Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Stress-intensity factors for V-notches in orthotropic plates using singular finite elements Lum, Conroy 1986

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

Item Metadata

Download

Media
831-UBC_1986_A7 L85.pdf [ 5.89MB ]
Metadata
JSON: 831-1.0062720.json
JSON-LD: 831-1.0062720-ld.json
RDF/XML (Pretty): 831-1.0062720-rdf.xml
RDF/JSON: 831-1.0062720-rdf.json
Turtle: 831-1.0062720-turtle.txt
N-Triples: 831-1.0062720-rdf-ntriples.txt
Original Record: 831-1.0062720-source.json
Full Text
831-1.0062720-fulltext.txt
Citation
831-1.0062720.ris

Full Text

STRESS-INTENSITY FACTORS FOR V-NOTCHES IN ORTHOTROPIC PLATES USING SINGULAR FINITE ELEMENTS by CONROY LUM B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE 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 1986 © Conroy Lum, 1986 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of C i v i l Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date A p r i l 30. 1986 •7Q ^ ABSTRACT A f i n i t e element program u t i l i z i n g singularity-enriched elements was developed to determine the stress-intensity factors for an arbitrary V-notch i n a linear elastic orthotropic beam. The special elements allow computation of the stress-intensity factors without recourse to an extremely fine element mesh. An application of the program was made to wood. A pilot experiment involving rectangular end-notched wood beams verified the program as well as demonstrated the u t i l i t y of the program i n predicting failure loads due to rapid crack propagation emanating from V-notches. The effect of varying wood properties on the V-notch stress-intensity factor was examined. A procedure to establish the failure surface encompassing V-notches and cracks under a l l loading conditions using the program i s presented. i i TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENT ix 1.0 INTRODUCTION 1 1.1 The Problem 1 1.2 Previous Research 2 1.3 Purpose and Scope of the Thesis 5 2.0 BASIC CONCEPTS AND ASSUMPTIONS 7 2.1 Concept of the Stress-Intensity Factor 7 2.2 Zero Notch Root Radius Assumption 9 3.0 FORMULATION OF THE SINGULAR SOLUTION 12 3.1 Derivation of the Governing D i f f e r e n t i a l Equation 12 3.2 Solution of the Governing D i f f e r e n t i a l Equation 14 3.3 Singular Stresses and Stress Boundary Conditions 19 3.4 Determining Values of A 24 3.5 Singular Strains and the Associated Displacement F i e l d . 35 4.0 IMPLEMENTATION IN A FINITE ELEMENT PROGRAM 42 4.1 Regular Quadratic Isoparametric Element 42 4.2 Formulation of the Singular Element 45 4.2.1 F u l l y Singular Element S t i f f n e s s Matrix 45 4.2.2 Singular T r a n s i t i o n Elements 49 4.2.3 Overlapping S i n g u l a r i t i e s 50 4.2.4 Body Force and I n i t i a l S t r a i n Vectors .. 57 4.2.4.1 I n i t i a l Strains 58 4.2.4.2 Body Forces and Surface Tractions 59 4.3 V-notch St r e s s - I n t e n s i t y Factors 60 4.4 Program V e r i f i c a t i o n and Convergence 64 4.4.1 Sharp Cracks 64 4.4.2 V-Notches i n Isotropic Bodies 68 4.5 F i n i t e Element Modelling of a Notched Beam 71 5.0 APPLICATIONS 89 5.1 Load Test on Rectangular End-notched Wood Beams 89 5.1.1 Procedure 89 5.1.2 Treatment of Data 91 5.1.3 Results and Discussions 92 5.2 Rectangular End-notch Stress-Intensity Factors 96 5.3 E f f e c t of Slope of Grain and Propagation D i r e c t i o n .... 97 5.4 E f f e c t of Varying the E l a s t i c Constants 102 5.5 Cracks at the Tip of Rectangular End-notches 104 5.6 V-notch Stres s - I n t e n s i t y Factor Interaction Curve 106 6.0 C0NCULSI0NS 115 BIBLIOGRAPHY 116 i i i TABLE OF CONTENTS - C o n t ' d Page APPENDIX I 120 APPENDIX I I 121 APPENDIX I I I 122 i v LIST OF TABLES TABLE Page I S t r e s s - i n t e n s i t y factors f o r various sharp crack problems .. 65 II S t r e s s - i n t e n s i t y factors for V-notched aluminium s t r i p s .... 69 III E f f e c t of slope of grain on rectangular end-notched beam s t r e s s - i n t e n s i t y f a c t o r s 99 IV E f f e c t of a cross-grain propagation on rectangular end-notched beam s t r e s s - i n t e n s i t y f a c t o r s at various slope of grain 101 V E f f e c t of varying e l a s t i c constants on rectangular end-notched beam s t r e s s - i n t e n s i t y factors 103 v LIST OF FIGURES FIGURE Page 2.1 Mode I and Mode II crack displacement modes 8 3.1 Local notch coordinate system 20 3.2 X vs notch angle for various notch geometries 26 3.3 Notch configuration (A) for Figure 3.2 26 3.4 Notch configuration (C) for Figure 3.2 26 3.5 F i r s t primary singular stress f i e l d f o r a 0°:360° crack ... 28 3.6 Second primary singular stress f i e l d f o r a 0°:360° crack .. 28 3.7 F i r s t primary singular stress f i e l d for a -45°:315° crack .. 29 3.8 Second primary singular stress f i e l d f o r a -45°:315° crack . 29 3.9 F i r s t primary singular stress f i e l d f o r a -90°:270° crack i n the l i m i t as V-notch becomes a crack 30 3.10 Second primary singular stress f i e l d f o r a -90°:270° crack i n the l i m i t as V-notch becomes a crack 30 3.11 F i r s t primary singular stress f i e l d f o r a -90°:270° crack when so l v i n g d i r e c t l y f o r a crack 31 3.12 Second primary singular stress f i e l d f o r a -90°:270° crack when so l v i n g d i r e c t l y f o r a crack 31 3.13 Primary singular stress f i e l d for a rectangular end-notch . 32 3.14 Primary singular stress f i e l d f o r a tapered notches 32 3.15 Secondary singular stress f i e l d f o r a rectangular end-notch 33 4.1 F i n i t e element global and l o c a l coordinate systems 44 4.2 T r a n s i t i o n elements between f u l l y singular and regular elements 51 4.3 Overlapping singular and t r a n s i t i o n elements 52 4.4 Sharp crack K-j- specimen 66 4.5 Sharp crack specimen (end-split beam) 66 vi LIST OF FIGURES - Cont'd FIGURE Page 4.6 Center cracked i s o t r o p i c s t r i p under tension 66 4.7 Inclined crack i n a square orthotropic plate under tension 66 4.8 Isotropic V-notched shear specimen ( K J - J - ) 70 4.9 Isotropic centerpoint loaded V-notched bending specimen (Kj) 70 4.10 Isotropic V-notched tension specimen (Kj) 70 4.11 Rectangular end-notched centerpoint loaded beam configuration 72 4.12 P a r a l l e l - t o - g r a i n stress contours for an unnotched beam ... 73 4.13 Perpendicular-to-grain stress contours for an unnotched beam 74 4.14 Shear stress contours f o r an unnotched beam 75 4.15 P a r a l l e l - t o - g r a i n stress contours f o r a 10-mm end-notched beam under centerpoint loading 76 4.16 P a r a l l e l - t o - g r a i n stress contours for a 10-mm end-notched beam under centerpoint loading 77 4.17 Perpendicular-to-grain stress contours f o r a 10-mm end-notched beam under centerpoint loading 78 4.18 Perpendicular-to-grain stress contours f o r a 10-mm end-notched beam under centerpoint loading 79 4.19 Shear stress contours f o r a 10-mm end-notched beam under centerpoint loading 80 4.20 Shear stress contours f o r a 10-mm end-notched beam under centerpoint loading 81 4.21 P a r a l l e l - t o - g r a i n stress contours f o r a 50-mm end-notched beam under centerpoint loading 82 4.22 P a r a l l e l - t o - g r a i n stress contours f o r a 50-mm end-notched beam under centerpoint loading 83 4.23 Perpendicular-to-grain stress contours f o r a 50-mm end-notched beam under centerpoint loading 84 v i i LIST OF FIGURES - cont'd FIGURE Page 4.24 Perpendicular to grain stress contours f o r a 50-mm end-notched beam under centerpoint loading 85 4.25 Shear stress contours f o r a 50-mm end-notched beam under centerpoint loading 86 4.26 Shear stress contours f o r a 50-mm end-notched beam under centerpoint loading 87 5.1 Rectangular end-notched centerpoint loaded beam configuration 90 5.2 F a i l e d 10-mm end-notched 2 x 4 Hem-Fir beam under centerpoint loading 93 5.3 Typ i c a l l o a d - d e f l e c t i o n curves showing P ^ ^ Pq, and 5% of f s e t l i n e . < C a s e : Pmax " V Pmax > Pq» a n d W » V 9 4 5.4 s t r e s s - i n t e n s i t y f a c t o r f o r various notch depths and notch lengths (E = 12400 MPa, E = 620 MPa, G = 690 MPa, and v)___ = 0.02) 98 5.5 Slope of grain at V-notches 100 5.6 Across-grain crack propagation at V-notches 100 5.7 S t r e s s - i n t e n s i t y factors f o r sharp cracks emanating from rectangular end notches 105 5.8 Typ i c a l V-notch K-j- specimens 109 5.9 K j specimen primary singular stress f i e l d f o r X = 1.500 ... I l l 5.10 KJJ- specimen secondary singular stress f i e l d f o r X = 1.518 111 5.11 K j specimen primary singular stress f i e l d for X = 1.518 ... 112 5.12 Mixed mode V-notch configuration, X = 1.518, K-j-rKj-j- = 0.3959 112 5.13 Mixed mode V-notch configuration, X = 1.518, K-J-IK-^ = 1.6432 113 5.14 Mixed mode V-notch configuration, X = 1.518, K j i K j j - 0.9818 113 5.15 K j and K-Q specimens f o r X = 1.518 114 5.16 Specimen f o r K j i K j j = 1.6432 114 5.17 Specimen f o r K j t K j j = 0.3959 114 5.18 Specimen f o r K j t K j j = 0.9818 114 v i i i ACKNOWLEDGEMENTS I owe a continuing debt to my advisor, Dr. Ricardo Foschi, for h i s patience and Invaluable assistance i n the preparation of t h i s t h e s i s . Special thanks go to Dr. Mervyn Olson and Prof. Borg Madsen for reading t h i s t hesis and f o r t h e i r h e l p f u l comments. My sincere gratitude to Forintek Canada Corp. for awarding me the 1985 Forintek Canada Corp. Fellowship i n Timber Engineering and f o r the use of t h e i r t e s t i n g f a c i l i t i e s . I also thank the s t a f f at Forintek Canada's Western Laboratory f o r t h e i r assistance during the t e s t i n g phase. ix INTRODUCTION 1 1.1 The Problem V-notches occur as corners i n cutouts or i n open butt j o i n t s or l a p - j o i n t s i n laminated beams. These sudden changes i n geometry lead to s t r e s s concentrations and s t r e s s conditions which cannot be determined by standard engineering formulae. Detailed analysis tend to be cumbersome. As a r e s u l t , one simply designs f o r notches based on past experience. The primary objective of t h i s t h e s i s i s the development of a general procedure to analyze notched beams. As an a p p l i c a t i o n , consider the case of notched timber beams. The governing Canadian design code, CAN3-086-M84, deals only with rectangular end-notched beams and does so simply by reducing the allowable shear stresses. The reductions depend on the notch length as well as notch depth but i s the same regardless of whether the notch i s rectangular or tapered. If the notch i s located i n a portion of the beam where the shear force i s zero, e.g. t h i r d point loading, then according to the current design procedure, the notch has no e f f e c t on strength other than reducing the depth of the sect i o n . E l a s t i c i t y solutions f o r cracked and V-notched bodies are known only f o r a l i m i t e d number of simple geometries under simple loading d i s t r i b u t i o n s . Therefore, numerical techniques such as the F i n i t e Element method are required. Used i n conjunction with l i n e a r - e l a s t i c f r a c t u r e mechanics concepts, a f a i l u r e c r i t e r i o n based on the rapid propagation of a crack emanating from the notch root can be est a b l i s h e d . But due to the stress s i n g u l a r i t y , an extremely f i n e mesh i s required i n the v i c i n i t y of the notch root to accurately determine stresses and 2 s t r a i n s ; t h i s makes the analysis c o s t l y not only i n computer time, but also i n data preparation time. Furthermore, f a i l u r e o r i g i n a t i n g at the notch root i s generally due to the i n t e r a c t i o n of shearing and t e n s i l e stresses, thus complicating f a i l u r e p r e d i c t i o n . 1.2 Previous Research P r i o r to the development of fr a c t u r e mechanics, research on the e f f e c t of notches or rectangular holes i n orthotropic materials consisted of determining values of stress concentration f a c t o r s f o r various notch geometries and loading conditons. A n a l y t i c a l and photoelastic studies by Savin (1961) provided some i n s i g h t i n t o the stresses at reentrant corners. Stieda (1964), using the photostress method, investigated semi-circular notches i n timber beams. He l a t e r conducted an experimental study of notches of various depths and lengths located on the tension edge of small c l e a r wood beam specimens (Stieda, 1965). Notched beam strengths were expressed as a f r a c t i o n of the unnotched beam strength. These solutions not only were l i m i t e d to the configurations analyzed, but also lacked a c l e a r d e f i n i t i o n of f a i l u r e unless one was w i l l i n g to t e s t every conceivable notch configuration i n every material. L i n e a r - e l a s t i c fracture mechanics, using the concept of the s t r e s s - i n t e n s i t y factor, has been shown to be an e f f e c t i v e method of pre d i c t i n g f a i l u r e of a cracked material (e.g. Broek, 1982; Hellan, 1984; M a l l et a l , 1983). Fracture mechanics studies on cracks i n i s o t r o p i c materials i s abundant i n the l i t e r a t u r e , while work on cracks i n o r t h o t r o p i c plates i s not as common, p a r t i c u l a r l y f o r mixed mode conditions. In an orthotropic body such as wood beams, crack propagation involves two e s s e n t i a l l y b r i t t l e f a i l u r e modes: shear and tension 3 perpendicular to the grain. Research on V-notches i n orthotropic materials i s even more l i m i t e d . Except f o r a few problems with simple geometry and boundary conditions, most solutions require numerical techniques. Various d i r e c t and i n d i r e c t numerical methods have been employed to c a l c u l a t e s t r e s s - i n t e n s i t y factors for sharp cracks i n orthotropic and i s o t r o p i c materials. D i r e c t methods based on the F i n i t e Element method incorporate the s t r e s s - i n t e n s i t y factors i n the solution vector. Indirect methods, such as those based on the energy-released rate, use known displacement or stress f i e l d s (e.g. from the F i n i t e Element method) to calculate the s t r e s s - i n t e n s i t y f a c t o r s (Cook, 1981; Walsh, 1972). A d i r e c t method suggested by Benzley (1974), u t i l i z i n g q u a d r i l a t e r a l isoparametric f i n i t e elements enriched with the s i n g u l a r part of the e l a s t i c i t y s o l u t i o n , has been shown to be e f f i c i e n t and r e l i a b l e . The f i n i t e element mesh consists of p l a c i n g singular elements around the crack t i p and regular elements elsewhere. This permits the computation of the s t r e s s - i n t e n s i t y f a c t o r s using a r e l a t i v e l y coarse mesh. A b i l i n e a r t r a n s i t i o n function was used to maintain compatibility between the singular and regular elements. Foschi and Barrett (1976), using quadratic isoparametric elements, extended the Benzley approach to the anisotropic case. This approach gave s i m i l a r r e s u l t s as Gandhi (1972), who used a modified mapping-collocation technique to c a l c u l a t e the s t r e s s - i n t e n s i t y f a c t o r s . G i f f o r d and H i l t o n (1978) implemented the Benzley approach with the 12-node isoparametric element for cracks i n i s o t r o p i c bodies thus permitting even coarse meshes near the crack t i p . Haymann (1980) reviewed a number of these s p e c i a l crack t i p elements. A common method of modelling a s i n g u l a r i t y using f i n i t e elements i s 4 to collapse the 8-node isoparametric element i n t o a t r i a n g l e and s h i f t the side nodes to the quarter points to produce a singular Jacobian at the crack t i p (Cook, 1981; Haymann, 1980). Element c o m p a t i b i l i t y i s maintained and modifications to the f i n i t e element program i s minimal. Although t h i s i s an elegant approach, i t i s r e s t r i c t e d to sharp crack - problems and cannot be used f o r V-notches because i t i s only able to model a l / / r type s i n g u l a r i t y . As we w i l l see l a t e r , s t r e s s - s t r a i n s i n g u l a r i t i e s at the root of V-notches are not, i n general, of the 1//t~ type. L e i c e s t e r (1971; 1982) proposed the extension of the s t r e s s - i n t e n s i t y factor concept to problems involving V-notches i n orthotropic materials. This was l a t e r used i n conjunction with the F i n i t e Element method to i n d i r e c t l y c a l c u l a t e s t r e s s - i n t e n s i t y f a c t o r s , and K^. Since singular elements were not used, an extremely f i n e mesh was required. Unlike the basic s t r e s s - i n t e n s i t y f a c t o r s , K^ . and K^^ f o r sharp cracks, K. and K do not correspond to any basic displacement mode A o such as symmetric crack opening and antisymmeteric s l i d i n g shear. In i s o t r o p i c materials, V-notches have been analyzed by Gross and Mendelson (1972) i n bending, shear, and tension using a boundary c o l l o c a t i o n procedure, and by L i n and Tong (1980) using singular f i n i t e elements. Studies to determine 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 (the value of the s t r e s s - i n t e n s i t y factor at which rapid crack propagation occurs) i n orthotropic materials have p r i m a r i l y been f o r sharp cracks i n wood. Schniewind and Pozniak (1971) found 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 f o r the opening mode, KJQ» i n Douglas F i r ; and using an e n d - s p l i t beam, Barr e t t and Foschi (1977) established K values f o r Hemlock. Since 5 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 are species dependent, a great deal of experimental work remains to be performed for cracks. Therefore, i t i s not s u r p r i s i n g to f i n d very l i t t l e i n the area of V-notches. In addition, Barrett (1976) found a s i z e e f f e c t due to crack-front width on Kj.