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Experimental studies on fracture of notched white spruce beams Lau, Wilson Wai Shing 1987

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EXPERIMENTAL STUDIES ON FRACTURE OF NOTCHED WHITE SPRUCE BEAMS by WILSON W.S. LAU B.A.Sc University 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 thesis as conforming . to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1987 © Wilson W.S. Lau, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C i v i l Engineering The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date A p r i l , 1987 ABSTRACT The fracture problem due to the singular stresses a r i s i n g from the sudden change of geometric properties around cracks and notches was studied both a n a l y t i c a l l y and experimentally. The f a i l u r e models of the cracked and the notched specimens were derived by using linear e l a s t i c fracture mechanics methodology, which led to the determination of the c r i t i c a l stress intensity factors. Experiments were conducted to determine fracture toughness for dif f e r e n t modes as well as the effect of variations in the crack-front width, specimen size and moisture content. Subsequently, fa i l u r e 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 v e r i f y the r e l i a b i l i t y of these experiments, the results obtained were compared with the published l i t e r a t u r e . 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 f a i l u r e loads due to the rapid crack propagation under different loading conditions. i i Table of Contents ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS xi 1. INTRODUCTION 1 1.1 The Problem 1 1.2 Previous Research 3 1.3 Objective 7 1 . 4 Scope 7 1.4.1 Introduction 7 1.4.2 Mode I Fracture Toughness 8 1.4.3 Mode II Fracture Toughness 8 1.4.4 Specimen Size Effect 8 1.4.5 Moisture Content Effect 9 1.4.6 Interaction Curves for Mixed Mode Loading 9 1.5 Summary of Objectives 9 2 . THEORY 11 2.1 Introduction 11 2.2 Application of LEFM on wood structures 11 2.3 Formulation of the Stress Intensity Factors for Cracks 17 2.4 Stress Intensity Factors for Sharp Notches 26 2.4.1 Introduction 26 2.4.2 Formulation of the problem for notches ...27 2.4.3 Implementation in a F i n i t e Element Program 38 2.5 Effect of Specimen Size on Stress Intensity Factors 43 2.5.1 Effect of Material Heterogeneity 43 2.5.2 Effect of the Change of the Stress State .44 2.6 Effect of Moisture Content on Fracture Toughness 49 2.7 Mixed Mode Fracture in White Spruce 51 Experiment Parameters 54 3.1 Introduction 54 3.2 Crack Orientation and Propagation 54 3.3 Materials 56 3.4 Specimen Preparation 57 3.5 Experimental Measurement of Load and Displacement 59 3.6 Treatment of Data 62 EXPERIMENT DESCRIPTIONS AND RESULTS 65 4.1 Introduction . 65 4.2 Experiment No.1, Mode I Fracture Toughness 65 4.2.1 Experimental Design and Procedure 65 4.2.2 Results 66 4.3 Experiment No.2 Crack-front Width Effect on Fracture Toughness 75 4.3.1 Experiment Design and Procedure 75 4.3.2 Results 77 4.4 Experiment No.3, Crack-front Length Variation ..84 4.4.1 Experiment Design and Procedure 84 4.4.2 Results 84 4.5 Experiment No.4 Moisture Content Effect 89 4.5.1 Experiment Design and Procedure 89 4.5.2 Results 91 4.6 Experiment No.5, Mode II Fracture Toughness ....94 iv 4.6.1 Experiment Design and Procedure 94 4.6.2 Results 95 4.7 Experiment No.6, Mixed Mode Mid-cracked Beams .103 4.7.1 Experiment Design and Procedure 103 4.7.2 Results 105 4.8 Experiment No.7, DCB under Mixed-mode Loading .114 4.8.1 Experiment Design and Procedure 114 4.8.2 Results 119 4.9 Experiment No.8, Notched Beam Specimens 122 4.9.1 Experiment Design and Procedure 122 4.9.2 Results 125 4.10 Summary 127 5. DISCUSSION 128 5.1 Introduction 128 5.2 Stress Intensity Factor Interaction Curve for Cracks 128 5.3 Stress Intensity Factor Interaction Curve for Notches 131 5.4 Application 138 6. CONCLUSIONS AND RECOMMENDATIONS 149 6.1 Conclusions 149 6.2 Recommendations for future research 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 properties of structural materials 32 2 Effect of the annual rings angle on 71 3 Effect of crack-front width on K , ^ for compact tension specimens of white spruce for longitiidinal propagation 78 4 Effect of crack-front length on for compact tension specimens of white spruce 86 5 The size c o e f f i c i e n t for CTSs and DCB specimens 90 6 Effect of moisture content on K ^ ^ , for CTSs i n longitudunal propagation 92 7 E f f e c t of the annual rings angle on K-J.-J.^ 100 8 Results of the mid-cracked beam specimens 107 9 Results of the two-loads beam tests 121 10 Results of the notched beam specimens 126 11 Stress intensity factors for various sharp crack problems v i LIST OF FIGURES FIGURE Page 1 Three basic modes of crack surface displacement 13 2 Three-letter system for designing orientation relations i n wood 15 3 Coordinate system used for describing the stresses ahead of a crack 19 4 F i r s t primary singular stress f i e l d for a 0° crack...25 5 Second primary singular stress f i e l d for a 0° crack 25 6 Notation for r e a l space transformed coordinates 33 7 Notation and coordinates for.notch roots 33 8 Determinant of eigenequation for notch i n Douglas f i r 36 9 Stress state i n specimen 4-6 10 A steel crack starter with crack t i p radius less than 0 . 5um 58 11 LVDT gage for centerline d e f l e c t i o n measurement 60 12 Modified LVDT gage for measuring longitudinal s l i d i n g displacement 60 13 Typical load vs displacement curves 61 14 Relations between k T _ / k i c a n d K u / K I I C • 6 4 15 Configuration of compact tension specimen (CTS) 67 16 Configuration of double cantilever beam specimen (DCB) 67 17 The experimental setup of the compact tension specimen test °8 18 F i n i t e element mesh for the CTS 7 0 19 F i n i t e element mesh for the DCB specimen 70 20 Mode I fracture toughness v a r i a t i o n with annual rings angle v i i LIST OF FIGURES - Cont'd FIGURE Page 21 Cumulative d i s t r i b u t i o n curve of the compact tension specimen ' ^  22 Cumulative d i s t r i b u t i o n curve of the double cantilever beam 74 23 Effect of s p e c i f i c gravity on mode I fracture toughness 76 24 Effect of crack-front width on mode I fracture toughness.(Based on the Weibull weakest l i n k model)..80 25 Effect of crack-front width on mode I fracture toughness . (Based on the stress-state model) 82 26 Specimen configuration of the crack-front length specimen 85 27 Effect of the crack-front length on mode I fracture toughness 88 28 Effect of the moisture content on mode I fracture toughness 93 29 Experimental setup of the mode II fracture toughness specimen 96 30 F i n i t e element mesh for the mode II fracture toughness specimen 96 31 Applied load versus longitudinal displacement 97 32 The apparatus and the experimental setup of the mode II fracture toughness test specimen 98 33 Mode II fracture toughness v a r i a t i o n with annual rings angle 101 34 Cumulative d i s t r i b u t i o n curve of the end-split beam specimen 102 35 Specimen configuration of the 45 deg. beam 104 36 Specimen configuration of the 90 deg. beam 104 37 F i n i t e element mesh for the mid-cracked beam specimen J- u° 38 Experimental and corrected cumulative probability curves of the mode I fracture toughness for the 90 degrees beam 109 v i i i LIST OF FIGURES - C o n t ' d 3 9 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 -curves o f the mode I I f r a c t u r e toughness f o r the 9 0 degrees beam. 1 1 0 4 0 Method of n o r m a l i z i n g the c u m u l a t i v e p r o b a b i l i t y curves of the 4 5 degrees m i d - c r a c k e d beam I l l 4 1 K J [ " ^ T I 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 beam r e s u l t 1 1 3 4 2 Specimen c o n f i g u r a t i o n o f the two- loads beam specimen 1 1 5 4 3 Experiment setup o f the two- loads beam specimen 1 1 6 4 4 F a i l u r e envelope o f the two- loads beam specimen 1 2 0 4 5 T y p i c a l l o a d - d i s p l a c e m e n t curve o f the two- loads beam specimen 1 2 3 4 6 Specimen c o n f i g u r a t i o n f o r notched beam 1 2 4 4 7 I n t e r a c t i o n between K ^ K ^ and K ^ / K ^ f o r c r a c k e d beam specimens 1 2 9 4 8 I n t e r a c t i o n curves d e r i v e d between K ^ / K ^ a n d ^ T l ^ H C ^ o r c r a c ^ - e d beam specimens 1 3 0 4 9 I n t e r a c t i o n models between K ^ / K ^ and K ^ / K ^ ^ f ° r cracked beam specimens 77. . . . V 1 3 2 5 0 I n t e r a c t i o n r e l a t i o n s between K-j. and f o r notches 1 3 4 5 1 I n t e r a c t i o n curve f o r notches w i t h A = 1 . 6 0 1 3 6 5 2 I n t e r a c t i o n curve f o r notches w i t h A = 1 . 7 0 1 3 7 5 3 D imens ion les s SIFs f o r pure moment and pure shear 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 degrees c r a c k e d beam 1 3 9 5 4 S e v e r a l sharp c r a c k problems 1 4 1 5 5 D e s i g n ' c u r v e s f o r 9 0 degrees c r a c k e d beam as a 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 1 4 3 i x LIST OF FIGURES - Cont'd FIGURE Page 56 K T / P versus notch length for various notch depth of a 2"x8" beam • 146 57 C r i t i c a l load versus notch length for various notch depth of a 2 "x8 " beam 148 x ACKNOWLEDGEMENTS The author would like to express his sincere gratitude to Dr. R.O. Foschi and Dr. J.D. Barrett for their guidance and invaluable advice in the preparation of this thesis. The author would also like to thank Conroy Lum for his assistance in using the finite 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 of C i v i l Engineering and Wood Science and Harvesting of the University of British Columbia for the use of their laboratories and equipment. xi 1. INTRODUCTION 1 . 1 THE PROBLEM In pr a c t i c a l situations wood i s sawn, chopped, chipped, sli c e d , dried, d r i l l e d , flaked, and fastened. Flaws or defects unavoidably occur as a result of each of these processes. In other cases, gaps between ends of boards in laminated timber (which are usually denoted by the term "butt-joint"), notches formed by a sawcut, re-entrant notches occurring at open butt or lap joints in laminated beams lead to a sudden change in geometry and stress concentrations. A l l of these cases result in a stress singularity formed at the notch root which can not be analyzed by ordinary stress formulae. The Timber Design Manual, as well as the governing Canadian design code, CAN3-086-M84: Code for Engineering Design in Wood, use a reduced net depth to account for the present of a notch in a beam, without consideration of the intense stress concentration at the notch corner. Furthermore, the code does not cover s p l i t e d specimens. Obviously, a the o r e t i c a l understanding of these s i n g u l a r i t i e s in wood should be of p r a c t i c a l importance. Application of fracture mechanics to wood is concerned with structural f a i l u r e by catastrophic crack propagation. In many applications, fracture mechanics techniques are used to eliminate such f a i l u r e s by special material control tests or by defining maximum crack or flaw sizes that can be 1 2 tolerated in the structure. As mentioned above, flaws in a wood structure cannot be t o t a l l y controlled, so fracture mechanics methods must be used to assess the allowable load on the structure. Near the v i c i n i t y of a notch root or a crack t i p , the stress at every point in this region i s subjected to plane stress or st r a i n conditions which can be expressed in terms of the stress intensity factors. Therefore, the determination of the allowable load i s the same as obtaining the c r i t i c a l values of the stress intensity factors. Depending on the mode of crack t i p deformation, the stress intensity factors K are designated with a subscript I, II, III for the cracked specimen (zero notch angle cracks) which corresponds to the opening, forward shear and tranverse shear modes of deformation. For notches, the subscript A, B are used to define the stress intensity factors for the primary and the secondary stress f i e l d s respect i v e l y . C r i t i c a l values for the pure mode I and mode II for a cracked specimen have been studied and published for different species of wood. There i s , however, a lack of information about the c r i t i c a l values for the mixed mode fa i l u r e of a cracked specimen. 3 1.2 PREVIOUS RESEARCH Studies of the cracks or notches were initiated by Inglis (1913) who made the stress analysis of an e l l i p t i c a l hole in a uniformly stressed elastic plate. A crack can be represented by an infinitesimally narrow ellipse. Based on Inglis' theory, G r i f f i t h (1921) formulated his well known energy criterion for b r i t t l e fracture which was extended by Irwin (1948) and Orowan (1955) to apply to metallic solids where plastic deformation takes place at crack tips. Later, Savin (1961), using his photoelastic method, analyzed the stress at re-entrant corners. Up to that moment, the study of cracks and notches was mainly based on the energy method or individual experiment results to determine the stress concentration factors for specific geometries. Linear elastic fracture mechanics (LEFM) is based on the elastic solution of the crack tip stress f i e l d where the yielding has been highly localized at the crack front. Williams (1957) has solved the problem of a cracked plate, expressing the stresses a i j / the strains e i j ' a n t^ the displacements u — in terms of infinite series of singular and regular terms of r and 6, where r and 6 are polar coordinates, with the crack tip at the origin. 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 tip elastic stresses, strains and displacements, using the stress intensity factor 4 method, was studied by Liu (1965&-1966) . For the case of small scale y i e l d i n g , SSY, L i u has shown that K i s capable of characterizing the crack t i p stresses, strains and displacements within the e l a s t i c f i e l d zone which forms the fundamental basis of the linear e l a s t i c fracture mechanics. The application of linear e l a s t i c fracture mechanics (LEFM), to cracked material using the stress intensity factors has been shown to be an e f f e c t i v e method by Knott (1973), Broek (1982), Hellan (1984) but r e s t r i c t e d to isotropic material. The extension of LEFM to orthotropic body was f i r s t made by Sih et at (1965) who derived formulae for the stresses in a small region surrounding a crack t i p in an orthotropic body. In the application of LEFM to the strength of structures, i t is not only the form of s i n g u l a r i t y but also the magnitude of the stresses near the root of the fractured surface what i s needed. This requires the computation of the so-called stress intensity factors for the s i n g u l a r i t y . And except for some simple geometry and boundary conditions, the solutions always require a numerical method. Walsh (1971,1972,1974) introduced a c a l i b r a t e d f i n i t e element method to compute the stress intensity factors using LEFM, which incorporated the singular terms in the displacement f i e l d . This method also s a t i s f i e s the equilibrium and compatibility conditions at the interface between the conventional f i n i t e elements and the modified elements. This method was extended to apply to orthotropic materials and 5 right angle notches. However, the method does not account for compatibility at the elements interfaces, which enhances monotonic convergence. A method suggested by Benzley (1974), using a compatible displacement formulation for a f i n i t e element, with singular 'enrichment' terms, solves this problem thoroughly. The method has been shown to be e f f i c i e n t and r e l i a b l e . This conformable displacement model was extended by Foschi and Barrett (1976) for the anisotropic case and used by G i f f o r d and Hilton (1978) to analyze cracks in isotropic bodies with 12-node isoparametric elements and a coarser mesh. The application of LEFM to the analysis of the stress f i e l d at the root of a sharp notch of arbitrary notch angle was f i r s t proposed by Leicester (1971). He introduced the stress intensity factors for notches, K A and Kfi, which correspond respectively to the primary and the secondary stress f i e l d of the eigenfields governing the notch t i p stress d i s t r i b u t i o n . However, the terms "opening mode" and " s l i d i n g mode" which are commonly used in connection with sharp cracks are applicable to notches only i f the notches are symmetrical with respect to the axes of e l a s t i c symmetry. These factors were incorporated in the f i n i t e element method to analyze notches in Leicester's paper with Walsh (1982). The V-notch, which i s a special case of notches, has been studied by Gross and Mendelson (1972) 6 using the boundary collocation procedure and also by Lin and Tong (1980) using the singular f i n i t e element method. According to the linear fracture mechanics model, fracture i s assumed to occur when the stress intensity factor attains a c r i t i c a l value — the c r i t i c a l stress intensity factor. This factor i s material dependent and has to be determined experimentally for di f f e r e n t species of wood. Schniewind and Centeno (1971) presented KI(-, values for the six p r i n c i p a l systems of propagation for Douglas-fir wood; using the end-split beam method, Barrett and Foschi (1971) found the K I I C values for Hemlock. Hunt and Croager (1982) also established the mode II fracture toughness for b a l t i c redwood by a mixed mode test method assuming an interation curve existed between mode I and mode I I . The studies on the interaction curve for isotropic and orthotropic materials has been r e s t r i c t e d to cracked elements due to the fact that the notch root can have different types of s i n g u l a r i t i e s , whereas the sharp cracks always have a 1 /Vr s i n g u l a r i t y . A method suggested by Lum (1986), which incorporates the eigenvalue X as a f a i l u r e parameter, has proved to be a reasonable approach to specify the c r i t i c a l notch root stress conditions and may be used to describe interaction curves for notches as well as for cracks. Studies on the crack-front width effect have shown that i t influences the prediction of the f a i l u r e load for large structures. Barrett (1976) found a size effect due to 7 crack-front width on Kj^. for cracks using the Weibull's theory whereas Ewing (1979), has also shown the existence of this effect based on his stress-state model. Leicester (1969,1973) has studied the magnitude of a size effect on notches by considering a size c o e f f i c i e n t implied on the nominal stress. The moisture content effect was studied by Ewing & Williams (1979) on compact tension specimen and by Dolan & Madsen (1985) on the shear strength. 