974, (1967). APPENDIX I Test Data of Experiment No. 1 Compact Tension Specimen Spec. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 M.C. „/ % 9.92 9.92 10.05 10.18 10.05 8.87 10.10 10.00 9.80 9.60 9.60 9.80 9.70 9.70 9.80 9.60 6.60 9.95 9.70 9.40 9.20 9.50 9.10 8.45 8.70 9.10 8.95 9.30 9.20 8.70 8.40 8.95 9.40 8.80 9.30 9.00 8.70 7.70 9.20 9.40 S.G. 0.372 0.324 0.411 0.408 0.377 0.450 0.330 0.414 0.374 0.378 0.369 0.405 0.412 0.375 0.410 0.328 0.408 0.417 0.373 0.386 0.384 0.386 0.389 0.370 0.378 0.386 0.386 0.394 0.394 0.374 0.441 0.390 0.411 0.389 0.381 0.400 0.399 0.402 0.388 0.359 a - :r e j e c t e d Angle to ° . RL degree P . max N N 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 80 76 75 65 60 60 248.0 216.1 307.2 324.2 199.1 492.5 238.8 291.2 341.3 335.6 244.5 238.8 261.6 329.9 307.2 270.7 432.2 238.8 318.5 261.6 204.7 204.7 298.0 227.5 398.1 394.6 409.5 301.5 394.7 420.8 272.9 369.7 261.6 364.0 184.3 382.2 449.3 466.3 537.0 278.8 227.5 208.1 290.1 312.9 147.9 489.2 227.5 272.9 332.1 329.9 248.0 236.6 252.5 295.8 284.4 260.5 415.2 213.9 304.8 254.9 184.3 199.1 275.3 224.1 389.1 378.7 407.3 278.7 381.1 415.2 255.9 364.0 252.5 345.8 177.5 364.0 434.5 443.6 536.8 263.9 because P max P„ Q > 1.2P Q' 1 55 ' P /P„ max Q K IC MPa/m 1.09 1.04 1.06 1.04 1.35 1.01 1.05 1.07 1.03 1.02 1.00 1.01 1.04 1.11 1.08 1.04 1.04 1.12 1.05 1.03 1.11 1.03 1.08 1.02 1.02 1.04 1.01 1.06 1.04 1.01 1.07 1.02 1.04 1.05 1.03 1.05 1.03 1.05 1.00 1.06 a 0.1570 0.1436 0.2002 0.2159 0.1021 0.3376 0.1570 0.1883 0.2296 0.2277 0.1711 0.1633 0.1743 0.2041 0.1963 0.1798 0.2865 0.1476 0.2103 0.1759 0.1272 0.1374 0.1900 0.1547 0.2685 0.2613 0.2811 0.1923 0.2630 0.2865 0.1766 0.2512 0.1743 0.2386 0.1225 0.2512 0.2998 0.3061 0.3704 0.1635 156 No. 1 - Cont'd Test Data of Experiment Compact T e n s i o n Specimen Spec. no. M.C. b S.G. % 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 8.08 7.40 9.65 9.90 9.67 8.40 8.70 7.80 7.90 8.57 8.60 9.40 9.92 9.00 9.00 7.20 8.57 7.20 7.00 9.70 9.00 9.50 7.00 10.00 9.95 8.95 9.60 9.10 9.40 9.30 9.60 9.80 9.95 9.20 b c Angle to RL degree C 0.390 0.383 0.350 0.350 0.457 0.364 0.356 0.386 0.455 0.453 0.368 0.456 0.359 0.353 0.377 0.357 0.356 0.355 0.354 0.349 0.356 0.452 0.356 0.336 0.364 0.348 0.353 0.350 0.347 0.355 0.366 0.337 0.335 0.355 57 55 54 53 52 51 50 50 50 49 49 48 47 47 47 46 45 44 40 40 39 33 32 23 16 16 16 12 7 5 2 0 0 0 P max N P n Q P max /P„ N 473.2 460.7 445.9 437.9 420.8 406.1 445.9 436.8 403.8 392.4 374.2 364.0 364.0 344.7 464.1 460.7 523.3 494.8 523.3 489.1 369.2 364.0 475.5 460.7 434.5 426.6 361.7 358.3 398.1 381.1 483.5 483.4 434.5 420.9 489.2 483.4 517.6 517.6 386.7 386.7 352.6 341.8 517.6 492.5 551.7 546.0 563.1 531.2 574.4 517.6 529.0 517.6 500.5 500.5 585.8 580.1 506.2 '502.8 557.4 541.5 602.9 591.5 532.3 530.1 625.6 620.0 602.9 580.1 Q IC MPav^ii 