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Fatigue cracking of lumber bandsaw blades Lu, Hongtao 1993

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FATIGUE CRACKING OF LUMBER BANDSAW BLADESByHongtao LuB. E. The Northern Jiao-Tong University, Beijing, P.R. ChinaA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESMECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1993© Hongtao Lu, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of  Mechanical EngineeringThe University of British ColumbiaVancouver, CanadaDate  Ap;L)/^) /P'” DE-6 (2/88)AbstractThe focus of this research is on the fatigue cracking problem in the tooth gullet region ofwide handsaw blades used in the lumber industry. This thesis presents both the theoryand the experiment aspects of this research in terms of the tasks defined as follows.The current literature was reviewed to critically examine the current state-of-the-art in smooth and cracked body analysis techniques and their application in the failureanalysis of a bandsaw blade. The advantage of the fracture mechanics approach insolving the problem was recognized and useful information was identified. Experimentaltesting was conducted to obtain the mechanical properties of the blade material and toestablish the cracking resistance curve which defines the stability threshold of cracks inthis material. Experimental testing was also conducted to determine crack propagationbehaviour in handsaw blade material subjected to representative loading conditions.Based on the experiment data obtained, a failure analysis of a blade under assumedloading conditions was performed. Critical crack lengths which a blade can tolerate underthe assumed loads were calculated. A fatigue crack propagation model has been proposedand used in the failure analysis of commonly used blades. The results offer explanationsfor cracking problem in the tooth gullet.The failure analysis also shows that careful handling of the blade resharpening processcan substantially prolong the fatigue life of a saw blade.Table of ContentsAbstractList of Tables^ viList of Figures^ viiNomenclatureAcknowledgement^ xiii1 Introduction 12 Literature Review 82.1 Introduction ^ 82.2 Component loads on the bandsaw blade ^ 92.3 Simplification of the loads ^ 122.4 Stress concentration factor at the toothed edge of the blade ^ 142.5 An approximation of stress concentration factor for a blade ^ 162.6 Two approaches to the fatigue cracking problem ^ 193 Material Property Determination—Strength and Fracture Resistance 253.1 Introduction ^ 253.2 Bandsaw material composition ^ 263.3 Strength properties determination 263.4 R-curve experimental tests ^ 273.4.1^Specimens and material ^ 323.4.2^Cracking opening devices and fixture ^ 333.4.3^Double-calibration method ^ 353.4.4^Establishment of the calibration relations ^ 373.4.5^R-curve Tests ^ 443.4.6^Experimental results ^ 453.4.7^Critical stress intensity factor assessment ^ 484 Crack Initiation Tests 534.1 Introduction ^ 534.2 Experiments 554.2.1^Specimen, load and equipment ^0 554.2.2^Test, results and analysis 595 Mode I Crack Growth Rate Test 645.1 Introduction ^ 645.2 Mode I crack growth rate experiment ^ 675.2.1^Specimen and equipment 675.2.2^Tests and data analysis ^ 696 Crack Propagation Test for Out-of-Plane Bending 766.1 Introduction ^ 766.2 Experiment 786.2.1^Basic considerations ^ 786.2.2^Specimens, equipment and devices ^ 786.2.3^Test results and analysis ^ 791 V7 Fatigue-Life Prediction of a Bandsaw Blade^ 887.1 Introduction  ^887.2 Critical crack length of a bandsaw blade  ^907.3 Propagation life of a bandsaw blade  ^917.3.1 Fatigue loads experienced by a blade  ^917.3.2 Analysis of growth life for a blade  ^958 Conclusions^ 116Bibliography^ 120Appendix A^ 123A Calculations in the Failure Analysis^ 123A.1 Bandsaw parameters used in the calculations ^  123A.2 K-formula used in the calculations ^  125A.3 Calculations of critical crack length  132A.4 Residual stress consideration ^  133A.5 Calculations of crack growth life  133List of Tables3.1 The chemical composition of saw steel ^ 264.2 Load parameters chosen ^ 584.3 The crack initiation test results ^ 617.4 The assessment of critical crack length ^ 907.5 Crack-growth life prediction in a bandsaw blade (A) ^ 1007.6 Crack-growth life prediction in a bandsaw blade (B) 101A.7 The calculated loads ^ 124A.8 The crack growth calculation for the out-of-plane bending testing^.^. 135viList of Figures1.1 Bandsaw geometry  ^21.2 Change of load on the blade  ^31.3 Cracks in the blade  ^52.4 Stress in the blade: Eschler's work [6]  ^132.5 Stress concentration at the gullet region: Jones' work [4]  ^152.6 SCF at a elliptical notch  ^183.7 Orientation of the specimens in the blade  ^283.8 The geometry of a specimen for a strength test ^  293.9 Test results of strength for 9 specimens  ^303.10 Critical condition in load control case  ^313.11 Critical condition in displacement control case  ^323.12 Geometry of CLWL specimen ^  333.13 The loading system for CLWL specimen ^  343.14 Loading wedge and dies ^  363.15 locations and orientations of strains  ^383.16 Fixture, specimen and sensor devices for experimental calibration^423.17 FEM vs experiment—ei/e3 calibration ^  433.18 FEM vs experiment—EBWe3IP calibration  443.19 Fractomat crack monitoring device and Krak-gage ^  463.20 The test set-up for R-curve test  ^473.21 Comparison between the crack lengths from different methods ^ 49vii3.22 Graphical assessment of critical stress intensity factor  ^504.23 Resharpening process—stone wheel and tooth ^  544.24 Close-up of a typical scratch produced by grinding process  ^554.25 Crack initiation life specimen  ^564.26 Photo of the surface roughness of the specimen  ^564.27 Fatigue testing machine working mechanism ^  604.28 The crack initiation test results  ^625.29 Three Modes of cracks ^  655.30 Definition of stress intensity range  ^665.31 The specimen and fixtures for crack growth rate test  ^685.32 The calibration relation for CT specimen ^  695.33 A typical crack length versus cycle record  ^715.34 The crack growth rate: R = 0.4  ^715.35 The crack growth rate: R = 0.295  ^725.36 The crack growth rate: R = 0.243  ^725.37 The crack growth rate: R = 0.2  ^735.38 Comparison of crack growth rates  ^736.39 Schematic relation of crack surfaces interacting: saw blade and crackedgullet position  ^776.40 Geometry of the specimen for out-of-plan bending load  ^796.41 A four-point bending fixture  ^806.42 A test set-up for out-of-plane bending test  ^806.43 Crack growth: R=0.313 ^  816.44 Crack growth: R=0.227  816.45 Crack growth: R=0 ^  826.46 Crack growth: R=-1  826.47 Crack growth: R=-0.373  ^836.48 Crack growth: R=-0.49 ^  836.49 Comparison of different crack growth rate ^  846.50 A typical crack front profile  ^856.51 The trace of crack advance ^  867.52 Transition of crack growth ^  947.53 Definition of a notch field  ^957.54 Schematics of crack growth in a blade ^  977.55 Crack-growth life: as a corner crack  1027.56 Crack-growth life: as a through-thickness crack ^  1037.57 Total crack-growth life: no residual stress ^  1047.58 Total crack-growth life: 10 ksi. residual stress  1057.59 Crack-growth life distribution (%): no residual stress ^ 1067.60 Crack-growth life distribution (%): 10 ksi. residual stress  1077.61 Total fatigue life of a blade: fine surface ^  1087.62 Total fatigue life of a blade: rough surface  1097.63 Total fatigue life of a blade: surface roughness effect ^  110A.64 Surface crack in a finite plate ^  127A.65 Surface-cracked plate subjected to tension or bending loads ^ 128A.66 A long plate with a crack subjected to bending ^  131ixNomenclatureThe symbols are listed in the order of their appearance.a = generic symbol for stress.o-T = nominal average stress produced in a blade by tensile load.F = pre-strain force used in a bandsaw machine.w = effective blade width.t = the blade thickness.o-B = maximum stress induced by bending.E = Young's modulus of elasticity.bandsaw machine wheel radius.0- cent — stress produced by centrifugal force.p = density of bandsaw steel.v = velocity of bandsaw blade.= plane strain fracture toughness.AK = stress intensity factor range.kT = stress concentration factor for pure tensile load.kB = stress concentration factor for out-of-plane bending.kT+B = stress concentration factor for the combined load of tension and bending,reference stress is (UT o-B).el = strain at location 1 on the Crack-Line-Wedge-Loaded specimen.e3 = strain at location 3 on the Crack-Line-Wedge-Loaded specimen.a = generic symbol for crack length .B = thickness of the Crack-Line-Wedge-Loaded specimen.W = width of Crack-Line-Wedge-Loaded specimen.P = symmetrical force acting at point A and B on the Crack-Line-Wedge-Loadedsepcimen.C2 = constants.K = generic symbol for stress intensity factor (SIF) at the crack tip, or the so-calledcrack driving force.Kc = critical stress intensity factor (plane stress condition).= the so-called crack growth resistance, material property with same dimension asK.dK/da = the change rate of the driving force with respect to crack growth da.dRrlda = the change rate of the resistance with respect to crack growth da.ao , a, = initial crack length.Aa = physical crack growth at crack tip.ry = plastic zone adjustment to Aa.ay = 2% offset yield strength.N = number of fatigue load cycles.daldN = crack propagation (or growth) rate.R = stress ratio (Or mzn I Crtnax)-= strain rate.T = ambient temperature.C,b = material constants.an = notch depth.pn = notch bottom curvature radius.U = elastic energy contained in the body.Fe = work performed by external force system.xiEp = energy consumed by crack propagation.G = d(Fe — U)Ida = elastic energy release rate.R,. = dEpl da = crack growth resistance force.GIG — critical energy release rate, a material property.v = Possion's ratio.KR = crack resistance force in terms of stress intensity factor.AcknowledgementI would like to thank everyone who helped during this research program. In particular,I would like to extend my appreciation to my supervisor, Dr.D.P. Romilly, for his val-ued advice, guidance and patience; Mr.Leonard Drakes, for his valued work in makingthe experiment equipment; and Mr.Robert Connop, for his valued assistance in makingspecial specimens.I would like to express my special thanks to Mr.John Crowden for spending a lot oftime reading the manuscript of this thesis and making valued comments.Finally, I would also like to thank my wife for her constant support and patience.Chapter 1IntroductionThe bandsaw is a wood cutting machine which consists of two large wheels of 5-8 ft.