1.3 OBJECTIVE The objective of this thesis is to investigate the fracture behavior of white spruce under the mode I, mode II and the combined mixed mode loading conditions. The effects-such as the variation of the moisture content and the crack-front width w i l l also be studied. 1 .4 SCOPE 1.4.1 INTRODUCTION In order to develop a design procedure for cracks and notches under loading, experiments were car r i e d out to establish the c r i t i c a l values for the stress intensity factors and results were compared with published ones. In part i c u l a r , the experiments were designed to examine the following features. 8 1.4.2 MODE I FRACTURE TOUGHNESS To determine experimentally the c r i t i c a l value of the opening mode stress intensity factor for white spruce using a compact tension specimen and a double cantilever beam specimen. Effect of crack orientation with respect to the grain was also considered. 1.4.3 MODE II FRACTURE TOUGHNESS To determine experimentally the c r i t i c a l value of the sl i d i n g mode stress intensity factor for White Spruce by the end-split beam method. Effect of the crack orientation was also considered for cracks propagating along the grain. 1.4.4 SPECIMEN SIZE EFFECT The strength of complex structures are frequently assessed from laboratory tests on scaled models. This i s most simply done by use of the 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 stress l e v e l . However, for structures containing singular stress f i e l d s , this may not be the real case. Experiments were carried out to study the dependence of the mode I stress intensity factor on the size of the specimen and also to study the a p p l i c a b i l i t y of the crack-front width theory developed by Barrett (1976) on the mode I stress intensity factor. 9 1 .4 .5 MOISTURE CONTENT EFFECT Many published papers have shown that the fracture toughness of wood depends on the moisture content as well as the temperature, but design formula are lacking to account for this e f f e c t . Experiments were conducted to study the moisture content effect on the mode I c r i t i c a l stress intensity factor and a model i s proposed in terms of residual stresses. 1.4 .6 INTERACTION CURVES FOR MIXED MODE LOADING Since wood is often subjected to combined loading producing both opening and s l i d i n g deformation modes, interaction curves are necessary to predict the f a i l u r e strength of the structures. Experiments were carr i e d out to establish the interaction curves for both cracks and notches. 1.5 SUMMARY OF OBJECTIVES 1. To determine the mode I c r i t i c a l stress intensity factor for white spruce. 2. To determine the mode II c r i t i c a l stress intensity factor for white spruce. 3. To study the specimen size effect on the mode I fracture toughness. 4 . To study the moisture content effect on the fracture toughness of the white spruce. 5 . To establish the family of mixed mode f a i l u r e 10 i n t e r a c t i o n curves f o r c r a c k s and notches of white spruce. 6 . To d i s c u s s the r e s u l t s , t h e i r a p p l i c a t i o n to design p r a c t i c e and suggested d i r e c t i o n s f o r f u r t h e r r e s e a r c h . 2. THEORY 2.1 INTRODUCTION This chapter explains the basic theory of linear e l a s t i c fracture mechanics as applied to cracked (zero angle notch) and notched wood specimens. The theory requires the computation of the stress intensity factors, which are associated with the singular stresses near the crack t i p or notch root. For s i m p l i c i t y , the theory w i l l be introduced s t a r t i n g from a sharp crack in an isotropic and orthotropic specimen. This w i l l be extended to the more general cases of notches. This information, combined with the experimentally determined c r i t i c a l stress intensity factors, makes the prediction of the ultimate strength of fracture structures possible. The size and the moisture content of the specimen had been proven to have pronounced ef f e c t on the strength of the specimen and w i l l be discussed in the following sections. This chapter w i l l end with the theory of interaction curves of stress intensity factor. 2.2 APPLICATION OF LEFM ON WOOD STRUCTURES The concept of linear e l a s t i c fracture mechanics for determining the fracture f a i l u r e mode and strength in wood have received a abundant amount of attention in the past several years. Fracture mechanics is concerned with 1 1 12 structural f a i l u r e by catastrophic crack propagation at average stress below the normal f a i l u r e stress l e v e l . In many applications, fracture techniques are used to eliminate t h i s kind of f a i l u r e by controlling the flaw size or in the cases that these defects are unavoidable, by considering the effect of the flaw on the allowable load on the structures. Complete studies of fracture behavior cover both the stress analysis aspects and the resistance of the material to the stress imposed. In this chapter, the purpose i s to develop the s i g n i f i c a n t stress analysis d e t a i l s and relevant parameters, which w i l l govern the f a i l u r e strength of structures containing cracks or notches. The r e d i s t r i b u t i o n of stress in a body due to the existence of a crack or a notch w i l l be analyzed by the LEFM method. The greatest attention should be paid to the high elevation of stresses at the v i c i n i t y of the crack t i p which w i l l usually be accompanied by at least some p l a s t i c i t y and other non-linear e f f e c t s . The surfaces of a crack or a notch dominate the d i s t r i b u t i o n of stresses near or around the crack t i p since they are stress-free boundaries of the body. Other remote boundaries and loading forces affect only the intensity of the l o c a l stress f i e l d at the t i p . The stress f i e l d s near a crack or a notch t i p can be divided into three basic types, each associated with a l o c a l mode of deformation as i l l u s t r a t e d in Figure 1. In mode I propagation, the crack surfaces open normal to themselves; in mode II the surfaces s l i d e tangential to Figure 1 - Three basic modes of crack surface displacement CO 1 4 themselves l o n g i t u d i n a l l y ; and i n mode I I I the su r f a c e s s l i d e t a n g e n t i a l to themselves and p e r p e n d i c u l a r to the propagation d i r e c t i o n i n a t e a r i n g f a s h i o n . The s u p e r p o s i t i o n of these three modes i s s u f f i c i e n t to d e s c r i b e the most ge n e r a l 3-dimensional case of l o c a l c r a c k - t i p deformation and the a s s o c i a t e d s t r e s s f i e l d s . As we know, wood i s a h i g h l y a n i s o t r o p i c , heterogeneous and porous m a t e r i a l with v a r y i n g mechanical p r o p e r t i e s i n d i f f e r e n t d i r e c t i o n s . The l o n g i t u d i n a l - r a d i a l and the l o n g i t u d i n a l - t a n g e n t i a l planes are n a t u r a l cleavage planes. For specimen s u r f a c e f r e e of checks, crack propagation u s u a l l y occurs along the g r a i n . Due to the high o r t h o t r o p y , a system s p e c i f y i n g the p r i n c i p a l crack propagation d i r e c t i o n s w i t h i n the o r t h o t r o p i c planes of symmetry w i l l be necessary. For each of the three modes of propagation, s i x p r i n c i p a l systems of propagation e x i s t s as shown i n F i g u r e 2. A system of propagation i s i d e n t i f i e d with a p a i r of l e t t e r s , the f i r s t r e f e r s to the d i r e c t i o n normal to the f r a c t u r e s u r f a c e , and the second r e f e r s to the d i r e c t i o n i n which the crack plane propagates. The s o l u t i o n s of the i s o t r o p i c c r a c k problem r e q u i r e the computation of s t r e s s i n t e n s i t y f a c t o r s . A v a r i e t y of s o l u t i o n s are a v a i l a b l e f o r sharp c r a c k s , p a r t i c u l a r l y f o r symmetrically p l a c e d c r a c k s i n i s o t r o p i c m a t e r i a l s , and many of these s o l u t i o n s have been summarized by P a r i s and S i h (1957). u . r.l. l.r. t.r. r.t. l.t. F i g u r e 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, a c c o r d i n g to A . P . Schniewind and R . A . P o z n i a k 16 Since the formulation of this fracture theory is based on continuum mechanics where material differences are not considered, i t is expected that fracture mechanics should be equally applicable to orthotropic materials such as wood. Similar formulation of stress intensity factors in anisotropic materials has also been proposed by Paris and Sih (1965). Walsh (1972) had investigated the effect of orthotropy on computed stress intensity factors for several geometry and concluded that for rectangular specimen of su f f i c i e n t length, orthotropic and isotropic results agree closely. However, i t would appear only for the case of symmetrical and skew-symmetrical s e l f - e q u i l i b r a t i n g loading on cracked i n f i n i t e plates. Thus, to apply LEFM on wood products, one must aware the dependence of stress intensity factor on the material properties of the specimen. Another assumption that is necessary to apply LEFM i s the condition of small scale y i e l d i n g (SSY) around the crack t i p . Under this condition, K can characterize the crack t i p stresses and strains even within the p l a s t i c zone. This i s equivalent to the p l a s t i c zone radius ^ rp^ being much smaller that the radius of e l a s t i c f i e l d zone (r e).Since r e i s proportional to the specimen size, in p r i n c i p l e , the SSY condition can always be s a t i s f i e d , i f one uses a large enough specimen. The ASTM recommended size requirements for v a l i d K c measurements are: £ > 2.5 ( 2 > 1 ) y 1 7 where a i s the length of the crack, L i s the distance ahead of the crack front, o-y is the y i e l d i n g stress or that stress that results in gross deformation. The condition of r e » Tp i s a s u f f i c i e n t but not necessary condition for the v a l i d i t y of the LEFM. The condition could be unduly r e s t r i c t i v e in terms of specimen size requirements. The necessary condition for the v a l i d i t y of the LEFM i s that K would be able to characterize the crack t i p stress or st r a i n component at the location of the defined fracture process. The zero notch root assumption w i l l be imperative for applying LEFM on notch problem. In order to have SSY at the notch root, the notch radius should be small compare to other dimensions, so that variation of the notch radius does not influence the surrounding stress and s t r a i n f i e l d s . Ewing and Williams (1979) had studied the importance of the sharpness of the i n i t i a l notch on the fracture toughness of Scots Pine and found that the mode I fracture toughness tends to increase as the radius increases. 2.3 FORMULATION OF THE STRESS INTENSITY FACTORS FOR CRACKS The formulation of the solution of the stress and displacement f i e l d s associated with each mode using the LEFM methodology follows in the manner of Irwin (1957) based on the method of Westergaard (1939). Mode I and II can be analyzed as two-dimensional plane problems with the symmetric and skew-symmetric stress f i e l d s with respect to the crack plane. Mode III can be treated as a pure shear 18 problem. From the cracks handbook by Hiroshi Tada (1973), the resulting stress and displacement fields for the isotropic case are given as follows with the notation referred to Figure 3. Mode I : a = - cos £ [1 - s i n £ s i n |£] + a + 0(rl'2) (2.2a) x ( 2 n r ) 1 / 2 2 2 J xo o = — cos £ [ l + s i n £ s i n |£] + 0 ( r l / 2 ) (2.2b) y (2*r)l>2 2 2 2 T = — s i n i cos i c o s — + 0 ( r l / 2 ) (2.2c) X y (2Trr)l/2 1 1 • 2 and for plane strain (with higher order terms omitted) o = v(a + a ) (2.2d) z x y T = T = 0 (2.2e) xy yz The c o r r e s p o n d i n g d i s p l a c e m e n t s a r e : u = ^ [ r / ( 2 7 r ) ] 1 / 2 cos J - [ l - 2v + s i n |£] (2.2f) v = ^ [ r / ( 2 n ) ] l / 2 s i n f [2 - 2v - cos |£] (2.2g) F i g u r e 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 s t r e s s e s ahead of a c r a c k . From P . C . P a r i s and G . C . S i n 20 w = 0 (2.2h) Mode II : av — sin £ [2 + cos | - cos + a + 0(rl '2) x /o 2 2 2 xo (2*1-_ _ II . 9 9 39 , i / o s a = sin TT cos -r- cos + 0 ( r 1 / z ) y (2^)1/2 2 2 2 X y (2Trr)l/2 2 2 2 and for plane strain (with higher order terms omitted) (2.3d) T = T = 0 (2.3e) xz yz with dispalcements: u = — i - [r/(2Tr)]l/2 s i n 1 [2 - 2v + cos 29 2 (2.3f) v [-1 + 2v + sin Y~] (2.3g) f 21 w =0 Mode III : hn J e (2.3h) T XZ T (2Tvr)l/ 2 s i n f + T x z Q + 0(rl/2) ( 2 . 4 a ) ' y z (2Ttr)l/2 cos i + 0(rl/2) ( 2 > 4 b ) a = f f = a = t =0 (2.4c) x y z xy v w w -IiI[(2r)/,r]l/2 s i n |. ( 2 > 4 d ) u = v = 0 (2.4e) Equations (2.2) and (2.3) have been written for the case of plane strain, but can be changed to plane stress by taking a = 0 and replace v by v/(]+v). As seen from equations (2.2) and (2.3), the formulae include higher order terms such as uniform stresses parallel to cracks, oXQ and r X Z O f a n ^ terms of the order of square root of r, 0 ( r 1 / / ^ ) . But these terms can be neglected since as the value of r approaches 0, (i.e. close to the crack tip), the singular term 1/Vr becomes the governing term in the equations. 