1.03 1.02 1.04 1.02 1.03 1.03 1.06 1.01 1.09 1.07 1.01 1.03 1.02 1.01 1.04 1.00 1.03 1.01 1.00 1.00 1.03 1.05 1.01 1.06 1.11 1.02 1.00 1.01 1.01 1.03 1.02 1.00 1.01 1.04 0.3179 0.3022 0.2802 0.3014 0.2708 0.2512 0.2379 0.3179 0.3415 0.3375 0.2512 0.3179 0.2944 0.2473 0.2630 0.3336 0.2905 0.3336 0.3572 0.2669 0.2359 0.3399 0.3768 0.3666 0.3572 0.3572 0.3454 0.4003 0.3470 0.3737 0.4082 0.3658 0.4279 0.4003 - moisture content recorded by r e s i s t a n c e type m o i s t u r e meter. - RL i s the r a d i a l - l o n g i t u d i n a l system of c r a c k i n i t i a t i o n . 157 Test Data of Experiment No. 1 Double Cantilever Beam Specimen Spec. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 M.C. „, % 9.56 9.79 9.51 9.83 9.85 9.78 9.61 9.52 9.33 9.41 9.21 8.78 9.14 9.26 9.03 8.98 9.75 S.G. Angle t o _ RL degree P max 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 1059.7 1100.2 1049.9 802.1 983.0 925.9 928.8 1040.8 848.0 1021.5 1080.2 852.3 1208.8 1242.8 826.3 1037.4 1006.9 T 0.367 0.392 0.373 0.335 0.389 0.401 0.377 0.362 0.388 0.389 0.374 0.368 0.332 0.387 0.396 0.332 0.327 N P„ Q x N P /P„ max Q x 981.2 1047.8 963.2 722.6 954.4 890.3 910.6 1020.4 839.6 954.7 1038.4 796.5 1140.4 1230.5 802.2 1007.2 987.2 1.08 1.05 1.09 1.11 1.03 1.04 1.02 1.02 1.01 1.07 1.04 1.07 1.06 1.01 1.03 1.03 1.02 K IC MPa/m 0.397 0.424 0.390 0.293 0.386 0.360 0.369 0.413 0.340 0.387 0.420 0.322 0.462 0.498 0.325 0.408 0.400 Test Data of Experiment 2 E f f e c t of C r a c k - f r o n t width on K Spec. no. S.G. P Width mm M.C. 1 2 3 4 5 7 7 7 7 7 8.98 8.94 9.21 8.68 8.83 0.335 0.335 0.335 0.338 0.338 69.5 70.8 70.0 76.3 69.4 66.7 68.6 68.6 70.6 66.7 1.04 1.03 1.02 1.08 1.04 0.2511 0.2585 0.2585 0.2659 0.2511 6 7 8 9 10 11 12 15 15 15 15 15 15 15 8.77 8.46 8.98 9.02 9.02 8.76 8.84 0.389 0.379 0.388 0.367 0.354 0.378 0.392 155.0 127.4 148.5 118.8 127.5 121.2 133.9 149.1 122.6 147.1 117.7 127.5 117.7 127.5 1.04 1.04 1.01 1.01 1.00 1.03 1.05 0.2619 0.2154 0.2585 0.2068 0.2240 0.2068 0.2240 13 14 15 16 17 18 21 21 21 21 21 21 9.24 9.17 9.09 8.94 9.01 8.76 0.377 0.401 0.410 0.389 0.397 0.396 178.7 202.0 208.0 189.7 216.2 220.0 173.6 196.1 205.9 182.4 204.0 202.0 1.03 1.03 1.01 1.04 1.06 1.09 0.2179 0.2462 0.2585 0.2289 0.2560 0.2535 19 20 21 22 23 24 25 29 29 29 29 29 29 29 8.49 8.54 8.78 8.77 8.92 8.96 8.88 0.377 0.376 0.377 0.373 0.373 0.375 0.372 207.9 180.3 220.1 161.6 226.3 216.4 209.9 196.1 178.5 215.7 156.9 219.7 215.7 207.9 1.06 1.01 1.02 1.03 1.03 1.00 1.01 0.1783 0.1622 0.1961 0.1426 0.1996 0.1961 0.1890 26 27 28 29 30 31 32 33 34 35 38 38 38 38 38 38 38 38 38 38 8.92 8.92 9.05 9.18 8.87 9.10 9.00 9.80 9.60 9.80 0.372 0.324 0.411 0.408 0.450 0.330 0.414 0.374 0.378 0.369 213.8 186.3 178.7 279.5 393.4 251.0 205.9 289.3 340.3 344.3 196.1 179.4 173.5 269.7 374.6 235.3 184.4 284.4 328.5 334.3 1.09 1.04 1.03 1.04 1.05 1.07 1.12 1.02 1.04 1.03 0.1353 0.1238 0.1197 0.1861 0.2585 0.1624 0.1272 0.1976 0.2267 0.2307 % P max N Q P max /P Q N n IC MPa/m' K 159 Test Data o f Experiment No. 3 E f f e c t of C r a c k - f r o n t Length on K^^ Spec. Board no. M . C ' S.G. % L mm Angle t o RL degree P P max max /P Q n N K IC MPa/m N 1 2 3 4 5 A A A A A 8.50 8.36 8.49 8.71 8.65 0.41 0.42 0.42 0.40 0.43 75 . 