(1.5-2.4m) in diameter and one closed-loop blade (Fig.1.1). The blade used on the bandsawmachine is a toothed closed-loop strip of high-strength alloy steel, which is driven byone of the two wheels. The so-called cutting region on the blade is a straight section ofthe blade supported between two guides (D-D'in Fig.1.1). The cutting process is carriedout by feeding logs laterally into the downward moving cutting edge. Because of its longand straight cutting region and its high edge stability, bandsaws are especially suitablefor deep cuts. Therefore bandsaws of various blade sizes and types are widely employedin the primary and secondary breakdown of logs. The high edge stability of bandsawblades are a result of the high stiffness of the cutting edge, which is achieved directly byapplying a tensile axial pre-load into the blade. This pre-load is imposed by a loadingsystem (Fig.1.1) that lifts the top wheel of the machine to stretch the blade. As thepre-load increases, the stiffness of the cutting region of the blade increases. Therefore itis desirable to increase the pre-load as much as possible to obtain better cutting accuracywhich strongly depends on the edge stability of the blade. The recently introduced 'high-strain system' is based on this idea. However, practical experience shows that increasingpre-load is limited by the increasing probability of the onset of cracks in the blade, ascracking problems occur much more often in high-strain systems. These cracks usuallyoccur in the tooth gullet or in the mid-section of the blade.However, the pre-load is not the only load acting on the blade. There are others, the1Chapter 1. Introduction^ 2Figure 1.1: Bandsaw geometryChapter 1. Introduction^ 3LEGEND: ^ STRESS IN THE OUTER FIBRE- - - STRESS IN THE INNER FIBRE BENDING STRESS—1 [ - -1 PRE-STRAIN FORCE STRESSA^Bd^C^B' A^LOCATIONIL _ _ _J^L _ _ _JONE REVOLUTIONFigure 1.2: Change of load on the blademost prominent one is the bending load. The bending load is suddenly induced on thefast moving blade when it comes into contact with the back of the wheel (at points Aor C' in Fig.1.1). When the blade is in motion and travelling through the arcs, ABC,C'B'A', the blade is subjected to a combined load, i.e. a bending load in addition to thepre-load. The bending load disappears when the blade leaves the back of the wheel (atpoints C or A' in Fig.1.1). The Fig.1.2 shows the schematic change of the regular loadon the blade.In addition, there is also a cutting force, a centrifugal force and possibly some ac-cidental loads. Therefore the blade actually experiences a dynamic fatigue loading. Interms of stress, the tensile axial stress caused by the pre-load is typically about one-thirdof the maximum stress, while magnitudes of the stress in the blade caused by cutting,Chapter 1. Introduction^ 4and centrifugal loads are only about 6% of the total stress [2]. The accidental loads (e.g.those caused by improper tilting or by objects caught between the blade and the back ofwheels) may induce a very high stress in the blade.One important 'load' that should be mentioned is the residual (or built-in) stressinduced by the blade during the manufacturing or reconditioning process, (such as rolling,tensioning, and levelling etc.). Stresses of this nature are hard to quantify and have notbeen generalized mathematically. Residual stresses can be either superimposed onto thestresses caused by the external loads or they may act to cancel or reduce externallyinduced stress. Thus residual stresses can greatly influence the cracking behaviour of theblade.There are two types of cracks often observed in handsaw blades after some period oftime in service (Fig.1.3). One is the so-called gullet crack, which is the most commonand therefore the focus of this research program. It is typically located at the bottomof the tooth gullet with a length varying from 1/16 in. (1.6mm), i.e. just visible, to 1.5in.(40mm) maximum. Another type is the center or quarter crack which consists of twocrack tips and is contained within the blade section. These are thought to be causedby local overloading induced by wood chips traped between the blade and the back ofthe wheel. The probability of the center/quarter cracks can be reduced by keeping thecutting area clean.After a gullet crack starts, it is driven by the cyclic loads and will continue to growslowly at first, then propagate quickly into the blade up to some critical length beyondwhich the crack will extend in a catastrophic manner, i.e. the blade will be separatedcompletely in the final failure. For a center or quarter crack, extension occurs from bothtips of the crack into the blade until the critical crack length is reached.Unlike problems which occur in the operation of a bandsaw system, which generallyresult in poor quality products (e.g. vibration of the blades), cracking of a blade alsoChapter 1. Introduction 5Figure 1.3: Cracks in the bladeChapter 1. Introduction^ 6produces damage to the blade itself. Furthermore, growth of a small crack can result ina catastrophic failure of the blade, which means not only a high operational costs (bladerepair cost, and downtime etc.) but a concern for human safety. The use of bandsawsystems has increased, especially in recent years since the 'high-strain system' methodwas introduced to increase cutting accuracy. Cracking problems have thus become moreevident. This has generated the need for a more specific qualitative and quantitativeunderstanding of cracking behaviour in bandsaw blades.The cracking problem is, to a large extent, beyond the traditional strength theoryand failure analysis, both of which are based on a smooth body assumption, i.e. theconventional strength theory can not explain why a component (like a saw blade) canfail under a nominal stress far below the yield strength of the material (less than 1/3of the yield strength). Conventional fatigue theory can not predict how a crack growsafter initiation and the critical crack length for this growth. Fracture mechanics, aconcept which has been developed since the early sixties, offers a consistent framework forunderstanding the cracking phenomena. Fracture mechanics theory is based on a crackedbody assumption. A new parameter, the stress intensity factor (SIF for abbreviationor denoted as K), is employed to characterize the stress field around the crack tip. Anassociated material property, the fracture toughness, or its counterpart in the case ofthin sheet material, the crack growth resistance curve (R-curve) is utilized in afracture analysis. From a fracture mechanics point of view, final failure of a componentis considered as the ultimate consequence of continuous growth of a crack which initiallymay be a micro crack. This growth process is first stable and then becomes unstable at acritical point, the stable part being represented by a crack growth curve. The fracturetoughness (or R-curve) defines a critical point which is the boundary between stable andunstable crack growth. Therefore if the information concerning fracture properties for amaterial available, crack growth behaviour in a component made of that material can beChapter 1. Introduction^ 7predicted.The primary effort of this investigation is to employ fracture mechanics concepts incooperation with experimentally obtained fracture properties of saw blade material toexplain and predict the nature and behaviour of gullet cracks in the blade. Therefore thefollowing are the objectives of this research program:1. To review the current literature to critically examine the state-of-the-art in smoothand cracked body analysis techniques and information related to determining band-saw blade integrity.2. To identify the major loading components responsible for gullet cracking.3. To perform experimental testing to establish the resistance curve and define thestability threshold of cracks in typically employed saw material.4. To perform experimental testing to determine the crack propagation behaviour inbandsaw blade material subjected to representative loading conditions.5. To provide recommendations for the development of improved prediction techniquesrelated to blade integrity.Chapter 2Literature Review2.1 IntroductionBandsaw blade cracking problems came under examination in the wood production indus-try in the early 70's. Up to the present, investigators have suggested some possible causesfor the problems, and some operational measures to prevent cracks from happening havebeen investigated. In the previous studies, the typical loads on the bandsaw blade wereidentified, some initial physical models were put forward, stress analysis was attemptedusing elasticity theory, and local stress distributions as well as stress concentration fac-tors at the tooth gullet were investigated by both experimental and numerical modellingtechniques. only limited basic mechanical properties of the material have been obtainedby some researchers. Some researchers made life assessments for the blades, based on thesmooth body design philosophy. However, the analysed or predicted bandsaw life weremuch longer than that found in practical service. Two factors could be responsible forthe substantial difference between the theoretical prediction and real service life. First,there is a lack of specific information on saw blade