22 The parameters Kj , K I I f i - n these equations are called the stress intensity factors for the three modes respectively, as shown in Fig 1. It is found that these parameters are coordinate-independent, so they can be thought of as the magnitude of the stress fields surrounding the crack tip. The parameters, K, are determined by the other boundaries conditions and the imposed loads. Consequently, formulae for their evaluation come from a complete stress analysis for the specimen configuration and loading. A crack stress fi e l d for certain loading and geometry is represented by a unique combination of the three stress intensity factors. Since they are correlated parameters , the failure criterion w i l l depends on a l l three. From the Equations (2.2), (2.3) and (2.4), we observe that the stress intensity factors have units of (ForceJxtLength) -^/ 2. Since they are linear factors in a linear elastic stress solution, the stress intensity factors are linearly related to the applied loads. Sih et al (1965) derived formulae for the stresses in a small region surrounding a crack tip of orthotropic material using a complex variable formulation for the LEFM method. Their results for a crack parallel to a material axis and coincident with the negative x-axis (Figure 3) are : Symmetric (about x-axis) plane loading, Kj - 6 , 6 , 6 / 2 n r n : "1 P2 "2 H K , K I 1 31 B2 /2Trr • 6 r 6 2 ^ B l •xy ~ B1 S2 ,1 1 and Skew-symmetric (about x-axis) plane loading, *II r 1 , B? B l c = " X /2 ir i KII D r 1 r 1 1 M 0? "^ R e [*V * ? ( * 7 KII 1 B l B2 where B. = /cos 9 + ie . sin 9 i = 1,2 J J i = /^T Re = real part 24 B? =• e 2 [« + (,c 2-l)l/2] -1 -1 B2. - E 2 [< - ( < 2 - l ) ! / 2 ] > for plane stress xy (2.6g) (2.6h) (2.61) And directly ahead of the crack the stresses are : II /2TTX xy • 2 TTX (2.7) where the K's values are dependent on remote boundaries conditions, highly dependent on geometry, and slightly dependent on orthotropic parameters (for finite bodies). Some typical symmetric and skew-symmetric stress fields are shown in Figure 4 and Figure 5. The stresses in polar co-ordinates are obtained by doing the transformation as follows : a = a cos 29 + o sin 2 6 + 2a cos8 sin9 r x y xy (2.8a) a = a sin 29 + a cos28 - 2a cos9 sln9 o x y xy (2.8b) o r 6 - (o y - a x) cos8 sin9 + o x y ( c o s 2 9 - sin 2 9 ) (2.8c) The symmetric and skew-symmetric stress fields w i l l only exist i f the crack is parallel to an axis of elastic material symmetry. Then, the stresses around the crack t i p 25 so* Primary s tress f i e l d E : E . . = 20.00 lao* i T 270* Figure 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 for a 0° crack Primary s tress f i e l d E : E = 20.00 E X : G y = 0.90 = 0.02 P o s i t i v e Negative F igure 5 - Second primary s i n g u l a r s t r e s s f i e l d for a 0° crack 26 will be the superposition of some linear combination of these two fields. 2.4 STRESS INTENSITY FACTORS FOR SHARP NOTCHES 2.4.1 INTRODUCTION The stress intensity factors have been shown to characterize the stress fields surrounding the fracture plane. However, the literature on stress intensity factors is mostly concerned with sharp cracks. Although there are few papers on notches, the method was based on the nominal fracture stress combined with limited experimental results. Leicester(1971) has presented a new method for the analysis of the stress fields at the root of a mathematically sharp notch of arbitrary notch angle. The application of this method indicates that in general there are two stress singularities of stress at the roots of notches in typical orthotropic structural materials. The magnitude of these stress fields are noted as the stress intensity factors for notch angle. He used the indices of A and B referring to the two stress fields and these two stress fields are different from the symmetry and skew-symmetry stress fields in most cases. The significance of the stress intensity factors lies in the fact that a criterion for crack propagation from the notch root may be formulated as follows: 27 where K A C and Kg^, the c r i t i c a l stress intensity factors, and the interaction equation (2.9) are determined by direct measurement. For the specific case of a zero angle notch, i.e., a crack, the notation Kj , Kj£, K J J and ^HQ will replace the notation KA , KA^, Kg and Kg^. 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 3x _xy_ 3y = 0 3a 3? 3a + - S L 3x = 0 (2.10a) (2.10b) The strain-displacement kinematic relationships are: X 3u 3x 3y_ 3y xy 3u t 3v 3y 3x (2.11a) (2.11b) (2.11c) 28 and the stress-strain equations for an orthotropic material with plane stress condition are : o u x yx E x = F T _ E — ° y (2.12a) x y J a u y xy G y E ~ E a x (2.12b) y x a xy Gxy = G (2.12c) xy From strain-energy considerations i t is known that : u u xy _ yx E E x y (2.13) Equations (2.11a,b,c) may be shown to result in the following equation of compatibility : 3 2e 32e 3 2e , , x y + £ (2.14) 8 x 3 y 3y 2 3x 2 Equations (2.l0a,b) are satisfied when the stress components are expressed by Airy's stress function 0 through 32<b (2.15a) 3x 2 7 29 (2.15b) 3y 2 A | % - = -a (2.15c) 3x3y xy ^ ' Substitude (2.l5a,b,c) into (2.12a,b,c) we get the expressions for strains in term of Airy's stress functions : £ x - ^ - ( f ) " ^ ( ^ ) (2.16a) X 3y 2 E x E y 3x 2 J 3x z y x 3y z Exy " " f x V ^  ( 2 ' 1 6 C ) Substituding (2.l6a,b,c) into the equation of compatibility (2.14), one finally obtains : 3x^ e 2 3x 23y 2 3yI* (2.17a) 30 where E e E y < - I ^ y ^ 2 t ( 1 / G xy) - < W " <W- <2'17b) In order to solve equation (2.17a), i t may also be written as : t 1 T + a I 1 T K i i - + « I I ^ - ] * =0 (2.18a) 3x2 1 3y2 3x2 1 1 dy2 where 1 r o I / 2 , aI = — [< + (<2-l) ] e 2 2 1 1 ' J (2.18b) a l l ? LN ^ ~ x ' J (2.18c) Since 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 fields will be the constants e and K defined as (2.17a,b). 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 elastic properties of structural materials published by Leicester (1971) is shown in Table 1. The solution of the fie l d equation with subscripts I and II indicates that two further transformed co-ordinates may be defined. These are x = x = x (2.19a) y " V l = a I l y I I (2.19b) and shown in Figure 6. The f i e l d equation (2.18), written in the new co-ordinates, is : [ j l . + _!!_][_>!_ • _ » ! _ ] » . 0 (2.20) 3V V »V ' - I I * For the case a i ~ a n > equation (2.20) reduces to a biharmonic equation, and in terms of polar co-ordinates : V 2 (v 2 ($)) = 0 (2.21) where the operator V 2 is T a b l e 1 E l a s t i c P r o p e r t i e s o f S t r u c t u r a l M a t e r i a l s Type of M a t e r i a l e K I s o t r o p i c s o l i d 1.0 1.0 T y p i c a l Wood ( LR & LT p lanes ) 2.0 2.0 T y p i c a l plywood of ba lanced c o n d i t i o n 1.0 4.0 T y p i c a l f i b r e - r e i n f o r c e d p l a s t i c 2.0 4 .0 From R . H . L e i c e s t e r (1971) 33 F i g u r e 6 N o t a t i o n f o r r e a l s p a c e and t r a n s f o r m e d 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 s p a c e , (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 . From R.H. L e i c e s t e r (1971) F i g u r e 7 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 r o o t s . From R.H. L e i c e s t e r (1971) 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 1 (2.24) where X is a constant. A suitable pair of harmonic functions for the solutions is the following : $ = A^* cos(X6 I) + A 2 r ^ sin(X6 ].) (2.25a) ' n = V I I c o s C x e n ) + V i i s i n ( x e n > (2.25b) where A1 to A^ are arbitrary constants. Substituting (2.25) into (2.l5a,b,c) leads to the following : a x = A(A-l)r^" 2(l/a2)[-A 1cos(X0 I-29 1) - A 2 s i n ( X 6 ^ 2 6 r ) j + X ( X - l ) r X . I 2 ( l / a 2 I ) [ - A 3 c o s ( X 9 I I - 2 e i I ) - A^slnUe^-Ze ) ] (2.26a) 35 o y = X(X-l)r^ 2 [A 1 cos(X8 I-26 I) + A 2sin(Xe ].-29 i)] + X ( X " l ) r ^ 2 [A 3 cos (Xe ] ; I -2e i I ) + A l t sin(Xe i ].-26 I I)] (2.26b) o = X ( X - l ) r ^ ~ 2 ( l / a T ) [ A , s i n ( X 9 - 2 9 T ) - A,cos(X9 -29 T)] xy I I 1 1 I I 2 I I J + X ( X - l ) r ^ ~ 2 ( l / a I I ) [ A 3 s i n ( X 9 I ] . - 2 9 I I ) - A ^ c o s U g ^ g ^ ] (2.26c) The constants A1 to A^ are obtained by substitution of eqnations (2.26a,b,c) in the boundary conditions OQ= ° R Q = Q at the notch edges along 0=0A and 6=9^, as shown in Figure 7. These four conditions lead to a matrix equation : a!2 a!3 a 2 i 3 2 2 a23 aZk a. a_ _ a 31 32 33 . a m \ l • a-3 a ^ = 0 (2.27) For any particular notch, the eigenequation (2.27) is satisfied when a value of X has been found such that the determinant |a m n| is zero. Figure 8 is a plot of this determinant for a notch in Douglas f i r . It can be shown that the eigenvalues of a singular eigenfield are limited to the range 1 < X < 2. Within this -3-0 I I I I I I I I 1-0 1-5 2-0 2-5 3-0 3-5 4-0 Parameter A F i g u r e 8 - D e t e r m i n a n t o f e i g e n e q u a t i o n f o r n o t c h i n D o u g l a s f i r . From R.H. L e i c e s t e r ( 1 9 7 1) CO cn 37 range i t is found that there is always at least one eigenvalue, and if the notch angle is small there will be two eigenvalues. Denote these two eigenvalues, XA and X B with XA < Xg. The eigenfields associated with these eigenvalues will be defined as 'primary' and 'secondary' eigenfields respectively. In general, the primary eigenfield will dominate at the notch root as r approaches 0 except in pure shear mode where the secondary eigenfield w i l l govern. Since equation (2.27) is a homogeneous equation, an additional constraint must be applied besides the four boundary conditions. The magnitude of the stress fields will f u l f i l this requirement. The stress intensity factors corresponding to the primary and the secondary stress fields will be denoted by KA and KB respectively. The definitions of KA and KB are quite arbitrary but in most cases of orthotropic material, they are defined as follows : oe(e=n) = K. /(2-jrr) 2-A, cr9(e=fi.) - K B / (2*r) 2-X, (2.28a) (2.28b) where fi is the angle of crack propagation and is usually the grain angle in the wood. For the special case of a sharp crack, KA = Kj and Kg = K T T . However, the terms "opening mode" and "sliding mode", 38 which are usually associated with a sharp crack, are applicable to notches only if the notches are symmetrical with respect to the axes of elastic symmetry. The failure criterion characterizing the onset of propagation of a crack from a sharp 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 A C and K B C respectively. From the equations of stress fields, we can derive the equations of strain and displacement f i e l d in terms of the stress intensity factors. The method for obtaining these factors is discussed in the next section. 2.4.3 IMPLEMENTATION IN A FINITE ELEMENT PROGRAM A singular finite element method is manipulated to compute the stress intensity factors. The finite element mesh consists of three regions — t h e elements around the notch tip, the elements remote from the notch tip, and the transitional elements in between to assure the compatibility of the elements. Around the notch tip, the quadratic displacement f i e l d is enriched by the singular terms as follows : u i = a i + a 2 c + a 3 n + ak^2 + a 5 C n + a 6 n 2 + a ? c 2 n + a 8 ? n 2 + K ^ U . n ) + KBg.(c,n) (2.29) where $ and TJ are natural co-ordinates. Substituting the nodal displacements (2.29), we get : {u.} = [M]{a} + KA{f.} + KB{g.} or {a} = [M]-l[{ui} - K A{f i } - KB{g.}] Substituting back into equation (2.29), ui(?.n) = [N(c,n)]{u.} + KA [^(Cn) " [N ( C ,Ti)]{f .}] + Kg [ g ^ . - l ) - [N(c,n)]{gi}] Using the virtual work principle results in / « T {<5}[B]T [[D][B]{6} + {K}T[D]{J2}]dV = « T {6}{R} V for a virtual change in {6}, and 40 / 6 T (K}{n} T [[D][B]{6> + {K}T[D}{fi}]dV = 6 T {K}{R } V k (2.34) for a virtual change in {K}. Combining equations (2.33) and (2.34), we get : [S] {Pj} {P2} " {V* [ C n ] [c21] L { P 2 } T [c 1 2] [c22] {6} {R} (2.35) where [S] = / [B] [D][B]dV V (2.36) { P J = / [B] [D]{Q }dV V (2.37) [C ] = / {Q.} [D]{n.}dV J v (2.38) 41 The element stiffness matrix can be assembled to form the global stiffness matrix, and the element load vector, the global load vector. The elements far from the notch tip will be ordinary quadratic, regular elements, whereas the transitional elements are introduced in a transition zone between fully singular and regular elements. The displacement f i e l d of the transitional elements will be : u.(?,n) = [NU,r,)]{u.} + R(c,n){K A [f i (c,n) - [N( c,n){f.}] + K B [g.(c.,n) - [N(c,n)]{g i}]} (2.39) where R($,TJ) is bilinear transition function with R = 1 at singular-singular boundary and is equal to 0 at singular-regular boundary. The use of the bilinear transitional elements ensures conformity between elements. After forming the stiffness matrix, the solution method will be the same as for an ordinary fi n i t e element problem. From the calculated KA and Kg values, which indicate the magnitude of the eigenfields, we can get the Kj and K J J stress intensity factors by assigning the amounts of opening and shear-sliding in each eigenfields. Thus, for a direction of crack propagation ^, — 2 " A A K = /2TT lim (r A o Q ( r , 9 = 1 r*0 9 — VAA = /2TT lim ( K A f A ( 0 = *) + Kg r f B(6 - i | 0 + ... (2.40) r+0 42 - 2 _ X A K n = /2TT lim ( r T r Q(r,e= 1( l) r+0 = /2TT lim (KAgA(9=*) + K Br ° gB(9=*) + ... (2.41) r+0 then K I = / 2 L F K A V 9 = * ) f 0 r X A < XB (2.42a) K l = / 2 7 [ K A F A ( 9 = * ) + K B f B ( H ) l f o r A A = XB (2.42b) and K I I = V 2 i r KA 8A ( e"* ) for 1 < X B (2.43a) K I I = /2ir [KAgA(9=*) + KRgw(6-*)] B°B> for A, = A (2.43b) For notches, the primary eigenfield will always dominate the stress and strain fields around the notch root except loading in pure shear-sliding mode. Lum (1986) has developed the program NOTCH, which utilized this method and has shown good results in comparison with the collocation method used by Gandhi (1972). 43 From equation (2.28) we can observe that the dimension 2-XA of KA and Kg is (Force)x(Length) which depends on the material elastic properties and grain angle. Therefore, for different angle of notches, we have different dimensions of stress intensity factors. Thus, in order to derive a failure criterion for sharp notches, we must include the parameter X in our criterion. 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 full-size structure w i l l f a i l at the same nominal stress. However, failure of large structures at a lower nominal stress indicates that there may be a specimen size effect. Most of the previous literature has applied a st a t i s t i c a l model based on the weakest link principle to account for the strength of reduction. Such a theory assumes that failure of a single volume element leads to the failure of the whole specimen. This has an obvious analogy with the strength of a chain in which weakest link govern the strength. 2.5.1 EFFECT OF MATERIAL HETEROGENEITY Barrett (1976) has applied this method to compare Mode I fracture toughness data obtained from specimens of differing thickness as multiple crack-fronts. Using the 44 Weibull analogy, Barrett defined the cumulative distribution function for c r i t i c a l stress intensity factor with total crack-front width B assuming a l l cracks have the same nominal stress intensity factor : F ( K I C ) = 1 - e x p [ - ( K I C / m ) k B] ( 2 . 4 4 ) where k and m are the shape and scale parameters respectively. An expression relating the fracture toughness of two specimens with the crack-front widths, B and B*,is : KIC = ,B* 1/k K*c B ; (2.45) Obviously, if we plot Kj^, against specimen width on a log-log scale, the slope of the regression line should be -1/k. 2.5.2 EFFECT OF THE CHANGE OF THE STRESS STATE Another model to explain the crack-front width effect is proposed by Ewing (1979) based on the stress state in the specimen, and in particular the residual stresses induced by the drying which increases the toughness and constraint stresses which cause a decrease. 45 A useful model for describing plane stress-plane strain effects has been used in other materials and has some u t i l i t y here. Figure 9 shows a cross section diagram of a specimen, the region near the surface within the area H is in a plane stress condition. The region in the middle is in the plane strain stress condition. Let denote and be the c r i t i c a l fracture toughness for the plane strain and plane stress condition respectively. Then the average c r i t i c a l stress intensity factor K'r for D>2H w i l l be : DK' c 2HK C 2 + (D-2H) K c l K' c KC1 + D 2H ( KC2 " K C 1 } (2.46a) (2.46b) Usually K C 1 < K C 2 so that we'll have a positive slope if we plot K'c versus inverse thickness and an intercept of K C 1 at D - 1 =0 . The depth of the value H of the plane stress region can be computed from : H (2.47) where is the yield stress or the stress which results in gross deformation. The plane strain condition can only exists as long as there is enough thickness to provide the constraint and in general : F i g u r e 9 - S t r e s s s t a t e i n specimen 47 D > 4H (2.48) In general cases, if D<2H, then we'll have a complete plane stress condition and K'c = K c 2. For D>2H, then plane strain condition will result in the middle that decrease the average stress intensity factor. For the notched specimen, there is a similar size effect on the c r i t i c a l stress intensity factor proposed by Leicester (1973). From equations (2.26a,b,c), we can rewrite it for the plane stress at failure due to the primary stress singularity f i e l d as follows : where a^j is the stress component around the notch root and Af is a constant that depends on the notch root angle, material and detail of the notch root. From the dimensional analysis, we get : s = A f (2.49) (2.50) where a is the normal stress, 48 L is a characteristic length denoting the size of the element, B is a dimensionless constant depends on the geometry and loading. Hence for a structural element of specific shape and loading, we can write : CTof " V L S S > 0 (2.51) where aQ£ is the value of o Q at failure and A1 is a constant. Similarily for the secondary stress f i e l d : ° o f L q = A2 q > 0 (2.52) where A 2 is another constant. Since s>q i t follow that (2.51) always predominates provided the structural element is sufficiently large. We can also write (2.51) for two similar specimen with different size : 0 . . L„ — - ( r - f (2.53) ° f 2 L2 where 1 and a£2 a r e t n e nominal stress at failure and Lj and 1>2 are the corresponding characteristic lengths respectively. 