75 100 125 125 90 90 90 90 90 538.1 545.1 635.2 680.1 803.4 "511.5 489.3 622.7 645.0 778.4 1.05 1.11 1.02 1.05 1.03 ' 0.2355 0.2252 0.2275 0.2014 0.2430 6 7 8 9 10 11 B B B B B B 8.43 9.19 9.37 9.35 8.98 8.96 0.41 0.41 0.42 0.42 0.41 0.41 50 50 75 75 100 125 60 60 55 58 60 65 444.1 416.6 650.5 694.9 772.0 841.4 427.0 389.2 600.5 667.2 756.2 822.9 1.04 1.06 1.08 1.04 1.02 1.03 0.2947 0.2686 0.2764 0.3072 0.2762 0.2569 12 13 14 15 16 17 C C C C C C 9.70 9.69 9.68 9.75 9.74 9.54 0.34 0.34 0.33 0.32 0.33 0.34 50 50 75 100 100 125 75 75 72 68 75 79 284.6 303.2 445.3 674.0 592.4 581.2 278.0 300.2 411.4 645.0 556.0 578.2 1.02 1.01 1.08 1.04 1.06 1.01 0.1918 0.2072 0.1894 0.2356 0.2031 0.1805 18 19 20 D D D 8.99 8.98 8.93 0.42 0.41 0.41 50 100 125 60 60 72 408.3 695.5 1063.9 382.5 689.4 1007.5 1.07 1.01 1.05 0.2640 0.2518 0.3145 160 Test Data of Experiment No. 4 E f f e c t of Moisture Content on K Spec. no. M.C. „ % S.G. Angle to „ RL degree T P max P„ 0 N N P max /? n Q K IC MPa/m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.388 0.357 0.396 0.403 0.357 0.394 0.372 0.356 0.439 0.410 0.372 0.366 0.379 0.387 0.375 0.371 90 90 90 70 90 90 90 90 90 90 90 90 90 90 90 90 207.9 280.4 214.2 205.6 282.1 274.4 344.1 337.9 355.2 225.8 252.3 322.1 311.6 206.9 207.1 186.9 196.1 269.7 213.8 204.0 255.0 262.8 335.4 326.5 351.1 219.7 235.3 313.8 298.1 196.1 204.0 179.4 1.06 1.04 1.00 1.01 1.11 1.05 1.02 1.04 1.01 1.03 1.07 1.02 1.05 1.05 1.01 1.04 0.1353 0.1861 0.1475 0.1407 0.1759 0.1813 0.2314 0.2253 0.2422 0.1516 0.1724 0.2165 0.2057 0.1353 0.1407 0.1238 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 6.2 6.1 6.3 6.1 6.3 6.1 6.6 6.8 6.3 6.1 6.1 6.7 6.3 6.5 6.4 6.7 0.371 0.365 0.351 0.344 0.342 0.340 0.327 0.334 0.387 0.312 0.314 0.386 0.343 0.317 0.357 0.369 90 90 90 90 60 90 90 90 90 90 90 90 90 90 90 90 274.2 231.9 351.1 252.7 333.4 302.1 264.9 260.1 279.2 330.7 343.2 220.6 224.2 378.9 360.3 359.2 262.8 219.7 326.5 240.3 313.8 298.1 262.8 255.0 235.3 313.8 328.5 213.8 217.7 362.8 351.1 335.4 1.04 1.06 1.07 1.05 1.06 1.01 1.01 1.02 1.09 1.05 1.04 1.03 1.03 1.05 1.02 1.07 0.1813 0.1516 0.2253 0.1658 0.2165 0.2057 0.1813 0.1759 0.1624 0.2165 0.2267 0.1475 0.1502 0.2504 0.2422 0.2314 161 Test Data o f Experiment No. 5 Mode I I E n d - s p l i t Beam Specimen Spec. no. M.C. % 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Note : 13.3 13.5 13.7 13.9 11.3 13.3 14.0 12.6 13.8 11.7 14.5 13.9 13.7 14.5 12.8 13.3 12.9 