material as manufacturers considerthe material properties to be confidential. Second, and more fundamentally, there is theinherent shortcoming in the analysis based on the smooth body design philosophy. Thenew methods and techniques of analysis based on fracture mechanics can be consideredas a more promising approach to understanding, both qualitatively and quantitatively,the nature of the cracking. To utilize these new techniques, it is essential that some basic8Chapter 2. Literature Review^ 9experimental work be carried out to obtain the important material properties related tothe fracture characteristics of the material.2.2 Component loads on the bandsaw bladeSome work has been done by previous investigators [1] [2] [6], to identify every type ofload acting on the blade during its service. The following is a brief description of theresults. The important point is that different loads are acting during different periodsof time in each revolution. To indicate when (or at what position) each individual loadacts, Fig.1.1 Chapter 1 can be referred to, with the label A, B,... etc. indicating thecorresponding positions.Pre-strain LoadIn order to transmit the power of the motor into cutting force, as well as to obtainhigh edge stability to ensure a high cutting accuracy, a pre-load, or the so-called 'pre-strain force' F is applied to the top wheel through its shaft so that a proper contactforce between the circumference of the wheel and the bandsaw blade can be achieved.This tensile loading is one of the dominant loads on the blade and remains essentiallyconstant throughout blade revolution. The tensile stress induced in the entire blade bythe pre-strain loading is calculated as [11]aT 2^tw(2.1)Whereo-T— the nominal average stress produced in the blade.F— the pre-strain force.w— the effective blade width.t— the blade thickness.Chapter 2. Literature Review^ 10Bending loadsBending stresses found in the blade result from bending over the wheel and the supportguides. These are defined below (refer to Fig.1.1 (Chapter 1 ) for specific positions):1. Longitudinal bending caused by the wheel:When the blade comes into contact with the circumference of the wheel (at pointA or C', the stress induced in the outer and inner fiber of the band due to itsbending over the wheel isEtuB = ± 2Rli, (2.2)whereoB— maximum stress induced in outer fiber (tensile) and inner fiber (compression).t— bandsaw thickness.E— Young's modulus of elasticity.Rev— bandsaw machine wheel radius.This stress remains until the blade arrives at point C or A' respectively.2. Longitudinal bending caused by the guides:This component of loading is introduced by the offset of the guide from the unal-tered path of the blade and strongly depends on the contour of the guide surface.Experience shows that the guide loading can be reduced by up to 50% if the guidesare contoured properly. For example, a typical stress of 4-6 ksi (27-40 MPa) canresult from guide displacement in a blade with a thickness of 0.065 in.(1.65mm) ona 5 ft. diameter (1524mm) sawmill [5]. This stress acts locally on the blade sectionin the blade guide region D and D'.Chapter 2. Literature Review^ 113. Transverse bending:This component of loading is caused by the crown-shape of the back of the wheel( given by r in Fig.1.1) as the pre-strain force is applied. It is difficult to quantifythe stress level produced. However, as the curvature is relatively large, the bendingstresses produced are considered to be small compared to other loading components.This loading only occurs in the blade section in contact with the wheel.Cutting forceThe parallel and normal in-plane cutting forces acting upon the teeth are, in general,proportional to kerf width (on the order of 50 lbs.(222 N) per tooth for kerf of 0.15in.(4mm)[5]). These loads are only applied to the blade section within the cutting region(D-D' in Fig.1.1) and depend on the material which is being cut. Lee's study }3} showsthat not only the stress produced by cutting is small but also that the stress field producedby cutting offsets from the tooth gullet. Therefore, the cutting force has a very smallcontribution to the gullet cracking problem when compared to the pre-strain force andbending.Centrifugal loadingThis blade stress results from the centrifugal force experienced by the moving bladearound the wheel, i.e. a centrifugal force is produced when the mass of the blade istravelling on a circular path defined by the wheel circumference, a normal accelerationproduces a centrifugal force which induces blade stress. This blade stress only occursduring blade-wheel contact, and generally is about 4-6% of the bending stress [2, 3} andtherefore is not major factor in blade cracking. The stress caused by centrifugal forcecan be given by [1]:Cr cent = pv2^(2.3)Chapter 2. Literature Review^ 12whereCrcent^stress produced by the centrifugal force.p— density of the saw steel.v— velocity of the blade.Accidental loadingIt should be mentioned that in actual service, bandsaw blades may experience acci-dental loads introduced either through improper use or adverse operating conditions. Forexample, a typical overload condition may result from an object becoming temporarilytrapped between the wheel surface and the blade. This can greatly increases the bladestress level. It is clear that this type of loading happens in a random manner, and itsprobability can be reduced by proper maintenance and operation. No attempt has beenmade to consider this effect on blade integrity in this research.2.3 Simplification of the loadsIn recognition of the relative magnitudes of the various types of applied loads, the loadcondition experienced by a blade have been idealized by considering only the two mostimportant loads in following experimental tests and calculations, these loads are:• the pre-strain force obtained by equation (2.1)• the longitudinal bending caused by the wheel, obtained by equation (2.2)Loads other than the above two are ignored because of either their insignificancein magnitude or their random nature. The static measurements made by Eschler [6](Fig.2.4) can be used to support the validity of this simplification.SOO4---ABLADE OUTSIDE^PRESTRESS SIG0(x)=67.6 N/mm2BLADE INSIDEABC^DEF G H I J K^L^m N•^• •^•^•^I^I^I^I^I I . 200 _(N/mm2)100 _400 -Chapter 2. Literature Review^ 130^2000^4000^6000^8000STRAINGAGE POSITION AROUND THE SAW (mm)Figure 2.4: Stress in the blade: Eschler's work [6]Chapter 2. Literature Review^ 142.4 Stress concentration factor at the toothed edge of the bladeIt is common knowledge in engineering practice that a local stress field will be severelydistorted when a notch is introduced into an originally regular section having an uniformstress field. The magnitude of the stress at a certain part of the edge of the notch willrise sharply to multiples of the uniform stress level. This feature can be described by aratio between maximum stress at the edge of the notch and nominal stress. The ratio iscalled the stress concentration factor (or SCF). In a handsaw blade, hundreds of teeth arepunched out on one or both sides of the strip of raw material, thus a stress concentrationexists at every tooth gullet when the blade is loaded. The very high stress concentrationat the gullet bottom is considered to be one of the many factors which are responsiblefor blade cracking.In 1965, experimental tests on a blade were conducted by Jones [4]. His static pho-toelasticity results show that under a pure tensile load the SCF near the gullet regionranges from 1.5 to 2.5, and reaches its maximum at the bottom of the gullet. Differenttooth profiles have an appreciable effect on the stress concentration (Fig.2.5).In 1988, a finite element analysis was conducted by Lee et al [3] to quantify the localstress distribution near the gullet region. Their results show that under a pure tensileloading, the SCF is about 2.35 for the tooth profile studied. Lee et al also proposedtwo methods to modify the geometry of the gullet region—'shaving' and 'flattening' thegullet region— to reduce the SCF. Approximately 21% reduction was achieved.As indicated in their report, the amount of material around the gullet region availablefor shaving and flattening is very limited and the overall performance of the blade mustbe taken into account since removal of material from the bottom of the gullet weakensthe tooth and reduces its stiffness.Chapter 2. Literature Review^ 15f= Nominal uniform stress at two boundaries.Figure 2.5: Stress concentration at the gullet region: Jones' work [4]Chapter 2. Literature Review^ 162.5 An approximation of stress concentration factor for a bladeIn the previous research work [1, 4, 2, 3], the difference between SCF's under pure tensionand pure bending loads were not mentioned. In some investigations [1, 2], the SCF fora pure tension load was directly used for the assessment of fatigue life of saw blades.However, theoretical research [21] shows that SCF's are very different under pure tensionand pure bending loads even if the geometry of the components are identical. In the caseof the bandsaw blade, in one revolution a blade experiences two different load conditions,a pure tension and a combination of tension and bending. Therefore, a separate treatmentof different SCF's is not only necessary but especially important for crack propagationstudies.The combined SCF at the blade tooth can be approximated by using the SCF's at theedge of an elliptical hole in a infinite plate subjected to tension and cylindrical bendingrespectively. The following outlines the details of this analysis. If it is assumed thatthe maximum stress produced by stress concentration is still within elastic region, thesuperposition of the loading can be applied. Therefore, at the gullet bottom of the toothof a blade, the maximum stress should be identical if calculated in two different ways,i.e.,kTCTT kgClg kT+B(aT aB)^(2.4)Rearrange the equation,kT-Fg^kT(aT UB^ )-"I- kg(^ )^ (2.5)uT