49 There are also two conditions required to apply the size coefficient factor on two different specimens. Fi r s t l y , the detail of the notch roots must be identical (not scaled) and secondly, the predicted fracture stress must be less than about 70% of the nominal proportional limit 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 show thickness variation in the kiln dried state, but the toughness also varies with moisture content. In green timber, water is 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 is removed from the ce l l walls, resulting in shrinkage and significant property changes. It was expected that strength would increase with decreasing moisture content below 27%. Since the drying process increases strength, then i t is 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 I T L C of Western white pine varied according to : GITLC = 1 , 2 8 + ° ' 1 1 2 5 ( e " 2 4 3 > ^ " e ~ M / 6 ) (2.54) 50 where 6 = temperature (0°K) M = moisture content (% ovendry weight) GITLC = c r ^ t i c a 1 strain energy release rate for the TL system We can relate G I T L C to K I C for TL system by : a. . a. r = vl J 1 1 ~ i z I T L C - K I ' — * ( 2 . 5 5 ) where 2a ,„+a, • _ J 3 2 2 " 1 2 ^ 6 6 ^ / 2 • " 1- a 7 7 + 2 a n J ( 2 - 5 6 ) and a 1 1 , a 2 2 f e t c . , 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 . 2 8 + 0 . 1 1 2 5 ( 9 - 2 4 3 ) ( l - e ~ M / 6 ) ] ( 2 . 5 7 ) c where a.. a C = 1 / ' / _ 1 1 " 2 2 ( 2 . 5 8 ) 5 1 A similar equation can be derived for the c r i t i c a l fracture toughness of white spruce along the TL system. 2 . 7 MIXED MODE FRACTURE IN WHITE SPRUCE In recent years, attention has been dedicated to the fracture of wood in the opening mode (mode I). It has been shown that fracture toughness, K J C, is a geometry independent material property of wood characterizing the stress f i e l d around cracks, which govern the initiation of a crack propagation. The fracture failure of wood for sliding mode, mode II, has also been investigated by Barrett and Foschi ( 1 9 7 7 ) . However,wood structural members are often subjected to complex loading condition that result in mixed mode fracture. Available information under this mixed mode is limited, and will be discussed briefly 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 mode loading tests on balsa wood and proposed the f i r s t interation curve for the mixed mode fracture in the form: K I KII ( i r ^ ) 2 = 1 (2.59) Similar experiments had been conducted by Leicester ( 1 9 7 4 ) on pine and an interaction curve for Kj and K J J stress intensity factors is in the form : 52 I II v—+ y 1 (2.60) KIC KIIC Although most of the investigators favor the idea that an interaction relation exists, Williams and Birch (1976), based on their experiments on Scots pine, concluded that there is no effect on the opening mode failure due to the sliding mode caused by the shear stress. Their proposed failure criterion is : K l K ^ - l ( K I I > 0 ) <2'61) Recently, Woo and Chow (1979) investigated the mixed mode fracture in Kapur and Gagil using the single-edge notch and center-crack specimen. Their results has shown that there is some interaction relation between the mode I and mode II stress intensity factors under combined loading conditions. More recently, Mall and Murphy (1983) have studied the mixed mode fracture failure of eastern red spruce by means of single-edge notch and center-crack specimen with various crack inclinations in the TL system. Their results, have shown that the criterion for the mixed mode fracture failure of red spruce is same as equation (2.59). A more general form for the interaction curve for mixed mode fracture may be proposed : 53 K T K T T K IC n c (2.62) where constants a is determined experimentally. The interaction phenomenon will also be valid for mixed mode loading of sharp notches. As we know, the dimension of the c r i t i c a l stress intensity factor for sharp notches depends on the parameter X, which is material and geometry dependent. In order to generate the interaction curve for notches, we need to have the same dimensions. A rational method is to generate a family of interaction curves each with a different X, where X has a range of 1.5 to 2.0. Then each curve will 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, for example, X =1.5, 1.6, 1.7, 1.8, 1.9 and 2.0. For specific geometry and loading condition, the value of X is determined and the appropriate curve to estimate the KT and K T T values at failure. 3. EXPERIMENT PARAMETERS 3.1 INTRODUCTION The aim of carrying out experiments is to verify the applicability of the theories to real practice. Different experiments were designed to study the validity of c r i t i c a l stress intensity factors methodology in predicting the failure mode and strength of the real 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 is 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 orientation (see sec.2.2). Within a plane, the crack may propagate in different directions and in a combination of the three different fundamental modes (see Figure 1). Therefore, in preparing the specimen for the experiments, attention must be paid to the crack orientation. Also, due to the large number of possibilities of fracture orientation, a set of appropiate experiments must be designed in order to simulate the fracture behavior in real structures. First of a l l , the determination of mode I fracture toughness may lead to six different sets of experiments for 54 55 each different crack orientation, namely TL,RL,LT,LR,RT or TR, where the f i r s t letter refers to the direction perpendicular to the crack surface and the second letter refers to the crack propagation direction. However, the fracture toughness results of Schniewind and Centeno (1973) have shown that the fracture strength for the plane in longitudinal direction has values approximately one-tenth the values where propagation is across 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 failure planes need to be made. On a macroscopic level, most of the commercial boards under bending are observed to f a i l in the radial longitudinal plane. However, a closer look at the crack initiation will reveal that the failure has been initiated in the tangential-longitudinal plane. Therefore i t was decided to conduct the experiments with specimen in this category and i t is postulated that i t would be representative of the initiation of failure 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 sliding mode fracture toughness. 56 3.3 MATERIALS Thirty two kiln-dried nominal 2-in by 8-in White Spruce (Picea glauca) boards were purchased from one sawmill. For studies of between board variation of K j . c compact tension specimens, 125mm by 120mm by 38mm, were cut randomly from 10 different boards. From another eleven board, sixty two one-metre long end-split beam specimens were used to study the K J Q , K H C a n <^ mixed mode loading. Sixty-six one-metre mid-crack specimens were cut to study the mid-cracked beams under the condition of mixed mode loading and another twenty-four one-metre long specimens were cut and used for the study of between board variations of notched beams. Twenty of the one-metre long specimen were tested for the average modulus of elasticity (MOE) by stressing them under three-point loading and measuring the displacement. Due to the high bearing stresses induced at the supports, the specimens were tested about their weak axis — on the flat -- to eliminate the bearing effect. The average MOE obtained was approximately 9000 MPa. Since obtaining local modulus of elasticity and Poisson ratios is d i f f i c u l t , published values of these constants were used in the analysis. (E L =10163 MPa, ET=494 MPa, ER=830 MPa, GLT=663 MPa, GLR=700 MPa, ^LR=0.337, *>LT=0.40, vRL=0.0275, U T L ~ 0.0194 ). These elastic parameters were computed from regression equations as proposed by Bodig and 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 which maintained a nominal 9% (±1%) equilibrium moisture 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. The cracks were f i r s t cut by a bandsaw which produced a kerf of about 1/8" so as to have enough room to accomodate the crack starter (see Figure 10) and the smooth plastic plate for the mode II fracture toughness testing. Then just prior to testing, the bandsawn notch was sharpened with the crack starter to extend the sawn notch approximately 0.1 in. An importance consideration of the specimens prepared is the sharpness of the i n i t i a l crack. As we assumed a zero notch root hypothesis, we need to have some control of the crack sharpness. Leicester (1974) has shown that the influence of root diameter on the fracture strength of drilled notches is less than 10% for notch roots below 5 mm in diameter. Thus a single point crack starter with an included angle of approximately 10° and a tip radius of less than 0.5Mm was used. It is postulated that this would give the same result as the cracks made by a razor blade. 58 Figure 10 A steel crack starter with crack t i p radius 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 differential transducer (LVDT) in a l l cases (see Figure 1 1 ) , while a modified LVDT (as shown in Figure 1 2 ) was used to measure the crack . opening displacement (COD) and the relative longitudinal displacement of the crack in a cracked beam. In obtaining the c r i t i c a l stress intensity factors, some criterion was employed to determine the c r i t i c a l failure load during the testings. Load-deflection plots were generated at the time of testing on an X-Y recorder. Since these curves were similar to those encountered in the ASTM fracture tests, a similar method was adopted to check the validity of the results. Small amounts of slow crack growth are allowed in the ASTM test, but growth is limited to approximately 5 % of i n i t i a l crack length in a valid K I C test. Three different curves were encountered during the experiments (as shown in Figure 1 3 ) , and the load P Q corresponding to a 5 % offset from the i n i t i a l slope was used to compute the c r i t i c a l stress intensity factors. The compact tension specimens were done in the humidity room with an Instron universal testing machine which provided autographic recordings of the load and displacement. Cross head speed of 0 . 5 mm/min was selected which produced failures in about 1 minute. Al l the beam tests were done at room temperature (approx. 2 0 ° C ) with a Satec 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 d e f l e c t i o n measurement connect 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 measuring l o n g i t u d i n a l s l i d i n g d i s p l a c e m e n t Q < O P =P Q max P„<P Q max P„<<P Q max DISPLACEMENT (COD o r M i d s p a n D e f l e c t i o n ) F i g u r e 13 - T y p i c a l L o a d Vs D i s p l a c e m e n t C u r v e s : ( a ) P Q = P ( a c c e p t a b l e ) ; (b) 1 . 2 P Q * p m a x ( a c c e p t a b l e ) ; ( c ) 1.2P Q<P m a xTunSgeeptable ) 62 plotting of the load and displacement. Cross head speed was controlled so as to produce failures 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 as the variance were calculated for each experiment. Moisture content data were also collected to assure that a l l the specimens were tested under uniform conditions and provide a basis for adjusting data as required. The values of the deflection is not significant in these experiments except that the load deflection curve is used for determining the failure loads. As mentioned earlier, the stress intensity factor is linearly related to the applied load in a linear elastic material, so the failure loads were used to compute the c r i t i c a l stress intensity factors by the following equation : K i c = F K x 1 = 1.2 (3.1) where PQ is the failure load; P * is the arbitrary load entered into the finite element computer program to compute the corresponding stress intensity factor Kj*, and Kj^ is the c r i t i c a l stress intensity factor for the specific specimen geometry and loading. 63 Calculated stress intensity factors were collected and used to generate the interaction curves for cracks and notches. The curves were expected to have a form similar to a quadrant of an ellipse as proposed by many investigators as shown in Figure 14. In order to verify the equation or derive a new empirical formula, we need to have evenly spreaded failure points on the curves. Different experiments were designed to generate the curve.s as well as to study the moisture constant effect and the specimen size effect. The details of each experiment as well as the results will be described in the next chapter. 64 4. EXPERIMENT DESCRIPTIONS AND RESULTS 4.1 INTRODUCTION This chapter gives a f u l l description of the procedures and results of the experiments. The eight experiments conducted will be discussed in separate sections, namely No.1 to No.8. Emphasis was put on studies of mode I c r i t i c a l stress intensity factors which tend to dominates in wood fracture failure. Effects of variation of the moisture content and crack-front width on the mode I fracture toughness was also studied. At the beginning of this thesis, i t was expected that the c r i t i c a l stress intensity factors would be a constant for specific geometry, material and crack orientation. However, as the experiments progressed, i t was found that K J C may also depends on the actual size of the specimen. Experiments were conducted to study the effect of the variation of the uncracked length 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 the mode I fracture toughness for white spruce. Effects of the grain orientation and specific gravity were also investigated. Two type of specimens were used to study the fracture toughness --the ASTM compact tension specimen (CTS) and the double 6 5 66 cantilever beam specimen (DCB) as shown in Figure 15 and 16. The length to width ratio of the compact tension specimen was as recommended by the ASTM Standard with thickness of 38 mm. A total of 75 specimens were machined from two 20ft boards with the dominant system of propagation being TL. The CTS is also designed to study the effect of the grain orientation on the mode I fracture toughness as well as the between board variation of K I C . 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 which produced failures in about 1 minute. The experiment setup is shown in Figure 17. For the double cantilever beam test, a total of 18 specimens were cut from three 20ft kiln-dried boards and were previously conditioned to approximately 10 per 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 failure 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 both the CTS as well as the DCB specimens. This was accomplished with the finite element program NOTCH, using quadratic 67 100 N K = 0.06901 MPa^ KII= ° D = 1 i n . Figure 15 Configuration of Compact Tension Specimen (CTS) P = 100 N 50 mm —* Kx•••= 0:0405 MPa/ra Figure 16 Configuration of Double Cantilever Beam specimen (DCB) 68 F i g u r e 17 The experimental setup of the compact tension specimen. 