13.5 12.5 13.5 12.9 13.5 12.0 12.6 P m a x S.G. 0.376 0.388 0.378 0.360 0.473 0.387 0.370 0.378 0.375 0.380 0.539 0.360 0.381 0.423 0.324 0.349 0.347 0.398 0.384 0.349 0.313 0.345 0.310 0.314 Angle t o RL degree 0 7 12 18 32 38 39 43 45 46 48 55 60 64 75 88 90 90 90 90 90 90 90 90 i s u s u a l l y g r e a t e r than 2P^ by the bending s t r e n g t h . P K Q I N 13344 12899 14856 12454 16013 15034 13033 13789 18682 17792 19794 18904 12232 19571 12588 13033 14678 12632 12677 14812 12899 13878 15568 15012 since I C MPa/m 2.0522 1.9837 2.2847 1.9153 2.4626 2.3121 2.0043 2.1206 2.8731 2.7362 3.0441 2.9072 1.8812 3.0098 1.9359 2.0043 2.2573 1.9427 1.9496 2.2779 1.9837 2.1343 2.3942 2.3087 specimens' a r e governed 162 Test Data of E x p e r i m e n t No. M i x e d Mode M i d - c r a c k e d 6 Beam c Phase 1 - Center p o i n t l o a d i n g Spec. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 M.C. % S.G. 8. 94 9. 03 9. 02 8. 87 8. 96 8. 98 9. 23 8. 46 8. 79 8. 90 8. 37 8. 38 8. 42 8. 57 8. 69 8. 94 9. 00 9. 02 8. 89 8. 78 8. 98 9. 03 8. 65 8. 75 0.396 0.394 0.374 0.388 0.401 0.403 0.416 0.369 0.388 0.404 0.399 0.396 0.393 0.385 0.344 0.346 0.384 0.388 0.374 0.376 0.392 0.377 0.354 0.346 Angle to Crack RL angle degree degree 41 55 65 60 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 a - Loading at the c e n t e r l i n e , i . e . , b - P c m a x a Load at P„ ^ center center center center center center center center center center center center center center center center center center center center center center center center 4448 5560 58715894 4670 4448 3425 6227 3647 3514 3174 5338 4782 3336 3625 4003 4226 3892 5894 3781 3447 3505 3394 3336 450 mm b K ^ MPa/m K „ ^ MPaAi 0.3360 0.4199 0.4435 0.4451 0.3527 0.3360 0.2587 0.4703 0.2755 0.2654 0.2397 0.4031 0.3612 0.2520 0.2738 0.3024 0.3192 0.2940 0.4451 0.2856 0.2603 0.2647 0.2563 0.2520 0.8330 1.0413 1.0996 1.1038 0.8747 0.8330 0.6414 1.1662 0.6831 0.6581 0.5831 0.9996 0.8955 0.6248 0.6789 0.7497 0.7914 0.7289 1.1038 0.7081 0.6456 0.6564 0.6356 0.6248 from the support. i s g r e a t e r than 1.2 P^, beam s t r e n g t h governed by bending, - phase 1 i s 45 degrees beam t e s t s . 1 63 Test Data of Experiment No. 6 - Cont'd Phase 1 - 4 5 deg. beam - S i x t h p o i n t !pec. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 20 21 22 23 24 25 M.C. S.G. % 8.44 8.64 8.76 9.04 8.56 8.94 8.48 8.52 8.79 9.12 9.08 9.12 9.26 8.96 8.94 9.04 9.02 8.59 8.54 9.34 9.33 9.31 9.16 9.22 0.375 0.386 0.398 0.401 0.375 0.336 0.384 0.386 0.383 0.396 0.399 0.391 0.405 0.386 0.377 0.389 0.403 0.380 0.377 0.411 0.408 0.412 0.402 0.398 Angle to RL degree 11 40 51 58 73 87 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 Crack angle degree 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 Load at 16ading P Q N 6th 6 th 6th 6th 6th 6th 6 th 6th 