£ for cracks, and i t i s not unreasonable to assume that there i s also a s i m i l a r e f f e c t i n V-notches. Although the s t r e s s - i n t e n s i t y f a c t o r approach has been researched for cracks and V-notches i n i s o t r o p i c materials, no attempt has been made to u n i f y s t r e s s - i n t e n s i t y f a c t o r s for cracks and V-notches. Studies on the i n t e r a c t i o n between the two basic modes i s l i m i t e d both for Isotropic and orthotropic materials. Wu (1976) obtained an i n t e r a c t i o n curve f o r and f o r balsa wood and f i b r e g l a s s with the assumption that the crack was c o l i n e a r with the g r a i n d i r e c t i o n . A s i m i l a r study by Mall et a l (1983) suggested the same i n t e r a c t i o n r e l a t i o n f o r Kj. and . Perhaps due to the lack of an e f f i c i e n t a n a l y t i c a l t o o l , an i n t e r a c t i o n equation f o r V-notches has yet to be established. 1.3 Purpose and Scope The objective of t h i s t h e s i s i s the development of the t h e o r e t i c a l background to a design procedure for the e f f e c t of V-notches. Thus the t h e s i s w i l l focus i n an a n a l y t i c a l and numerical procedure to compute the s t r e s s - i n t e n s i t y f a c t o r s , and not on an experimental program to e s t a b l i s h the c r i t i c a l values f o r these f a c t o r s . The thesis w i l l focus on 1) the formulation of the s i n g u l a r i t y function f o r an a r b i t r a r i l y oriented V-notch, with any notch angle, i n a plane, homogeneous, l i n e a r - e l a s t i c orthotropic body; and 2) the 6 implementation of the s o l u t i o n i n a F i n i t e Element program to enable one to analyze V-notches with a r e l a t i v e l y coarse mesh. The program w i l l d i r e c t l y determine equivalent opening and shear mode s t r e s s - i n t e n s i t y f a c t o r s , which can then be compared to the basic and K^.^ obtained from standard specimens to e s t a b l i s h an i n t e r a c t i o n r e l a t i o n . Although the theory applies to a l l orthotropic materials, s p e c i f i c a p p l i c a t i o n s w i l l be made to wood. The "grain" d i r e c t i o n w i l l be the s t i f f e s t of the two axis of e l a s t i c symmetry and w i l l a lso be assumed to be the d i r e c t i o n of crack propagation unless s p e c i f i e d othwerwise. Two rectangular notch depths i n timber beams w i l l be analyzed and a small test w i l l be conducted to v e r i f y the program. The s e n s i t i v i t y of the s t r e s s - i n t e n s i t y f a c t o r to notch geometry and mat e r i a l properties w i l l be studied as well. F i n a l l y , a procedure to generate i n t e r a c t i o n curves f o r mixed mode conditions w i l l be presented. This procedure w i l l u nify s t r e s s - i n t e n s i t y factors f o r sharp cracks and V-notches. 7 BASIC CONCEPTS AND ASSUMPTIONS 2.1 The Str e s s - I n t e n s i t y Factor Before introducing the 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 notches, i t i s useful to review the t h e o r e t i c a l background to sharp cracks. The s t r e s s - i n t e n s i t y f a c t o r , K, i s e s s e n t i a l l y the c o e f f i c i e n t associated with the singular term i n the e l a s t i c i t y s o l u t i o n . Upon closer examination, one fi n d s that the sin g u l a r term corresponds to the s o l u t i o n of the crack i n an i n f i n i t e body. Thus, the form of the singular term, which w i l l be c a l c u l a t e d i n Chapter 3 f o r a V-notch, i s independent of the o v e r a l l geometry and load d i s t r i b u t i o n and can be determined given the notch geometry, notch o r i e n t a t i o n , and e l a s t i c constants. For cracks i n a plane i s o t r o p i c body, there are two independent crack displacement modes: one representing the opening mode, or Mode I, and the other the shear mode, or Mode I I . The two modes are shown In Figure 2 . 1 . With the s t r e s s - i n t e n s i t y f a c t o r , one i s able to completely describe the stress and displacement f i e l d s i n a small region around the notch root; thus, the advantage of using the s t r e s s - i n t e n s i t y f a c t o r — i n any given material, cracks with the same s t r e s s - i n t e n s i t y factor have the same notch root stresses and s t r a i n s . As we approach the notch root, the stresses and st r a i n s are more accurately described by the singular s o l u t i o n . But because of the s i n g u l a r i t y , one i s unable to quantify the stresses at the notch root i t s e l f . Since no material can su s t a i n i n f i n i t e stresses, the e l a s t i c s o l u t i o n suggests that a l l notched structures w i l l f a i l except under zero loads. In r e a l i t y , some form of str e s s r e l a x a t i o n such as y i e l d i n g w i l l 8 Figure 2.1 - Mode I and Mode II crack displacement modes 9 occur at the notch root even f o r seemingly " b r i t t l e " m aterials. The s t r e s s - i n t e n s i t y f a c t o r approach, although based on l i n e a r - e l a s t i c f r a c t u r e mechanics, w i l l t o l e r a t e some l o c a l y i e l d i n g i f i t i s l i m i t e d to "small scale y i e l d i n g " (SSY). Irwin showed that, i f the c h a r a c t e r i s t i c s i z e of the zone of p l a s t i c deformation or energy d i s s i p a t i o n around the crack, front i s small when compared to the length of the crack, then the energy being converted to surface tension energy (thus allowing the crack to propagate) w i l l come prim a r i l y from the e l a s t i c body (Erdogan, 1983). Therefore, the behaviour of the cracked body at the onset of rapid crack propagation i s e s s e n t i a l l y e l a s t i c . Consequently, even d u c t i l e materials such as s t e e l can also be analyzed using the s t r e s s - i n t e n s i t y factor approach, but only i f the region of p l a s t i c deformation i s small compared to the specimen s i z e and crack length. It i s postulated that crack propagation occurs when the s t r e s s - i n t e n s i t y f a c t o r reaches a c r i t i c a l value. For cracks loaded i n pure opening or pure shear, 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 are designated and K ^ re s p e c t i v e l y . These c r i t i c a l values are material properties and are a measure of the material's toughness. For the most general loading, the crack i s subjected to mixed mode conditions and the i n t e r a c t i o n of the two modes r e s u l t s i n f a i l u r e at s t r e s s - i n t e n s i t y f a c t ors l e s s than the i n d i v i d u a l c r i t i c a l values. 2.2 Zero Notch Root Radius Assumption S t r e s s - i n t e n s i t y f actors can only be obtained from e l a s t i c i t y solutions corresponding to notches with zero notch root radius and thus 10 r e s u l t i n g i n singular stresses at the notch root. There are two reasons why we may assume a zero notch root radius f o r most notches. As outlined i n the previous section, y i e l d i n g i s permitted at the notch root i f i t i s l i m i t e d to "small scale y i e l d i n g " (SSY). This reasoning can a l s o be applied when considering the notch root radius. That i s , i f the radius i s such that i t i s small when compared to the length of the crack, v a r i a t i o n s i n the root radius w i l l have n e g l i g i b l e e f f e c t on the stresses and s t r a i n s around the notch root. Consequently, fo r small notch root radius, i t i s acceptable to assume a zero value. Secondly, when analyzing a n i s o t r o p i c bodies, we usually assume that i t i s a homogeneous continuum. Notches or cracks are e s s e n t i a l l y flaws i n the structure, and i f the s i z e of these flaws i s large i n comparison to the c h a r a c t e r i s t i c m i c r o s t r u c t u r a l dimension of the material, we can assume the medium to be a homogeneous continuum. In other words, i f the assumption of a homogeneous medium i s v a l i d , then the sharp notch root assumption i s also v a l i d provided the d e t a i l s of the notch root have c h a r a c t e r i s t i c dimension of the same order of the magnitude as the c h a r a c t e r i s t i c microstructure dimensions. In wood, th i s may, f o r example, be the thickness of the growth rings or the f i b r e diameters. When the c h a r a c t e r i s t i c s i z e of the notch approaches the growth r i n g thickness then the o r i e n t a t i o n of the growth r i n g has a s i g n i f i c a n t e f f e c t on the ultimate load and d e f l e c t i o n s (Palka and Holmes, 1973). Gerhardt (1984) analyzed notched p a l l e t s t r i n g e r s with non-zero f i l l e t r a d i i and found, as expected, that the maximum hoop stress increased as the f i l l e t radius decreased. However, he a l s o found that the f i l l e t radius had an i n s i g n i f i c a n t e f f e c t on the mean crack i n i t i a t i o n moment and mean ultimate moment capacity. L e i c e s t e r (1974) and Stieda (1966) also found the strength of sing l e edge notched wood beams to be independent of the notch root d e t a i l . So, although a zero notch root radius i s p h y s i c a l l y impossible, assuming such a notch i s not unreasonable or overly conservative. In f a c t , the problem i s s i m p l i f i e d i n that we are now able to describe the notch root stresses and s t r a i n s i n terms of the singular e l a s t i c solutions and t h e i r associated s t r e s s - i n t e n s i t y f a c t o r s . 12 FORMULATION OF THE SINGULAR SOLUTION 3.1 Derivation of the Governing D i f f e r e n t i a l Equation The s t r e s s - s t r a i n r e l a t i o n s f o r an orthotropic material subject to plane stress conditions and undergoing small displacement are: a v e = # - a (3.1a) x E E y x y J v a E = ^ a + =2- (3.1b) y E x E ' ' x y where v v xy _ yx E = E y x The A i r y stress function, i s defined such that: 3 2$ rxy = ~ 3x3y (3.Id) 3 2$ a„ = — (3.2a) 3y 5 3 2$ a - — (3.2b) y 3x2 (3.2c) 13 Equilibrium i s then s a t i s f i e d since a , a and x under zero body forces x y xy are now solutions to the equilibrium equations i n the XY plane: and 3a 3T 3T 3a The strain-displacement or kinematic r e l a t i o n s : C3.4a) _ 3u 3v Y x y " 1 y + 3 ^ ( 3 * 4 c ) can be manipulated to form the compatibility equation: 3 2e 3 2e 3 2y JL + y = xy 3y 2 3x 2 3x3y (3.5) Substituting equation (3.2a-c) i n t o equation (3.1) we can express the s t r a i n s as: 14 1 3 2 $ \y 3 2$ , v e = — - — * — •=ru- T~T (3.6a) x E 3y* E 3 x 2 v ' x y Vyx 3 2$ , 1 3 2 $ ,~ e y = " E 3y2~ E~ ai* ( 3' 6 b )  7 x J y 1 3 2 $ Y = ~ " X — r (-5. be) 'xy G 3x3y v F i n a l l y , u t i l i z i n g equation (3.6), equation (3.5) can be written i n terms of the Airy stress function: 3H , f V 3H . i _ „ x 3 H _ 0 n ? , 3 ^ ty* ^ 2 v x y j ^ 3 y T " ( } Since equation (3.7) was derived using kinematic, equilibrium and comp a t i b i l i t y r e l a t i o n s , a s o l u t i o n to equation (3.7) w i l l s a t i s f y a l l equilibrium and compatibility requirements. 3.2 Solution of the Governing D i f f e r e n t i a l Equation Substituting the i s o t r o p i c case ( E = E , v = v = v and G = ° c x y xy yx E / 2(1 + v ) ) , equation (3.7) reverts to the biharmonic equation — the governing d i f f e r e n t i a l equation f o r plane e l a s t i c i s o t r o p i c bodies. Therefore, solutions to equation (3.7) may then be obtained following methods analogous to solving the biharmonic equation. Although s t r i c t l y speaking, the term "biharmonic" only applies for the i s o t r o p i c case, equation (3.7) w i l l , f o r the purpose of t h i s t h e s i s , be r e f e r r e d to as the biharmonic equation for plane orthotropic bodies. 15 In s o l v i n g the biharmonic equation (equation 3.7), we f i r s t look f o r solutions to the harmonic equation. If we set: E £ 2 = g—" (3.8a) y and -klh t / " 2 v x y ^ ( 3 * 8 b ) y *2 the biharmonic equation may be written as two "harmonic" operators: L l L 2 = + + a 2 ^ p ) (3.9) where a x = |- 2 [tc + - 1 ] (3.10a) a 2 = |"2 [< - /K z - 1 ] (3.10b) For an i s o t r o p i c material, 04 = a 2 = 1. For wood, 04 and a 2 are approximately 0.9 and 0.6 r e s p e c t i v e l y with the g r a i n along the x axis. Changing from cartesian to polar coordinates using the transformation: x = cos ifi^ = r cos 0 (3.11a) 16 Ya^ i T i / (3.11b) where p^, represent a transformed polar coordinate system to an "equivalent i s o t r o p i c " material, the harmonic operators can be written for i = 1, 2: Li " "3P7 + T± W± + P , 2 3*7 (3-12) Points between coordinate systems can be related by: p ± = r [ c o s 2 6 + l i H 2 ! ] 2 (3.13a) 1 tan = , — tan 6 . o u . i Ya^ (3.13b) Note that equation (3.12) i s simply the Laplace operator i n polar coordinates. The s o l u t i o n to equation (3.12) i s then of the form: * i ( p i * V = p i A F ( V ( 3 , 1 4 ) where X i s some constant. In general, there w i l l be an i n f i n i t e number of possible X. When X i s non-zero, s u b s t i t u t i n g equation (3.14) i n t o equation (3.12), we f i n d : 17 d 2F( i|0 + A 2 F ( ^ ) = 0 (3.15) Thus, the portion of the stress function obtained from the harmonic operators, f o r i = 1, 2, and X = X^ can be written as: X n X n $ n = A n p l C 0 S X n *1 + B n p l s i n V 1 X n X + C n p 2 cos A n ^2 + D n p 2 s ^ - n ^ n ^2 (3.16a) where A , B , C , and D are constants to be determined from stress n n n n boundary conditions at the notch faces. When X = 0, we get: d 2F(i|, ) 2 0 (3.17) die, or $ = constant (a t r i v i a l solution) when F ( ^ ) = 1, and: * • K > co,~li.T.U,,,,) <3-18*> (cos^e H ) , z *= K2 cos_1' 2.7.U.1/2i <3-i8b> (cos^e H ) i / z a 2 18 when F ( = ty^-For solutions independent of ty: d 2F . 