69 isoparametric, singular-enriched elements. An arbitrary load of 100 N was selected in obtaining the stress intensity factors and the c r i t i c a l fracture toughness were easily obtained by eqn. (3.1). The finite 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 different grain orientations. A trend of decreasing mode I fracture toughness from TL system to RL system is apparent. A plot of Kj^ versus annual rings angle is shown in Figure 20 and a line was fitted by least-squares to the data. Thus, K I C = 0.3933 - 2.1323 x 10"3 e (4.1) where Kj^ is the mode 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 22 shows the cumulative distribution functions of the K J Q for the CTS and DCB specimens. They were best-fitted by the Weibull distribution curves which are often used to model the strength distribution of wood products. From Table 2, i t is obvious that the Kj^ values for the CTS is lower than the DCB specimens. This implies there is some relationship between the KJQ and the specimen size, and this was investigated in the later phase of the experiments. 70 s t e e l elements s i n g u l a r e l e ments F i g u r e 18 F i n i t e Element Mesh f o r t h e CTS P \ V singular elements P Figure 19 F i n i t e Element Mesh for the DCB specimens T a b l e 2 E f f e c t o f t h e A n n u a l R i n g s A n g l e on K S p e c i m e n A n g l e t o s a m p l e M . C . S . G . P m x / ^IC T y p e R L ( d e g ) n o . (7„) P p ( M p a / m ) (psi/Tn) 0 - 10 6 9 . 5 4 0 . 349 1. 02 0 . 3 8 7 2 3 5 2 . 3 1 11 - 20 4 9 . 4 0 0 . 354 1 .04 0 . 3 6 5 0 3 3 2 . 1 7 21 - 30 1 10 . 0 0 0 . 336 1 .06 0 . 3 6 6 6 3 3 3 . 6 0 31 - 40 5 8 . 4 4 0 . 374 1 .05 0 . 3 1 5 3 2 8 6 . 9 6 41 - 50 12 8 . 49 0 . 386 1 . 0 3 0 . 2 9 7 2 2 7 0 . 4 4 51 - 60 8 8 . 9 6 0 . 380 1 . 0 3 0 . 2 8 2 2 2 5 6 . 8 0 61 - 70 1 9 . 0 0 0 . 4 0 0 1 . 0 5 0 . 3 0 6 1 2 7 8 . 5 5 71 - 80 3 9 . 0 0 0 . 3 9 3 1 .03 0 . 2 2 4 5 2 0 4 . 3 0 81 - 90 33 9 . 3 3 0 . 388 1 .05 0 . 2 0 5 1 2 2 8 . 4 1 90 17 9 . 4 3 0 . 370 1 .05 0 . 3 8 7 9 3 5 2 . 9 9 0.50-S6 a , 20 30 40 50 60 70 A N N U A L R I N G S A N G L E ( DEG. ) Figure 20 t Mode I Fracture Toughness Variation with Annual Rings Angle MODE I CUMULATIVE DISTRIBUTION Compact T e n s i o n Spec imen Kic-° F = 52ILE: 5CUILE: MEAN: 95* ILE : OF DISP: DATA (SOLID L INE) : N=33 DATA ST.DV. WE I BULL 0. 133 0.010 0.138 0.192 0.010 0.196 0.205 0.051 0.205 0.315 0.038 0.301 0.250 0.249 5 INTERVAL CHI-SQUARE F IT : 1.39 3-PAR WE I BULL (DASHED): SHAPE = 1 .6554 SCALE = 0.0920 LOCA. = 0.1227 ST.DV. 0.007 0.011 0 .051 0.027 o.o 1 1 1 1 1 1 1 0.15 0.2 0.25 0.3 0.35 0.4 0.^5 0.5 K I C > MPa/m Figure 21 Cumulative Dis t r i b u t i o n Curve of the Compact Tension Specimen MODE I CUMULATIVE DISTRIBUTION Doub le C a n t i l e v e r Beam F = 1 8.2095 0 .4099 DATA (SOLID L INE) : N=17 DATA ST.DV. WE 1 BULL 5XILE: 0.291 0.020 0.285 50* ILE : 0.391 0.010 0.392 MEAN: 0.387 0.051 0.386 95J ILE : 0.502 0.035 0.468 OF DISP: 0.132 0.144 INTERVAL CHI-SQUARE F IT : 1.52 2-PAR WE I BULL (DASHED): SHAPE = 8.2095 SCALE = 0.4099 0.2 i— 0.3 —1— 0.4 ST .DV 0 .037 0.016 0 .056 0 .020 oo K IC' 0.5 MPa /in 0.6 0.7 0.8 0.9 I .0 F i g u r e 22 C u m u l a t i v e D i s t r i b u t i o n Cu rve o f 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, KI(, , for the CTS of white spruce in the TL and RL system of crack propagation were 0.205 MPa/m and 0.387 MPa/m respectively, (see Table 2) This species of wood has not been previously tested for this fracture mechanics parameters. However, present values of K I C are comparable with their counterparts obtained for other species; K J C for the TL mode of western white pine is about 0.190 MPa/m and for western red cedar is about 0.185 MPa/m obtained by Johnson (1973). The mode I fracture toughness for the DCB spscimens in the TL system of crack propagation is approximately 0.387 MPa/m . Average moisture content recorded for a l l tested specimens is 9.1% and average specific gravity is 0.38, based on the ovendry weight-ovendry volume. An example of the dependence of K J C on specific gravity is shown in Figure 23 for the CTS and i t is apparent that there is no corelation between the specific gravity and the K I C . 4.3 EXPERIMENT NO.2 CRACK-FRONT WIDTH EFFECT ON FRACTURE  TOUGHNESS 4.3.1 EXPERIMENT DESIGN AND PROCEDURE Experiment No.2 was designed to study the effect of the variation of the crack-front width on the mode I fracture toughness. A total of thirty-eight CTSs were cut into five different widths, nominally B = 7,15,21,29,38 mm and mostly 0.30 0.25 H (TJ a, 0.20 H 0.15 C O M P A C T T E N S I O N B = 3 8 m m o. io- i r 0.32 0.34 A A A A A A A A^ 0.36 0.38 0.40 0.42 S P E C I F I C G R A V I T Y 0.44 0.46 Figure 23 Effect of Specific Gravity on Mode I Fracture Toughness. cn 77 i n the TL system. The specimen geometry and the l o a d i n g procedure was same as used i n Experiment No.1. I t i s expected t h a t the s t r e s s i n t e n s i t y f a c t o r i s i n v e r s e l y p r o p o r t i o n a l t o the w i d t h of the specimen, so i f we double the w i d t h of the specimen, the s t r e s s i n t e n s i t y f a c t o r w i l l be h a l f of b e f o r e . A l l the specimens were t e s t e d i n the temperature and h u m i d i t y c o n t r o l l e d room w i t h the I n s t r o n T e s t i n g Machine. 4.3.2 RESULTS The c r i t i c a l s t r e s s i n t e n s i t y f a c t o r s o b t a i n e d f o r each w i d t h are g i v e n i n Table 3. Shown are the average v a l u e s of c r i t i c a l s t r e s s i n t e n s i t y f a c t o r s , KI(-. , c o e f f i c i e n t of v a r i a t i o n (COV) i . e . , r a t i o of s t a n d a r d d e v i a t i o n t o average v a l u e s , sample s i z e as w e l l as the shape (k) and s c a l e (m) parameters f o r a 2-parameters W e i b u l l model. The l a r g e v a r i a t i o n of the k and m v a l u e s can be e x p l a i n e d by the l i m i t e d t e s t s f o r each c a s e . The i n f l u e n c e of c r a c k - f r o n t w i d t h on the f r a c t u r e toughness can be d e r i v e d by the W e i b u l l weakest l i n k model. The c r a c k - f r o n t w i d t h , B ,mm, i s a l s o i n c o r p o r a t e d i n the d e r i v a t i o n and the r e l a t i o n s h i p o b t a i n e d i s : F ( K K ) = 1 - e Xp[-(K I C/m) kB] (4.2) where k and m a r e the shape and s c a l e paramters r e s p e c t i v e l y . T a b l e 3 E f f e c t o f c r a c k - f r o n t w i d t h on f o r c o m p a c t - t e n s i o n s p e c i m e n s o f w h i t e s p r u c e f o r l o n g i t u d i n a l p r o p a g a t i o n W i d t h , B S a m p l e S p e c i f i c P . A v e r a g e K I C C . V . S h a p e S c a l e ? a „ p a r a m e t e r p a r a m e t e r mm s i z e g r a v i t y P Q M P a / m % k m 7 5 0 . 3 3 5 1 .08 0 . 2 5 7 0 2 . 1 5 4 7 . 9 8 0 . 2 5 9 8 15 7 0 . 3 7 8 1 .05 0 . 2 2 8 2 9 . 3 3 1 0 . 9 5 0 . 2 3 8 4 21 6 0 . 3 9 5 1 .03 0 . 2 4 3 5 6, . 18 2 2 . 3 1 0 . 2 5 0 1 29 7 0 . 3 7 5 1 .04 0 . 1 8 0 5 1 0 . .87 1 2 . 9 3 0 . 1 8 8 7 38 10 0 . 3 8 9 1 .05 0 . 1 7 6 8 2 7 . ,15 4 . 2 3 0 . 1 9 4 0 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 line will be -1/k. The least-squares method was used to f i t the regression line to the data. The relationship between KI(-. and crack-front width obtained is : log K I C = -0.3795 - 1/4.423 log B R 2 = 0.735 (4.4) where B is the crack front width,mm; KJ Q is the c r i t i c a l stress intensity factor, MPa/m . A plot of this relationship is shown in Figure 24 and results obtained by Barrett (1976) for Douglas f i r is also presented in the graph. The parameters k and m obtained by Barrett (1976) are 7.41 and 0.41 respectively. The discrepany between the two curves is due to the difference in the elastic properties of two species of wood. From equation (4.2) we get the cumulative distribution for K I C : 1 o 0.14 1 l o g i n K . r = -0 .379 l o g 1 0 B R =0.735 • s ' ° , c 4.423 1 0 i i i i i 1 1 1 10 B m m i i i 1 1 1 — 100 I I I I I I I I Legend A WHITE SPRUCE CTS O DOUGLAS FIR CTS 1000 Figure 24 Effect of Crack-front Width on Mode I Fracture Toughness (Based on the Weibull Weakest Link Model) CO o 81 F(K I C) = 1 - exp[-(K IC / 0 . 3 8 4 )4 - 4 2 3 B] (4.5) which is comparable to the results of Douglas-fir. The results of these tests confirmed that a relationship exists between the fracture toughness and the crack-front width. It has been shown that K J C deceases as the crack-front width increases which can be explained by the weakest-link principle. Another consistent model of the crack-front width effect is based on the stress state in the specimen. From equations (2.46b) and (2.47) we have : KC - KC1 + 2 H / D ( KC2 " K C 1 } H 1_ 2TT where OQ is the yield stress or that stress which results in gross deformation. A graph of Kj^ versus inverse thickness is shown in Figure 25 and a regression line is plotted based on the four data points. From the intercept at 1/D = 0 and equations (2.46b) and (2.47), we get : Figure 25 Effect of Crack-front Width on Mode I Fracture Toughness (Based on the Stress-state Model) 83 K = 0.1414 MPa/m K_» = 0.435 MPa/m C2 H = 6.95 mm The transition thickness from plane stress to plane strain will be D = 2H which is equal to 13.90 mm. When D < 13.90 mm, a plane stress condition will be maintained, so the basic plane stress value, will be achieved. For D > 13.90 mm, the plane strain condition w i l l result in a decline in Kj£ . As seen from Figure 24 and Figure 25, both the Weibull theory model and the stress-state model fitted very well with the data points. However, if a specimen has a width less than the transition thickness, according to the stress-state theory, i t is postulated that the width has no effect on the fracture toughness. For the Weibull model, i t is proposed that the effect will exist no matter what the specimen thickness i s . Since the transition thickness is usually less than 10 mm, the deviation between the Weibull model and the stress-state model is not significant at a l l and both models behaved well in the experiments conducted. 4.4 EXPERIMENT NO.3, CRACK-FRONT LENGTH VARIATION 84 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 L = 50,75,100,125 mm, where L is defined in Figure 26. The crack is maintained at the same length as in experiment no.1. Five specimens were prepared for each case and a total of twenty specimens were cut from four boards. The experiments were conducted in room temperature with a •average moisture content of 9%. The load was plotted against the COD recorded by the modified LVDT on a X-Y plotter. The design failure load were determined in the same way as ASTM Standard E399 and the time to failure were about 1 minute for a l l cases. 4.4.2 RESULTS Table 4 summarize results of within-board experiments designed to determine the relation between KI(, and the crack-front length, L. Average values of KJQ , C.V., moisture content and specific gravity for each length are Figure 26 Specimen Configuration of the Crack-front Length Specimen. oo T a b l e 4 E f f e c t o f c r a c k - f r o n t l e n g t h on K^^ f o r compact t e n s i o n s p e c i m e n s o f w h i t e s p r u c e B o a r d L e n g t h Sample No. L, mm s i z e A 75 2 100 1 125 2 B 50 2 75 2 100 1 125 1 C 50 2 75 1 100 2 125 1 D 50 1 100 1 125 1 A v e r a g e A v e r a g e A v e r a g e C.V. M. ,C. % S.G. K I C , MPa/m / o 8. .43 0 .413 0. .2304 2. , 24 8. .49 0 .420 0. .2275 0. .00 8, .68 0 .414 0. .2222 9. .36 8. .81 0 .412 0. .2817 4. ,63 9. .36 0 .419 0. .2918 5. , 28 8. .96 0 .410 0. .2762 0. ,00 8. .96 0 .408 0, .2569 0. .00 9. .69 0 .335 0. .2076 3. ,86 9. .68 0 .334 0. .1894 0. ,00 9. .75 0 .322 0. .2193 7 . 41 9. .54 0 .342 0. . 1805 0. ,00 8. .99 0 .416 0. .2640 0. ,00 8, .98 0 .413 0. .2518 0. ,00 8, .93 0 .405 0, .3145 0. ,00 87 also shown and the results are plotted in Figure 27. The results do not seem too helpful as i t is not obvious any trend exists for a l l cases except for board A, for which the values of KI(, decrease as the length increases. The lack of consistent trends can be attributed to the limited number of specimens tested. However, comparing the average fracture toughness between the CTSs and the DCB specimens, surprisingly, the DCB specimens gave fracture toughness values twice as great as the CTSs. This may have different causes. F i r s t l y , the CTSs were prepared from many different boards which implies a higher degree of variability, whereas the DCB specimens were prepared from only three boards. This can be observed by examining the coefficient of variations for both cases, as shown in Fig. 21 and 22. The difference of the fracture toughness between CTSs and DCB specimens may also be explained by the stress-state condition in the specimens. As mentioned before, the fracture initiation constant, K J Q , is thickness dependent. The amount of material which yields at the crack tip must also be small. To ensure this, specimens must be of sufficient thickness so that a tria x i a l state of stress can exist at the flaw tip. It is postulated that Kj^ decreases as the crack-front width increases. The lower limit occurs as the width approaches infinity and i t is called the plane strain fracture toughness. The K j C at this point is considered to be a geometric invariant material property. The size requirements recommended by Liu (1983) for this SSY Legend A BOARD A O BOARD B © B O A R D C O BOARD D 1 1 1 1 1 1 1 1 50 60 70 80 90 100 110 120 130 L (mm) Figure 27 Effect of Crack-front Length on Mode I Fracture Toughness 89 and plane strain condition are : (4.6a) or (4.6b) where t is the specimen thickness, m; a,l are defined in Fig. 26. This means that if the left-hand term is less that one, then a plane stress condition w i l l exist; otherwise,' the plane strain condition will result in a decline in K J C . Table 5 shows the data for the cases of CTSs and DCB specimens, the left-hand term for the CTSs is 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 Kj^ , which agrees with the results obtained. Further analysis of this difference is included in Appendix II. 4.5 EXPERIMENT NO.4 MOISTURE CONTENT EFFECT 4.5.1 EXPERIMENT DESIGN AND PROCEDURE This experiment uses Kj^, to examine the variation of fracture toughness in white spruce with the practically T a b l e 5 The s i z e c o e f f i c i e n t f o r CTSs and DCB spec imens Specimen Sample a L t t / ^ ^ 5 (—!£.) Q - ^ t ^ C Y ) O y a K-j.^ , t ype s i z e mm mm mm m MPa L CTS 33 50 50 38 0 .038 2 . 8 0 0 .01288 2 .9503 DCB 17 250 700 38 0 .1064 2 . 8 0 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. Sixteen were placed in a dry air oven at 105°C 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 Kj^ at different moisture content. Specimen size and experimental procedures were exactly the same as Experiment No.1. The tests were conducted in the same humidity room. 4.5.2 RESULTS Table 6 summarized the results from each set of experiments and a plot is given in Figure 28. As can be seen in the figure, the ovendried specimens failed at lower loads than the wet. The results indicate that the effect of moisture content on KI(- shows a decrease 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 the results of the Newsletter of Technical Research Centre of Finland (1986) for spruce. According to the results obtained by P.D.Ewing (1979), the T a b l e 6 E f f e c t o f mo i s ture content on K T r f o r CTS i n l o n g i t u d i n a l p r o p a g a t i o n M o i s t u r e Sample Average K-j-^ C . V . S p e c i f i c content s ± z & MPa/m % g r a v i t y /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 i C 0.20 a, 0.18 C O M P A C T T E N S I O N S P E C I M E N A 0 . 1 6 H K I C 2 = 0.03051 + 0.01362 ( 1 - e M / 6 ) - r 6 2 4 6 8 M O I S T U R E C O N T E N T ( % ) 10 Figure 28 Effect of Moisture Content on Mode I Fracture Toughness CO 94 effect of moisture content on K J C generally shows an increase as the moisture content decreases from 20% to a maximum at around 10% and then the Kj^ values tend to decrease. This agrees with the results obtained from Experiment No. 4. An exponential curve was fitted to the data points using the least-squares technique and the following relationship is obtained for M^ 10 % : K 2 C = 0.0305144 + 0.013619 (1 - e~ M / 6) (4.7) R2 = 0.9688 where K J Q is the mode I fracture toughness, MPay/m ; M is the moisture content, %. This equation shows good agreement with the experimental data points, as shown in Figure 28. However, this equation shows a limitation in that the fracture toughness will increase as long as the moisture content increases. According to Ewing's theory, this equation should only be valid 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 failure; or to study the wood structure under the mode II failure 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 is shown in Figure 29 and the mesh for the finite element program is shown in Figure 30. During the experiments, the relative 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. Friction induced by the crack closure was avoided by placing a smooth plastic plate between the sawn notch as shown in Figure 29. Twenty-four specimens were prepared and the bandsawn notches were extended by a steel knife crack starter just before the testing. The specimens were tested in the Satec Testing Machine under room temperature and the crack orientation was recorded for each specimen. The experimental setup is shown in Figure 32. 