6th 6th 6th 6th 6 th 6th 6th 6th 6th 6th 6th 6 th 6th 6 th 6th 6 th 4670 4 5115.2 6004.8 4781.6 5560.0 6227.2 5782.4 4448.0 3647.4 3558.4 4670.4 6004.8 5115.2 5560.0 6227.2 5560.0 6449.6 6672.0 4448.0 4448.0 5449.0 3447.6 5115.3 4782.0 K IQ K IIQ MPa/m MPa/m 0.3255 0.3566 0.4186 0.3333 0.3876 0.4341 0.4031 0.3100 0.2542 0.2480 0.3255 0.4186 0.3566 0.3876 0.4341 0.3876 0.4496 0.4651 0.3100 0.3100 0.3798 0.2403 0.3565 0.3333 0.4768 0.5222 0.6130 0.4882 0.5676 0.6357 0.5903 0.4541 0.3724 0.3633 0.4768 0.6130 0.5222 0.5676 0.6357 0.5676 0.6585 0.6812 0.4541 0.4541 0.5563 0.3519 0.5222 0.4882 164 Test Data of Experiment No. 6 - P h a s e 2 - 9 0 d e g . beam - c e n t e r l i n e >pec. no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 M.C. S.G. % 8.98 8.73 8.16 8.48 8.81 9.04 9.08 8.48 8.63 8.71 8.68 8.70 8.65 8.68 8.72 8.71 0.356 0.398 0.388 0.359 0.328 0.396 0.400 0.377 0.369 0.399 0.396 0.394 0.390 0.388 0.392 0.392 Angle to RL degree Crack angle degree 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 Cont'd loading K N 5393 4559 5627 6505 5338 4782 4559 3447 4537 6283 3781 5894 4337 3892 4782 4537 IIQ MPav^ni MPa/m 0.3402 0.2876 0.3550 0.4104 0.3367 0.3017 0.2876 0.2175 0.2862 0.3937 0.2385 0.3718 0.2736 0.2455 0.3017 0.2862 0.7246 0.6125 0.7560 0.8740 0.7172 0.6425 0.6125 0.4631 0.6096 0.8441 0.5080 0.7919 0.5827 0.5229 0.6425 0.6096 Note: Crack I n i t i a t i o n occured ±90 degrees to t h e i n i t i a l c r a c k simultaneously. 165 Test Data of E x p e r i m e n t No. DCB u n d e r Mixed-mode Spec, no. M.C. S.G. % 7 Loading Angle to RL degree P 2 N K IQ K IIQ MPav/iii MPav^m 1 2 3 4 5 6 7 8 9 10 8.42 8.73 8.71 8.76 9.26 8.16 8.96 8.94 8.89 8.75 0.379 0.386 0.384 0.379 0.359 0.403 0.396 0.394 0.382 0.374 90 90 90 90 90 90 90 90 90 90 1925 1925 1925 1925 1925 1925 1925 1925 1925 1925 3825 3336 4448 4003 4448 3670 4003 3558 4114 4448 0.4009 0.4493 0.3393 0.3833 0.3393 0.4163 0.3833 0.4273 0.3723 0.3393 0.5761 0.5024 0.6699 0.6029 0.6699 0.5526 0.6029 0.5359 0.6196 0.6699 11 12 13 14 15 16 17 18 19 20 8.96 8.94 9.07 9.01 8.92 8.96 8.46 8.29 8.45 8.65 0.369 0.346 0.358 0.398 0.374 0.368 0.348 0.376 0.368 0.384 90 90 90 90 90 90 90 90 90 90 2800 2800 2800 2800 2800 2800 — 2800 2800 2800 9341 8229 8674 7784 8006 8562 — 8006 8674 9118 0.2095 0.3195 0.2755 0.3635 0.3415 0.2865 — 0.3415 0.2755 0.2315 1.4067 1.2393 1.3062 1.1723 1.2058 1.2895 * 1.2058 1.3062 1.3732 * S p l i t occured i n the specimen b e f o r e t e s t i n g . 166 Test Data of Experiment No. 8 Notched Beam Specimens Spec. type Spec. no. M.C. % S.G. A Angle t o RL degree P K K I N i Q MPam MP am A A A A A A A A A 1 2 3 4 5 6 7 8 8.84 8.76 8.42 9.24 9.80 7.78 8.12 9.13 0.382 0.369 0.384 0.321 0.402 0.416 0.332 0.303 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 90 90 90 90 90 90 90 90 4203 4559 2335 4114 5226 4504 4915 4114 0. 