1 dF + — T T " 0 (3.19) or $ = constant (a t r i v i a l solution) when F(ij^ ) = 1, and; $ = K 3 {In r + |- In [co S 2 e + ^ ^ ] } (3.19a) $ = K 4 {In r + I ln [cos2 6 + l i s i i ] } (3.19b) cx 2 when = In ( P ± ) F i n a l l y , since: L i L 2 ( P l 2 ) = 0 (3.20) and L X L 2 (p 22) = 0 (3.21) then we also have solutions of the form: $ = K 5r2 (cos2e + (3.22a) a l 19 4 = K 6r2 (cos2e + s 1 ^ 6 ) (3.22b) Thus, the complete stress function s a t i s f y i n g equation (3.7) i n polar coordinates i s : X X * = I l A n p l n c 0 8 <* n * l > + B n P l n S i n ^ n * ^ n=l x n x n + C n p 2 n cos ( X n ^ 2 ) + D n p 2 n s i n (X nip 2)} + K Q + + K 2^ 2 + K 3 In ( P l ) + In ( p 2 ) + K 5 P l 2 + K 6p 22 + K ? In ( p ^ + K e In ( p 2 ) * 2 (3.23) 3.3 Singular Stresses and Boundary Conditions Stresses i n the polar coordinate system are obtained from the st r e s s function, equation (3.23), by u t i l i z i n g : 1 34> , 1 3 2$ . a = — — - + —rr A (3.24a) r r 3r r 2 392 v ' 6 3r 2 v ' 1 3* 1 3 2$ fr, 0 / . Tre = 72" Te " 7 3rTe ( 3 , 2 4 ) Enf orcing stress free notch faces ( a ^ and = 0 at 9 = 6 j , 8 = 6 2, as defined i n Figure 3.1) and neglecting the t r i v i a l solutions, i t may be shown that Kg through Kg must equal zero. At the notch root (r - 0) we also require that, while the stresses and strains can be singular, the displacements must remain f i n i t e . That i s , i f : 21 * - r X F(6) (3.25a) so X - 2 E = r G(6) (3.25b) x-i Therefore, u = 77-1". G(6) + H(6), X * 1 (3.26a) ( X — 1 ) u = l n ( r ) G(6) + H(6), X = 1 (3.26b) Examining equation (3.25b), singular stresses occur only f or X < 2.0 and from equation (3.26), f i n i t e displacements are possible only i f X > 1.0. Therefore, we consider only: X X * = A p, n cos (X \|>,) + B pi n s i n (X i|>,) n n 1 n 1 n 1 n 1 X X + C p, n cos (X + D p, n s i n (X (3.27) n ^ n ^ n * n ^ as the stress function i n the i n t e r v a l 1.0 < X < 2.0. n U t i l i z i n g equation (3.24), the singular stresses at a notch root f o r X = X^ may be expressed as: 22 \ 2 + B n [ Q l n s i n + T l n C O S a n * l ) J + C n [ Q 2 n cos ( A n * 2 ) - T 2 n s i n ( A ^ 2 ) ] + D [Q, s i n ( A + , ) + T 9 cos (A i|> 9)] (3.28a) n L *n n * z n n * J A _ 2 n A ( A -1) r n { A F ^ P cos (X J i^e ) ) n n n n A n (e) - 2" s i n (xn*!(e)) A n + C F , ( e ) - 2 " cos ( A i|)o(6)) n * n i A n + D nF 2 ( e)~2" s i n (A n i|» 2(9)) (3.28b) " " A n ' V ^ ^ <AJMln C O S ( A n * l > - N i s i n (A • , ) ] + B [M, s i n (A + , ) x n n 1 J n L xn n A + N l n cos ( A ^ ) ] + C n [ M 2 n cos ( X n « 2 ) - N ^ s i n ( X n * 2 ) ] + D n [ M 2 n s i n (X n*. 2) + N 2 n c o s &n*2>] 23 where f o r i = 1, 2: and A . , X n X A n _ • 4 i n i 2 v2 ; i ^ i ; n X 2 X A n 1 X„ n _ + - | F ^ " 1 F ± - - B - F ^ " 2 ( 3.28d) A / A , x , n _ N • = °( n - 1) F (TT ~ 2 ) F ( 3.28e) n a i M ± = — 2 ~ (3.28f ) n N ± - — (3.28g) n / F ± = c o s 2 6 + S ^ n 2 Q (3.28h) denotes d i f f e r e n t i a t i o n with respect to theta once. For str e s s f r e e notch faces, we set equations (3.28b) to (3.28c) to zero at 6 = 6^ 6 = 6 2. This r e s u l t s i n a homogeneous set of four l i n e a r equations f o r each value of n: 24 - r ' a l l a12 a13 A n a21 a22 a23 a29 B n - {0} (3.29) a31 a32 a33 a3*t C n a«*2 a<t3 a»t«» D ti -n For a n o n - t r i v i a l s o l u t i o n , we need to solve for values of A that make the determinant of the matrix of c o e f f i c i e n t s equal to zero. Since the a ^ j are nonlinear functions of A^, the problem may be e a s i l y solved by employing techniques used f o r f i n d i n g zeros of nonlinear equations such as the b i s e c t i o n method. Note that t h i s problem i s s i m i l a r i n nature to the c l a s s i c a l eigenvalue problem, Ax = Ax. For the purpose of t h i s t h e s i s , A n w i l l be referred to as the "eigenvalue" of the singular s o l u t i o n and the associated s t r e s s f i e l d w i l l be r e f e r r e d to as the " e i g e n f i e l d " . 3.4 Determining Values of A Enforcing zero normal and shear stresses at the notch faces r e s u l t s i n f i n d i n g the roots of a nonlinear equation (3.29). This can be e a s i l y solved using the b i s e c t i o n method and stepping through the i n t e r v a l 1 < A < 2, which give singular stresses and s t r a i n s but f i n i t e displacements. We search f o r values of A such that the determinant of A i s zero. For v a l i d notch angles ( i . e . angle between notch faces less than 180°), there 25 i s at l e a s t one eigenvalue and at most two. Cracks have a repeated eigenvalue of 1.5 or a l / / r * s i n g u l a r i t y regardless of the a n i s o t r o p i c nature of the body or the i n c l i n a t i o n of the crack to the axis of e l a s t i c symmetry. For V-notches, X values are i n the range 1.5 < X < 2.0 and are found to be functions of the e l a s t i c properties as w e l l as notch geometry and o r i e n t a t i o n (see Figure 3.2). For large V-notch angles, there i s only one v a l i d eigenvalue since A 2 i s greater than 2.0. Note that at t h i s point i n the s o l u t i o n of the notch problem, no reference has been made to the external loads. As the notch angle approaches 180°, the eigenvalues approach 2.0 or a l/r° s i n g u l a r i t y r e s u l t i n g i n a weaker s i n g u l a r i t y . It i s i n t e r e s t i n g to note that f o r notch angles greater than 180°, the singular s o l u t i o n becomes that of the wedge problem. Thus i t i s important how one expresses the angle of the notch faces when enforcing the s t r e s s boundary conditions. If the notch faces were defined at 6j and 8 2 where the absolute values of and 6 2 are l e s s than 90°, then the eigenvalue problem w i l l become that of a wedge. The angles must describe a v a l i d domain, that i s , varying theta between these two l i m i t s should define the notched body and should sweep an angle of more than 180° and l e s s than 360°. The same also holds for cracks. Su b s t i t u t i n g the eigenvalues back i n t o equation (3.29), we can then solve for the eigenvectors or e i g e n f i e l d s . These describe the singular s t r e s s and s t r a i n f i e l d around the notch root; and l i k e mode shapes i n v i b r a t i o n a n a l y s i s , the magnitude of the stresses and s t r a i n s are undetermined. For i s o t r o p i c bodies or notches positioned symmetrically along one of the axis of e l a s t i c symmetry, the two e i g e n f i e l d s correspond to the opening and s l i d i n g shear modes; only under these conditions do 26 Notch angle fi 0 Figure 3.2 - A vs notch angle f o r various notch geometries (B : notch i n an i s o t r o p i c plate) GRAIN Figure 3.3 -Notch configuration (A) f o r F i g . 3.2 Figure 3.4 -Notch configuration (C) f o r F i g . 3.2 27 independent opening and s l i d i n g shear displacement modes e x i s t . With the exception of cracks, the opening mode for symmetrically placed V-notches w i l l be associated with the stronger s i n g u l a r i t y or the smaller eigenvalue. In general, i f we denote the two eigenvalues as A^ and Ag, where A^ < A^ (except when they are equal for cracks), then the stress f i e l d corresponding to A^ w i l l always dominate at the notch root as we approach r = 0. Therefore, the e i g e n f i e l d due to A^ w i l l be r e f e r r e d to as the primary f i e l d (A_ corresponding to the secondary f i e l d ) . For i s o t r o p i c is or nearly i s o t r o p i c V-notched bodies, the opening mode i s represented by the primary f i e l d . This suggests that for most general loadings, V-notches i n i s o t r o p i c bodies w i l l f a i l i n the opening mode. Figures 3.5 to 3.15 are plots of the e i g e n f i e l d s or singular s t r e s s f i e l d s f o r various crack and notch geometries at a u n i t radius from the notch root. They are presented i n a manner analogous to plots of the pressure around a c y l i n d e r i n a f l u i d . The distance ( i n the r a d i a l d i r e c t i o n ) from the unit c i r c l e marked "datum" to the s o l i d curve i s a p o s i t i v e s t r e s s while the distance from the u n i t c i r c l e to the dashed curve i s a negative s t r e s s . The magnitude and sign of the stresses, however, are only r e l a t i v e ; thus the p l o t only gives the s t r e s s "pattern" around the notch root. That i s , the actual stress l e v e l s are only known a f t e r the f i e l d i s m u l t i p l i e d by a signed constant - the s t r e s s - i n t e n s i t y f a c t o r . At t h i s point i n the s o l u t i o n , only the stress pattern around the notch root i s known. The p l o t s are of the t a n g e n t i a l and shear stresses from the singular stress s o l u t i o n at a unit distance from the notch root (I.e. r = 1). Figure 3.5 - F i r s t primary singular stress f i e l d f o r a 0:360 degree crack Primary Stress Field Ex:Ey = 20.000 _ Ey:G = 0.900 MUxy = 0.020 / ' Lambda = 1.500 / — / / » i i y A >< i * X j \ / 1 ^ v / Datum 90" Face angles: 01 = 0.00 " 02= 360.00* \ \ \ \ \ V \ / • \ / ' \ ' 1 i I / / v. / / \ / / v \ ' y >* \ ' — v iao* i i i i ! i / \ p \ \ P o s . N e g . I S . i i i i i o * 270° J Figure 3-6 - Second primary singular stress f i e l d f o r a 0 O 6 0 degree crack Primary Stress Field Ex:Ey = 20.000 Ey:G = 0!900 MUxy = 0.020 Lambda = 1.500 / ~ / / ; / CE' 7 I r r \ \ / Datun 90° Face angles: 01 = -45.00° 0 2 - 315.00° s \ \ \ \ > \ i \ i \ ' IT \ / / \ / / >v / / — ' 180' 1 1 1 1 \ 1 \ I — r*-*—-P o s . N e g . V i/i 1 1 1 1 1 ° 0 Jr/ 270° X Figure 3«? - F i r s t primary singular stress f i e l d f o r a - 4 5 5 3 1 5 degree crack Primary Stress Field Ex:Ey = 20.000 Ey:G = 0.900 / MUxy = 0.020 JT Lambda = 1.500 / ~~ T„[ \ ^ j( Datun 90° Face angles: 01 = -45.00 0 "^v. © 2 = 315.00° W * V v' h X / ' \ / ' \ / / 180° 1 1 1 1 1 X l ' P o s . N e g . v i i ' > o° 270° Figure 3 . 8 - Second primary singular stress f i e l d f o r a - 4 5 0 1 5 degree crack Primary Stress Field Ex:Ey = 20.000 _ Ey:C = 0.900 s^"^ MUxy = 0.020 / 90* Face angles: ©1 = -90.00* © 2 = 270.00* Lambda = 1.500 / ~ ( \ > \ \ / \ ,'~ /„ \ / 1 Datum * ' 1 —' \ N \ \ 160* 1 1 1 \l { 1 I ) 1 ! L / I 1 1 0* P o s . N e g . 270* Figure 3.10 - Second primary singular stress f i e l d f o r a -90:270 degree crack i n the l i m i t as a v-notch becomes a crack Primary Stress Field Ex:Ey = 20.000 Ey:C = 0.900 ^ MUxy = 0.020 Lambda = 1.500 / ~ yf Datun 90* Face angles: 01 = -90.00* 02 = 270.00 ° W — \ A' iao° i i i i i r4 * \ Pos. Neg. J y i i i i i o° / / V 270° Figure 3.11 - F i r s t primary singular stress f i e l d f o r a -90:270 degree crack when solving d i r e c t l y f o r a crack Primary Stress Field Ex:Ey = 20.000 _ Ey:G = 0.900 MUxy = 0.020 • Lambda = 1.500 / I ^ / \ r \ \ f Datun 90° Face angles: 01 = -90.00 ° 02= 270.00° \ \ \ \ \ / \\. ' 1 \ 1 \ ^ / / x / / >. / y — ^ i k 180° 1 1 1 1 I Pos. Neg. r) 1 I i i i o° Jy 270° Figure 3«12 - Second primary singular stress f i e l d f o r a -90:270 degree crack when solving d i r e c t l y f o r a crack Face angles: 01 = 0.00 ' 02= 270.00' Figure 3 - 1 3 - Primary singular stress f i e l d f o r a rectangular end-notch 1 Primary Stress Field Ex:Ey = 20.000 Ey:G = 0.900 ~ MUxy = 0.020 S Lambda = 1.611 / ~ / ; i i — \/- \ , (\ r \ \ 1 Datum 1 \ V [ J 90s Face angles: 01=0.00 • 02 = 225.00* \ i ' I T / / loo* i i i i K \ l sP o s . / ^ N e g . 1 ) I I l 1 1 0 * j 270* Figure 3 ' ! ^ - Primary singular stress f i e l d f o r a tapered end-notch Secondary Stress Field Ex:Ey = 20.000 Ey:G = 0.900 MUxy = 0.020 X Lambda= 1.898 / ~~~ TJ \ N / Datun 90* Face angles: 01 =0.00 * ^ \ 02= 270.00* ^ -I h X / 1 \ / ' y\ / / — \/ 180° 1 1 1 1 1 • "Si Pos . Neg . I 1 1 1 1 l o * 270* Figure 3 - 1 5 - Secondary singular stress f i e l d f o r a rectangular end-notch 34 Notch face angles are al s o given and i t can be e a s i l y v e r i f i e d that the sol u t i o n s a t i s f i e s the stress boundary conditions at the notch faces -tan g e n t i a l stresses and shear stresses equal to zero. Since the two eigenvalues f o r cracks are equal, there are two primary f i e l d s representing the two modes instead of a primary and secondary f i e l d . For a crack positioned along the grain, the singular stress f i e l d s represent the opening (Figure 3.5) and shear (Figure 3.6) modes. Cracks oriented along l i n e s other than l i n e s of e l a s t i c symmetry, such as the crack shown i n Figures 3.7 and 3.8, do not have independent opening and shear modes. For cracks along a l i n e of e l a s t i c symmetry, a symmetric and antisymmetric stress patterns are expected (Figures 3.9 and 3.10). But as shown i n Figures 3.11 and 3.12, the two e i g e n f i e l d s may not be symmetric and antisymmetric. This i s because no attempt has been made to separate the even and odd terms i n the singular stress function; thus, the two e i g e n f i e l d s may not represent the opening and shear modes. However, because of the two eigenvalues are equal, the two solutions can be superimposed and the f i n a l s o l u t i o n w i l l be cor r e c t . By adding the two f i e l d s , the r e s u l t i n g f i e l d represents the shear mode and by subtracting the two f i e l d s , the r e s u l t i n g f i e l d represents the opening mode. Mixed modes may be obtained from other l i n e a r combinations. The l i m i t i n g case of V-notches i n i s o t r o p i c materials creates some numerical d i f f i c u l t i e s . If we enter i s o t r o p i c constants ( i . e . E = E = x y E and G = E / ( 1 + 2 * v) ) , and since the equations are formulated f o r xy or t h o t r o p i c materials, degeneracies occur (Hoenig, 1982) and equation (3.29) becomes independent of X. However, i t can be shown that the formulation i s s t i l l v a l i d i n the l i m i t . Numerically, we can solve f o r 35 e i g e n f i e l d s i n i s o t r o p i c b o d i e s by e n t e r i n g n e a r i s o t r o p i c m a t e r i a l c o n s t a n t s ( i . e . E : E = 1 + e , E :G = 2 * (1 + v) + e , where e i s , f o r x y y xy ' ' e x a m p l e , 0 . 0 0 0 1 ) k e e p i n g i n m ind t h e p r e c i s i o n o f t h e s o l v e r ( v a l u e s o f e l e s s t h a n 0 . 0 0 0 1 w i l l be l o s t due t o r o u n d - o f f ) . 3 . 5 S i n g u l a r S t r a i n s and t h e A s s o c i a t e d D i s p l a c e m e n t F i e l d D i s p l a c e m e n t s may be o b t a i n e d by i n t e g r a t i n g t h e s i n g u l a r s t r a i n s . E q u a t i o n s ( 3 . 2 8 a - h ) may be c o n v e n i e n t l y w r i t t e n a s : r \ T r e y = K r A - 2 P l ( 6 ) \ { P 2 ( 6 ) > ( 3 . 3 0 ) f o r A = A n« A l t h o u g h t h e e l a s t i c c o n s t a n t s a r e g i v e n w i t h r e s p e c t t o t h e ( x , y ) s y s t e m , i t i s s t i l l c o n v e n i e n t t o work i n t h e ( r , 6) s y s t e m . S i n c e : X V x y J ' r 1 (3.31) and 36 'r6 [ T ] " xy (3.32) we can write: r6 [ T ] T [C] [T] K r A ~ 2 Pl(6) < P 2 ( 0 ) P 3(6) (3.33) where [T] i s the transformation matrix: [T] = cos H sin20 j i n 2 6 cos 2e sin 2e -sin26 -sin28 sin26 cos26 (3.34) and [c] i s the matrix of compliances for an orthotropic body under plane stress conditions: 37 [c] 1_ E -v _J2L E x -v x y E I G ( 3 . 3 5 ) Let, f o r convenience, Rl(6) Pl(6) <U(e)> = [ T ] T [ C ] [ T ] ^ R 3(6) P 2(6) P 3 ( 8 ) ( 3 . 3 6 ) Manipulating equations (3.33) and (3.36) and sub s t i t u t i n g into the kinematic r e l a t i o n s : 3u _ i e r ~ 3r (3.37a) u , 3u n r , 1 6_ e 9 ~ r r 36 (3.37b) Y r 9 3r r r 36 (3.37c) we get, from i n t e g r a t i n g equation (3.37a): 38 u r = r p r y r X _ 1 R i ( e ) + K o ( e ) ( 3 - 3 8 ) where K 0(6) i s a constant of i n t e g r a t i o n . D i f f e r e n t i a t i n g equation (3.38) and u t i l i z i n g equation (3.37c) we get: 3ufl U R x(e) K (6) where • denotes d i f f e r e n t i a t i o n with respect to 6. We then f i n d : Q-1) R i O ) u e = ( n ) " [R 3(e) - 7j=iy] + Ki(e)r + K o(e) (3.40) To solve f or K Q(0) and K 1 ( 6 ) , we d i f f e r e n t i a t e and substitute equations (3.39) and (3.40) i n t o equations (3.33) and (3.36). Thus: K x(e) - 0 (3.41a) and K Q(6) + K Q(9) -.0 (3.41b) Therefore: K ^ O ) = K x (3.42a) K (6) = K 2 cos 6 + K 3 s i n 6 (3.42b) 39 But K,r represents r i g i d body r o t a t i o n and since: u = u cos 6 - u„ s i n 6 = K 9 (3.43a) r 9 z v = u r s i n 9 + u Q cos 6 = K 3 (3.43b) K 2 and K 3 are r i g i d body tra n s l a t i o n s i n the x and y d i r e c t i o n s r e s p e c t i v e l y . Thus, dropping the terms associated with r i g i d body r o t a t i o n and t r a n s l a t i o n , and transforming to the x-y coordinate system, we f i n d : R (6) „. A - l rCos9 , Q X s i n e r _ , o N 1 i I / R> , , \ u = K r { (A^ i ) R i ( 9 ) - o=2 ) [ R 3 ( 9 ) " o=iy^ ( 3 , 4 4 a ) v - K r W R l (e) + | ^ | ) [ R 3 ( e ) - ^ ] } (3.44b) D i f f e r e n t i a t i n g equation (3.36) with respect to 6 keeping i n mind that [ T ] i s also a function of 6. {*} - | e ^ W ™ W + ( [ T ] T [C] [T]) - ^ i (3.45) Note that, i f we l e t [w] = [ T ] T [C] [ T ] , then: 3_ 36 ( [ T ]T [C] [T]) - [w] + [ w ] T (3.46) 40 To f i n d {p}, we use the equilibrium equations: 3 a r . 