crB^GrT cfBWherekT—SCF under pure tension load (reference stress UT).kB—SCF under pure bending load (reference stress o-B).Chapter 2. Literature Review^ 17kT+B—SCF under combination of tension and bending load (reference stress (o-T-I-o-B).o-T—nominal tensile stress produced by pre-strain force.o-B—maximum bending stress produced by the wheel.From the above equation, it can been seen that the SCF for the combined stresscondition depends on both the tension and bending loads. Therefore a conclusion can bemade that Lee and his co-workers' result [3] of combined SCF is only valid for the specificload magnitudes used. For the same wheel radius, the combined SCF should be differentfor the different pre-strain forces which range from 20,000 to 13,000 lb.. Therefore aseparate treatment of the SCF's for tension and bending seems more convenient.The SCF at the edge of an elliptical hole in an infinite plateIn order to approximate kT and kB at the gullet region of a saw blade, the standardformula for SCF at the edge of an elliptical hole in an infinite plate can be used [21] (seeFig.2.6). For a pure tension load, the saw tooth can be approximated by an ellipticalnotch at the edge of a half infinite plate, whose SCF can be obtained by multiplying theSCF for an infinite plate by factor of 1.12, this being the free edge geometry factor whichis usde in stress-intensity-factor (SIF) calculation [9]. For the bending load, this factoris 1 [27].SCF for a elliptical hole ^ tensile loadThe stress concentration factor for an infinite plate with elliptical hole subjected totensile load is following (see Fig.2.6),aTmax — (1 + 2—/) (2.6)aTWhereUT—far field stress level.aTmax—peak stress at edge of the elliptical hole.aT^M^0-T■......\. 0-T.,M-Chapter 2. Literature Review^ 18Figure 2.6: SCF at a elliptical notchChapter 2. Literature Review^ 19/—half major axis.h—half minor axis.SCF for a elliptical hole^bending load= 1 + 2(1 + v)(1 — m)(3 + v)(1 + m)(2.7)WherecrB— peak stress at edge of the elliptical hole.a Bn— maximum nominal stress produced by bending moment.m—(/ — h)/(/ + h). 1 and h are defined as in Fig.2.6Assessment of SCF for a bandsaw bladeIn order to show the effectiveness of equation (2.5), Lee's results were used for com-parison.Consider the standard formula of SCF for and elliptical hole in an infinite platesubjected to far field tension (eqn.(2.6)), an equivalent 1/ h ratio for a blade tooth profilecan be obtained. In Lee's analysis, kT = 2.35. From equation (2.6), kT = (1+2h/01.12 =2.35, the equivalent ratio can be obtained as hil 0.55. By inserting this ratio intoequation (2.7), We have kB = 1.43. Finally, the equation (2.5) can be used for thecombined SCF, i.e. kT+B = 1.68. There is virtually no difference between the aboveresult and Lee's. This example shows that as long as kT for blade is known, which isrelative easy to obtain, kB and kT+B can be easily assessed by equation (2.7) and (2.5)for specific load conditions.2.6 Two approaches to the fatigue cracking problem(TBa BnThere are two basic approaches in both design and failure analysis to deal with crackingproblems. One is the so-called smooth body approach—a conventional method very wellChapter 2. Literature Review^ 20documented in reference [8]. The other is the fracture mechanics approach or crackedbody approach.Smooth body approachThis approach makes no assumptions about crack length in the specimen or compo-nent studied. A very important phase in crack behaviour, i.e. crack propagation, can notbe quantified. The information for this approach is based on the data obtained by testingstandard small polished specimens (e.g. Goodman Diagram and S-N curves). Furthermodifications counting for surface conditions and stress concentration etc. are made byapplying modifying factors. In this approach the Goodman diagram is often used. AGoodman diagram is constructed by plotting stress amplitude versus mean stress for fail-ure at 2 x 106 cycles (equivalent 97.5 hrs for 5.7 Hz. used in this research). Generally thedata for the Goodman diagram is obtained by testing standard small, smooth specimens,and should be modified accounting actual surface conditions and the SCF of the com-ponent studied from which a reduced Goodman diagram can be obtained. The reducedGoodman diagram can be used to predict, with a desired safety factor, maximum stressamplitude the that material can sustain at a certain mean stress without fatigue failure.Ingema's work [1] is a typical treatment of this kind. He modified the Goodmandiagram by considering a) surface condition; b) stress concentration factor; and c) residualstress. Ingemar's approach is based on achieving an infinite life of two million cycles,which is unlikely to be achieved in practice. It is difficulty to use the safety factor heobtained to explain an actual failure in more specific details.Lehmann [5] made use of the Allene's modified Goodman diagram [10] and S-Ndiagram to do a life prediction. He predicted 18.7 hours of blade life in the case considered,which was an improvement on Ingemer's and Allene's work, but is still far longer thanthe actual blade life, which is sometimes less than four hours. Although smooth bodytheory predictions can be made close to the actual life of the blade by including everyChapter 2. Literature Review^ 21factor which is likely to be involved, there are some fundamental questions which can notbe answered using smooth body theory, for example, why in practice, a blade actuallystill can carry on for a period of time after a crack occurs, which means that the bladecan tolerate a certain length of crack. Actual existence of cracks in a working blade raisesanother question, i.e. what is the critical crack size which a working saw blade can tolerateand how long it takes for a crack to reach this critical size. All these questions are beyondsmooth body theory. On the other hand, the fracture mechanics based fatigue theorycan give quantitative answers to these questions and can offer a consistent framework tounderstand and quantify the blade cracking process.Fracture mechanics approachFracture mechanics is based on the assumption of cracked body. The focus is on thebehaviour of a crack in the body or component being examined. The assumption is thatonce a crack is initiated, it will extend due to cyclic fatigue in a stable manner until somecritical crack length is reached beyond which the cracked body will fail catastrophically.Failure is ultimately a consequence of a continuous propagation of an initial smallercrack. In fracture mechanics the cracking process is divided into three phases : a) crackinitiation, i.e. a very short crack occurs ; b) crack propagation. the initial crack willgrow due to some external cyclic loading. (stable growth); c) failure. when the criticallength is reached, the crack will extend in an unstable manner until the component isseparated.The key concept in fracture mechanics is the stress intensity factor (SIF) [9] which isdefined by a combination of characteristic stress, crack length and geometric parameters(for example, an infinite plate with a transverse crack of length a is subjected to auniform stress a at far boundary parallel to the crack, the stress intensity factors atboth its crack tips are K aOra). The stress intensity factor is a parameter whichcharacterizes the stress field at the tip of the crack and is a measure of intensity of theChapter 2. Literature Review^ 22stress field at the crack tip. Based on the concept of a SIF, a material property, fracturetoughness (denoted as Kic), has been proposed for a material to measure its ability toresist cracking. From the fracture toughness the critical crack length or critical load canbe predicted for a specific case.In the case of fatigue propagation of cracks, the stress intensity factor range (denotedas AK) is the proposed parameter, which characterizes the stress field at the crack tipunder cyclic loading and controls the crack growth rate during the propagation stage ofthe cracking process.In 1988, Lehmann [5] tried to apply fracture mechanics to explain cracking in theblade. He predicted the critical crack length and life-time of the blade under assumedconditions. The predicted critical crack length is 0.11 in.(2.8mm) and blade life-timeranges from 2.2 to 23.7 hours depending on the surface roughness. The predictions areencouraging. It is worth mentioning that in his analysis he assumed that the surfaceroughness measurement, center-line-average (CLA), is the average dimension of micro-cracks produced by the re-sharpening process, therefore CLA can be treated as the initialcrack length. The surface condition, therefore, could be well defined and introduced asa quantity into the cracking analysis.The weak point of his analysis, as he himself noted, was the lack of experimentalinformation, especially that related to the cracking process. For example, he employed aplane strain Kw of AISI 4340 steel for the bandsaw blade material and had to use thegeneral material crack growth rate for crack propagation calculations. In fracture me-chanics theory, if a crack length is on the order of the surface roughness, the propagationbehaviour is much more complicated, and crack propagation behaviour may take a verydifferent form from that assumed by Lehmann [9]. These factors prevented his analysisfrom being used to develop more detailed and useful predictions.Chapter 2. Literature Review^ 23Crack resistance curve (R-curve): plane stress conditionPrevious research [9] has shown that within the so-called Linear Elastic Fracture Me-chanics (LEFM) framework, a crack behaves differently under a plane stress conditionfrom that under plane strain. Under a plane strain condition the crack growth resistance( in terms of SIF) is essentially constant, independent of crack length, and can be con-servatively estimated by a single parameter, Kw. In contrast while under a plane stresscondition the crack growth resistance increases