2676 0. 2902 0. 1487 0. 2619 0. 3327 0. 2867 0. 3129 0. 2619 0.4647 0.5041 0.2582 0.4549 0.5778 0.4980 0.5434 0.4549 B B B B B B B B 1 2 3 4 5 6 7 8 8.41 8.12 8.56 9.37 10.01 7.68 8.08 8.46 0.342 0.383 0.369 0.414 0.311 0.308 0.324 0.326 1.6 1.6 1.6 1.6 1.6 1.6 1.6 1.6 90 90 90 90 90 90 90 90 4114 6338 5894 5115 6227 5226 5449 5560 0. 1776 0. 2737 0. 2545 0. 2208 0. 2689 0. 2256 0. 2352 0. 2400 0.9025 1.3904 1.2928 1.1221 1.3660 1.1465 1.1953 1.2197 C C C C C C C C 1 2 3 4 5 6 7 8 8.11 8.43 8.56 8.64 7.13 6.84 8.42 7.88 0.414 0.308 0.388 0.384 0.369 0.342 0.324 0.336 1.7 1.7 1.7 1.7 1.7 1.7 1.7 1.7 90 90 90 90 90 90 90 90 8006 10342 9341 5338 8118 6116 10898 11898 0. 1940 0. 2506 0. 2263 0. 1293 0. 1967 0. 1482 0. 2640 0. 2883 0.6840 0.8835 0.7980 0.4560 0.6935 0.5225 0.9310 1.0165 APPENDIX III Difference between Mode I Fracture Toughness of CTSs and DCB spec imens. A. Influence on K -, of Assumed E l a s t i c Properties I( E (MPa) X Spec. type K* E (MPa/m) c E y E y y 12500 DCB CTS 0.0498 0.0693 0.0403 0.0688 0.0359 0.0681 10000 DCB CTS 0.0498 0.0693 0.0403 0.0688 0.0359 0.0681 7500 DCB CTS 0.0498 0.0693 0.0403 0.0688 0.0359 0.0681 *E„/G = 1.0 and v „ = 0.02 were used in generating A y the v Y chart. As can be influence on seen above, the fracture E :E._, E„:G and v „ v V A y y and K variation of toughness providing E has no the r a t i o s are the same as those used in generating A y compute the ratio J C T S ^ I D C B ' the chart. Thus, we can (DCB) the I C K k K IC (CTS) for the three cases : K. I(CTS) K I(DCB) K IC(DCB) ( M P a ^ ) K IC(CTS) ( M 10 1.3909 0.4760 0.2059 20 1.7047 0.3855 0.2044 30 1.8970 0.3429 0.2023 The difference between the CTSs and 167 P a ^ DCB specimens is 168 significant for various r a t i o s of E / E . This indicates that x the difference in the mode I y fracture toughness i s not affected much by the assumed e l a s t i c properties. B. T-test on the hypothesis that the two samples being compared are drawn from the same population. The test i s applied to the n u l l hypothesis that the two samples being compared are drawn and we calculate two means from the same the p r o b a b i l i t y of having a value as large population, the difference of the as, or greater than, observed. Sample Mean K •LL* (MPa m) Standard d e v i a t i o n n s X Sample s i z e 1. CTS 0.205 0.0510 33 2. DCB 0.387 0.0497 17 The pooled estimate of variance i s s 2 = n l s l + n 2 2 s and s c = 0.0506 MPa/m = 33 x 0.0510 + 17 x 0.0497 2 2 = 2.557x10 169 The standard deviation of the difference of mean i s thus : S. d = s c /l/n. + l/n. i 1 0.0506 / 1/33 + 1/17 = 0.01509 X l " 2 X 10.205 - 0.387| 12.06 0.01509 The number of degrees of freedom i s 33 + 17 - 2 = 48, and from the t - d i s t r i b u t i o n , i t gives t = 2.0126 at the 95 percent l e v e l of confidence. Therefore, the n u l l can be rejected, and we conclude that from different populations. hypothesis the two samples are APPENDIX III S.A.A. A u s t r a l i a n T i m b e r E n g i n e e r i n g Code AS 1 7 2 0 - 1 9 7 5 S t r e n g t h o f N o t c h e d Beams For a r e c t a n g u l a r beam o f d e p t h D, n o t c h e d on t h e edge as shown i n F i g . 