1 3 T r 0 . ( o r " g9> . W  + T - l e " + r 0 ( 3 ' 4 7 a ) and 3 T r e + T ^ J - + T - 0 (3.47b) 3r r 39 r r9 Using equations (3.47a) and (3.30), we get: P 3 (9) = P 2(9) - (X - 1) P x(9) (3.48a) and with equation (3.47b): P 2(9) - - X P 3(9) (3.48b) Only the P^ need be d i f f e r e n t i a t e d . After some manipulation, we f i n d : P x(9) = A {Vl cos X ^ - Wx s i n Xi^} + B {Vx s i n Xif^ + Wx cos Ai^x} (3.49a) + C {V 2 cos X<|»2 ~ W2 s i n AV2} + D {V 2 s i n X^ 2 - W2 cos Xiji2} 41 where: • T v = Q - 4 : r 1 1 Ja± i w i = T i + ( 3 . 4 9 b ) (3.49c) and F , T , Q are defined i n equation (3.28). 42 IMPLEMENTATION IN A FINITE ELEMENT ANALYSIS A f i n i t e element program, NOTCH (Lum, 1986), u t i l i z i n g s i n g u l a r i t y enriched eight-node quadratic isoparametric elements was w r i t t e n to analyze V-notched orthotropic plates. 4.1 Regular Quadratic Isoparametric Element An eight-node quadratic isoparametric element was selected to model the regular displacements; however, any high order element should perform s a t i s f a c t o r i l y f o r t h i s a p p l i c a t i o n . I t s d e r i v a t i o n i s w e l l known and can be found i n any standard text on the F i n i t e Element Method (see f o r example, Bathe, 1982; Cook, 1981). The shape functions have been included i n Appendix I for completeness. Formulation of the element s t i f f n e s s matrix i s s t r a i g h t forward except that a transformation to the l o c a l "grain" coordinated system i s made. With respect to the l o c a l coordinate system, the matrix of e l a s t i c s t i f f n e s s f o r plane st r e s s conditions, can be written as: M = (1-v' v ) xy yx where, for symmetry, v E = v E . It should be emphasized that, unlike xy x yx y most l i n e a r e l a s t i c f i n i t e element formulations, p l a i n s t r a i n conditions v E xy x v E yx y E 0 y (4.1) xy cannot be considered simply by replacing the [D] matrix. This w i l l become more apparent i n the following discussions. Since the nodal displacements are given r e l a t i v e to the g l o b a l system, s t r a i n s w i l l also be given r e l a t i v e to the global system. That i s : Wxy " [«] Wxy . <4'2> Therefore, i n order to ca l c u l a t e the s t r a i n energy using'equation (4.1), the s t r a i n s must be transformed to the gr a i n coordinate system (x*, y') shown i n Figure 4.1, by wr i t i n g : Wx.y. - [T] [B] {6}^ (4.3) where [T] i s the transformation matrix: [T] -cos 2e s i n 2 6 sin26 s i n 2 6 cos 2e sin26 -sin26 sin2 9 cos28 (4.4) and 6 i s the angle the x' axis makes with the x axis. This permits us to form the element s t i f f n e s s matrix using nodal displacements from the g l o b a l coordinate system. The global s t i f f n e s s matrix i s symmetric and banded, and i s assembled from element matrices using standard procedures. Only the lower half-band need be assembled and a Cholesky equation solver, f o r example, can be u t i l i z e d . 4 .2 Formulation of the Singular Element 4 . 2 . 1 F u l l y Singular Element S t i f f n e s s Matrix In formulating the singular quadratic element, we follow a procedure not unlike that f o r the regular elements. Let U j = u and u 2 v. To the polynomial of generalized coordinates {a} f o r the regular displacements, u^, we add the displacement function from the singular stress s o l u t i o n : u A = a x + a 2 K + a 3 n + a ^ 2 + a 5 £ TI + a 6 n 2 + a 7 n + a 8 5 n 2 ( 4 . 5 ) + K A f ± (c , n ) + Kfi 8 l (e,n) where §, TI are natural coordinates. Then f or some a r b i t r a r y point (5, within the element: u ± ( 5 , n) = [M ( 5 , n ) ] {a} + K A f ± ( c , n) + K B g ± ( c , T I ) ( 4 . 6 ) Evaluating the displacements at the nodes using equation ( 4 . 6 ) , we get \u±) = [A] {a} + K A {f ±} -r-Kg {g±} ( 4 . 7 ) and s u b s t i t u t i n g for the generalized coordinates, {a}: {a} - [A]"1 [{»±1 " K A {f±} - K g { g j ] ( 4 . 8 ) Thus we have a r e l a t i o n s h i p between the nodal displacements, K , and K A with the displacements at some a r b i t r a r y point within the element. So u ± U . n) -[M U , n)] [M ] " l [{u ±} - K A {f ±} - K {g ±}] ( 4 . 9 ) + K A f ± U , n) + K g g t U , n) However, [M ( 5 , TI)] [M] i s a c t u a l l y the matrix of shape functions, [ N ( C , TI ) ] . So rewriting equation ( 4 . 9 ) , we f i n a l l y get: u ± U , n) = [N(£, n)] {ui} + K A ^ f i n ) " f N ( 5 , t f ± } ] ( 4 ' 1 0 > + Kg [ 8 ± ( 5 , n) - [N(c, n ) ] {g±}] Therefore, the t o t a l displacement i s the sum of the regular quadratic displacements and the singular part. From equation ( 4 . 10 ) we can c a l c u l a t e the s t r a i n s : {e} = [B] {6} + [{fi A} { % } ] {K} (4.11) where [B] = strain-displacement matrix: {6} = vector of element nodal displacements and u 2 {K} = vector of s t r e s s - i n t e n s i t y f a c t o r s : {ft^}, {flg} = singular s t r a i n components for a unit and Kg and {ft } w i l l be given i n more d e t a i l i n a l a t e r s e c t i o n . From equation (4.11) we can c a l c u l a t e the stresses. Using the v i r t u a l work p r i n c i p l e , we get: / 6 T {&} [ B ] T [[D] [B] {6} + {K} T [D] {fl}]dv = 6 T {«} {R} (4.12) f o r a v i r t u a l change i n {6} and: / « T W !"}T [ID] IB] {«} + {K} T [D] {Q} ]dv = '6 T {K} {R,} (4.13) f o r a v i r t u a l change i n {K}. The system of equations from equations (4.12) and (4.13) can then be written and p a r t i t i o n e d as follows: CA! [ C M ] K„] ABJ M 1 [CBA1 ISBI r > * » -(4.14) with: [S] = / [ B ] T [D] [B] dv (4.15) = regular element s t i f f n e s s matrix [ P i J = / [ B ] T [D] dv (4.16) [ C i j ] = / ( " i } T [D] {B.} dv (4.17) {R} = regular part load vector (4.18) {R^} = singular part load vector (4.19) Equation (4.14) can then be assembled to form the global s t i f f n e s s 49 matrix and g l o b a l load vector. Although the g l o b a l s t i f f n e s s matrix w i l l equations w i l l increase the bandwidth s u b s t a n t i a l l y since i t couples a l l the elements associated with a s i n g u l a r i t y together. Solving equation (4.14) d i r e c t l y using a Cholesky solver would therefore be i n e f f i c i e n t since the cost of the s o l u t i o n increases with the square of the bandwidth f o r a given number of unknowns. A technique to solve the system of equations as presented i n equation (4.14) w i l l be discussed i n a l a t e r 4.2.2 Singular Element T r a n s i t i o n Function The enriched displacement f i e l d would not be continuous across element boundaries j o i n i n g singular and regular elements. To r e t a i n inter-element c o m p a t i b i l i t y of displacements, we adopt the Benzley t r a n s i t i o n function and i n s e r t i t i n equation (4.10). We can then write: be banded, assembly of the element [P 1 into the global system of s e c t i o n . w i l l also be developed i n a l a t e r s e c t i o n . u ± U , n) = [ N U , ii)] {u±} + z u , TI) {KA [ f ± U , n) - [NU, n)]{f i}] (4.20) + K B [g ±U, n) - [NU, n)] { g i}]} where Z(£, n) i s b i l i n e a r t r a n s i t i o n function written i n terms of the natural coordinate system such that: 50 Z - 1 sing u l a r - s i n g u l a r boundary (4.21) Z = 0 , singular-regular boundary and proceed as i n Section 4.2.1. For f u l l y singular elements, the . t r a n s i t i o n f u nction can be set to u n i t y . Since Z i s a l s o a function of C and n, i t must be d i f f e r e n t i a t e d when c a l c u l a t i n g the s t r a i n s . Using t r a n s i t i o n elements w i l l ensure monotonic convergence as the mesh i s r e f i n e d . Also, because the planes of e l a s t i c symmetry do not coincide with the global coordinate system, we must ensure we are d i f f e r e n t i a t i n g with respect to the co r r e c t coordinate system. In general, {u^} w i l l be nodal displacements with respect to the global coordinate system and f ^ , {f^}, g^, and {g^} w i l l be r e l a t i v e to the grain coordinate system. In addi t i o n , [N] and Z w i l l be d i f f e r e n t i a t e d with respect to the n a t u r a l coordinate system and then using the Jacobian w i l l be transformed to the global system. The Jacobian i s part of the isoparametric formulation. Since we wish to c a l c u l a t e the stresses and s t r a i n s with respect to x' and y' to c a l c u l a t e the s t r a i n energy, we must not only transform the d i f f e r e n t i a l operators, but also the Jacobian. Although i t i s possible to surround the s i n g u l a r i t y with only t r a n s i t i o n elements, i t i s more desirable to use f u l l y singular elements around the s i n g u l a r i t y and surround the singular elements with t r a n s i t i o n elements as shown i n Figure 4.2. 4.2.3 Overlapping S i n g u l a r i t i e s For narrow notches, such as i n Figure 4.3, we may f i n d that the singular elements from two s i n g u l a r i t i e s are overlapping. It may then be Figure 4.2 - Transition elements between singular and regular elements 52 - T R A N S I T I O N E L E M E N T - O V E R L A P P I N G T R A N S I T I O N - S I N G U L A R E L E M E N T r O V E R L A P P I N G S I N G U L A R Figure 4 . 3 - Overlapping singular and t r a n s i t i o n elements advantageous to have singular elements enriched by more than one s i n g u l a r i t y . A l t e r n a t i v e l y , we may re f i n e the mesh and set the t r a n s i t i o n functions f o r each s i n g u l a r i t y such that the singular solutions are forced to zero at element boundaries common to two s i n g u l a r i t i e s . This, however, may r e s u l t i n a complicated mesh p a r t i c u l a r l y for higher order elements. An element with N overlapping s i n g u l a r i t i e s may be formulated by generalizing equation (4.6) to: u±(e,n) = [MU,n)] {a} + N I ( K A n f £ (Cn) + K B n g . n (C,n)) (4.22) n=l Proceeding as i n Section 4.2.1, we get: u 1 ( C , n) = [N(C, T O ] {ui} + N I {K A n [f u , T,) - [N (5 , n)] { f ± n } ] (4.23) n=l + Kg1 [ g i n a, n) - [N(?, r,)] { 8 i n}]} Z n ( c , n) Using the same notation as equation (4.11), the s t r a i n s within the sing u l a r element may be expressed as follows: {e} = [B] [6} + ^ ( K / {ft/} + K B n {ftg}) (4.; 54 where: „ n n 8 8 3N, 3Z r n r „ n T , „n r *n r i n i - [u " i I i N i u 1 ] + Z [ e y a T « i ] 3Z n r n 8 8 3N ot r u v „ n i _ n r ~n r , ay- L v " I « v ] + Z e - J a 7 n n 8 n ^ 8 r n v _T n-i . 3Z r n 3y i n, V i ] 3y i = l i=l * 8 3N, 8 3N, , n r *n v i n v i n-i ( 4 . 2 5 ) For the nth s i n g u l a r i t y : n n u , v n n V v i *n *n *n e , e , y x y xy = singular s o l u t i o n ( " A " f i e l d ) displacements at U, n) = singular s o l u t i o n ( " A " f i e l d ) displacements f o r node i = singular s t r a i n s ( " A " f i e l d ) at (£, n ) = shape fu n c t i o n = 1 when at node i = b i l i n e a r t r a n s i t i o n function S i m i l a r l y , {fi R n} can be calculated for the "B" f i e l d . 55 Stresses can then be expressed as: {0} = [D] {e} (4.26) using {e} as defined i n equation (4.24). U t i l i z i n g the concept of v i r t u a l work as i n Section 4.2.1, we get the element s t i f f n e s s matrices. These can then be superimposed to form the following system of l i n e a r equations with n displacement degrees of freedom and N s i n g u l a r i t i e s : nxn AB J2xn [ P N 1 T L ARJ ABJ 2xn N AB Jnx2 [ C H ] 2x2 [ c 1 N ] T 2x2 2x2 [ c M ] 2x2 nxl {Kl} 2x1 2x1 r *\ (4.27) [ s ] {Kh} = regular global s t i f f n e s s matrix = vector of glo b a l nodal displacements, = vector of K A > Kfi s t r e s s - i n t e n s i t y factors for s i n g u l a r i t y h. = regular part global load vector = singular part global load vector for s i n g u l a r i t y h As seen i n equation (4.27), coupling i s v i a [C 1^] and [ P ^ ] , i * j which re s u l t s i n a bandwidth that i s generally larger than that i n [ s j . To avoid t h i s , we solve equation (4.27) i n two parts. 56 From equation (4.27), we have: [S] {6} + [Pl ] {Kl} +....+ [P»j {KN} - W (4.28a) and: [^1 {«} + [ C 1 1 ] {^} + [ C « ] IK*} + [C 1 N] {KN} - {Rl} (4.28b) Rewriting equation (4.27a) for {6}, we then substitute f o r {6} i n equation (4.27b) and get: [*1JT ([S]" 1 {*} - [S]" 1 { [ P l J {Kl} +...+[P?J {KN}}) + [ C11} { K1} + [ C 1 2 } {K2} +... +[C l N} {KN} = {RlR} (4.29) i1 i T — 1 Setting [G j} = [P A B} [s] 1 [ P A B} w e then get another system of equations which may then be p a r t i t i o n e d as follows: [G 1 1] [Gl*] [G 1 N] [021] [ G22] [ G N 1 ] {Kl} < • • • > • • (4.30a) where: (4.30b) 57 Note that the matrix of [G may be banded i f the s i n g u l a r i t i e s are numbered following r u l e s not u n l i k e that f o r numbering nodes. By doing so, a Cholesky type solver may be u t i l i z e d . The s o l u t i o n method consists of f i r s t s o l v i n g f o r the regular displacements, that i s : {d} = [ S ] " l {R} (4.31) and employing {d} i n equation (4.30b). By using a Cholesky type s o l v e r , the LU decomposition can be retained f o r evaluating the inverse i n equation (4.29) to form equation (4.30). After solving for the s t r e s s - i n t e n s i t y f a c t o r s , they may be used i n equation (4.28a) to correct the regular displacements, {d}, for the s i n g u l a r i t i e s . Since the formulation s a t i s f i e s a l l the conditions of completeness and com p a t i b i l i t y , the displacement adjustments for the s i n g u l a r i t i e s w i l l generally increase the displacements ( i . e . make the structure l e s s s t i f f ) . Overlapping s i n g u l a r i t i e s have not been implemented i n the NOTCH program. However, I t may benefit higher order elements such as the 12-node cubic element. 4.2.4 Body Force and I n i t i a l S t r a i n Vectors The consistent load vector, {R^}* due to body for c e s , surface t r a c t i o n s , and temperature or shrinkage e f f e c t s w i l l be developed i n t h i s s e c t i o n . 58 A.2.4.1 I n i t i a l Strains To derive a consistent load vector to introduce i n i t i a l strains from temperature or shrinkage, we calculate the total potential energy, n: / {e} T [D] {E }dv- {}*)} { R V = N (4.32) where {eQ} is the vector of i n i t i a l strains and the singularity enriched strains are given by: [*} = [IB] I [Q]] { ( KU (4.33) Application of the principle of virtual work yields: 6 {|K(}T / [ m l ^ ] ] T P>] {eo} dv - 6 {JM}T {jjh} (4.34) or f i n a l l y : 1 ^ } } - / [ [ B l l t f i ] ] 1 [D] {,o} dv (4.35) Here: [n] = [{QA} {fig}] (4.36) 59 where {fi^} i s defined as i n equation (4.11) for independent s i n g u l a r i t i e s and equation (4.25) f o r overlapping s i n g u l a r i t i e s . Using equation (4.35) w i l l give the correct displacements due to an i n i t i a l s t r a i n . Stresses may be calculated from {a} - [D] {e} = [D] ( [ B ] T {&} - {e }). 4.2.4.2 Body Forces and Surface Tractions As i n the previous section, we c a l c u l a t e the p o t e n t i a l energy and applying the p r i n c i p l e of v i r t u a l work, we have: V v [M] (4.38) where [N] = matrix of shape functions; [M] = matrix of s i n g u l a r i t y displacement corrections as defined i n equation (4.10) = Z(C,n) 8 8 U A "\L N i ( C * n ) U A i U B "j[ N i U * n ) U i=l 8 8 Bi 'A N i ( 5 » ^ A i - B ^ " i Bi (4.39) and Z(£, n) i s the b i l i n e a r t r a n s i t i o n function. For surface t r a c t i o n s , {T}, we introduce i n equation (4.37) the Dirac d e l t a function such that i t s i n t e g r a l i s unity along the loaded surface. Thus: 60 R } l l s - / [\3 IT} ds ( 4 . 39 ) KKJ s M where s is the running coordinate along the element edge. Since the singularity consistent load vector i s a linear function of [M], {R^l i s a null vector i f [M] = 0. This occurs when loads are directly applied at the nodes, - the shape function is zero except at node i where = 1. In general, the contribution from {R^ } to the global load vector w i l l be small since the volume occupied by the enriched elements i s usually small i n comparison to the total volume. 4 . 