as crack length increases. This differenceshows its influence more clearly in the dynamical aspect of critical crack length and crackpropagation. In the case of thin sheet materials, e.g. saw blades (sometimes as thin as0.065 in. (1.65mm) in thickness), a plane stress condition exists. Therefore, the fractureresistance can be described fully only by a crack resistance curve (R-curve) which canbe applied to predict the instability condition of crack growth.In 1961, Krafft et al [13] postulated as a working hypothesis that for a given materialand thickness, there is a unique relationship between the amount of stable crack growthand the stress intensity factor applied to the structure, this relationship being the crackgrowth resistance curve or R-curve. A significant amount of work conducted since thenhas been devoted to the development of experimental methods intended to verify thishypothesis. ASTM E561 standard [14], in which the above hypothesis has been adopted,has been established for determination of a material R-curve.In this research program, in order to understand and quantify the behaviour of gulletcracks in handsaw blades, fracture mechanics theory is employed to guide experimentaltests to obtain various properties regarding cracking behaviour, such as the crack growthresistance curve ,or R-curve, for bandsaw steel, its crack initiation and crack propaga-tion rates, which are not currently available. Once all the above information is available,failure analysis and bandsaw life predictions are made. It is also expected that this inves-tigation can offer more useful information for further studies into the cracking problemChapter 2. Literature Review^ 24in bandsaw blades.Chapter 3Material Property Determination—Strength and Fracture Resistance3.1 IntroductionCurrently, little information is available on the causes and quantitative treatment ofcracking in handsaw blades. Certain data, such as the chemical composition and me-chanical properties of blade steel are considered to be confidential by manufacturers.material properties, such as the critical stress intensity factor or crack growth resistancecurve, according to this author's survey, are not available at all. There have been someindependent investigators who have experimentally obtained some chemical compositions[5] and basic mechanical strength properties [1]. However, in order to collect first-handinformation about the material used in this research, its chemical composition was ob-tained via an independent analysis [7]. A series of tests were also performed to obtainthe yield strength and ultimate strength of the bandsaw material.As discussed in Chapter 2, in the case of thin material, fracture resistance can befully described only by the crack growth resistance curve. For handsaw blade thicknessesof about 0.1 in.(2.5mm), an entire R-curve is necessary to obtain complete informationrelated to its fracture behavior.There are several different methods recommended in ASTM E561 [14] to determinethe R-curve for the material. The displacement control and CLWL (Crack-Line-Wedge-Loaded) specimen method were chosen because of the relative ease of control. In thistesting, a series of three specimens were employed to minimize uncertainty in the results.25Chapter 3. Material Property Determination—Strength and Fracture Resistance^26The chemical composition of the bandsaw blade steelIn PercentageElements C Cu Cr Ni Si MoContents 0.38 0.15 0.16 2 0.23 0.01Elements Mn V Al Ti P SContents 0.35 <0.01 <0.01 <0.01 0.05 0.05Table 3.1: The chemical composition of saw steelThis chapter is devoted to the experimental work used to determine the saw steelstrength and crack growth resistance properties.3.2 Bandsaw material compositionThe chemical composition of the saw blade steel used in this research have been analyzedby an independent institute [7] and are reported in Table 3.1. The composition is, asLehmann noted [5], close to that of AISI 4340 steel.3.3 Strength properties determinationIt has been noted by some researcher that saw steels from different manufacturers aresimilar both in composition arid mechanical properties [5]. In order to have first-handChapter 3. Material Property Determination^Strength and Fracture Resistance^27information regarding the mechanical properties of the blade material used in this re-search, tests on the saw steel were conducted. The following is the brief description ofthis testing.SpecimenTen rectangular specimens were cut from a used blade in such a way that the longside of each specimen is parallel to the rolling direction (Fig.3.7). The dimensions of thespecimens are shown in Fig.3.8. Therefore the properties obtained relate to the rollingorientation of this material.Equipment, test and resultsThe tension tests were performed on a Tinius Olsen testing machine. The load-strain curves were automatically recorded by the built-in Model 51 electrically drivenrecorder. The strain was monitored by a S-1000-2A series clamp type extensometer, andthe electrical signal was sent to the recorder. The yield strength was determined by the0.2% strain offset method. The test results are shown in Fig.3.9The average yield strength was found to be 184 ksi.(1270 MPa), with a standarddeviation of 9.49 ksi.(65.5 MPa) The average ultimate strength was found to be 206 ksi.(1420 MPa)), with a standard deviation of 2.21 ksi. (15.2 MPa). The Young's moduluswas found to be 26650 ksi.(184000 MPa). The above results agree closely with the resultsby Porter, i.e. a yield point of 179 ksi.(1230 MPa), 207 ksi.(1430 MPa) of the ultimatestrength [2].3.4 R-curve experimental testsR-curve establishment is based on an energy balance concept. According to the energybalance concept [9], there is a continuous balance between released and consumed energyduring slow stable crack growth. In other words, a stable-cracking event occurs only whenChapter 3. Material Property Determination—Strength and Fracture Resistance^28SPECIMEN NO.9SPECIMEN NO.8SPECIMEN NO.7SPECIMEN NO.6SPECIMEN NO.5SPECIMEN NO.4SPECIMEN NO.3SPECIMEN NO.2SPECIMEN NO.1Figure 3.7: Orientation of the specimens in the blade(67mm)2-5/8"(216mm)-1/2"(57mm)2-1/4"(67mm)2-5/8"3/4" (19 mm)1/2" (12.5mm)1/2" RADIUS(12.5mm)Chapter 3. Material Property Determination—Strength and Fracture Resistance^29Figure 3.8: The geometry of a specimen for a strength testthe energy supply rate from the stress field around the crack tip is just enough for theenergy demand rate for material cracking. Therefore the energy consumed rate representsa material ability (or its toughness) to resist cracking. If there were no balance, theneither crack growth would arrest or become unstable. Consequently, during slow stablecrack extension, the energy release rate equals the energy consumption rate at the cracktip. The instantaneous values of the energy release rate will indicate how the energyconsumed rate depends upon crack size and applied stress. An R-curve is a continuousrecord of the toughness development in terms of 14 plotted against crack extension inthe material as a crack is driven in a stable manner with a continuously increasing stressintensity factor, K.In principle, there are two methods to deliver crack extension force to the material.One is the load control method, the other is the displacement control method.Chapter 3. Material Property Determination—Strength and Fracture Resistance^30Figure 3.9: Test results of strength for 9 specimensChapter 3. Material Property Determination—Strength and Fracture Resistance^31Figure 3.10: Critical condition in load control caseLoad control methodIn this method, the load is controlled and the corresponding crack length at variouspoints are obtained. This method usually uses a machine with a closed-loop feedbackcontrol. A constant load, however, is still very difficult to maintain. Also, the data pointsbeyond a certain point A can not be obtained. (Fig.3.10).Displacement control methodIn this method, the specimen crack opening displacement is controlled and the corre-sponding load and crack length are monitored over the range of interest.. The advantageof this method is that the displacement control is achieved easily and every point is sta-ble. Therefore, the entire R-curve can be obtained (Fig.3.11). The experimental testmethod employed in this research is of the displacement control type.RESISTANCE CURVEVsDRIVINGCURVES^CONSTANT DISPLACEMENT LEVELV < V2 < V3 < V4V2Chapter 3. Material Property Determination—Strength and Fracture Resistance^32CRACK LENGTHFigure 3.11: Critical condition in displacement control case3.4.1 Specimens and materialTo perform the R-curve tests, an ASTM E561 [14] standard CLWL specimen was used.CLWL is the abbreviation for Crack Line Wedge Loaded which is shown in Fig.3.12. Thespecimen is designed such that it can be loaded at point A and B on a horizontal basewith a vertical wedge. This loading mechanism has several advantages 1) it is easy toachieve displacement control; 2) it eliminates machine stiffness involvement; and 3) itis easy to prevent the specimen from buckling when the load is high. A starting notch,1/16in.