5 7 , t h e n o m i n a l maximum ^b for = tension bending stress ScO ^ l maximum s h e a r s t r e s s f = calculated n n the net s e c t i o n s h a l l comply w i t h t h e f o l l o w i n g i n t e r a c t i o n a n c n o m : L n a g formula : 0.3f,+0.7f D S « 1 3 sj c where C i s a c o n s t a n t 3 0 F tabulated p e r i m i s s i b l e shear s t r e s s f o r strength for the s p e c i e s of in table joint 11 and details F • sj or the shear interest, v M F i g u r e 57 Notation f o r Table P a r a m e t e r C^ f o r notch slope b/a = 0 b/a = 2 b/a = 4 Notch 11 selected notch a ^» O0 . 1l DD • 3/D , 2.6/D? 2.2/D" % 170 is angles a<0.1D l/a 1.2/a % 1.3/a' 1 the block
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Experimental studies on fracture of notched white spruce beams Lau, Wilson Wai Shing 1987
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Title | Experimental studies on fracture of notched white spruce beams |
Creator |
Lau, Wilson Wai Shing |
Publisher | University of British Columbia |
Date Issued | 1987 |
Description | The fracture problem due to the singular stresses arising from the sudden change of geometric properties around cracks and notches was studied both analytically and experimentally. The failure models of the cracked and the notched specimens were derived by using linear elastic fracture mechanics methodology, which led to the determination of the critical stress intensity factors. Experiments were conducted to determine fracture toughness for different modes as well as the effect of variations in the crack-front width, specimen size and moisture content. Subsequently, failure surfaces for cracks and notches were developed based on the experiments undertaken, describing in each case the interaction between mode I and mode II fracture toughness. To verify the reliability of these experiments, the results obtained were compared with the published literature. As an application, design curves for a 90 degrees-cracked beam and a 90 degrees-notched beam are presented. These curves allow the prediction of the failure loads due to the rapid crack propagation under different loading conditions. |
Subject |
Girders -- Testing Strains and stresses Wooden beams |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-07-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062927 |
URI | http://hdl.handle.net/2429/26714 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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