3 V-notch, K^-K^j Stress-Intensity Factors V-notches i n orthotropic materials do not, i n general, have independent opening and sliding shear displacement modes. Although and K are stress-intensity factors, they cannot be directly compared to D the c r i t i c a l K^ . and values obtained from symmetrically or asymmetrically loaded cracked specimens. A more useful arrangement would be to find the equivalent amounts of K T and K T T in K. and K„. It is 1 X I A a convenient to designate the stress-intensity factors obtained from and Kg as Kj^, Kj-j-A» and Kjg» Kjjg respectively. We then have four stress-intensity factors describing the stress-strain state at the notch root. In the limit as r + 0, we find the primary stress f i e l d dominates at the notch root except under certain situations and establishes the conditions for rapid crack propagation. An exception to this occurs when the primary f i e l d i s suppressed, K A = 0, and the singular stresses are due to the secondary f i e l d . If we define a and K^^ as: 61 then: and r+o K X = / 2 ? K A f A (e=ep), for xA < xB K X - /ET ( K A f A (e-e p) + K B f B ( e - e p ) . for X A = X R (4.41a) K I X = /2TT K A g A (e=ep), for xA < xB K X I = m ( K A g A (6=6 p) + K f i g B ( 8 - B p ) ) , ( 4 .41b) for X A - X B where 0_ = 9 From t h i s d e f i n i t i o n , the opening and shear P propagation mode s t r e s s - i n t e n s i t y f a c t o r s are s t r i c t l y functions of K for V-notches; they give an i n d i c a t i o n of the magnitudes of the stresses causing crack propagation. Only f o r cracks, where A^ = Ag, are the KJ-KJ . -J . s t r e s s - i n t e n s i t y factors functions of both K, and K,,; and i f the crack A B l i e s along one of the axis of e l a s t i c symmetry then we also f i n d that e i t h e r K T. = 0 and K T T_ = 0, or K T D = 0 and K T T. = 0. IA IIB IB IIA If the primary f i e l d i s suppressed,Kj. and IC^ w i l l be s t r i c t l y functions of the K„ stress s i n g u l a r i t y . When a V-notch i s positioned symmetrically and loaded anti-symmetrically with respect to an axis of e l a s t i c symmetry, the s t r e s s - i n t e n s i t y factor i s found to be zero. In f a c t , we are suppressing K^ or loading i n such a way that we get only mode II type displacements at the notch root - t h i s i s analogous to problems i n s t a b i l i t y analysis where higher buckling modes may be achieved under c e r t a i n load d i s t r i b u t i o n s . For t h i s case, equations (4.41a,b) should be redefined as: Kj = /2ir K r f B ( 9 = 0 p ) , i f K A = 0 (4.42a) K J J = /2TT K b g B ( 6 = 0 p ) , i f K A = 0 (4.38b) Looking at the same example, we can see that unless the notch i s pe r f e c t l y loaded i n shear (thus f o r c i n g and to zero) the notch w i l l always f a i l when K^ = K-j-,-.' In other words, even the smallest d e v i a t i o n from pure shear w i l l change the f a i l u r e c r i t e r i o n from Kg = KJ-J-^ to K^ = 63 K._ (a mixed mode co n d i t i o n ) . A c t u a l l y , there should be some t r a n s i t i o n when becomes small or when the two eigenvalues are almost equal. Unlike K^-K^ f o r cracks, using t h i s d e f i n i t i o n , equation (4.41a,b), the K J - K J J s t r e s s - i n t e n s i t y f a c t o r s are now functions of the propagation d i r e c t i o n . K j - K ^ f o r cracks i n i s o t r o p i c bodies and ^ " K g f ° r V-notches do not change with the propagation angle. Fortunately i n materials such as wood, cracks w i l l always propagate along the grain; thus the s t r e s s -i n t e n s i t y f a c t o r s c o n t r o l l i n g f a i l u r e can be determined. In f a c t , the r e l a t i v e amount of and K^j. can be determined independent of the loading. From the above comments, we then conclude that the ASTM s i n g l e edge-notched specimen f o r metals and the associated equation for (ASTM-E399-83) i s not applicable to wood i f the grain runs p a r a l l e l to the span. In the case of a wood beam with a s i n g l e edge notch, the crack does not propagate i n l i n e with the crack, but perpendicular to i t . Although the singular stress f i e l d of a s i n g l e edge-notched or V-notched beam correspond to the opening mode, i t i s not s u f f i c i e n t to predict the d i r e c t i o n of crack growth. The d i r e c t i o n of propagation i s determined not only by the magnitude of the stresses, but also by the d i r e c t i o n of the planes of weakness. In wood, the t e n s i l e strength p a r a l l e l to the grain i s approximately 20 to 40 times the t e n s i l e strength perpendicular to the gr a i n and approximately 10 times the shear strength p a r a l l e l to the grain (Stamm, 1964). Consequently, the planes of weakness, both i n tension and shear, are always oriented p a r a l l e l to the grain. Even a crack perpendicular to the grain 6uch as an edge notch, i s not s u f f i c i e n t to cause crack to propagate across the gr a i n - t h i s resistance to propagation can be explained by the Cook-Gordon e f f e c t (Gordon, 1976). Therefore, c a l c u l a t i n g the s t r e s s - i n t e n s i t y based on stresses d i r e c t l y 64 forward of the crack t i p as outl i n e d by ASTM-E399 i s not v a l i d . S t r e s s - i n t e n s i t y factors for wood should, instead, be determined using equations (4.41) or (4.42) with the propagation angle equal to the grain angle. Another c h a r a c t e r i s t i c of V-notch s t r e s s - i n t e n s i t y f a c tors i s that t h e i r dimensions depend on the notch geometry and material properties. Thus, changes i n the slope of gr a i n or moduli r a t i o s w i l l cause changes i n the s t r e s s - i n t e n s i t y factor dimensions. The question i s whether i t i s proper to compare numbers which have d i f f e r e n t u n i t s and whether K^-K^.^ can be compared to K^-K^.^ so that an i n t e r a c t i o n curve can be developed. By examining the singular stress r e l a t i o n s , one finds that the stress state around the notch root can uniquely be defined i n terms of three parameters: Kj^* k H A ' A N C * T n i s concept w i l l be used i n a l a t e r chapter to develop an i n t e r a c t i o n curve f o r V-notch s t r e s s - i n t e n s i t y f a c t o r s . 4.4 Program V e r i f i c a t i o n and Convergence Published r e s u l t s of 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-notches i n orthotropic materials i s lim i t e d to the and Kg values obtained by Leic e s t e r (1982) f or rectangular notches. Sharp crack K^-K^ re s u l t s are r e a d i l y a v a i l a b l e i n the l i t e r a t u r e for both i s o t r o p i c and orthotropic bodies. V-notch K^-K^ s t r e s s - i n t e n s i t y f a c t o r s have only been determined f o r symmetrically loaded i s o t r o p i c bodies by Gross and Mendelson (1972). Consequently, program v e r i f i c a t i o n was r e s t r i c t e d to the l i m i t i n g cases of sharp cracks and V-notches i n i s o t r o p i c materials. 4.4.1 Sharp Cracks Three sharp crack problems i n orthotropic bodies were analyzed: 1) 65 a compact tension specimen of wood loaded perpendicular to the grain, 2) an end-slotted beam which subjects a crack to type loading, and 3) an i n c l i n e d crack i n an anisotropic plate loaded i n tension. Case 1 and 2 r e s u l t s were compared to those using the FEM program (Foschi, et a l , 1976) which was written to determine s t r e s s - i n t e n s i t y factors f o r cracks i n o r t h o t r o p i c materials. The t h i r d case was analyzed by Gandhi (1972). A center cracked i s o t r o p i c s t r i p was also analyzed (Broek, 1982). The r e s u l t s are summarized i n Table I . TABLE I Stress Intensity Factors f o r Various Sharp Crack Problems (Normalized to NOTCH s o l u t i o n unless zero **) Source Figure 4.4 Figure 4.5 Figure 4.6 Figure 4.7 NOTCH K I 1.0000 (0.0522) 1.0000 1.0000 K I I 0.00 1.0000 0.0 1.0000 FEM K l 1.0000 (0.0377) K I I 0.00 1.0000 Exact K I 0.9841 K I I 0.0 Gandhi (1972) K I 0.9805 K I I 0.9684 ** Values i n parenthesis have not been normalized since they should be zero. When entering nodal coordinates f o r sharp cracks, one must be c a r e f u l not to enter the same coordinates for nodes on opposite faces of F i g u r e 4 . 4 H = W = 2 a S h a r p c r a c k K j s p e c i m e n 3 i n c h e s ? P = 100 l b s . F i g u r e 4 . 6 - C e n t e r c r a c k e d i s o t r o p i c s t r i p u n d e r t e n s i o n W = 4 a = 4 i n c h e s ; (7= 100 p s i . L *• u * t D -—GRAIN | F i g u r e 4 . 5 - S h a r p c r a c k KJJ s p e c i m e n ( e n d s p l i t b e a m ) L = 20 i n c h e s , a = 8 i n c h e s , D = 4 i n c h e s ? P = 10 l b s . rrrm / 5* n rm c B L cr 5 i n c h e s 5 i n c h e s 1 i n c h 1000 p s i B F i g u r e 4 . 7 - I n c l i n e d c r a c k i n a s q u a r e o r t h o t r o p i c p l a t e u n d e r t e n s i o n 67 the crack. Singular displacements and s t r a i n s are cal c u l a t e d on the basis of the nodal coordinates and i f the nodes have the same coordinates, then the value of the singular function w i l l be the same f o r both nodes. To avoid t h i s , one must enter coordinates that are s l i g h t l y more clockwise on the counterclockwise face. The axis of the crack should then l i e between the nodes. For the following examples, the coordinates were changed i n the f i f t h or higher s i g n i f i c a n t f i g u r e to open the crack s l i g h t l y and avoid t h i s problem. As ind i c a t e d i n Table I, the NOTCH program gave the same r e s u l t s as the FEM program (E = 12400 MPa, E :E = 20.0, E :G = 1.0, v = 0.02). r o x ' x y ' y xy ' Gandhi (1972), using a c o l l o c a t i o n technique found K j and K J J to be 90.2 and 91.8 p s i - i n 0 . 5 respectively (E = 7(10 6) p s i , E = 2.5(10 6) p s i , G x y xy = 1.0 (10 6) p s i , v = 0.29); these d i f f e r e d from the NOTCH so l u t i o n by xy approximately 3%. Refining the singular mesh improved the s o l u t i o n as d i d increasing the Gauss quadrature i n t e g r a t i o n g r i d from 5 by 5 to 9 by 9. We expect monotonic convergence i f we r e f i n e the singular mesh, or increase the Gauss quadrature. Increasing the number of singular elements but keeping the same element si z e s ( i . e . making more of the surrounding elements singular) did not improve the s o l u t i o n . But according to Tong and Plan (1973), the region occupied by singular elements should not be small i n comparison to the crack length. This was not evident for the rectangular notch which suggests that the s i z e of the s i n g u l a r i t y dominated zone does not change s i g n i f i c a n t l y with notch depth. In the ana l y s i s , one layer of f u l l y s i n g u l a r and one lay e r of t r a n s i t i o n elements performed adequately for a l l the cracks examined. 68 It should also be emphasized that although a s i n g u l a r i t y enriched element i s being used, i t only supplements the regular s o l u t i o n with the "singular s o l u t i o n of a notch i n an i n f i n i t e plate." The use of s i n g u l a r i t y enriched elements permits a course mesh to be used at the notch root since the singular s o l u t i o n takes care of the singular stresses and s t r a i n s ; however, the mesh must s t i l l be capable of accurately modelling the stresses and displacements i n the region surrounding the notch root since i t i s the surrounding stresses and s t r a i n s which a f f e c t the i n f i n i t e p l a t e s o l u t i o n . Therefore, the mesh around the notch root should be f i n e r than that used i n the rest of the structure. One may consider these stresses and s t r a i n s i n the surrounding region as "boundary conditions" for the e i g e n f i e l d . When u t i l i z i n g symmetry to reduce the s i z e of the problem such as for the s t r i p shown i n Figure 4.6, the so l u t i o n may have a component. There are two l i n e s of symmetry i n t h i s problem. If both l i n e s of symmetry are used (e.g. d i s c r e t i z i n g only one-quarter of the s t r i p ) , the sol u t i o n contains a component. However, i f the en t i r e s t r i p or h a l f the s t r i p were d i s c r e t i z e d , then the i s found to be zero as expected. Thus, i t i s necessary to e i t h e r suppress the c a l c u l a t i o n of e x p l i c i t l y or model as l e a s t h a l f of the structure (e.g. use only one l i n e of symmetry). 4.4.2 V-notches i n I s o t r o p i c Bodies Several V-notch configurations previously analyzed by Gross and Mendelson (1972) were selected. V-notch aluminium bars were tested i n three-point bending, u n i a x i a l tension and shear as shown i n Figure 4.8 to 69 4.10. NOTCH-calculated s t r e s s - i n t e n s i t y f a c t o r s using a 9 x 9 quadrature rule were within 3% of the Gross and Mendelson (1972) r e s u l t s . Again, e i t h e r r e f i n i n g the singular element mesh or increasing the quadrature rule produce better r e s u l t s . Refining the singular mesh consisted of rep l a c i n g the crack t i p elements (Meshl) with element approximately ha l f as large (Mesh2). Near i s o t r o p i c properties were used to avoid numerical problems from degeneracies E x = 10.4 (10^) p s i , : E x : E y = 1.0001, E y:G = 2.6001, and v = 0.3. xy TABLE I I St r e s s - i n t e n s i t y Factors f o r V-notched Aluminium S t r i p s (Normalized to Gross and Mendelson (1972) unless SIF i s zero) Source B30 B90 T30 T90 S30 NOTCH Meshl (5x5) K I 1.0484 0.9684 1.0512 0.9690 0.0 K I I 0.0 0.0 0.0 0.0 1.0655 NOTCH Meshl (9x9) K I 0.0 K I I 1.0100 NOTCH Mesh2 (5x5) K I 1.0360 0.9709 1.0393 0.9829 0.0 K I I 0.0 0.0 0.0 0.0 1.0673 NOTCH Mesh2 (9x9) K I 0.9739 0.9750 0.9766 0.9871 0.0 K I I 0.0 0.0 0.0 0.0 1.0145 Gross et a l (1972) K I 1.0000 1.0000 1.0000 1.0000 0.0 K I I 0.0 0.0 0.0 0.0 1.0000 B30 - bending specimen (Figure 4.9) with 30° V-notch B90 - bending specimen (Figure 4.9) with 90° V-notch T30 - tension specimen (Figure 4.10) with 30° V-notch T90 - tension specimen (Figure 4.10) with 90° V-notch S30 - shear specimen (Figure 4.8) with 30° V-notch 3 1 = 6 2 = 0.5 * (V-notch angle) 70 F i g u r e 4 . 8 - I s o t r o p i c V - n o t c h e d shear specimen ( K - J - J ) D N:H = 1 , 2H:W = 0 . 6 , W = 4 i n c h e s ; CT = 1 0 0 0 p s i -H-T w F i g u r e 4 . 9 - I s o t r o p i c c e n t e r p o i n t loaded V-notched bending specimen (K T) DN:W = 0 . 5 , W = 2 i n c h e s , H = 4 i n c h e s P = 1 0 0 l b s i 1 N cr •H-W F i g u r e 4 . 1 0 - I s o t r o p i c V-notched t e n s i o n specimen (K T) D NsW = 0 . 5 , W = 2 i n c h e s , H = 4 i n c h e s ; P = 1 0 0 l b s 71 4.5 F i n i t e Element Model of a Notched Beam From St. Venant's p r i n c i p l e , we know that the e f f e c t s of a notch on the stresses diminishes quite r a p i d l y as we move away from the notch (Timoshenko, 1934). The implications are that i n modelling a notched beam, we need not consider the e n t i r e length of the beam, instead, only _ the notched portion can be d i s c r e t i z e d . The actual dimensions of the c r i t i c a l section, however, depends on the type of problem and are not given by St. Venant's p r i n c i p l e . Savin (1961) used the photoelastic technique to show that the zone of disturbance of a rectangular hole i n the d i r e c t i o n of the neutral axis i s about twice the side length of the rectangle (measured from the center of the hole) and i n the perpendicular d i r e c t i o n , approximately one-third of the side length. A rectangular end-notched ortho t r o p i c beam w i l l be considered here. Two simply supported end-notched beams with configuration as shown i n Figure 4.11 were analyzed - symmetry was taken advantage of i n the a n a l y s i s . Both were subjected to a 1-KN centerpoint reference load (2-KN t o t a l load on beam). The NOTCH program was run i n the CONTOUR mode to generate f i l e s of p a r a l l e l and perpendicular-to-grain and shear stresses at the Gauss quadrature points (3 by 3 per element). Contour maps of stresses were then generated using the UBC-SURFACE package (Mair, 1978). The p a r a l l e l and perpendicular to grain, as w e l l as the shear stress contour diagrams for the two notch geometries are given i n Figures 4.15 to 4.26. For comparison, the s t r e s s contours f o r an unnotched beam are also included (Figures 4.12 to 4.14). The diagrams, p a r t i c u l a r l y the p a r a l l e l to g r a i n stresses, reveal that the length of the zone of influence i s approximately equal to 10 to 20 mm measured from the notch root and i s independent of the notch depth. The shear stress disturbance gure 4 . 