(1.6mm) wide, was made by saw cutting. The specimens were the pre-cracked upto 0.35 to 0.45 times the specimen width by cyclic tension loading on a Sonntag fatiguetesting machine.The specimens were made from a strip of new bandsaw blade (not toothed) with athickness of 0.073 in (1.85mm) in the as-received condition, having the chemical compo-sition and mechanical properties as shown in the previous section of this Chapter.Chapter 3. Material Property Determination—Strength and Fracture Resistance^33/4" (1 24mm )(102mm)Figure 3.12: Geometry of CLWL specimen3.4.2 Cracking opening devices and fixtureCorresponding to the CLWL specimen configuration, a special set of loading fixturesare needed. The basic idea is 1) to obtain displacement control; 2) to prevent the thinspecimen from buckling. To this end, a loading system was designed and constructed,which consists of several parts as follows.Die and cover platesTo hold the specimen in a horizontal position, a massive mild carbon steel die wasdesigned to serve as a base (Fig.3.13). Its stiffness minimizes its deformation when thespecimen is loaded, and its weight ensures stability of the system when the test is inprogress. A group of parallel groves were cut in its top surface to hold the lubricant andto reduce the direct contact area between the specimen and die surface so as to minimizethe friction which may induce some uncontrolled loading onto the specimen.To keep the specimen from buckling under high loading, a pair of cover plates weregrooved—surfaceloading wedgecover platehold—downframesspecimenpad•rollerssliding blockbase dieChapter 3. Material Property Determination—Strength and Fracture Resistance^34Figure 3.13: The loading system for CLWL specimenChapter 3. Material Property Determination—Strength and Fracture Resistance^35employed (Fig.3.14). The cover plates were held down onto the loaded specimen throughrollers and pads by five holding frames so that the necessary pressure was maintained toprevent the specimen from buckling while leaving the covers free to move laterally withthe specimen to reduce friction between the contact surfaces.Loading systemIn this system, the crack loading is accomplished by using the loading wedge andtwo sets of small dies (Fig.3.14). The wedge was hardened and lubricated, with a totaltaper angle of 3 degrees on its working sides. To maintain an accurate load line as anangle between the crack surfaces becomes larger when the specimen crack extends, twosets of intermediate dies were designed and constructed. One pair consists of circularsegments which make contact with the specimen, while the other pair has one of itssurfaces relieved, which contacts with the working sides of the wedge. The vertical loadon the wedge is transmitted through the contacts between dies providing a horizontalforce acting at the loading points A and B on the specimen via the circular segments.3.4.3 Double-calibration methodThe information required to establish an R-curve from experimental data includes a)the current crack length ; b) the corresponding loads. For the CLWL specimen and thewedge loading system discussed above, the current load value must be obtained via anindirect method, i.e. calibration. The current crack length can be obtained by eitherdirect monitoring or calibration. In this research, both monitoring and calibration wereemployed to obtain the crack length so that the results could be checked against orcompared with each other, while the current loads were obtained by calibration.The calibration relationship between the relevant quantities must be obtained by oneof three methods:Loading WedgeLoading DieIntermediate DieCover Platenon-circular diescircular segmentsrChapter 3. Material Property Determination—Strength and Fracture Resistance^36BoltHolding FramePadRollerSpecimenSliding SupportBlockBase DieFigure 3.14: Loading wedge and diesChapter 3. Material Property Determination—Strength and Fracture Resistance^37• analytical method (if mathematically possible)• numerical method• experimental methodBecause of its complex geometry (especially its discontinuity at the boundaries), aclosed-form solution is not available, therefore a finite element approach was employedto numerically obtain the physical quantities necessary for the calibration relations. Anexperiment was also conducted to confirm the finite element results.3.4.4 Establishment of the calibration relationsThe physical quantities for calibration relationsThe calibration relationship is the relationship among relevant quantities in the formof dimensionless groups. How to choose the quantities largely depends on the particularcase. For the CLWL specimen configuration, the quantities which are ultimately requiredare load P and crack length a. The experimental quantities used to determined thesequantities can be the crack open displacement (COD), strains at particular locations onthe specimen or some other relatable quantities. The COD related calibration relationsare available in ASTM E561 [14]. However, when the specimen is large, acquisition ofthe COD information requires large clip-gauges which are more sensitive and more easilydisturbed. Also, the large clip-gauges will present more difficulties for installation in avery limited space. In contrast to the COD, strains at chosen locations are much easierto obtain and handle. Also, common strain gauges are accurate enough to make theinformation more reliable. Therefore, in this research the strains at particular locationsand orientations were used as calibration quantities, i.e. the calibration relations areaccordingly strain related. Fig.3.15 shows the locations and orientations where the strainswere measured.Chapter 3. Material Property Determination—Strength and Fracture Resistance^38Figure 3.15: locations and orientations of strainsMathematically, in order to determine the two unknowns, P and a, at least twoequations are required, while to establish the two relations, two strains el and e3 arerequired. Based on the assumption that the specimen under load remains elastic, thefollowing relationships are valid:EBWei af ( ^)P WEBWe3 af( p^w) C2whereel— strain in Y direction at location 1.e3— strain in X direction at location 3.B— thickness of the specimen.P— symmetrical force acting at point A and B.Chapter 3. Material Property Determination—Strength and Fracture Resistance^39E— Young's elastic modulus.C1, C2- constants, their values depends on the particular specimen.The above equations can be expressed in more convenient way as follows:EBWei EBWe3 = a^ ^f2( j )Obtaining a ratio between the two strains el and e3 provides a further relationship:ii(-4`7)^a—f(—)/24,1^we3Among these three equations, only two are independent. In R-curve determination,any two can be used as calibrating relationships to determine the crack length a and loadP.To evaluate the above equations for the specific CLWL specimen, data is required.This data includes:1. The geometry and properties of the specimen i.e. W, B, elastic modulus E andload P. These are all given for the CLWL specimen.2. The crack lengths corresponding to strains el and e3 over the range of crack growth.These are to be found either numerically or experimentally.Numerical results—finite element methodIn order to define the calibration relations, a finite element model was used to solvethe problem numerically. The finite element method [20] (FEM) is a numerical procedurefor discretizing continuum problems in order that they may be solved on digital comput-ers. This method models a structure as an assemblage of small parts (elements). EachChapter 3. Material Property Determination—Strength and Fracture Resistance^40element has a simple geometry and is therefore much easier to analyze than the structure.In essence, a complicated solution consists of a series of piecewise-continuous simple so-lutions. Elements are called "finite" to distinguish them from differential elements usedin calculus. This model was implemented using the software program ANSYS[26] andwas used in establishing the calibrations required in the R-curve test.In the FEM modelling, the following assumptions were made:1. The specimen is in plane stress.2. No plastic deformation occurs.In order to achieve the desired accuracy within a reasonable computing time, onlythe area where the strains el and e3 are located were densely meshed. In this model aload P=10,000 lb.(44.5 kN) was applied at point A and B. Young's modulus is 26000 ksi.(184000 MPa) as experimentally obtained. The strains el and e3 were calculated for 12different a/W values covering a range from 0 to 0.60 so that it was easy to establish theentire curves. The final results were then least square fitted as follows:w-3 = 114.08 - 2125.79(wa ) 4439.79(ii-va-)1.5 -^a-5365.63(^)2.5 + 4530.35( -(1-- )3.5 -W-1591.79(^)4.5 (3.8)EBWe3=- 10.30 + 163.24( vv-a ) - 63.70( wa )°.5 - 137.484)1.5 ++33.854)2-5^ (3.9) Chapter 3. Material Property Determination—Strength and Fracture Resistance^41Calibration relations^experimental resultsExperimental measurements were conducted as confirmation of the FEM results.These experiments were performed using three identical specimens cut from the sameband strip as the CLWL specimens for the R-curve testing. The Sonntag fatigue testmachine was utilized to provide the load since it could be controlled manually. A specialloading fixture (Fig.3.16) was designed and constructed to accommodate the specimento the Sonntag. Two 300 ohm foil-type resistance strain gauges, a multi-channel switchbox and a strain indicator were used to measure the strains at the required points.The load levels were set manually through the scaled loading bolt in the machine,while the different crack lengths were provided via saw cutting. The whole process wascontrolled manually, i.e. load setting, crack cutting and strains recording. The procedurewas the same for each crack length, i.e.1. saw cut the crack to a desired length.2. load the specimen to the desired level.3. take the strain readings.4. remove the specimen, and cut the new crack length...Three different load levels were chosen for each crack length and three specimenswere used to minimize the uncertainty of the data. Nine different crack lengths weremeasured for each specimen. The mean of the values for each crack length was used forthe least-square fitting. The final fitted formulas are as follows:el^a— = 1.018 — 4.999(w) + 17.380()—ae3 W (3.10)EBWe3^a^ = 2.686 + 2.172(—w ) (3.11)GRIP GRIPFOIL-TYPESTRAIN GAGEGRIPLOADINGPOINTYSPECIMENGRIPChapter 3. Material Property Determination—Strength and Fracture Resistance^42ASSEMBLYFigure 3.16: Fixture, specimen and sensor devices for experimental calibrationChapter 3. Material Property Determination—Strength and Fracture Resistance^4310864200.35^04^0.45^05^0.55relative crack length