4.6.2 RESULTS A typical load versus crack displacement curve is also shown in Figure 31 which indicates some slow crack growth prior to the crack starting to propagate. The failure load were determined using the 5% offset line as suggested in ASTM Standard E399. The experiment results also indicated a dependence of mode II fracture toughness on the crack 9 6 Applied Load i 92 mm 300 mm -500 184 38 mm 1 smooth p l a s t i c plate 900 mm 50 mm F i g u r e 29 Experiment Setup o f the Mode I I F r a c t u r e Toughness Specimen. Fi g u r e 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. 9 7 smooth p l a s t i c plate Figure 31 Applied Load versus Longitudinal 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 I C values were plotted against the annual rings angle, 6 in Fig 33. The best-fitted polynomial curve calculated by the least-squares method is also shown in the figure, and the following relationship was derived : K = 1.8798 + 2.5114 x 10 _ 29 - 2.4828 x KT^e 2 (4.8) where K J J ^ is the mode I I fracture toughness, MPa/m ; 0 is the annual rings angle, degree. The curve shows a maximum at a rings angle of 50 degrees which indicates a mode of cross-grain plane failure. This is reasonable as the cross-grain failure always require a higher failure load. The average mode I I fracture toughness values obtained for white 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 Barrett (1981) for white spruce -- K n c for the TL mode is 1.89 MPa/m which is in the same order of magnitude. Figure 34 shows the cumulative distribution funmction for K I I C based only on a l l the specimens tested in the TL system. The coefficient of varience is 16% with the f i f t h percentile being 1.925. MPa/m . T a b l e 7 E f f e c t o f t h e a n n u a l r i n g s a n g l e on K I I C Specimen A n g l e t o Sample M.C. S.G. K n c K I I C t y p e , g i z e % MPa/m p s i / T n " d e g r e e E n d - s p l i t Beam 0 -- 10 2 13. 38 0. 384 2.0180 1836. 33 11 -• 20 2 13. 75 0. 369 2.1000 1911. 00 21 -- 30 0 - - -31 -- 40 3 12. 83 0. 410 2.2597 2056. 30 41 -- 50 4 13. 13 0. 418 2.6935 2451. 09 51 -- 60 2 13. 78 0. 371 2.3942 2178. 72 61 -- 70 1 14. 50 0. 423 3.0098 2738. 90 71 -- 80 1 12. 75 0. 324 1.9359 1761. 70 81 -- 90 8 12. 91 0. 345 2.1561 1962. 01 4 -3-a . 1 -• 90 deg re 0 degre cs Is equ Iv e is equivo • elent to T lent to RL sys tem sys tem • • • *~] • • • < > • 10 20 30 40 50 60 70 A N N U A L R I N G S A N G L E ( DEG. ) 80 90 Figure 33 Mode II Fracture Toughness Variation with Annual Rings Angle MODE II CUMULATIVE DISTRIBUTION End-split Beam K 15.229 F = 1 -( IIC 2.2329 5JILE 50ZILE MEAN 952ILE OF DI5P DATA (SOLID L I N E ) : N=8 DflTR ST.UV. WEIBULL 5T.DV I .925 0.014 1 .837 0 . 190 2.163 0.151 2.179 0.073 2.156 0.178 2.157 0.173 2.428 0.047 2.399 0.081 0.082 0.080 INTERVAL CHI-SQUARE F IT : 0.00 2-PAR WE I BULL (DASHED) : SHAPE = 15.2286 SCALE = 2.2329 .0 0.5 1 .0 1 .5 1 2.0 K IIC ~ l 2.5 I— 3.0 MPa/m 3.5 4 . 0 -1.5 Figure 34 Cumulative Distribution Curve of the End-split Beam Specimens. 1 03 4.7 EXPERIMENT NO.6, MIXED MODE MID-CRACKED BEAMS 4.7.1 EXPERIMENT DESIGN AND PROCEDURE Since wood structural members are often subjected to complex loading conditions that result in mixed mode fracture, a failure criterion should be expressed in terms of combinations or interations of the mode I and mode II failure modes. Experiment No.6 was designed to investigate the fracture strength of mid-cracked beams under the mixed mode loading condition. Di f f i c u l t i e s were encountered in designing the specimen geometry in order to produce the desired ratio of Kj to Kjj . This had to be done by t r i a l and error, and finally the 45 degrees and the 90 degrees mid-cracked beams were adopted for testing. This investigation had two principal phases. In the 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 the support as shown in Figure 35. Twenty-four specimens were cut and prepared for each case with mostly in the TL system of crack propagation. The second phase consists of the 90 degrees mid-cracked beams loaded at the centerpoint as shown in Figure 36. Sixteen specimens were prepared for the TL propagation and the tests were performed in the Satec Testing Machine at room temperature. 104 P = 100 N K. = 7.55xl0~ MPa/ni K__ = -0.0187 MPa/m Type A P = 100 N Kj= 6.97x K „ = 0.0102 MPa/m " MPa^ II A Figure 35 Type B Specimen Configuration of the 45 deg. Beam (a) Loading at the centerpoint. (b) Loading at 200mm from the left support. P = 100 N Kx= 6.309x10 -3 MPa/m K II 0.0134 MPa>/m A Figure 36 cr Type C Specimen Configuration of the 90 deg. Beam 1 0 5 Prior to testing, the band-sawn notches were also sharpened with the crack starter 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 fini t e elements meshes shown in Figure 37. The predicted direction of propagation was determined by comparing the Kj and K J J values for both directions from the computer output. A higher value of Kj and K J J indicates a lower failure load during the testing. The predicted failure direction is 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 of the i n i t i a l crack, the mid-span deflections were measured instead. 4.7.2 RESULTS Table 8 summarizes the results for each phase of experiment with the average K I C , K I I C values shown as well. As can be seen from Figures 3 5 , 3 6 and Table 8 , there are two potential directions of propagation for the 9 0 degrees mid-cracked beams. Since cracks started at the weaker direction of the two, i t implies a double chance of failure. The experimental curves were corrected based on the following formulae : 106 Figure 3 7 F i n i t e Element Mesh f o r the Mid-cracked Beam Specimen. (a) 45 degrees Mid-cracked Beam. (b) 90 degrees Mid-cracked Beam. Table 8 Results of the mid-cracked beam specimens K J J C . V . R * K I I C "/O 7 ° Mean 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 , 180 0.8496 3.8 0.3160 10.7 100 Specimen Sample Average Average C . V . Predicted C . V . K T„ K T T O prop. type size I C I I C „ , /. K _ . V A / MPa/m MPa/m '* E r e c t i o n K I C L 'Mean •k _ R i s the percentage of the specimens f a i l e d i n the predicted d i r e c t i o n . 1 0 8 F £ - 1 - [1 - F £]2 ( 4 . 9 ) F - 1 ~ / l - F ( 4 . 1 0 ) c E where F E is the experimental cumulative probability. F c is the corrected cumulative probability. These cumulative distribution curves are shown in Figure 3 8 and Figure 3 9 together with the best-fitted 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 intensity factors of the 9 0 degrees beams, the K J C has a mean value of 0 . 3 0 8 4 MPa/m for the experimental cumulative distribution and a value of 0 . 3 5 5 6 MPa/m for the corrected cumulative distribution. In order to determine the normalized interaction curve between mode I and mode II fracture toughness, the and K J J values for mixed mode failure are normalized by the c r i t i c a l mode I and mode II stress intensity 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 different percentiles. One method to tackle this problem is to normalize the Kj and K J J values at failure by the K I C and K I I C values which correspond to the same percentile. This method is shown diagramatically in Figure 4 0 . The solid curve and the dashed curve correspond to the pure mode Kj ( MPaVm) Figure 38 Experimental and Corrected Cumulative P r o b a b i l i t y Curves of the Mode I Fracture Toughness for the 90 degrees beam. o K J J ( M P a V m ) Figure 39 Experimental and Corrected Cumulative P r o b a b i l i t y Curves of the Mode II Fracture Toughness for the 90 degrees beam. o Figure 40 Method of Normalizing the Cumulative Pr o b a b i l i t y Curves for the 45 degrees mid-cracked beam. 112 I and mode II fracture toughness Weibull curves respectively. Each data point is divided by the value which refers to the same level of cumulative probability. Refering to Figure 40, we have : K I X l K I I X2 IC X l c K I I C X 2 C where Kj is the mode I stress intensity factor at failure. K J J is the mode 11 stress intensity factor at failure. Kj^ is the pure mode I fracture toughness. • KIIC * s fc^e P u r e m°de 1 1 fracture toughness. X 1 , X ^ , X 2 , X 2 Q are as shown in the figure. Both the 45 degree and the 90 degree mid-cracked beam data points are treated by this method. For the 90 degrees beams data, the corrected cumulative distribution functions were used to calculate the required ratios. The average values for the Kj / K I C and K J J / K J I C for this experiment are listed in Table 8 and are plotted in Figure 41. 1.2 I A 0.8-0.6 0 . 4 -0.2-0.2 0.4 0.6 K „ / K Legend A COMPACT TENSION O DOUBLE CANTILEVER a 45 DEG BEAM O 90 DEG BEAM 0.8 -Q-1 1.2 ric Figure 41 K^-K^ Interaction Diagram based on the Mid-cracked Beams Result. 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 failure for different geometries. The main di f f i c u l t y in generating a interaction curve occurs because of the limited range of Kj / K J J (or ( Kj / KIC K I I / KIIC ^ * n fc^e s P e c i m e n design. It was found that for most beam geometries, the mode I stress intensity factor often dominates the fracture failure. This is equivalent to the failure points on the upper region of the interaction curve or a high value of 9 as shown in Figure 41 . The practical diff i c u l t y was overcome by applying two separate loads on the DCB specimens which could be controlled to produce different mode I and mode II stress fields around the crack tip. This is 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 is possible to control the ratio of Kj to K J J at fracture. The test method is 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 calibrated load c e l l . The experimental setup is shown in Figure 43. Figure 42 Specimen Configuration of the Two-loads Beam Specimen. 160 ran 1 r — i o ' 3/4 In. place • 3/8 l n . place To Hand pump 3/4 I n . , • 3/« l n . p l a t e 'xz.zr.n-r i ZJ. * B e a r i n g b a l l - Load c e l l - M e t a l p l a t e - H y d r a u l i c J a c k A p p l i e d Load CL F r o n t 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 - 4.0488 x lo" 1* ? 1 - 9.894 x io"5 P,, (4.12a) K I I = - 1 ' 5 0 6 0 x 1 0 ~ * p 2 (4.12b) Using the results from before for the TL system : K = 0.388 MPa/m K = 2.13 MPa/m •L V' 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^O) P = = 14143.5 N 1.5060 x I O - 4 Introducing the condition Kj >0 i.e., that the crack does not close, from Equation (4.12a) we have : 4.0488 x 10" 4 P, - 9.894 x 10~ 5 P = 0 or P2 < 4.1 ? i Assuming the interaction curve to be : Case I : K l K l l K i c K n c K I / K I C L e t r = —71? = t a n 9 K I I / K I I C we have, P = (SrZ+T - r)[1234.93r K + 811.29 K_ ] P2 = 3320.062 (/?2+4 - r)[KI I (,] Case II : ( i r - ) 2 + • 1 K i c K n c we have, P1 = 1//1+P" [2469.9 K I C • r + 1622.6 K ] P2 = 6640.12 /l+ r2 1 19 Since case I is more conservative than case II, in designing the fracture criterion, case I is adopted. The interaction failure envelope for the P1~?2 mi- xed mode loading is shown in Figure 44. Outside the curve is the failure region, where as inside, the specimen is intact. To obtain the desired ratio of fracture toughness, the following procedure was followed : 1. A ratio r was selected. 2. From equations (4.14a) and (4.14b), failure loads and P 2 were obtained. 3. The corresponding Kj and K J J values from equations (4.12a) and (4.12b) were calculated. 4. From Figure 44, a loading path was designed within the safe region to achieve the P1 and P 2 calculated. Experiment no.7 was carried out in determining the Kj and K J J value at failure for two different ratios as shown in Figure 44 together with the loading path. Ten specimens were prepared and tested for each case under room temperature with the dominant system of crack propagation being TL. The applied mid-span load and the COD were recorded by a X-Y plotter. 4.8.2 RESULTS The experiment results are summarized in Table 9 and Figure 44, with the failure load shown as well. This gives the ratio of Kj to KTl values of 0.64 and 0.229, with corresponding d values of 74° and 52°. These values were 15000 n 10000 H 5000 STAGE 1 K I I ~ K I I C 0-<f Legend • PREDICTED FAILURE ENVELOPE A EXPERIMENTAL RESULTS 1 1500 2000 P, ( N ) 2500 3000 3500 Figure 4 4 Failure Envelope of the Two-loads Beam Specimen O T a b l e 9 R e s u l t s o f the t w o - l o a d s beam t e s t s Spec imen Sample P , P 9 Ave rage C . V . A v e r a g e C . V . K x C . V . K J T C . V . A v e r a g e t y p e s i z e / / K ] . : % % * % " % MPa/m MPa/m i L i i L 5 - s t e p s 10 1925 .0 3985 .4 0 .3851 9 .6 0 .6002 9 .6 0 .998 5 . 3 0 .282 9 . 3 3 .525 9 - s t e p s 9 2800 .0 8488 .3 0 .2938 16 .7 1.2783 16 .7 0 .754 5 . 4 0 .600 5 .9 1.260 1 22 used to generate the interaction curve later. Theoretically, i t is possible to attain any stress intensity factors ratio even for d close to zero, however, in practice i t may require the specimen loaded close to the mode II fracture toughness, which may cause the specimen failed in mode II instead of the mixed mode condition wanted. The onset of the crack propagation were determined by a change of the slope of the load verses COD curve in the final phase as shown in Figure 45. Some slow crack growth was observed during the tests. A 5% offset method is used here to determine the failure load. 4.9 EXPERIMENT NO.8, NOTCHED BEAM SPECIMENS 4.9.1 EXPERIMENT. DESIGN AND PROCEDURE The final phase of the experiment concerned the notched beam specimens under the mixed mode loading condition. As mentioned before, the stress intensity factor for notches 2-X has a unit of (Load)x(Length) , which depends on the eigenvalue X. Therefore, the interaction curve for notched specimens include three variables, i.e., Kj / K J C » K T . T / K H C and X. Experiment no.9 was designed to produce data points for notches with X = 1.6 and 1.7. Three types of specimens were used in these experiments as shown in Figure 46. 123 \ STAGE 2 STAGE 1 COD Figure 45 Typical Load-Displacement curve of Two-loads Beam Specimen. 124 (a) Applied Load 1 92mm 50mm TYPE A 130' \ 7S~ •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 were prepared for each case and the tests were carried out with the Satec Testing Machine at room temperature. The i n i t i a l notches were not extended by the crack starter and the loads were plotted against the midspan deflection for a l l cases. The pure mode I and mode II fracture toughness tests for these Xs were not done due to the d i f f i c u l t y of obtaining feasible specimen dimensions and loading method. For example, to obtain the K J C and K I I C for X=1.6, we need to test a 77° notch in the compact tension specimen, and a 13° notch in shear. 4.9.2 RESULTS The results were summarized in Table 10. Since the specimens are symmetrical with respect to the midspan neutral axis, there is a equal probability of crack initiation at both notches. Similar treatment of the experiment data were preformed according to the equations (4.9) and (4.10). The average corrected Kj and K J J values are given in Table 10 as well. The failure of specimens type A and type C was very brit t l e and the rate of crack propagation was extremely fast while the failure 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 is expected that the interaction curves will Table 10 Results of the notched beam specimens Specimen Sample M.C. type size % 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 S.G. Average Average C.V. MPam 2 _ X MPam 2 _ A 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 of the experiment design, procedures and results. Each experiment contributed to a better understanding of how wood specimens performed under different geometries and loading conditions. The 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 This chapter f i r s t explains the method of generating an interaction curve for a cracked beam, and then presents the results obtained. The method i s then extended to a notched specimen, to establish the family of interaction curves. Following is a detailed discussion of the treatment of data in order to obtain the curves, including assumptions that had been made during the process and the limitations of the experiments. F i n a l l y , rules for the design of cracked beams and notched beams are suggested. 