1 1 - Rectangular end-notched centerpoint loaded beam configurati C o n t o u r s are l a b e l l e d w i t h the f i r s t t h r e e d i g i t s o f the c o r r e s p o n d i n g B t r e s s l e v e l ( e x c l u d i n g s i g n ) igure 4.12 - P a r a l l e l - t o - g r a i n stress contours f o r an unnotched beam (P = 1000 N) Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) Figure 4.13 - Perpendicular-to-grain stress contours f o r an unnotched beam (P = 1000 N) Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) - 4 S 4 -36.838' - 1 67.822--1 98.806* 129.79" -I •160.78' Stress Level Label (MPa) -0.81*5 81>5 -0.i»i9 l»19 O.0745 71*5 kjk 0.860 B60 1.287 129 1.713 171 2.139 214 •191.76' —1 •222.74« —I 253.73 315.7 X (MM) Figure 4.l4 - Shear stress contours f o r an unnotched beam (P = 1000 N) C o n t o u r s a r e l a b e l l e d w i t h t h e f i r s t t h r e e d i g i t s o f t h e c o r r e s p o n d i n g s t r e s s l e v e l ( e x c l u d i n g s i g n ) S t r e s B L e v e l L a b e l ( M P a ) -5.667 567 -3.653 365 -1.639 164 0.376 376 2.390 239 k.liOU UUO 6.418 642 8.432 843 •4.961-1 r 3 6 . 0 3 8 — 67.116* '98.194 -\ 1—r 129.27—* 160.35 1 T 191.43 222.5 253.5B 284.66 315.74 X (MM) Figure 4 . 1 5 - Para l l e l - t o - g r a i n stress contours f o r a 10-mm end-notched beam under centerpoint loading (P = 1 0 0 0 N) to Contours are labelled with the f i r s t three digits ™ of the corresponding stress level (excluding sign) Stress Level Label (MPa) ~ ~ l 1 1 1 1 —I 1— 1 1 1 1 J02.92 113.38 123.84 134.3 144.77 155.23 165.69 176.15 186.61 197.08 207.54 Figure 4 . 1 6 - Par a l l e l - t o - g r a i n stress contours f o r a 10-mm end-notched X (n beam under centerpoint loading (P = 1 0 0 0 N) Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) 3 Q -- i 1 1 — — — i — 1 3 6 . 0 3 8 — » 6 7 . 1 1 6 — » 9 8 . 1 9 4 — — 1 2 9 . 2 7 — • 1 "1 160 Stress Level Label (MPa) -2.321 232 -1.882 188 -1.442 144 -1.002 100 -0.562 562 -0.122 122 0.318 318 0.757 757 35 191.43 222.5 -1 253.58 r284.66 315.74 X (MM) Figure 4 . 1 7 - Perpendicular-to-grain stress contours f o r a 10-mm end-notched beam under centerpoint loading (P = 1 0 0 0 N) Contours are labelled with the f i r s t three digits of the corresponding B t r e s s level (excluding sign) Stress Level (MPa) -0.122 O.OU26 0.208 0.373 0.537 0.702 0.867 1.032 Label 122 U26 208 373 537 702 867 103 102.92 113.38 123.84 134.3 144.77 155.23 165.69 176.15 186.61 197.08 Figure 4 . 1 8 - Perpendicular-to-grain stress contours f o r a 10-mm end-notched beam under centerpoint loading (P = 1 0 0 0 N) 207.54 X (MM) Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) gure 4.19 - Shear stress contours f o r a 10-mm end-notched beam under centerpoint loading (P = 1000 N) Of) to Y S n Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) Stress Level Label (MPa) 0.0272 272 0.219 219 0.411 411 0.603 603 0.796 796 0.988 988 1.180 118 1.372 137 102.92 113.38 123.84 134.3 144.77 155.23 165.69 176.15 166.61 197.08 F i g u r e 4.20 - Shear s t r e s s contours f o r a 10-mm end-notched beam under c e n t e r p o i n t l o a d i n g (P = 1000 N) 207.54 X (MM) OO Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) Stress Level Label (MPa) -5.080 508 0.431 431 5.9^ 2 594 11.45 115 16.96 170 22.48 225 27.99 280 33-50 335 284.66 315.74 X (MM) Figure 4.21 - Pa r a l l e l - t o - g r a i n stress contours f o r a 50-mm end-notched beam 00 under centerpoint loading (P = 1000 N) 9--308--<J1 ' 102.92 Figure 4 . 2 2 Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) Stress Level Label (MPa) -5.080 508 0.431 1*31 594 11. b5 115 16.96 170 22.48 225 27.99 280 33.50 335 113.38 123.84 134.3 144.77 155.23 165.69 176.IS 166.61 191.08 207.54 P a r a l l e l - t o - g r a i n stress contours f o r a 50-mm end-notched beam X (MM) under centerpoint loading (P = 1000 N) oo Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) h $ 10/ 8-1 I 10 f 4.961" —I ' 36.038* I •67.116' 754 Stress Level Label (MPa) -2.211 221 -1.068 107 0.0754 754 1.219 122 2.362 236 3.505 4.648 465 5.791 579 '98.194 129.27* 160.35 ~T 191.43 I 222.5 —1 253.58 284.66 315.74 X (MM) gure 4 . 2 3 - Perpendicular-to-grain stress contours f o r a 50-mm end-notched be under centerpoint loading (P = 1 0 0 0 N) IO cn I at 1 on i n . to tn. in i n 9 to 3 102.92 ~I 113.38 ~1 123.84 Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) Stress Level (MPa) 0.536 1.335 2.135 2.935 3.735 5.535 5.335 6.135 Label 536 13*» 214 29<» 371* $5«» 53* 614 134.3 144.77 155.23 165.69 ~I 176.15 I 166.61 197.06 Figure 4.24 - Perpendicular-to-grain stress contours f o r a beam under centerpoint loading (P = 1 0 0 0 N) 50-mm end-notched 207.54 X (MM) 00 Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) Stress Level Label (MPa) X (MM) Figure 4 . 2 5 - Shear stress contours f o r a 50-mm end-notched beam under centerpoint loading (P = 1000N) oo Y sh O l r-in i n cu 3 Contours are labelled with the f i r s t three digits of the corresponding stress level (excluding sign) Stress Level Label (MPa) -0.299 299 0.712 712 1.722 172 2.732 273 3.742 374 *.752 *75 5.762 576 6.772 677 102.92 ~I 113.38 -1 123.84 "1 134.3 144.77 _)  155.23 "I 165.69 I 176.15 I 186.61 "I 197.08 Figure 4.26 - Shear stress contours f o r a 50-mm end-notched beam under centerpoint loading (P = 1000 N) 207.54 X (MM) oo 88 zones, unfortunately, are not as c l e a r but seem to be s i m i l a r to the p a r a l l e l to grain stresses. On the other hand, perpendicular to grain stresses are n e g l i g i b l e at approximately 0.5D^ from the notch root. A subsequent analysis u t i l i z i n g a larger s i n g u l a r i t y enriched region d i d not r e s u l t In any s i g n i f i c a n t change i n the str e s s pattern. From the contour diagrams, the lengths of disturbances from the reaction points can a l s o be q u a n t i f i e d - these are approximately one beam depth both i n the reduced and unreduced sections. Thus, the minimum length of beam which can be modelled should approximately be twice the unreduced beam depth plus one notch depth (governed by perpendicular to grain stresses and e f f e c t s from substitute f o r c e s ) . This w i l l ensure that the s o l u t i o n w i l l not be influenced by the substitute forces acting on the segment to maintain equilibrium. Furthermore, the diagrams show that the s i n g u l a r i t y dominated region i s r e l a t i v e l y small. Thus, singular elements need only be u t i l i z e d i n a small region approximately 0.5DN from the notch root. Tong and Plan (1973) showed that convergence of displacements i s co n t r o l l e d by the order of the s i n g u l a r i t y rather than the order of the polynomial used f o r the i n t e r p o l a t i o n . They maintain that the rate of convergence may be improved by incl u d i n g the proper s i n g u l a r i t i e s i n the f i n i t e element approximation function, which i s consistent with the method presented i n th i s t h e s i s . I t was also suggested that the siz e of the s i n g u l a r i t y region be retained even as the mesh i s being refined; i f the s i n g u l a r i t y enriched region were confined only to the elements adjacent to the s i n g u l a r i t y when the mesh i s re f i n e d , the method w i l l revert back to a f u l l y regular f i n i t e element a n a l y s i s . 89 APPLICATIONS 5.1 Load Test on Rectangular End-notched Wood Beams A small experiment was conducted to v e r i f y as w e l l as demonstrate the use of the program i n p r e d i c t i n g f a i l u r e loads due to cracks propagating from V-notches i n wood. 5.1.1 Procedure Twenty c l e a r k i l n - d r i e d nominal 2 i n by 4 i n Hem-Fir boards were purchased l o c a l l y . Two 2 foot long specimens were cut from the ends of each board. The average modulus of e l a s t i c i t y (MOE) was estimated f o r each specimen by s t r e s s i n g them up to approximately 800 p s i under centerpoint loading and measuring the displacement. Due to the short span, the specimens were bent about t h e i r weak axis - on f l a t . They were then ranked according to t h e i r average modulus of e l a s t i c i t y and d i v i d e d into two groups of twenty such that each group had approximately the same MOE-distribution. Rectangular end notches were cut at both ends of the specimen using a table saw for the v e r t i c a l cut, and a bandsaw for the h o r i z o n t a l cut. One group was notched (Djg) 10 mm and other 50 mm from the bottom edge as i n Figure 5.1 (1 inch = 25.4 mm). A l l notches were placed (e) 6 inches from the ends to avoid r e a c t i o n point e f f e c t s . The notched specimens were then stored for several days i n the t e s t i n g room of the Timber Engineering s e c t i o n at the Forintek Canada Corp. Western laboratory. Since obtaining l o c a l modulus of e l a s t i c i t y , modulus of r i g i d i t y , and Poisson r a t i o s were d i f f i c u l t , average values of these moduli were used i n the f i n i t e element a n a l y s i s (E = 1.2(10 6) p s i = 8274 MPa, I GRAIN D Figure 5>1 - Rectangular end-notched centerpoint loaded beam configuration 91 E x:Ey = 20.0, Ey.:G = 0.9, v x y = 0.02). A s e n s i t i v i t y a n a l y s i s showed that s t r e s s i n t e n s i t y f a c t o r r e s u l t s were not s e n s i t i v e to changes i n the moduli r a t i o s i n the ranges a n t i c i p a t e d . The specimens were simply supported on a 22 inch simple span with the load applied at the midspan. Testing was performed on an MTS t e s t i n g machine at a cross-head speed of 0.1 inch per second f o r both specimen types. This resulted i n f a i l u r e s i n approximately 1 to 2 minutes. The load at which a v i s i b l e crack at the surface formed and the ultimate load was recorded. Moisture contents were obtained at the time of t e s t i n g with a resistance-type moisture meter (since the specimens were at room temperature, the readings did not have to be corrected f or temperature). The average propagation angle from the two faces was a l s o noted. Small blocks were then cut from near the f a i l e d notch to determine density. 5.1.2 Treatment of Data Although the material was claimed to be k i l n - d r i e d , moisture contents ranging from 8.5% to 25% were measured. The measured value, however, was not f e l t to be a true i n d i c a t i o n of the moisture content near the notch root. Specimens with moisture contents greater than 15% generally experienced more crushing or slow crack growth and were subsequently reje c t e d . There was no apparent c o r r e l a t i o n between the ultimate strength and measured moisture content f o r the samples accepted. The specimens accepted had moisture contents i n the range 9 to 12%. Load-deflection p l o t s were generated at the time of t e s t i n g on an X-Y recorder. De f l e c t i o n r e s u l t s were not r e l i a b l e because of crushing perpendicular to the g r a i n at the r e a c t i o n points, p a r t i c u l a r l y f o r the 10 mm notch specimens. Some specimens experienced slow crack growth but 92 continued to carry a d d i t i o n a l load. This was allowed so long as there was no noticeable drop i n load and the crack was r e l a t i v e l y short compared to the notch depth (10% of the notch depth was chosen as an a r b i t r a r y l i m i t ) . Specimens where long cracks had formed were r e j e c t e d . A f a i l e d 10 mm end-notched beam i s shown i n Figure 5.2 Since crushing did not occur with the 50 mm end-notched specimens, the ASTM-E399 procedure was adopted to determine the fr a c t u r e load. The procedure i s intended f or analyzing load vs crack-opening displacements (COD). COD i s d i f f i c u l t to measure f o r V-notches, so the procedure was used for load vs midspan displacement. Since the load-displacement curves from t h i s experiment were s i m i l a r to those encountered i n the ASTM fracture t e s t s , an analogous procedure was adopted to check the v a l i d i t y of the r e s u l t s . Load at the 5% o f f s e t was compared to the maximum load to determine whether the test was v a l i d . For the 50 mm end-notched beam, n o n - l i n e a r i t i e s were due to slow crack growth. T y p i c a l curves are shown i n Figure 5.3. This approach could not be applied to the 10 mm notched specimen since the n o n - l i n e a r i t i e s were primarily due to the large amounts of crushing at the supports. 5.1.3 Results and Discussions Calculations were based on a f i n a l sample s i z e of 11 f o r each notch depth. The average f a i l u r e loads f or the 10 mm deep notch and the 50 mm deep notch were 2501 lbs (CoV = 0.238) and 445 l b s (CoV = 0.183) res p e c t i v e l y (1 l b = 4.4482 Newton, 1 inch = 25.4 mm). This gives a load r a t i o of 5.62. From a f i n i t e element a n a l y s i s , a 100 l b load on a 10 mm rectangular end-notched beam r e s u l t s i n a K stress i n t e n s i t y factor of A 36.8 p s i - i n 0 . 1 * 5 . At 2504 l b s , the K would then be 922.0 p s i - i n 0 . 1 * 5 , Figure 5.2 - Cracked 10-mm end-notched 2x4 Hem-Fir beam under centerpoint loading 5 p e r c e n t o f f s e t l i n e MIDSPAN DISPLACEMENT Figure 5 .3 - Typical load-deflection curves showing P m a x » p q » a n d 5% o f f s e t l i n e < 0 a s e ! Pmax ' V p » « > V Pmax » V 95 which i s the average 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 f o r t h i s p a r t i c u l a r notch i n Hem-Fir. A f i n i t e element analysis of the 50 mm notched beam gave 185.8 p s i - i n 0 . 1 * 5 per 100 l b load. Therefore, the predicted average ultimate load for th i s specimen i s : 185.8°psl-ln'1.l»!> X 9 2 2 * ° P 8 1- 1" 0** 5 = 4 9 6' 2 L B S C5'1) a value which compares well (error approximately 10%) with the average measured value of 445 l b . No attempt was made to correc t f o r propagation angle, density, or moisture content. A table of the experimental r e s u l t s i s included i n Appendix I I . I t i s i n t e r e s t i n g to compare these r e s u l t s to CAN3-086-M84 design values f o r shear and fl e x u r e . If we consider a l l the specimens, the weakest ( i n a sample s i z e of 20 f o r each notch depth) was 220 l b and 1600 l b for the 50 mm and 10 mm notched beams re s p e c t i v e l y . CAN3-086-M84 does not allow the 50 mm notch since i t i s greater than (0.25 * depth), however, i f we ignore t h i s r e s t r i c t i o n , the allowable loads from CAN3-086-M84 are 1640 l b f o r the 10 mm deep notch and 398 l b f o r the 50 mm deep notch under short term loading and a safety factor of 1.0. The load based on the allowable shear s t r e s s on the reduced s e c t i o n are 1847 lb and 910 l b for the 50 mm and 10 mm notched beams resp e c t i v e l y ( a l l c a l c u l a t i o n s are based on the assumption of an uncracked beam. See Appendix I I I ) . Therefore, CAN3-086 i s capable of pr e d i c t i n g the strength of shallow rectangular end-notched beams (e.g. D J J / D = 0.10) but i s unconservative f o r deep notches (e.g. Dj/D = 0.50). Note that the notch geometry examined i s the only one considered by CAN3-086-M84. Had we analyzed a tapered notch or a notch i n a zero shear 96 region (middle t h i r d of a t h i r d point loaded beam), we would not have been able to take into account the notch. It i s also i n t e r e s t i n g to note that i f a notch i s unavoidable, a tapered notch i s suggested but there are no provisions i n the code to design such a notch. Although the t e s t r e s u l t s agreed reasonably w e l l with the computed r e s u l t s , a d d i t i o n a l tests are required under more c o n t r o l l e d conditions -es p e c i a l l y f o r small specimens f o r c r i t i c a l and K J J * However, from t h i s test we can conclude that: 1) K. and not K n or K.+K„, i s the governing parameter i n p r e d i c t i n g A JS A Jo rapid crack propagation, 2) The average 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 , K ^ , f o r a rectangular end notch i n Hem-Fir i s 922.4 p s i - i n 0 . 1 ^ . 5.2 Rectangular End-notch Stress-Intensity Factors Using the findings of Section 4.5, rectangular end-notched 2 x 8's were analyzed with various notch depths and lengths under center point loading (P = 2 kN) notched segment under constant shear and l i n e a r l y varying bending moment. The V-notch s t r e s s - i n t e n s i t y factor were plotted against the notch length for various notch depths (Figure 5.4) with: E - 12400 MPa, E - 620.0 MPa, G = 690.0 MPa and v = 0.02. x y xy Since the K^-K^ r a t i o i s constant f o r a given notch geometry and material, can be ca l c u l a t e d from the r e l a t i o n » 1.062(Kj) a f t e r obtaining from the p l o t . These values can then be compared to the c r i t i c a l values using: - 1.374KA(, MPa-mm0.4*5, and K I I C - 1.458KA(, MPa-mm0.1*5. K A^ can only be obtained from tests of a rectangular notch with the above pr o p e r t i e s . 