a/WFEM , expt.06^0.65^07Figure 3.17: FEM vs experiment—e1/e3 calibrationThe graphical representations are shown in Fig.3.17 and Fig.3.18. The equations(3.10) and (3.11) are valid for a/ W range from 0.42 to 0.62.Calibration results—FEM vs experimentFig.3.17 and Fig.3.18 show the comparison between the calibration curves from theFEM prediction and via the experiments. It is clear that ei/e3 curve for the experimentalmeasurement is slightly higher, while the EBWe3/P curve the experimental calibrationis slightly lower. These offsets may have been caused by some kind of system erroron the data of the strain at location 3 (e3). If the data of strain at location 3 (e3) arecorrected by a factor of 0.88, the difference between two sets of curves will disappear. Thecrack lengths obtained from the FEM calibration relation show agreement with the directmeasurement results (Fig.3.21) and were, therefore used in the subsequent computationsto establish the R-curve of the bandsaw material.Chapter 3. Material Property Determination—Strength and Fracture Resistance^44108203^0.35^04^0.45^05^0.55^06^0.65relative crack length a/W^ FEM , expt3Figure 3.18: FEM vs experiment—EBWe3/P calibration3.4.5 R-curve TestsAs discussed in the previous section, the R-curve test utilizes crack lengths and corre-sponding loads to compute the crack resistance (in the same unit as the stress intensityfactor).In addition to the strain measurement, a crack monitoring system, i.e. Fractomatand Krak-gage (Fig.3.19), was used so that a comparison with the result obtained viathe calibration relations could be made. The Fractomat uses a potential drop technique.The crack length sensor, i.e. Krak-gage, is a special foil-type gauge. It is mounted onthe specimen to cover the location where the crack will propagate so that it cracks thesame amount as the specimen. When the crack grows, the electrical potential betweenthe points A and B (Fig.3.19) will change proportionally. The monitoring device, i.e.Fractomat, measures the change in electric potential and directly relates it to crack lengthChapter 3. Material Property Determination—Strength and Fracture Resistance^45so that the current crack length can be read directly from the panel of the Fractomat.The wedge loading system and its support devices were specially designed and con-structed. Its working mechanism is described in detail in the previous section. Loading ofthe wedge was provided using the Tinus Olsen testing machine, with a cross-head speedof 0.025in/min. (0.64mm/min.), i.e. the minimum speed the machine can achieve). Astrain indicator with a multi-channel switch-box was used to monitor the strain changesel and e3 at the two different locations on each specimen. Fig.3.20 shows the test set-upfor the R-curve test.In order to gain more statistical confidence in the data a total of three identicalspecimens were tested. A minimum of 25 load steps were used to obtain a sufficientnumber of points to develop the entire continuous R-curve. For the first ten steps, aninterval of five minutes between steps was allowed for the crack tip to reach a stable state,while for the later 15 steps, a 10-20 minute interval was required to stabilize the cracktip. These intervals were necessary as the crack extended in a 'pop-in' fashion, despitethe fact that a slow quasi-static loading speed was used.3.4.6 Experimental resultsBased on the calibration curves, the experimental records were correlated into a crackresistance force, which according to the concept of energy balance [17], is quantitativelyequal to the current stress intensity factor K available at the crack tip as quantified bythe following expression [141:KBI/1729.6( IT-a )°.5 — 185.5( wa )1.5 ++655.7()2.5 — 1017.0( ^ )3.5 ++638.9(±)4.5 (3.12)00SIGNALINPUTSPECIMEN AND Krak—GAGEWIRESPECIMENKrak—gageREADOUTPANEL10.75Chapter 3. Material Property Determination—Strength and Fracture Resistance^46FRACTOMAT CRACK MOICORING DEVICEFigure 3.19: Fractomat crack monitoring device and Krak-gageloading wedgebase diestrain indicator• • ••.^.^_^ractomat crack monitor. ;^- • • ' Tinus Olsen testing machineChapter 3. Material Property Determination—Strength and Fracture Resistance^47Figure 3.20: The test set-up for R-curve testChapter 3. Material Property Determination—Strength and Fracture Resistance^48The least squares fitted expressions for the R-curve from calibrations were obtainedas follows (Fig.3.22):RBM710.484 + 90.168(--aW—) 61.678(-±)°.5) —a^a—1545.500( ^)2.5 + 3784.3301 -147. )3.5 (3.13)As expected, the R-curve for the bandsaw steel tested shows a behavior similar tothose of ultra-high strength sheet steels, i.e. the resistance increases, first steeply, thengradually.Fig.3.21 shows a comparison between the crack lengths from Krak-gage and thoseobtained from the FEM calibrations. From Fig.3.21 it can be seen that fairly goodagreement is obtained between the data using the two different methods (with the ex-ception of the first few points), which in turn verifies these validity of the relationships.Considering that the FEM non-plasticity model was used for the calibration relations,the agreement means that the plastic region at the crack tip has little or no effect on thecalibration results. The deviation shown by,of the first few points (maximum 2.6%) maybe a result of initial loose contact between the loading pieces which occurred at relativelylow load values.3.4.7 Critical stress intensity factor assessmentOne of the significant features of an R-curve is that it can be used to assess the fracturetoughness Kw in the case of plane strain or Ifc for the plane stress condition [9]. Thisassessment is based on an energy balance concept (if K refers to the cracking drivingforce, while R refers to the cracking resistance force, then the state of the crack is aresult of balance between K and R)—where the critical stress intensity factor is definedas the critical point when a crack starts to grow in a unstable manner. At this pointCV)mJs.G0.650.60.55" 0.5)..0.451-4c.)0.4C.) 0.35"a7) 0.3from different methods (no.6)Chapter 3. Material Property Determination—Strength and Fracture Resistance^4903^0.35^04^0.45^0.5^0.55^0.6^0.65relative crack length from Krak –gage (a/W)measured^• calibratedFigure 3.21: Comparison between the crack lengths from different methodsthe energy available at the crack tip supplied by the external force is not only equal tothe energy required for extension of a crack but that the energy rate provided exceedsthe energy rate that the material can consume through stable growth, i.e. at this criticalpoint, two conditions must be satisfied:K^R,. (3.14)dK df4.da^da(3.15)whereK—so-called crack driving force, i.e. the energy available for a crack extension.R,.—so-called crack growth resistance, the energy required for a crack extension.dKida—the rate of change of the driving force with respect to crack growth da.dR,./da—the rate of change of the resistance with respect to crack growth da.400350300250a.)200IX 1507100EZ4500K —Curves qs% upperaverage95% lower bondR —curv tspointstangent iChapter 3. Material Property Determination—Strength and Fracture Resistance^500^2^2_5^3^3_5^4Crack Length (in)Figure 3.22: Graphical assessment of critical stress intensity factorFig.3.22 shows the geometric representation of the above relations. It is clear that thecritical point is the point where the driving force curve and resistance curve are tangentto each other. Both numerical and graphical methods can be used to assess the criticalstress intensity factor. Here a graphical method was employed for the determination ofthe critical stress intensity factor of the bandsaw steel.In order to account for the range of scatter, all of the data from the three specimenswere used to develop a single curve for the saw blade steel with the 95% confidence level.The upper and lower bounds were also determined. The K-curves were plotted using theequation (3.12). The fracture toughness assessed is 220 ksi.irt112(242MPa.m112) with theupper bound 246ksi.in1/2(270MPa.m1/2) and the lower bound 210ksi.in1/2(231/11Pa.m1/2)(Fig.3.22).Chapter 3. Material Property Determination—Strength and Fracture Resistance^51The plastic zone correctionWithin the frame of Linear-Elastic-Fracture Mechanics (LEFM), a cracked body un-der study is assumed to deform in a purely elastic manner, even at the crack tip. However,a portion of the material within the tip region always behaves plastically. The extentof this deviation from fully linear elastic behavior varies as a function of material andconstraint conditions. For materials where a large plastic zone develops (relative to thecrack length) an Elastic-Plastic Fracture Mechanics (EPFM) approach should be em-ployed. For materials with a very high yield strength (> 150ksi(1035MPa)) the plasticzone at the tip of a crack can be considered as being very small such that a plastic cor-rection to elasticity based theory is enough to make the theory work. The correctionsavailable are mostly in the form of crack length, i.e. converting-the plastic zone intoan equivalent crack increment to the original crack length (or physical crack length) bymeans of an energy balance [9].The following relationship is widely used to account for limited plasticity at the cracktip:Irwin's CorrectionThe effective crack length:ae^+ Aa + rywhereao— starting crack length.Aa— physical crack growth at crack tip.ry— plastic zone adjustment.1 K2rY = (27r)(-47)(for plane stress)K— current stress intensity factor.Chapter 3. Material Property Determination—Strength and Fracture Resistance^52ay-2% offset yield strength.It should be mentioned that in many cases, Irwin's correction may over-estimated thetrue plastic zone effect [221.Plastic effectThe necessity of the correction can be seen clearly from the R-curve, namely theplastic correction changes the shape of the R-curve, ultimately changing the assessmentof critical stress intensity factor (Fig.3.22). For the saw blade steel tested, the