5.2 STRESS INTENSITY FACTOR INTERACTION CURVE FOR CRACKS Using the results of a l l the cracked specimens tested - DCB specimens, two-loads specimens, mid-crack beam specimens, i t is possible to establish the interaction curve for the cracked beams. Figure 47 shows a l l the experimental data points obtained which had been normalized with the c r i t i c a l mode I and mode II stress intensity factors. An interation curve has been derived by the least-squares technique based on equation (2.65) and is shown in Figure 47 with a l l the data points, and in Figure 48 with their average values. This gives the formula : K I . , KII .2.5587 — + (j7 ) - 1 (5.1) IC IIC 128 1.5 I f -0.5 -i A <£ A ° P" 8* 9> A.A" V ^ A (J<L ) + (IAL ) K i c K n c 2 .5587 = 1 0.2 0.4 0.6 K „ / K , I C L e g e n d a TWO-LOADS SPECIMENS o 45 DEG BEAMS a 90 DEG BEAMS 0.8 • A A -/<\A — A -1.2 Figure 47 Interaction between K ^ / K ^ and K I T _ / K j _ x c f o r c r a c k e d beam specimens. Figure 48 Interaction curve derived between K J / K T C A N ^ ^n/^xiC for cracked beam specimens. — CO o 131 This is close to the model proposed by Wu (1967),i.e.: Kic Sic <5-2> This formula is replotted in Figure 49 together with other suggested models. The linear relation of Leicester (1974) seems to be too conservative. The model proposed by Williams and Birch (1976) that the shear stress causing sliding mode has no effect on the mixed mode failure, does not follow the data trend. The present experiments on white spruce corroborate the findings of Wu (1976) on balsa. The question might be raised whether this interaction relation is applicable to other species of wood, but i t cannot be answered until further results are obtained. 5.3 STRESS INTENSITY FACTOR INTERACTION CURVE FOR NOTCHES As mentioned before, the stress intensity factors for notches have dimensions of stress*length , where X is a non-linear function of the material and notch orientation in orthotropic materials. Consequently, for any given value of X, there may be an infinite number of compatible notch geometries and material combinations. In a general case of a notch, a minimum of three parameters are essential to uniquely specify the stress fields at the notch root : Kj , K J J and X. Kj and K J J indicate the strength of each of the stress fields while X X = 1.50 c r a c k e d beam specimens. 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 is necessary to obtain interaction curves for X values from 1.5 to 2.0. This is rather tedious as compared to sharp cracks, for which the primary and secondary modes have equal eigenvalues. In order to obtain data points on the interaction curve, different notch geometries are required. The experiments contributed two data points for X=1.6 and one data 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 flat surface. Due to the limited number of the data points, the assumption that the shape of the interaction curves for notches would be the same as 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 best-fitted based on the following interaction relationship: A K x + B K ^ ' 5 5 8 7 = i ( 5 > 3 ) A/B = 17.84 (5.4) 0.4 0.3 H ctf cu 0.2 H X=1.50 X=1.60 o • •A-1.70 o.H o.o H Legend A TWO-LOADS SPECIMENS .O 45 DEG BEAM B 90 DEG BEAM • LAMBDA=I.6 LAMBDA=1.7 A K, + B K,, 2- 5 5 8 7 = 1 X B 1.5 2.5573 0.1445 1.6 3.2101 0.1799 1.7 4.2108 0.2360 0.5 1 1.5 Kn ( MPam2_A) 2.5 Figure 50 Interaction relations between K-j. and K-^ for notches 135 where Kj , K J J are the mode I and mode I I f r a c t u r e toughness 2 _X r e s p e c t i v e l y , MPam ; A, B a r e some c o n s t a n t s as shown i n F i g u r e 50. T h i s g i v e v a l u e s of K I C of 0.3880 MPa/m , 0.3115 MPam 0* 4, 0.2375 MPam 0 , 3 f o r X=1.5, 1.6, 1.7 r e s p e c t i v e l y ; K I I C of 2.183 MPa/m , 1.955 MPam 0 , 4, 1.758 MPam 0 , 3 f o r X=1.5, 1.6, 1.7 r e s p e c t i v e l y . A more g e n e r a l i n t e r a c t i o n e q u a t i o n was d e r i v e d f o r whit e spruce u s i n g the l e a s t - s q u a r e s method on the e x p e r i m e n t a l r e s u l t s : 2 SS87 Kj + 0.05605 K1Z = 1-8355 - 1.1525 X + 0.125 \2 (5.5) where Kj , K j J are the mode I and mode I I f r a c t u r e toughness 2 - X r e s p e c t i v e l y , MPam ; The i n d i v i d u a l i n t e r a c t i o n curve f o r notched beams f o r X=1.6 and 1.7 are a l s o showned i n F i g u r e 51 and F i g u r e 52 r e s p e c t i v e l y . A f t e r e s t a b l i s h i n g t he i n t e r a c t i o n c u r v e s , we a r e now a b l e t o compute the s t r e n g t h of any notched or c r a c k e d specimen of d i f f e r e n t g e o m e t r i e s , m a t e r i a l s and g r a i n o r i e n t a t i o n s under any k i n d of l o a d i n g . But u s i n g of t h e s e c u r v e s f o r beam d e s i g n seems t o be too troublesome as one needs the s t r e s s i n t e n s i t y f a c t o r s . However, we can a l t e r these c u r v e s i n o r d e r t o o b t a i n a d e s i g n method f o r the Figure 51 Interaction curve for notches with A=1.60 0.8 H 0.4 Figure 52 Interaction curve for notches with X=1.70 138 cracks and notches. 5.4 APPLICATION The application of linear elastic fracture mechanics methods for beam design allows estimates of the strength of notched beams. Murphy (1978) had shown that the recent code has underestimated the effect of the presence of a notch based on the net-section theory. A more rational method of treating the notches and cracks is presented herein which provides rules for the strength design of notched beams. These rules were based on theoretical 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 degree cracked beam problem under investigation is shown in Figure 53. The geometrical dimensions a, b, d are indicated therein as well as the sign convention of bending moment M and shear force V acting on the beam at the cross-section containing the crack 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 NOTCH and followed the "transformed stress intensity factors method" proposed by Murphy (1978). In application, one can obtain the stress intensity factors for specific configuration by evaluating the moment I I I I I 0.0 0.1 0.2 0.3 0.4 0.5 a / d Figure 53 Dimensionless Stress Intensity Factors for pure moment and pure shear loadings as a function of notch-to-depth r a t i o for 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 stress f i e l d surrounding the eigenfield, the accuracy of the proposed curves for computing stress intensity factors depends on how similar the nominal stress f i e l d of the beam being analyzed is to the stress f i e l d of the beam shown in Figure 53. To verify the validity of these curves, a comparison had been made between the results obtained by these curves and the the program NOTCH. Five cases, with two simple structural configurations under different loadings were investigated. The notch-to-depth ratio is 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 is larger in cases 3 and 5, which might be explained by the high discrepancy between the nominal stress f i e l d of the off-center cracked beam and the nominal stress fi 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 between the ultimate moment and shear force can be established by using the interaction curves for the stress intensity factors 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 is shown in Figure 55. The curves show a linear relationship between Case 1 P=100 N b = 38 mm Case 2 = 38 mm P = 100 N Case 3 b = 38 mm p = 1 N/mm H 1 I I I I i ) a = 92 mm t 800 mm d = 184 mm Case 4 b = 38 mm P = 100 mm d = 184 mm Case 5 b = 38 mm p = 1 N/mm T T i n i n .1 mm •600 mm Aa = 92 — t M 1 d = 184 mm L = 800 mm Figure 54 Several Sharp Crack Problems. T a b l e 11 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 v a r i o u s sha rp c r a c k p rob lems Case From c u r v e s From NOTCH A c c u r a c y * A c c u r a c y K I MPa/m K I I MPa/m MP a/ii i K I I l-IPa'/m o f o ia o f o K. o 1 7 , . 2 1 0 x l 0 " 3 0.0154 7 , 0 0 5 x l 0 ~ 3 0.0149 2, ,9 2. .7 2 6 . 6 0 6 x l 0 " 3 0.0107 6 , 3 7 2 x l 0 ~ 3 0 .0101 3. .7 5. .7 3 0 .0288 0.0614 0, .0317 0 .0672 8. ,9 8. .6 4 9 . 6 0 6 x l 0 " 3 0.0137 9, . 2 8 0 x l 0 ~ 3 0 .0133 3. .5 3. .3 5 0, .0216 0 .0461 0, .0241 0 .0502 10. , 1 8. .3 I I * A c c u r a c y i s b a s e d on the r e s u l t s o b t a i n e d f rom the p rog ram NOTCH. F i g u r e 55 D e s i g n c u r v e s f o r 90 deg rees c r a c k e d beam 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 . 144 moment and shear with d i f f e r e n t i n t e r c e p t s and slopes f o r d i f f e r e n t notch-depth r a t i o . These r e l a t i o n s h i p can be expressed i n formulae i n terms of the nominal maximum bending s t r e s s fk=6M/bd 2 and nominal maximum shear s t r e s s f s=3V/2bd c a l c u l a t e d f o r the t o t a l beam depth. T h i s g i v e s , For 90 degrees cracked beam ; af + f - J ^ f < 1 (5.6) where a, 0 are constants l i s t e d below; d i s the t o t a l beam depth, m; f ^ i s the maximum bending s t r e s s , MPa; f s i s the maximum shear s t r e s s , MPa. a/d a(dimensionless) B(/m/MPa) 0.5 0.0644 0.1271 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 (5.6) i s analogous to the design equations f o r notches from the S.A.A. A u s t r a l i a n Timber En g i n e e r i n g Code 145 CA65-1972 (S.A.A. 1972) : 0.3f. + f ——^- S-< 1 (5.7) C, F , 3 sj where 6M . 1.5V b lie min B d2 s Bd . C Q • min F • = shear block strength for the species of interest; C3 = constant for specific notch angle. For the 90 degrees notched beam, the stress intensity factors do not show a consistent relationship with the applied moment and shear. This means the stress distribution around the eigenfield of a notch is very sensitive and varies with different loadings and geometries. Consequently, the stress intensity factors for notches have to be computed by using singular finite element program, and general design curves for different loadings can not be obtained. One particular case of a one-metre rectangular end-notched white spruce 2x8's beam under centerpoint loading (P) was analyzed with various notch-to-depth ratio. The ratio Kj /P were plotted against the notch length for various notch-to-depth ratio, (see Figure 56) Since the Kj to K J J ratio is constant for a given notch geometry and 1 4 7 material, K J J can be obtained from the relation K^j = 1 . 0 9 6 Kj after obtaining the Kj value from the figure. Using equation ( 5 . 5 ) and substituting K J J by 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 4 5 2 1 . . MPam ' for the c r i t i c a l mode II stress intensity factor. Using these values and Fig. 5 6 , a plot of the c r i t i c a l applied load against the notch 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 distortion 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 relation between the c r i t i c a l stress intensity factors and the applied load seems to be very complicated for notches, design curves for other beam configurations and loadings can be established in a similar way based on the notch length and the notch-to-depth ratio. F i g u r e 57 C r i t i c a l Load v e r s u s N o t c h L e n g t h f o r v a r i o u s n o t c h d e p t h o f a 2 " x 8 " beam. 6. CONCLUSIONS AND RECOMMENDATIONS 6.1 CONCLUSIONS The linear elastic fracture mechanics method has been found to be appropiate and to apply well to the characterization of the fracture behavior of wood. The fracture toughness for white spruce has been shown to vary with the annual ring orientation in the specimens. It has also been shown that there is a dependence of the mode I fracture toughness, KIC ' o n t* i e width of the crack front and the moisture content in the specimens. The mode II c r i t i c a l stress intensity factor, K J J ^ , has been shown to govern the fracture of end-cracked wood beams and i t has also shown a dependence on the annual ring orientation. The mixed mode interaction curves, which incorporate the mode I and mode II fracture toughness, have been presented and applied successfully to the fracture behavior of white spruce. The criterion that the mode I fracture toughness is independent of forward shear effect has not been observed here while an interaction relationship between Kj and K J J in the mixed mode fracture of white spruce is more obvious. This relation applied equally well to the notched specimens, and a family of interaction curves for predicting the onset of rapid crack propagation has been established for white spruce. 149 1 50 Design methods has been provided herein for 90 degrees cracked beam based on the applied moment 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 Timber Design Code. Design curves for 90 degree notched beam has also been presented here for a particular case 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 possibility of applying on notches. The pursuit of a valid size effect on the mode I 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 simplified picture of mixed mode failure for notches which has not been studied before. A proposed design method has also been presented here which can be adopted by the timber design code. Further research should include performing experiments to study this interaction relation for other geometrical configurations and other species of wood. 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 intensity factors for Douglas-fir," IUFRO wood engineering meeting, Delft, Holland, (1976). 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 Finite Elements," Int. J. Numerical Methods in Engineering, 8, 537-545, ( 1 974). 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Fracture Mechanics, 1, 189-203, (1965). Tada, H., The Stress Analysis of Cracks Handbook, Del Research Corp., Penn., (1973). 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 Finite Element Technique," Int. J. Solids and Structures, 7(10), 1333-1342, (1971). Walsh, P.F., "Linear fracture mechanics in orthotropic 154 materials," Engineering Fracture Mechanics, 4, 533-541, ( 1972) . Walsh, P.F., "Linear fracture mechanics solutions for zero and right angle notches," CSIRO Aust. Div. of Building Research Technical Paper (Second series) No. 2, 1-16, Melbourne, (1974). Westergaard, H.M., "Bearing Pressures. and Crack," Transactions, Am Soc. Mechanical Engrs., /. Applied Mechanics, (1939). 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 distribution at the base of a stationary crack," /. Appl. Mech., 24(1), (1957). Woo, C.W., Chow, C.L., "Mixed Mode Fracture in Orthotropic Media," Proceedings, Fracture Mechanics in Engineering Appl i cat i on, India, 387-396, (1979). 