97 Rectangular end notches i n other materials can be obtained using fig u r e 5 .4 i f the r a t i o s E :E and E :G and v are the same as those 6 x y y xy used i n generating the chart. S t r e s s - i n t e n s i t y factors taken from the chart must be adjusted by multiplying by E / 1 2 4 0 0 where E i s the modulus of e l a s t i c i t y of the alternate material i n MPa. For short notch lengths, the e f f e c t and d e t a i l s of the r e a c t i o n point become s i g n i f i c a n t and therefore the stress i n t e n s i t y f a c tors cannot be summarized i n a simple p l o t . Notches wi t h i n approximately one reduced depth (D - D n) of the reaction may encounter end e f f e c t s . Also, when the notch depth or length approaches the c h a r a c t e r i s t i c s i z e of the microstructure, the analysis becomes i n v a l i d as discussed i n Section 2 . 2 . Consider a notched beam as shown i n Figure 5.1 with D • 89 mm, " 50 mm, e - 127 mm (5 in) and L - 279 mm (11 i n ) . Thus, e/L - 0 . 4 6 , D^/D - 0 . 5 6 . From Figure 5 . 4 , we f i n d K - 25 MPa-mm0.1*5. But t h i s i s for E I x - 12400 MPa. For a beam with E - 8300 MPa: x K j = ( 25 .0 MPa - T . 0 , . , 5 ) ( fgg^_) = 1 6 . 7 MPa-mm<M5. K - ( 16 .7 MPa - mm 0 . 1 * * ) ( 1 .062 ) - 17 .7 MPa-mm0. *»5. 5 .3 E f f e c t of Slope of Grain and Propagation D i r e c t i o n Slope of gra i n has a pronounced e f f e c t on the computed s t r e s s - i n t e n s i t y f a c t o r . Not only i s the propagation angle a l t e r e d , but al s o the s t r e s s f i e l d . The magnitude of the e f f e c t appears to be a function of the notch depth as well. Since i t depends on the slope of 98 0.4 0.6 e/L - K j s t r e s s - i n t e n s i t y f a c t o r f o r v a r i o u s n o t c h d e p t h s a n d n o t c h l e n g t h s ( E = 1 2 4 0 0 MP a , E = 620 MPa, G = 690 MPa, a n d V = 0.02) 99 grain, i t i s d i f f i c u l t to say whether the f a i l u r e load w i l l decrease proportionately with K^. Slope of grains of 10, 0, and -10 degrees were analyzed f o r the 10 mm and 40 mm deep notches (E - 8300 MPa, E - 415 MPa, G - 461 MPa, and v = 0.02, L = 279 mm, D = 89 mm) at the xy xy reference load of P = 2.0 KN (see Figure 5.1). TABLE I I I E f f e c t of Slope of Grain on Rectangular End-notched Beam  Stress-Intensity Factors (Normalized with respect to 0° grain angle case) Theta Lambda |KAI @ DN - 5 0 -10.0 1.5823 1.2670 0.9897 0.0 1.5493 1.0000 1.0000 10.0 1.5245 0.8067 1.0928 K A (@ D N - 10 mm, P - 2.0 KN) - 2.5090 MPa-mm0." K A (@ L\j = 50 mm, P= 2.0 KN) - 12.3600 MPa-mm0.1^ Slope or gra i n as defined i n Figure 5.5. Beam configuration as defined i n Figure 5.1 It i s i n t e r e s t i n g to note that the 50 mm notch r e s u l t s do not decrease with increasing 6 . Also, i t should be pointed out that the allowable general slope of grain for Select, No. 1, and No. 2 s t r u c t u r a l grades are 4.8 (1 i n 12), 5.7 (1 i n 10) and 7.1 (1 i n 8) degrees respectively, so the range of grain angles studies i s reasonable. L o c a l grain v a r i a t i o n s due to defects may be higher and thus should be kept away from the notch root or the notch should be reanalyzed using the l o c a l g r a i n angle. Note that f o r p o s i t i v e g r a i n angles, there are two possible along grain propagation d i r e c t i o n s ( f i g u r e 5.5). Figure 5-5 - Slope of grain at V-notches Figure 5.6 - Across-grain crack propagation at V-notches 101 Since wood i s a highly anisotropic material, crack propagation i s nearly always p a r a l l e l to the grain. For other composite materials, i t may be possible f o r the crack to propagate i n a d i r e c t i o n that i s not colinear with one of the axis of e l a s t i c symmetry. To consider t h i s case, l e t us assume zero slope of grain, and look at the s t r e s s - i n t e n s i t y factors when across-grain propagation occurs. Since i s the same for a l l values propagation d i r e c t i o n s , we need to look at and K^ .^. - these give an i n d i c a t i o n of the magnitude of the stresses causing propagation. TABLE IV E f f e c t of Across-grain Propagation on Rectangular  End-Notched Beam Stress-Intensity Factors  at Various Slope of Grain (Normalized with respect to 0° propagation angle at various slope of grain) Prop. Angle K I K I I K I K I I K I K I I -10.0 1.200 1.210 1.334 1.310 1.452 1.392 0.0 1.000 1.000 1.000 1.000 1.000 1.000 10.0 0.820 0.807 0.780 0.735 0.689 0.677 Lambda 1.5823 1.5495 1.5245 Grain Angle -10.0 0.0 10.0° K A (@ Dn = 10 mm, P - 2.0 KN) - 2.5090 MPa-mm0.1^ K A (@ Dn = 50 mm, P - 2.0 KN) - 12.3600 MPa-mm0.1*5 Kj = 1.3735 K A MPa-mmO.'+S; Kj., - 1.4581 K A MPa-mmO.'tS Slope of Grain as defined i n Figure 5.5 Propagation angle as defined i n Figure 5.6 Beam Configuration as defined i n Figure 5.1 102 In a l l cases, the crack w i l l have a tendency to propagate i n the - 1 0 d i r e c t i o n - the d i r e c t i o n c l o s e r to the mean of the face angle. The polar p l o t s of the singular stress s o l u t i o n , Figure 3 . 1 3 , a l s o shows th i s r e s u l t . At 6 * 1 8 0 ° (along the grain) the shear-tangential stress r a t i o or the K ^ i K ^ r a t i o i s approximately one. This r a t i o remains at one from about 6 « 1 5 5 ° to about 6 = 2 1 0 ° . However, f o r a given value of K , the absolute values of the s t r e s s - i n t e n s i t y f a c t o r s K A. X and K J J , increase as 9 i s decreased from 2 1 0 ° to 1 5 5 ° . From Figure 3 . 1 3 , the stresses at 0 = 1 5 5 ° are about 3 0 % higher than at 1 8 0 ° . In wood, the tension p a r a l l e l to the grain strength i s 2 5 0 0 % higher than the tension perpendicular to g r a i n so propagation across the g r a i n i s u n l i k e l y . But, for other composite materials, the difference i n strength may not be as large and the crack may propagate across the g r a i n . 5 . 4 E f f e c t of Varying the E l a s t i c Constants In most cases, the e l a s t i c constants can only be estimated because either they are d i f f i c u l t to f i n d or te s t i n g only provides approximate values. The e f f e c t s of varying the e l a s t i c constants on the s t r e s s - i n t e n s i t y factor for the rectangular notch of Figure 5 . 1 w i l l be examined. The reference e l a s t i c constants w i l l be taken as E = 8 3 0 0 x MPa, E = 4 1 5 MPa and G - 4 6 1 . 1 MPa. v w i l l be kept at 0 . 0 2 . Each y xy xy of E x , E y and G ^ w i l l be varied independently by plus or minus 2 0 percent. Although Poisson r a t i o s are perhaps the most d i f f i c u l t parameters to determine, the e f f e c t of varying them w i l l not be considered here. 103 TABLE V Effect of Varying Elastic Constants on  Rectangular End-notched Beam Stress-Intensity Factors (Normalized with respect to reference elastic constants case) E y (MPa) 9960 E x (MPa) 8300 6640 498 ^ ~""""""" 0.9964 415 0.8758 1.0000 1.1952 332 • 1.0234 • Beam configuration as shown in Figure 5.1 G x y (MPa) \ - 8300 MPa 553.30 1.1929 461.11 1.0000 368.89 0.8595 By changing each parameter independently, the moduli ratios change and, as a result, the X value w i l l change. However, i f we maintain the original ratios and simply vary E , then X w i l l not change and K. w i l l vary linearly with E^. As can be seen i n Table V, the stress-intensity factor appears to be quite sensitive to variations in E and G but not to variations i n E . x y This conclusion, however, Is highly dependent on the type of notch we are analyzing. For example, i f we were considering a Kj. type specimen, then the stress-intensity factor would be directly proportional to the E y value and almost insensitive to changes in E x and G. On the other hand, i f we were considering a type specimen, then the stress-intensity factor would be highly dependent on G and virtually independent of E . 104 5.5 Cracks at the Root of V-notches It i s not uncommon to f i n d cracks at the root of V-notches. Depending on the environment and the load h i s t o r y , such cracks may be due to shrinkage stresses or the r e s u l t of s u b c r i t i c a l crack growth. A seemingly l o g i c a l approach to p r e d i c t i n g rapid crack propagation from V-notches Is to analyze the notch structure assuming various crack lengths and extrapolating the r e s u l t s back to a zero crack length. An obstacle to t h i s approach, as mentioned previously, i s that X f o r V-notches and cracks are not equal. Consequently, the dimensions of a s t r e s s - i n t e n s i t y f a c t o r for an i n f i n i t e s i m a l crack and a V-notch are quite d i f f e r e n t . We must f i r s t determine whether an i n f i n i t e s i m a l crack at the root of a rectangular notch r e s u l t s i n the same singular s t r e s s - s t r a i n f i e l d as the same notch without a crack. Unless there i s a threshold at the onset of crack formation, i n t u i t i v e l y , we expect the stresses from both cases to converge. In f a c t , as the crack i s reduced to a length which f a l l s within the s i n g u l a r i t y dominated region of a notch, the crack becomes p h y s i c a l l y i n s i g n i f i c a n t . But even i f the stresses are i d e n t i c a l , the comparison i s only v a l i d i f the c r i t i c a l and K^ .^ and the stress state which they describe are the same for V-notches and cracks. A p l o t of the sharp crack 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 a crack emanating from an end-notched beam vs crack length i s given i n Figure 5 . 7 . The s t r e s s - i n t e n s i t y f a c t o r f o r the rectangular notch i s a l s o plotted at zero crack length, with an open c i r c l e , f o r comparison. The curve i n d i c a t e s that the crack s t r e s s - i n t e n s i t y f a c t o r converges to that of the notch i n magnitude under the same loading conditions. However, 105 3 CO o 0.0 10.0 20.0 30.0 40.0 50.0 60.0 C r a c k L e n g t h , a ( m m ) Figure 5»7 - St r e s s - i n t e n s i t y factors f o r sharp cracks emanating from rectangular end-notches E =12420 MPa, E :E =20.0, E :G=0.90, V =0.02 x x y y ' rxy D=184 MM, DN=60 mm, e=200 mm, L=4?5 mm, P=2 KN Beam configuration as shown i n Figure 5-1 106 because of the d i f f e r e n t X, the rate of increase of the stresses as we approach the notch root i s d i f f e r e n t . Thus the stress states are not equivalent even though the s t r e s s - i n t e n s i t y f a c t o r s are equal. This leads us to conclude that the s t r e s s - i n t e n s i t y factors are not s u f f i c i e n t to describe the s t r e s s - s t r a i n state and that the sharp crack f a i l u r e c r i t e r i o n cannot be r a t i o n a l l y applied to V-notches. 5.6 V-Notch S t r e s s - I n t e n s i t y Factor I n t e r a c t i o n Curve As derived i n Section 4.3, V-notch s t r e s s - i n t e n s i t y f a c t o r s have 2.-X dimensions of s t r e s s * length , where X i s the eigenvalue f o r a p a r t i c u l a r notch geometry. For cracks, both s t r e s s - i n t e n s i t y f a c t o r s , K and K , have dimensions of s t r e s s - l e n g t h 0 . 5 or X = 1.5 regardless of A B the crack o r i e n t a t i o n or material properties. For V-notches i n an i s o t r o p i c continuum, X i s a nonlinear function of the notch angle and a l i n e a r function of the modulus of e l a s t i c i t y . In o r t h o t r o p i c materials, however, X i s a nonlinear function of the e l a s t i c constants as well as the geometry of the notch. Consequently, f o r any given value of X, there w i l l be, t h e o r e t i c a l l y , an i n f i n i t e number of compatible V-notch geometries and material combinations. This, at f i r s t glance, does not seem to be too h e l p f u l . If we examine the singular stress terms, we f i n d : K YL^ 0 i i = _ A q r f A ^ + ~hrfB ( e ) ( 5 * 2 ) 1 J r AA A I j r AB B i j When c a l c u l a t i n g s t r e s s - i n t e n s i t y f a c t o r s , we f a c t o r out the s i n g u l a r i t y or " r " dependency: K = lim 2-A r A a i j (5.3a) r+o or K = lim 2-A r B o..t i f K. = 0 i j A (5.3b) r+o Since a l l cracks have l / / r ~ type s i n g u l a r i t i e s regardless of the crack o r i e n t a t i o n and material properties, we need not concern ourselves with the value of A when attempting to predict mixed-mode f a i l u r e . In other words, i f we are dealing only with cracks, then the s t r e s s - i n t e n s i t y factors are s u f f i c i e n t for describing the notch root s t r e s s - s t r a i n s t a t e , and mixed mode conditions may be compared, f or example, to the re s u l t s from a compact tension specimen and to the K^.^ r e s u l t s from an end-split beam. For the more general case of V-notches, a minimum of three parameters are required to uniquely describe the tangential and shear stress states at the notch root: K^, and A. and K ^ are calc u l a t e d from (or Kg i f K^ i s zero). From equation (5.1 ), i t can be shown that the notch root stress states are i d e n t i c a l f o r notches 108 which have i d e n t i c a l K j , K^ .^  and X values; Kj. and give an i n d i c a t i o n of the magnitude of the stresses and X gives an i n d i c a t i o n of the rate at which the stress increases as one approaches the notch root. To elaborate, i t i s obvious that the notch geometries i n Figure 5.8 w i l l a l l r e s u l t i n s t r i c t l y mode I type stresses and displacements. If we i n s i s t that the c r i t i c a l s t r e s s - i n t e n s i t y factors are independent of X, then a l l the notch geometries shown i n Figure 5.8 w i l l f a i l at the same s t r e s s - i n t e n s i t y f a c t o r . Furthermore, i f we plot values of s t r e s s - i n t e n s i t y f a c t o r s without considering the value of X, then we w i l l be comparing s t r e s s - I n t e n s i t y factors which have d i f f e r e n t dimensions. A more r a t i o n a l approach would be to develop i n t e r a c t i o n curves f o r constant values of X. Thus, computed s t r e s s - i n t e n s i t y f a c tors w i l l be compared to c r i t i c a l values with the same dimensions. The three parameters i m p l i c i t l y consider the material and geometric properties of the notch. To e s t a b l i s h the family of i n t e r a c t i o n curves, one requires the c r i t i c a l s t r e s s - i n t e n s i t y factors for modes I and II for d i f f e r e n t values of X between 1.5 and 2.0. This i s not as straight-forward as i s the case for cracks where the primary and secondary modes have equal eigenvalues. But l i k e cracks, i n order to get the independent opening and shear displacement modes, the V-notch must be positioned such that i t i s symmetric about the grain. Since V-notches have unequal eigenvalues two d i f f e r e n t notch configurations are required to determine the c r i t i c a l mode I and I I s t r e s s - i n t e n s i t y factors f o r a p a r t i c u l a r value of X i n a given m a t e r i a l . 110 For example, to e s t a b l i s h the K j - K j j i n t e r a c t i o n curve at X -1.518, we need to test a 40° notch i n the compact tension specimen configuration, and a 3° notch i n shear. The s i n g u l a r s t r e s s f i e l d s f o r Mode II and Mode I are shown i n Figures 5.10 and 5.11 r e s p e c t i v e l y . Note that the Mode II c o n f i g u r a t i o n i s a secondary s t r e s s f i e l d while the Mode I configuration i s a primary stress f i e l d . By loading i n shear, the primary s t r e s s f i e l d , shown i n Figure 5.9, i s suppressed. Mixed-modes configurations can be determined by solving the eigenvalue problem a number of times; these r e s u l t i n s t r e s s f i e l d s shown i n Figures 5.12 to 5.14. Note that the r e l a t i v e amounts of K^ . and f o r a given notch geometry i s f i x e d . Therefore, changing the loading conditions, f o r example, from uniform to concentrated or from bending to tension w i l l not change the K^-K^ r a t i o . In order to obtain points on the i n t e r a c t i o n curve, various notch configurations must be examined - not loading d i s t r i b u t i o n s as i s the case for cracks. The mixed-mode specimens are shown i n Figures 5.16 to 5.18. Since X has been shown not to be overly s e n s i t i v e to E :E , E :G, •> x y' y ' and v , the same notch configurations can be used to generate xy i n t e r a c t i o n surfaces f o r most species. A general r e l a t i o n s h i p f o r X i n terms of the notch geometry, o r i e n t a t i o n r e l a t i v e to the grain, and the moduli r a t i o s can be approximated with a m u l t i p l e regression. Obviously, the c r i t i c a l s t r e s s - i n t e n s i t y factors are species dependent since 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 are a f f e c t e d by changes In perpendicular to grain and shear strengths. I l l Primary Stress Field Ex.Ey =11.970 Ey:G = 0.920 MUxy = 0.020 J Lambda = 1.500 / t / /£. i j ! ' l 1 \ j Datun 90* Face anfles: 1 61= 150 * 62 = 358.50* \ \ s \ \ I \ \ \ t \ 1 1 — ^ • / 1 1 \ 1 / w\ / 180' I I 1 ! 