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. M.C. S.G. Angle to P P„ ' P /P„ K „/ ° . . max Q max Q IC no. % RL degree N N MPa/m 1 9.92 0.372 90 248.0 227.5 1.09 0.1570 2 9.92 0.324 90 216.1 208.1 1.04 0.1436 3 10.05 0.411 90 307.2 290.1 1.06 0.2002 4 10.18 0.408 90 324.2 312.9 1.04 0.2159 5 10.05 0.377 90 199.1 147.9 1.35 a 0.1021 6 8.87 0.450 90 492.5 489.2 1.01 0.3376 7 10.10 0.330 90 238.8 227.5 1.05 0.1570 8 10.00 0.414 90 291.2 272.9 1.07 0.1883 9 9.80 0.374 90 341.3 332.1 1.03 0.2296 10 9.60 0.378 90 335.6 329.9 1.02 0.2277 11 9.60 0.369 90 244.5 248.0 1.00 0.1711 12 9.80 0.405 90 238.8 236.6 1.01 0.1633 13 9.70 0.412 90 261.6 252.5 1.04 0.1743 14 9.70 0.375 90 329.9 295.8 1.11 0.2041 15 9.80 0.410 90 307.2 284.4 1.08 0.1963 16 9.60 0.328 90 270.7 260.5 1.04 0.1798 17 6.60 0.408 90 432.2 415.2 1.04 0.2865 18 9.95 0.417 90 238.8 213.9 1.12 0.1476 19 9.70 0.373 90 318.5 304.8 1.05 0.2103 20 9.40 0.386 90 261.6 254.9 1.03 0.1759 21 9.20 0.384 90 204.7 184.3 1.11 0.1272 22 9.50 0.386 90 204.7 199.1 1.03 0.1374 23 9.10 0.389 90 298.0 275.3 1.08 0.1900 24 8.45 0.370 90 227.5 224.1 1.02 0.1547 25 8.70 0.378 90 398.1 389.1 1.02 0.2685 26 9.10 0.386 90 394.6 378.7 1.04 0.2613 27 8.95 0.386 90 409.5 407.3 1.01 0.2811 28 9.30 0.394 90 301.5 278.7 1.06 0.1923 29 9.20 0.394 90 394.7 381.1 1.04 0.2630 30 8.70 0.374 90 420.8 415.2 1.01 0.2865 31 8.40 0.441 90 272.9 255.9 1.07 0.1766 32 8.95 0.390 90 369.7 364.0 1.02 0.2512 33 9.40 0.411 90 261.6 252.5 1.04 0.1743 34 8.80 0.389 90 364.0 345.8 1.05 0.2386 35 9.30 0.381 80 184.3 177.5 1.03 0.1225 36 9.00 0.400 76 382.2 364.0 1.05 0.2512 37 8.70 0.399 75 449.3 434.5 1.03 0.2998 38 7.70 0.402 65 466.3 443.6 1.05 0.3061 39 9.20 0.388 60 537.0 536.8 1.00 0.3704 40 9.40 0.359 60 278.8 263.9 1.06 0.1635 a - : rejected because P > 1.2P max Q' 1 55 156 Test Data of Experiment No. 1 - Cont'd Compact Tension Specimen Spec. M.C.b S.G. Angle to P P n P /P„ no. % RL C max Q max Q IC degree N N MPav^ii 41 8.08 0.390 57 473.2 460.7 1.03 0.3179 42 7.40 0.383 55 445.9 437.9 1.02 0.3022 43 9.65 0.350 54 420.8 406.1 1.04 0.2802 44 9.90 0.350 53 445.9 436.8 1.02 0.3014 45 9.67 0.457 52 403.8 392.4 1.03 0.2708 46 8.40 0.364 51 374.2 364.0 1.03 0.2512 47 8.70 0.356 50 364.0 344.7 1.06 0.2379 48 7.80 0.386 50 464.1 460.7 1.01 0.3179 49 7.90 0.455 50 523.3 494.8 1.09 0.3415 50 8.57 0.453 49 523.3 489.1 1.07 0.3375 51 8.60 0.368 49 369.2 364.0 1.01 0.2512 52 9.40 0.456 48 475.5 460.7 1.03 0.3179 53 9.92 0.359 47 434.5 426.6 1.02 0.2944 54 9.00 0.353 47 361.7 358.3 1.01 0.2473 55 9.00 0.377 47 398.1 381.1 1.04 0.2630 56 7.20 0.357 46 483.5 483.4 1.00 0.3336 57 8.57 0.356 45 434.5 420.9 1.03 0.2905 58 7.20 0.355 44 489.2 483.4 1.01 0.3336 59 7.00 0.354 40 517.6 517.6 1.00 0.3572 60 9.70 0.349 40 386.7 386.7 1.00 0.2669 61 9.00 0.356 39 352.6 341.8 1.03 0.2359 62 9.50 0.452 33 517.6 492.5 1.05 0.3399 63 7.00 0.356 32 551.7 546.0 1.01 0.3768 64 10.00 0.336 23 563.1 531.2 1.06 0.3666 65 9.95 0.364 16 574.4 517.6 1.11 0.3572 66 8.95 0.348 16 529.0 517.6 1.02 0.3572 67 9.60 0.353 16 500.5 500.5 1.00 0.3454 68 9.10 0.350 12 585.8 580.1 1.01 0.4003 69 9.40 0.347 7 506.2 '502.8 1.01 0.3470 70 9.30 0.355 5 557.4 541.5 1.03 0.3737 71 9.60 0.366 2 602.9 591.5 1.02 0.4082 72 9.80 0.337 0 532.3 530.1 1.00 0.3658 73 9.95 0.335 0 625.6 620.0 1.01 0.4279 74 9.20 0.355 0 602.9 580.1 1.04 0.4003 b c - moisture content recorded by resistance type moisture 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 crack i n i t i a t i o n . 157 Test Data of Experiment No. 1 Double Cantilever Beam Specimen Spec. M.C. S.G. Angle to P P„ P /P„ K „, _ T max Q max Q IC no. % RL x x degree N N MPa/m 1 9.56 0.367 90 1059.7 981.2 1.08 0.397 2 9.79 0.392 90 1100.2 1047.8 1.05 0.424 3 9.51 0.373 90 1049.9 963.2 1.09 0.390 4 9.83 0.335 90 802.1 722.6 1.11 0.293 5 9.85 0.389 90 983.0 954.4 1.03 0.386 6 9.78 0.401 90 925.9 890.3 1.04 0.360 7 9.61 0.377 90 928.8 910.6 1.02 0.369 8 9.52 0.362 90 1040.8 1020.4 1.02 0.413 9 9.33 0.388 90 848.0 839.6 1.01 0.340 10 9.41 0.389 90 1021.5 954.7 1.07 0.387 11 9.21 0.374 90 1080.2 1038.4 1.04 0.420 12 8.78 0.368 90 852.3 796.5 1.07 0.322 13 9.14 0.332 90 1208.8 1140.4 1.06 0.462 14 9.26 0.387 90 1242.8 1230.5 1.01 0.498 15 9.03 0.396 90 826.3 802.2 1.03 0.325 16 8.98 0.332 90 1037.4 1007.2 1.03 0.408 17 9.75 0.327 90 1006.9 987.2 1.02 0.400 Tes t Data o f Exper iment 2 E f f e c t of Crack-front width on K Spec. no. Width mm M.C. % S.G. P max N PQ N P /P n max Q K I C MPa/m' 1 7 8.98 0.335 69.5 66.7 1.04 0.2511 2 7 8.94 0.335 70.8 68.6 1.03 0.2585 3 7 9.21 0.335 70.0 68.6 1.02 0.2585 4 7 8.68 0.338 76.3 70.6 1.08 0.2659 5 7 8.83 0.338 69.4 66.7 1.04 0.2511 6 15 8.77 0.389 155.0 149.1 1.04 0.2619 7 15 8.46 0.379 127.4 122.6 1.04 0.2154 8 15 8.98 0.388 148.5 147.1 1.01 0.2585 9 15 9.02 0.367 118.8 117.7 1.01 0.2068 10 15 9.02 0.354 127.5 127.5 1.00 0.2240 11 15 8.76 0.378 121.2 117.7 1.03 0.2068 12 15 8.84 0.392 133.9 127.5 1.05 0.2240 13 21 9.24 0.377 178.7 173.6 1.03 0.2179 14 21 9.17 0.401 202.0 196.1 1.03 0.2462 15 21 9.09 0.410 208.0 205.9 1.01 0.2585 16 21 8.94 0.389 189.7 182.4 1.04 0.2289 17 21 9.01 0.397 216.2 204.0 1.06 0.2560 18 21 8.76 0.396 220.0 202.0 1.09 0.2535 19 29 8.49 0.377 207.9 196.1 1.06 0.1783 20 29 8.54 0.376 180.3 178.5 1.01 0.1622 21 29 8.78 0.377 220.1 215.7 1.02 0.1961 22 29 8.77 0.373 161.6 156.9 1.03 0.1426 23 29 8.92 0.373 226.3 219.7 1.03 0.1996 24 29 8.96 0.375 216.4 215.7 1.00 0.1961 25 29 8.88 0.372 209.9 207.9 1.01 0.1890 26 38 8.92 0.372 213.8 196.1 1.09 0.1353 27 38 8.92 0.324 186.3 179.4 1.04 0.1238 28 38 9.05 0.411 178.7 173.5 1.03 0.1197 29 38 9.18 0.408 279.5 269.7 1.04 0.1861 30 38 8.87 0.450 393.4 374.6 1.05 0.2585 31 38 9.10 0.330 251.0 235.3 1.07 0.1624 32 38 9.00 0.414 205.9 184.4 1.12 0.1272 33 38 9.80 0.374 289.3 284.4 1.02 0.1976 34 38 9.60 0.378 340.3 328.5 1.04 0.2267 35 38 9.80 0.369 344.3 334.3 1.03 0.2307 159 Test Data of Experiment No. 3 Effect of Crack-front Length on K^^ Spec. no. Board M.C % ' S.G. L mm Angle to RL degree P max N N P /P n max Q K I C MPa/m 1 A 8.50 0.41 75 90 538.1 "511.5 1.05 0.2355 2 A 8.36 0.42 . 75 90 545.1 489.3 1.11 0.2252 3 A 8.49 0.42 100 90 635.2 622.7 1.02 0.2275 4 A 8.71 0.40 125 90 680.1 645.0 1.05 0.2014 5 A 8.65 0.43 125 90 803.4 778.4 1.03 ' 0.2430 6 B 8.43 0.41 50 60 444.1 427.0 1.04 0.2947 7 B 9.19 0.41 50 60 416.6 389.2 1.06 0.2686 8 B 9.37 0.42 75 55 650.5 600.5 1.08 0.2764 9 B 9.35 0.42 75 58 694.9 667.2 1.04 0.3072 10 B 8.98 0.41 100 60 772.0 756.2 1.02 0.2762 11 B 8.96 0.41 125 65 841.4 822.9 1.03 0.2569 12 C 9.70 0.34 50 75 284.6 278.0 1.02 0.1918 13 C 9.69 0.34 50 75 303.2 300.2 1.01 0.2072 14 C 9.68 0.33 75 72 445.3 411.4 1.08 0.1894 15 C 9.75 0.32 100 68 674.0 645.0 1.04 0.2356 16 C 9.74 0.33 100 75 592.4 556.0 1.06 0.2031 17 C 9.54 0.34 125 79 581.2 578.2 1.01 0.1805 18 D 8.99 0.42 50 60 408.3 382.5 1.07 0.2640 19 D 8.98 0.41 100 60 695.5 689.4 1.01 0.2518 20 D 8.93 0.41 125 72 1063.9 1007.5 1.05 0.3145 160 Test Data of Experiment No. 4 Effect of Moisture Content on K Spec. M.C. S.G. Angle to P P„ P /?n K „ „ T max 0 max Q IC no. % RL degree N N MPa/m 1 0.0 0.388 90 207.9 196.1 1.06 0.1353 2 0.0 0.357 90 280.4 269.7 1.04 0.1861 3 0.0 0.396 90 214.2 213.8 1.00 0.1475 4 0.0 0.403 70 205.6 204.0 1.01 0.1407 5 0.0 0.357 90 282.1 255.0 1.11 0.1759 6 0.0 0.394 90 274.4 262.8 1.05 0.1813 7 0.0 0.372 90 344.1 335.4 1.02 0.2314 8 0.0 0.356 90 337.9 326.5 1.04 0.2253 9 0.0 0.439 90 355.2 351.1 1.01 0.2422 10 0.0 0.410 90 225.8 219.7 1.03 0.1516 11 0.0 0.372 90 252.3 235.3 1.07 0.1724 12 0.0 0.366 90 322.1 313.8 1.02 0.2165 13 0.0 0.379 90 311.6 298.1 1.05 0.2057 14 0.0 0.387 90 206.9 196.1 1.05 0.1353 15 0.0 0.375 90 207.1 204.0 1.01 0.1407 16 0.0 0.371 90 186.9 179.4 1.04 0.1238 17 6.2 0.371 90 274.2 262.8 1.04 0.1813 18 6.1 0.365 90 231.9 219.7 1.06 0.1516 19 6.3 0.351 90 351.1 326.5 1.07 0.2253 20 6.1 0.344 90 252.7 240.3 1.05 0.1658 21 6.3 0.342 60 333.4 313.8 1.06 0.2165 22 6.1 0.340 90 302.1 298.1 1.01 0.2057 23 6.6 0.327 90 264.9 262.8 1.01 0.1813 24 6.8 0.334 90 260.1 255.0 1.02 0.1759 25 6.3 0.387 90 279.2 235.3 1.09 0.1624 26 6.1 0.312 90 330.7 313.8 1.05 0.2165 27 6.1 0.314 90 343.2 328.5 1.04 0.2267 28 6.7 0.386 90 220.6 213.8 1.03 0.1475 29 6.3 0.343 90 224.2 217.7 1.03 0.1502 30 6.5 0.317 90 378.9 362.8 1.05 0.2504 31 6.4 0.357 90 360.3 351.1 1.02 0.2422 32 6.7 0.369 90 359.2 335.4 1.07 0.2314 161 Test Data of Experiment No. 5  Mode II End-split Beam Specimen Spec. M.C. S.G. Angle to P K no. % RL Q I I C degree N MPa/m 1 13.3 0.376 0 13344 2.0522 2 13.5 0.388 7 12899 1.9837 3 13.7 0.378 12 14856 2.2847 4 13.9 0.360 18 12454 1.9153 5 11.3 0.473 32 16013 2.4626 6 13.3 0.387 38 15034 2.3121 7 14.0 0.370 39 13033 2.0043 8 12.6 0.378 43 13789 2.1206 9 13.8 0.375 45 18682 2.8731 10 11.7 0.380 46 17792 2.7362 11 14.5 0.539 48 19794 3.0441 12 13.9 0.360 55 18904 2.9072 13 13.7 0.381 60 12232 1.8812 14 14.5 0.423 64 19571 3.0098 15 12.8 0.324 75 12588 1.9359 16 13.3 0.349 88 13033 2.0043 17 12.9 0.347 90 14678 2.2573 18 13.5 0.398 90 12632 1.9427 19 12.5 0.384 90 12677 1.9496 20 13.5 0.349 90 14812 2.2779 21 12.9 0.313 90 12899 1.9837 22 13.5 0.345 90 13878 2.1343 23 12.0 0.310 90 15568 2.3942 24 12.6 0.314 90 15012 2.3087 Note : P m a x i s usually greater than 2P^ since specimens' are governed by the bending strength. 162 T e s t Data o f Experiment No. 6 Mixed Mode M i d - c r a c k e d Beam c Phase 1 - Center point loading Spec. M.C. S.G. Angle to Crack Load P „ b K K „ % RL angle at ^ ^ ^ degree degree MPa/m MPaAi 1 8. 94 0.396 41 45 a c e n t e r 4448 0.3360 0.8330 2 9. 03 0.394 55 45 center 5560 0.4199 1.0413 3 9. 02 0.374 65 45 center 5871- 0.4435 1.0996 4 8. 87 0.388 60 45 center 5894 0.4451 1.1038 5 8. 96 0.401 90 45 center 4670 0.3527 0.8747 6 8. 98 0.403 90 45 center 4448 0.3360 0.8330 7 9. 23 0.416 90 45 center 3425 0.2587 0.6414 8 8. 46 0.369 90 45 center 6227 0.4703 1.1662 9 8. 79 0.388 90 45 center 3647 0.2755 0.6831 10 8. 90 0.404 90 45 center 3514 0.2654 0.6581 11 8. 37 0.399 90 45 center 3174 0.2397 0.5831 12 8. 38 0.396 90 45 center 5338 0.4031 0.9996 13 8. 42 0.393 90 45 center 4782 0.3612 0.8955 14 8. 57 0.385 90 45 center 3336 0.2520 0.6248 15 8. 69 0.344 90 45 center 3625 0.2738 0.6789 16 8. 94 0.346 90 45 center 4003 0.3024 0.7497 17 9. 00 0.384 90 45 center 4226 0.3192 0.7914 18 9. 02 0.388 90 45 center 3892 0.2940 0.7289 19 8. 89 0.374 90 45 center 5894 0.4451 1.1038 20 8. 78 0.376 90 45 center 3781 0.2856 0.7081 21 8. 98 0.392 90 45 center 3447 0.2603 0.6456 22 9. 03 0.377 90 45 center 3505 0.2647 0.6564 23 8. 65 0.354 90 45 center 3394 0.2563 0.6356 24 8. 75 0.346 90 45 center 3336 0.2520 0.6248 a - Loading at the c e n t e r l i n e , i . e . , 450 mm from the support. b - P m a x i s greater than 1.2 P^, beam strength governed by bending, c - 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 - Sixth point 16ading !pec. no. M.C. % S.G. Angle to RL Crack angle Load at PQ KIQ K I I Q degree degree N MPa/m MPa/m 1 8.44 0.375 11 45 6th 4670 4 0.3255 0.4768 2 8.64 0.386 40 45 6 th 5115.2 0.3566 0.5222 3 8.76 0.398 51 45 6th 6004.8 0.4186 0.6130 4 9.04 0.401 58 45 6th 4781.6 0.3333 0.4882 5 8.56 0.375 73 45 6th 5560.0 0.3876 0.5676 6 8.94 0.336 87 45 6th 6227.2 0.4341 0.6357 7 8.48 0.384 90 45 6 th 5782.4 0.4031 0.5903 8 8.52 0.386 90 45 6th 4448.0 0.3100 0.4541 9 8.79 0.383 90 45 6th 3647.4 0.2542 0.3724 10 9.12 0.396 90 45 6th 3558.4 0.2480 0.3633 11 9.08 0.399 90 45 6th 4670.4 0.3255 0.4768 12 9.12 0.391 90 45 6th 6004.8 0.4186 0.6130 13 9.26 0.405 90 45 6 th 5115.2 0.3566 0.5222 14 8.96 0.386 90 45 6th 5560.0 0.3876 0.5676 15 8.94 0.377 90 45 6th 6227.2 0.4341 0.6357 16 9.04 0.389 90 45 6th 5560.0 0.3876 0.5676 17 9.02 0.403 90 45 6th 6449.6 0.4496 0.6585 18 8.59 0.380 90 45 6th 6672.0 0.4651 0.6812 20 8.54 0.377 90 45 6th 4448.0 0.3100 0.4541 21 9.34 0.411 90 45 6 th 4448.0 0.3100 0.4541 22 9.33 0.408 90 45 6th 5449.0 0.3798 0.5563 23 9.31 0.412 90 45 6 th 3447.6 0.2403 0.3519 24 9.16 0.402 90 45 6th 5115.3 0.3565 0.5222 25 9.22 0.398 90 45 6 th 4782.0 0.3333 0.4882 1 6 4 T e s t Da ta o f Expe r imen t No. 6 - C o n t ' d Phase 2 - 9 0 deg . beam - c e n t e r l i n e l o a d i n g >pec. no. M.C. % S.G. Angle to RL Crack angle KIIQ degree degree N MPav^ ni MPa/m 1 8.98 0.356 90 90 5393 0.3402 0.7246 2 8.73 0.398 90 90 4559 0.2876 0.6125 3 8.16 0.388 90 90 5627 0.3550 0.7560 4 8.48 0.359 90 90 6505 0.4104 0.8740 5 8.81 0.328 90 90 5338 0.3367 0.7172 6 9.04 0.396 90 90 4782 0.3017 0.6425 7 9.08 0.400 90 90 4559 0.2876 0.6125 8 8.48 0.377 90 90 3447 0.2175 0.4631 9 8.63 0.369 90 90 4537 0.2862 0.6096 10 8.71 0.399 90 90 6283 0.3937 0.8441 11 8.68 0.396 90 90 3781 0.2385 0.5080 12 8.70 0.394 90 90 5894 0.3718 0.7919 13 8.65 0.390 90 90 4337 0.2736 0.5827 14 8.68 0.388 90 90 3892 0.2455 0.5229 15 8.72 0.392 90 90 4782 0.3017 0.6425 16 8.71 0.392 90 90 4537 0.2862 0.6096 Note: Crack I n i t i a t i o n occured ±90 degrees to the i n i t i a l crack simultaneously. 165 T e s t Data o f Experiment No. 7  DCB under Mixed-mode L o a d i n g Spec, no. M.C. % S.G. Angle to RL degree P2 KIQ K I I Q N MPav/iii MPav^ m 1 8.42 0.379 90 1925 3825 0.4009 0.5761 2 8.73 0.386 90 1925 3336 0.4493 0.5024 3 8.71 0.384 90 1925 4448 0.3393 0.6699 4 8.76 0.379 90 1925 4003 0.3833 0.6029 5 9.26 0.359 90 1925 4448 0.3393 0.6699 6 8.16 0.403 90 1925 3670 0.4163 0.5526 7 8.96 0.396 90 1925 4003 0.3833 0.6029 8 8.94 0.394 90 1925 3558 0.4273 0.5359 9 8.89 0.382 90 1925 4114 0.3723 0.6196 10 8.75 0.374 90 1925 4448 0.3393 0.6699 11 8.96 0.369 90 2800 9341 0.2095 1.4067 12 8.94 0.346 90 2800 8229 0.3195 1.2393 13 9.07 0.358 90 2800 8674 0.2755 1.3062 14 9.01 0.398 90 2800 7784 0.3635 1.1723 15 8.92 0.374 90 2800 8006 0.3415 1.2058 16 8.96 0.368 90 2800 8562 0.2865 1.2895 17 8.46 0.348 90 — — — * 18 8.29 0.376 90 2800 8006 0.3415 1.2058 19 8.45 0.368 90 2800 8674 0.2755 1.3062 20 8.65 0.384 90 2800 9118 0.2315 1.3732 * S p l i t occured i n the specimen before t e s t i n g . 166 Test Data of Experiment No. 8  Notched Beam Specimens Spec. Spec. M.C. S.G. A Angle to P K K type no. % RL I i Q degree N MPam MP am A A 1 8.84 0.382 A 2 8.76 0.369 A 3 8.42 0.384 A 4 9.24 0.321 A 5 9.80 0.402 A 6 7.78 0.416 A 7 8.12 0.332 A 8 9.13 0.303 B 1 8.41 0.342 B 2 8.12 0.383 B 3 8.56 0.369 B 4 9.37 0.414 B 5 10.01 0.311 B 6 7.68 0.308 B 7 8.08 0.324 B 8 8.46 0.326 C 1 8.11 0.414 C 2 8.43 0.308 C 3 8.56 0.388 C 4 8.64 0.384 C 5 7.13 0.369 C 6 6.84 0.342 C 7 8.42 0.324 C 8 7.88 0.336 1.6 90 4203 0. 2676 0.4647 1.6 90 4559 0. 2902 0.5041 1.6 90 2335 0. 1487 0.2582 1.6 90 4114 0. 2619 0.4549 1.6 90 5226 0. 3327 0.5778 1.6 90 4504 0. 2867 0.4980 1.6 90 4915 0. 3129 0.5434 1.6 90 4114 0. 2619 0.4549 1.6 90 4114 0. 1776 0.9025 1.6 90 6338 0. 2737 1.3904 1.6 90 5894 0. 2545 1.2928 1.6 90 5115 0. 2208 1.1221 1.6 90 6227 0. 2689 1.3660 1.6 90 5226 0. 2256 1.1465 1.6 90 5449 0. 2352 1.1953 1.6 90 5560 0. 2400 1.2197 1.7 90 8006 0. 1940 0.6840 1.7 90 10342 0. 2506 0.8835 1.7 90 9341 0. 2263 0.7980 1.7 90 5338 0. 1293 0.4560 1.7 90 8118 0. 1967 0.6935 1.7 90 6116 0. 1482 0.5225 1.7 90 10898 0. 2640 0.9310 1.7 90 11898 0. 2883 1.0165 APPENDIX III Difference between Mode I Fracture Toughness of CTSs and DCB  spec imens. A. Influence on KI(-, of Assumed Elastic Properties E (MPa) X Spec. type K*c (MPa/m) E y E y E y 12500 DCB 0.0498 0.0403 0.0359 CTS 0.0693 0.0688 0.0681 10000 DCB 0.0498 0.0403 0.0359 CTS 0.0693 0.0688 0.0681 7500 DCB 0.0498 0.0403 0.0359 CTS 0.0693 0.0688 0.0681 *E„/G = 1.0 and v v„ = 0.02 were used in generating the y A Y chart. As can be seen above, the variation of E has no influence on the fracture toughness providing the ratios EV:E._, E„:G and vv„ are the same as those used in generating A y y A y the chart. Thus, we can compute the ratio k J C T S ^ K I D C B ' KIC (DCB) and K I C (CTS) for the three cases : K. I(CTS) K I(DCB) KIC(DCB) ( M P a ^ ) KIC(CTS) ( M P a ^ 10 20 30 1.3909 1.7047 1.8970 0.4760 0.3855 0.3429 0.2059 0.2044 0.2023 The difference between the CTSs and DCB specimens is 167 168 significant for various ratios of E x/E y. This indicates that the difference in the mode I fracture toughness is not affected much by the assumed elastic properties. B. T-test on the hypothesis that the two samples being compared are drawn from the same population. The test is applied to the null hypothesis that the two samples being compared are drawn from the same population, and we calculate the probability of the difference of the two means having a value as large as, or greater than, observed. Sample Mean K (MPa m) •LL* Standard deviation Sample s i z e X s n 1. CTS 0.205 0.0510 33 2. DCB 0.387 0.0497 17 The pooled estimate of variance is s 2 = n l s l + n2 s2 = 33 x 0.05102+ 17 x 0.04972 = 2.557x10 and s = 0.0506 MPa/m c 169 The standard deviation of the difference of mean is thus : S. = s / l / n . + l / n . d c i 1 0.0506 / 1/33 + 1/17 = 0.01509 X l " X2 10.205 - 0.387| 0.01509 12.06 The number of degrees of freedom is 33 + 17 - 2 = 48, and from the t-distribution, i t gives t = 2.0126 at the 95 percent level of confidence. Therefore, the null hypothesis can be rejected, and we conclude that the two samples are from different populations. APPENDIX I I I S.A.A. A u s t r a l i a n T imber E n g i n e e r i n g Code AS 1720-1975 S t r e n g t h o f Notched Beams F o r a r e c t a n g u l a r beam o f dep th D, n o t c h e d on the t e n s i o n edge as shown i n F i g . 57 , the n o m i n a l maximum b e n d i n g s t r e s s ^b = ScO a n c ^ n o m : L n a l maximum s h e a r s t r e s s f g = c a l c u l a t e d n n f o r the ne t s e c t i o n s h a l l comply w i t h t he f o l l o w i n g i n t e r a c t i o n f o r m u l a : 0 . 3 f , + 0 . 7 f D S c 3 F s j « 1 where C 0 i s a c o n s t a n t t a b u l a t e d i n t a b l e 11 and F • i s the 3 s j p e r i m i s s i b l e shear s t r e s s f o r j o i n t d e t a i l s o r the shear b l o c k s t r e n g t h f o r the s p e c i e s o f i n t e r e s t , v M Figure 57 Notation for Notch T a b l e 11 Paramete r C^ f o r s e l e c t e d n o t c h a n g l e s n o t c h s l o p e a ^ O . l D a < 0 . 1 D b / a = 0 b / a = 2 b / a = 4  » 0 .1 D • 3 / D % , 2 . 6 / D ? 2 . 2 / D " l / a % 1 . 2 / a 1.3/a' 1 170 

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