1 1 A 1 1 1 \ V . Pos. Neg. k 1 1 1 I 1 0 * d / / * 270* Figure 5 . 9 - K j specimen primary singular stress f i e l d f o r X = 1 . 5 0 0 Secondary Stress Field 90* Face angles: Ex.Ey = 11.970 Ev:G = 0.920 MUxy = 0.020 61= 1.50 * 62 = 358.50' Lambda= 1.518 X / Datun / / 1 \ 1 \ I t / 1 / 1 / / / / / / / • 180* 1 1 1 1 • > , . 1 1 1 0 * X, , i \ / V / \ / \ » i i i t * / \ \ J.) / / / y / / Pos. Neg. 270* Figure 5 . 1 0 - KJJ specimen secondary singular stress f i e l d f o r X = 1 . 5 1 8 112 Primary Stress Field Ex-.Ey =11.970 Ey:C = 0.920 MUxy = 0.020 Lambda = 1.518 ^ t f t *r 1 S 1 / — 1 / 1 I • (1 Datum 80* Face angles: e i = 20.00 • 62 = 340.00* \ _^^*^^ \ \ ^^^^ \ \ \ \ * 1 _ ^ ^ * ^ 1 1 __^ *^"^  1 J — 1 / X 1 / _^^*^^ 180* • 1 1 1 A <*\ i < Pos. Neg. i i i i i o* ./ \ ^^^^ » ^"^^^ • ^*^^^ ' ^^*^^ <• 270* F i g u r e 5.11 - K j specimen primary s i n g u l a r s t r e s s f i e l d f o r X = 1.518 Primary Stress Field Ex:Ey = 11.970 ^ Ey.C = 0.920 ^ MUxy = 0.020 S Lambda = 1.518 / ~ \ \ / Datum 90* Face angles: 61= -38.50* 62= 218.50* / " \ X / ' " \ / ' 180* 1 1 1 1 1 \ \ l * Pos. N e e-V 1/V'1 1 1 1 1 o* 270* F i g u r e 5.12 - Mixed mode V-notch c o n f i g u r a t i o n , X = 1.518 K T t K T T = 0.3959 Primary Stress Field Ex:Ey = 11.970 Ey:G = 0.920 MUxy = 0.020 , ' ' Lambdas 1.518 ' ~ i i i i — » / i / i / \ ,* / \ /' f ~ \ ' / \ \ [ Datum 90* Face angles: 61 = 0.00 * 62= 307.00* S \ \ » \ 1 \ / / \ / / A ' / 180' 1 1 1 1 \ 1 \ I Pos. Neg. 270* \ Figure 5 . 1 3 - Mixed mode V-notch configuration, X = 1 . 5 1 8 K T : K T T = 1 . 6 4 3 2 Primary Stress Field Ex:Ey = 11.970 Ey:G = 0.920 ~ MUxy = 0.020 / ' t Lambda = 1.518 / ~ / / \ ( Datum 90* Face angles: 61 = -9.50 * " \ 62 = 279.50* » 180* 1 1 1 1 I \ l Pos. Neg. r~ |i" i i i i i o * 270*1 Figure 5 » l 4 - Mixed mode V-notch configuration, X = 1 . 5 1 8 K T s K T T = 0 . 9 8 1 8 -GRAIN-V Figure 5.16 - Specimen f o r KJMJJ ~ 1.6432 -GRAIN-< 51.5* .51.5* > Figure 5-17 - Specimen f o r K j i K X I = O.3959 - — G R A I N — t -^ Vf 1 Figure 5.18 - Specimen f o r K T:K T T = 0.9818 115 CONCLUSIONS A f i n i t e element program u t i l i z i n g s i n g u l a r i t y enriched f i n i t e elements has been shown to be an e f f i c i e n t and r e l i a b l e t o o l f o r c a l c u l a t i n g s t r e s s - i n t e n s i t y factors f o r V-notches i n an orthot r o p i c plate. Tests of rectangular end-notched wood beams showed that the c r i t i c a l s t r e s s - i n t e n s i t y factor can be used to predict the onset of rap i d crack propagation. A procedure to develop an i n t e r a c t i o n surface f o r p r e d i c t i n g r a p i d crack propagation i n orthotropic materials encompassing both V-notches and cracks has been presented. Such a surface used i n conjunction with the program w i l l allow designers to determine the strength reducing e f f e c t s of any notch i n an orthotropic plate. The timber design code i s an i d e a l candidate f o r adopting such an approach. Future research should include performing experiments to determine c r i t i c a l s t r e s s - i n t e n s i t y factors f o r V-notches. In wood, studies i n the area of s u b c r i t i c a l crack growth emanating from notches should be undertaken. 116 BIBLIOGRAPHY ASTM-D245-81. "Standard Methods for Establishing Structural Grades and Related Allowable Properties for Visually Graded Lumber." American  Society for Testing Materials, (1985) Sec. 4, Construction: Vol. 04.09 Wood. 135-158. ASTM-D2061-74. "Standard Test Methods for Moisture Content of Wood." American Society for Testing Materials, (1985) Sec. 4, Construction: Vol. 04.09 Wood. 431-44. ASTM-D2555-81. "Standard Methods for Establishing Clear Wood Strength Values." American Society for Testing Materials (1985) Sec. 4, Construction; Vol. 04.09 Wood. 511-534. ASTM-E399-B3. "Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials." American Society for Testing Materials (1985) Sec. 3, Metals Test Methods and Analytical Procedures; Vol. 03.01, Metals - Mechanical Testing; Elevated and Low Temperature Tests. 547-582. Barrett, J.D. "Fracture Mechanics and the Design of Wood Structures." Philosophic Transactions of the Royal Society, London Series A, 299 (1981) 17-226. Barrett, J.D. "Effect of Crack-Front Width on Fracture Toughness of Douglas-Fir." Engineering Fracture Mechanics 8 (1976), 711-717. Barrett, J.D. and R.0. Foschi. "Shear Strength of Uniformly Loaded Dimension Lumber." Canadian Journal of C i v i l Engineering 4(1) (1977) 86-95. Barrett, J.D. and R.0. Foschi. "Mode II Stress-Intensity Factors for Cracked Wood Beams." Engineering Fracture Mechanics 9 (1977) 371-378. Bathe, Klaus-Jurgen. Finite Element Procedurs in Engineering Analysis, New Jersey: Prentice-Hall, 1982. Benzley, S.E. "Representation of Singularities with Isoparametric Finite Elements." International Journal for Numerical Methods in  Engineering 8 (1974) 537-545. Broek, David. Elementary Engineering Fracture Mechanics 3rd ed., The Hague: Martinius Nijhoff Publishers, 1982. CAN3-086-M84. Engineering Design in Wood (Working Stress Design). Canadian Standards Association. 1984. Cook, Robert D. Concepts and Applications of Finite Element Analysis, 2nd ed., Toronto: John Wiley & Sons, 1981. 117 BIBLIOGRAPHY - cont'd Erdogan, F. "Stress Intensity Factors." Journal of Applied Mechanics,  Transactions of the American Society of Mechanical Engineers 50 (December 1983) 992-1002. Foschi, Ricardo 0. and J.D. Barrett. "Stress Intensity Factors i n Anisotropic Plates Using Singuklar Isoparametric Elements." International Journal f o r Numerical Methods i n Engineering 10 (1976) 1281-1287. Foschi, Ricardo 0., J.D. Barrett and Kenneth Lau. "FEM: Two-Dimensional F i n i t e Element Program, User's Manual and Input In s t r u c t i o n s " , Western Forest Products Laboratory, Can. Forestry Service, Vancouver, B.C. (1976). Gandhi, K.R. "Analysis of an Inclined Crack C e n t r a l l y Placed i n an Orthotropic Rectangular Plate." Journal of S t r a i n Analysis 3(3) (1972) 157-162. Gerhardt, Terry D. "A Hybrid/Finite Element Approach for Stress Analysis of Notched Ani s o t r o p i c Materials." Journal of Applied Mechanics,  ASME 51 (December, 1984) 804-810. Gerhardt, Terry D. Strength and S t i f f n e s s Analysis of Notched, Green Oak  P a l l e t S t r i n g e r s . United States Department of A g r i c u l t u r e , Forest Service, Forest Product Laboratory, Research paper FPL 452, December 1984. G i f f o r d J r . , L. Nash and Peter D. H i l t o n . "Stress Intensity Factors by Enriched F i n i t e Elements." Engineering Fracture Mechanics 10 (1978) 485-496. Gordon, J.E. The New Science of Strong Materials or Why You Don't F a l l  Through the Floor, 2nd ed., Markham, Ontario: Penguin Books, 1976. Gross, B. and A. Mendelson. "Plane E l a s t o s t a t i c Analysis of V-Notched Plates." International Journal of Fracture Mechanics 8 (1972) 267-276. Hellan, Kare. Introduction to Fracture Mechanics, Montreal: McGraw-Hill, 1984. Haymann, F.J. "A Review of the Use of Isoparametric F i n i t e Elements f o r Fracture Mechanics." Engineering Application of Fracture A n a l y s i s . Procedings of the F i r s t National Conference on Fracture Held In Johannesburg, South A f r i c a , 7-9 Nov. 1979. Ed. G.G. Garrett and D.L. M a r r i o t t . Toronto: Pergamon Press, 1980. Hoenig, Alan. "Near-Tip Behaviour of a Crack i n a Plane A n i s o t r o p i c E l a s t i c Body." Engineering Fracture Mechanics 16(3) (1982) 393-403. 118 BIBLIOGRAPHY - cont'd Le i c e s t e r , R.H. "Fracture Strength of Wood." F i r s t A u s t r a l i a n Conference of Engineering Materials, The U n i v e r s i t y of New South Wales, 1974. Lei c e s t e r , R.H. and P.F. Walsh. "Numerical Analysis f o r Notches or Arbi t r a r y Notch Angle". Proceedings of the International Conference  on Fracture Mechanics Technology Applied to Mat e r i a l Evaluation and  Structure Design. Melbourne, A u s t r a l i a , August 1982. Leice s t e r , R.H. "Some Aspects of Stress F i e l d s at Sharp Notches i n Orthotropic M a t e r i a l s . " D i v i s i o n of Forest Product Technology Paper No. 57. Commonwealth S c i e n t i f i c and I n d u s t r i a l Research Organization, A u s t r a l i a , 1971. L i n , K.Y. and Pin Tong. "Singular F i n i t e Elements f o r the Fracture Analysis of V-Notched P l a t e . " International Journal f o r Numerical  Methods i n Engineering 15 (1980) 1343-1354. L i u , H.W. "On the Fundamental Basis of Fracture Mechanics." Engineering  Fracture Mechanics 17(5) (1983) 425-438. Lum, Conroy. "Two-Dimensional S i n g u l a r i t y Enriched F i n i t e Element Program, User's Manual and Input Instructions Version 2.06." January 1986. Mall, S., Joseph F. Murphy and James E. Shottafer. " C r i t e r i o n for Mixed Mode Fracture i n Wood." Journal of Engineering Mechanics 19(3) (1983) 680-690. Palka, L.C. and B. Holmes. "Tangential F a i l u r e of Small Wood Cantilevered Beams with Square Notches." Wood Science 5(3) (1973) 172-180. Savin, S.G. Stress Concentrations Around Holes. Trans, from Russian by E. Gros. New York: Pergamon Press, 1961. Schniewind, A.P. and R.A. Pozniak. "On the Fracture Toughness of Douglas F i r Wood." Engineering Fracture Mechanics 2 (1971) 223-233. Sih, G.C., P.C. Paris and G.R. Irwin. "On Cracks i n R e c t i l i n e a r l y Anisotropic Bodies." International Journal of Fracture Mechanics 1(3) (1965) 189-203. Stamm, A l f r e d J . Wood and Cel l u l o s e Science, New Yrok: The Ronald Press Co., 1964. Stieda, C.K.A. "Stress Concentrations Around Holes and Notches and Their E f f e c t on Strength of Wood Beams." Journal of Materials 1(3) (1966) 560-582. 119 BIBLIOGRAPHY - cont'd Stieda, C.K.A. "Stress Concentrations in Notched Timber Beams." Transactions of the Engineering Institute of Canada 7(A-5) October 1964. Tlmoshenko, S.P. and J.N. Goodier. Theory of Ela s t i c i t y , Toronto: McGraw-Hill Book Company, 1934. Tong, Pin and T.H.H. Plan. "On the Convergence of the Finite Element Method for Problems with Singularity." International Journal of  Solids and Structures 9 (1973) 313-321. Tong, Pin and T.H.H. Plan. "The Convergence of Finite Element Method i n Solving Linear Elas t i c Problems." International Journal of Solids  and STructures 3 (1967) 865-879. Walsh, P.F. "Linear Fracture Mechanics in Orthotropic Materials." Engineering Fracture Mechanics 4 (1972) 533-541. Wu, E.M. "Application of Fracture Mechanics to Anisotropic Plates." Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 34(4) Series E., December 1967. 120 APPENDIX I Shape Functions for Plane Quadratic Element With the nodes numbered as in Figure A-I.l, the shape functions can be expressed as: » 1 8 ' -d - o < :i - n) (1 + C + Tl) / 4 N 2 : (1 > o [i - Tl) / 2 N 3 : • -d " o a - n) (1 - S + n) / 4 N 4 : (1 + o ci - Tl) / 2 N 5 5 • -(1 + o (i + Tl) (1 - 5 " n) / 4 (1 - o (i + n) / 2 N 7 : • - d " a u + n) (1 + 5 - n) / 4 N 8 : (1 - o u - n) / 2 Figure A-I.l 121 APPENDIX - II  Experimental Results ™ - •- — -------- —------— —= = ========• ======== ========== Spec. HOE-flat Width Depth Notch P.q P.iax P.q : Prop. MCI Density Angle to Nuiber (10 A6) psi (••) (••) Depth (••) Load (lb) Load (lb) P.iax (deg.) (g/c§A3) TL (deg) 1A 1.056 39.00 88.60 50.0 220.0 291.0 » 1.3227 2.0 B.5 0.388 85.0 IC 1.233 38.90 89.55 50.0 275.0 292.0 1.0618 0.5 8.5 0.396 85.0 2A 0.965 38.55 B9.35 50.0 670.0 677.0 1.0104 0.0 10.5 0.621 45.5 2C 1.269 38.80 89.45 50.0 380.0 387.0 1.0184 2.5 10.0 0.507 35.0 3A 1.356 39.10 88.70 10.0 2670.0 0.0 12.5 0.461 B5.0 3C 1.458 38.95 89.05 50.0 367.0 390.0 1.0627 0.0 9.0 0.453 80.0 4A 1.326 38.70 89.75 10.0 2100.0 0.0 8.5 0.426 60.0 4C 1.226 38.85 89.50 10.0 1600.0 1.0 8.5 0.396 80.0 5A 0.834 38.65 89.70 50.0 450.0 450.0 1.0000 4.5 9.0 0.442 50.0 5C 0.641 38.65 10.0 2940.0 25.0 8.5 60.0 6A 1.589 38.70 89.20 50.0 526.0 570.0 1.0837 3.0 10.5 0.540 65.0 6C 1.256 38.70 88.35 10.0 2925.0 0.5 14.5 0.53B 55.0 7A 1.239 39.10 89.00 10.0 2370.0 * -. 0.5 13.0 0.451 70.0 7C 1.124 38.40 89.70 50.0 390.0 420.0 1.0769 0.5 15.5 0.465 B5.0 BA 1.245 38.50 88.95 10.0 2580.0 * -. 0.5 19.0 0.462 55.0 BC 1.279 38.00 88.85 50.0 370.0 442.0 * 1.1946 0.5 20.0 0.463 60.0 9A 1.193 38.45 88.55 50.0 430.0 434.0 1.0093 2.0 15.5 0.471 60.0 9C 1.459 38.50 87.80 10.0 2150.0 » -. 3.5 18.0 0.49B 90.0 10A 1.437 38.65 89.95 10.0 3000.0 0.0 10.0 0.474 40.0 IOC 1.356 3B.40 89.00 50.0 330.0 502.0 * 1.5212 0.0 15.0 0.471 45.0 lift 1.180 38.45 89.50 10.0 2900.0 ™" * — — — 1.0 12.0 0.511 85.0 11C 1.318 38.45 89.45 50.0 452.0 595.0 * 1.3164 0.0 14.5 0.540 90.0 12A 1.070 38.35 87.45 50.0 365.0 425.0 * 1.1644 0.0 25.0 0.468 85.0 12C 0.926 38.60 87.80 10.0 1830.0 • —~ ~" "* 5.0 22.0 0.438 80.0 13A 1.009 38.50 89.60 10.0 2600.0 0.0 10.0 0.403 40.0 13C 1.149 38.45 B7.15 10.0 2B75.0 0.0 9.5 0.441 10.0 14A 1.242 38.30 89.25 50.0 405.0 432.0 1.0667 5.0 16.0 0.502 65.0 14C 1.209 3B.55 88.80 50.0 280.0 400.0 * 1.4286 1.0 19.0 0.480 90.0 15A 1.327 38.75 89.50 50.0 395.0 395.0 1.0000 2.0 9.5 0.394 60.0 15C 1.276 38.65 89.55 10.0 2300.0 1.0 B.5 0.397 60.0 16A 1.44B 38.70 89.20 50.0 262.0 322.0 * 1.2290 2.0 10.5 0.468 90.0 16C 1.457 38.50 89.10 10.0 3150.0 * -. 0.0 10.5 0.462 90.0 17A 1.194 38.45 88.35 10.0 2875.0 ~ ~ — * -. 2.5 20.0 0.484 75.0 17C 1.352 38.40 87.50 50.0 315.0 405.0 * 1.2857 2.0 1B.0 0.483 85.0 18A 1.064 38.50 89.50 10.0 2350.0 -0.5 B.5 0.408 75.0 18C 1.246 38.50 89.30 50.0 430.0 430.0 1.0000 2.5 9.0 0.419 75.0 19A 1.172 38.55 88.65 50.0 485.0 595.0 t 1.2268 2.0 22.0 0.487 55.0 19C 1.332 38.30 88.60 10.0 2575.0 * -. 0.5 1B.0 0.486 45.0 20A 1.294 38.70 B9.30 10.0 2525.0 3.5 10.5 0.533 75.0 20C 1.123 38.60 89.35 10.0 2820.0 1.5 13.5 0.526 70.0 (t) - rejected either because P„:P..» > 1.10 (or excessive s u b c r i t i c a l crack growth) <**) - resistance type i o i s t u r e i e t e r reading at root teiperature TL - tangential-longitudinal plane; plane p a r a l l e l to the grain and tangential to the growth rings 122 APPENDIX I I I CSA Standard CAN3-086-M84 Design Loads Shear i n rectangular members i s designed according to Sec. 4.5.1. Although the material tested was graded as Standard and Better, the allowable shear s t r e s s i s i d e n t i c a l f o r a l l Hem-Fir grades: 0.50 MPa. For members notched at the lower (tension) face at the support: bd - 1.5 V K„ / F ( A - I I I . l ) m N v b = width; d s depth; V m = e f f e c t i v e shear force; notch factor = [d/(d - <* N)] 2; dN = notch depth < 0.025d; e = length of notch from inner edge of support to i t s f u r t h e s t edge; F V = allowable working stress i n l o n g i t u d i n a l shear. A l l other modification factors (load duration K^, service c o n d i t i o n K c „ , treatment K„, and load sharing K u have been taken to be u n i t y . oV T n The calculated values have been reduced by a fac t o r of 1/4.1 (Sec. 6.2, ASTM-D245-81) which includes an adjustment f o r normal duration of load (accumulation of 10 years of loading to design l e v e l s ) and a fact o r of safety. Also, the beam i s assumed to contain an end crack. If no crack e x i s t s , then the allowable shear stress can be increased by a factor of 2.0 (CAN3-086-M84). Thus, the short term, uncracked 5th per c e n t i l e shear strength i s (2.0)(4.1) =8.2 times the design value. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0062720/manifest

Comment

Related Items