differenceof the assessed critical stress intensity factor is 10 ksi.in1/2(230ksi.in1/2). The correctionrp ranges from 2 — 11% of the current crack length.Chapter 4Crack Initiation Tests4.1 IntroductionCrack initiation in the gullet region of a bandsaw blade can be the result of contributionsfrom many factors such as the stress (or strain) level (i.e. function of the diameter ofthe wheel) or stress ratio (i.e. pre-strain force level), the micro-structure of the steel,the surface condition of the gullet region (introduced by the sharpening process) andthe environment (i.e. coolant, wood sap and temperature). Amongst these factors, thestress level, stress ratio and surface roughness in the gullet region are dominant. Asin many components in engineering practice, the surface roughness of a blade plays asignificant role in affecting its fatigue-life. The orientation and depth of scratches on asteel surface can severely affect the fatigue life. In routine sawmill practice, the saw bladeis typically removed from the machine for resharpening after one shift of four hours ofcontinuous running and cutting service. This process removes a very thin surface layerof the gullet region, grinding off most micro-cracks developed during the previous serviceperiod. However, at the same time, this process (Fig.4.23) can cause scratches in themost unfavorable orientation, some of which are very deep (Fig.4.24) and will serve asperfect crack starters in future service. High temperatures (i.e. red-hot) can be reachedduring the sharpening process, and the following rapid air-cooling can transform a thinlayer of the blade adjacent to the surface of the gullet region into a very brittle structurewhich makes the region more susceptible to cracking [1]. The combination of these two53TOP VIEW11111 1 1111 111111111111111111'1surfacc scratchesproducedGRINDING WHEELChapter 4. Crack Initiation Tests^ 54Figure 4.23: Resharpening process—stone wheel and toothunfavorable factors plus the severe stress concentration at the bottom of the gullet (1.5-2.5) [4] can greatly enhance the possibility of crack initiation.One phase of this study was to investigate the effect of surface roughness of a bladeand stress levels on its fatigue life. To gain some statistical information on the prob-lem, two groups of small specimens with different surface roughnesses were tested. Inorder to simulate the actual loading condition experienced by a saw blade in service, thestress levels and stress ratio were set equal or close to the service levels used in sawmills(recognizing the presence of stress concentration).Chapter 4. Crack Initiation Tests^ 55Figure 4.24: Close-up of a typical scratch produced by grinding process4.2 Experiments4.2.1 Specimen, load and equipmentSpecimen and materialsA series of small, flat, rectangular-sectioned specimens with two different qualitiesof surface roughness were produced from a new blade steel band as in the as-receivedcondition. The surface finish was prepared by an experienced sawfiler on a real bandsawgrinding machine using different feed speeds (Fig.4.26)..The first group was ground to asurface roughness ranging from 115 to 194 micro-inch (central line average). The secondgroup was ground to a surface roughness ranging from 300 to 400 micro-inch (central lineaverage). The roughness in the second group was believed, by the sawfiler, to be closeto the condition of a blade as used in a sawmill. The crack initiation was defined in thisresearch as a crack length of 1/25.4 in.(1mm).Chapter 4. Crack Initiation Tests^ 56Figure 4.25: Crack initiation life specimenactual saw blade surfacerough surface (specimen)fine surface (specimen)Figure 4.26: Photo of the surface roughness of the specimenChapter 4. Crack Initiation Tests^ 57Loads chosenIn order to simulate the actual loading conditions which a real blade experiences inservice, four tensile stress levels were chosen to simulate the stress fields produced byfour different levels of pre-strain forces generally used in sawmills. The specific stresseswere calculated by assuming that the blade has dimensions of 9.5 in.(241mm) in widthand 0.073 in (1.85mm) in thickness, on a 5 ft.(1524mm) diameter handsaw machine. Thefour pre-strain force levels chosen were:1. 20,000 lb.(90,000 N)—used on the so-called 'high-strain' systems.2. 18,000 lb.(81,000 N)—generally used in sawmills.3. 15,000 lb.(67,500 N)—generally used in sawmills.4. 13,000 lb.(58,500 N)—generally used in sawmills.As for the stress concentration at the bottom of the tooth gullet, a factor of 2.2,obtained by Lee [3] was used to modified the stresses so as to obtain some degree ofsimulation to the real load conditions in the gullet region of a blade (Table.4.2). Thetheory used to calculate the stresses were those summarized in Chapter 2.Equipment and devicesThe machine employed for this testing was the Sonntag universal fatigue testingmachine. Fig.4.27 shows schematically the working mechanism. The pre-load (staticload) is controlled by a loading spring, the deflection of which can be adjusted manually.The alternating load (dynamic load) is produced by rotation of a centrifugal mass. Themagnitude of the cyclic load is set by the adjustable distance from the mass center tothe motor shaft to which the mass is attached. The fixed speed of the machine is 1800rpm. The machine has a built-in mechanical counter which stops counting at the samemoment as the machine is automatically shut off if the specimen fails (separated). InChapter 4. Crack Initiation Tests^ 58Load Parameters Chosengroup first group second groupsurface * 100-200 (microin) 300-400 (microin)cyclic (ksi)load mean (ksi) cyclic (ksi) mean (ksi)20,000 lb 67.45 3533 67.45 353318,000 lb 64.23 35.33 64.23 35.3315,000 lb 59.43 35.33 59.43 35.3313,000 lb 56.7 35.33 56.7 35.33note: * inducates center line averageTable 4.2: Load parameters chosenChapter 4. Crack Initiation Tests^ 59order to get a more reliable load setting, the static and dynamic loads were set by astrain indicator with a foil-type 300 ohm strain gage mounted on the specimen.4.2.2 Test, results and analysisThe specimen was held using a stiff-bolted clamp/joint friction grips and cyclically loadedusing the Sonntag fatigue testing machine. The cycles elapsed were counted automaticallyby the built-in mechanical counter. To gain some statistical certainty, a minimum of threespecimens were tested for each load case. The cyclic life presented in the Table.4.3 isa statistical mean. The cyclic life was converted into service hours assuming a bendfrequency of 5.7 Hz, a reasonable estimate based on sawmill operation.ObservationsFrom Table.4.3 and Fig.4.28, the following observations can be made:1. The result at 20,000 lb.(90,000 N) and 18,000 lb.(81,000 N) pre-strain levels showa little variation in crack initiation time for the group with fine surface roughness.The 15,000 lb.(67,500 N) level has a life span of more than 4 times as that of the18,000 lb.(81,000 N) level.2. For the group with a coarse surface, the 20,000 lb (90,000 N) and 18000 lb. (81,000N) pre-strain force levels were found to have no difference in initiation time. The15,000 lb. (67,500 N) level initiation time results is slightly longer than those for20,000 lb and 18,000 lb levels. However, the 13,000 lb level has 50% longer life.3. The initiation life of the group with fine surface roughness is about, at least, morethan 10 times that of the group with rough surface roughness.4. In the coarse surface group, whose surfaces are believed to be closer to the realservice condition of a blade, the longest initiation life was just over one hour, i.e.Chapter 4. Crack Initiation Tests^ 60SpecimengripsSPECIMEN AND GRIPSFigure 4.27: Fatigue testing machine working mechanismChapter 4. Crack Initiation Tests^ 61Pre—strainfarcelevel20,000lb18,000lb15,000lb13,000lbmaximumstress102.78ksi99.56ksi94.76ksi92.03ksi8ac,c=."=—T,4.-2C...-"'uUo ,9Z5!.:-,) (,)N146.3 156not foiledafter686not failedafter700"la'P047.12 7.59 >33 >38.4000li00g,..., 07••-• v0 -o 0 15.3 15.3 16.3 22CI7i..g,40.74 0.74 0.78 1.07Table 4.3: The crack initiation test resultsChapter 4. Crack Initiation Tests^ 62Figure 4.28: The crack initiation test resultsChapter 4. Crack Initiation Tests^ 6325% of one shift. While in the fine surface group, whose surfaces were producedwith greater care, even the shortest life lasted well beyond one shift of 4 hours inall load cases.Possible explanationsAccording to Porter [2], the fatigue limit for handsaw steel is estimated to be lessthan 120 ksi. (828 MPa). The specimens in this crack initiation test were working undera maximum stress 92.03-102.78 ksi (635-703 MPa) (Table.4.2), which is within a narrowmargin of the fatigue limit. Previous investigations [8] have shown that surface roughnessbecomes a major degrading factor in component fatigue life when the maximum stress isnear the fatigue limit. A SEM (Scanning Electronic Microscope) examination of the twogroups shows that a poor surface finish often means more and deeper scratches whichwork as more severe stress raisers. Micro-cracks may also be formed, so that there aremore chances for stress-favourable scratches to grow into micro-cracks at faster rate.What is alarming is that SEM examination of a sample cut from a failed blade showsmore and deeper scratches than on the specimen tested. Recognizing that there are sev-eral hundred teeth punched in one closed loop blade, thus providing a higher probabilityfor deep scratches to be present, it is not unreasonable to expect that cracks can benurtured in as short as 45 minutes, consequently, a dangerous stage II, i.e. crack prop-agation, begins. If this kind of situation happens to a blade in actual service, it meansthat a cracked blade will be working for the rest (82%) of one shift of four hours!Chapter 5Mode I Crack Growth Rate Test5.1 IntroductionIn contrast to smooth body fatigue analysis in which the crack growth stage is notseparated, fracture mechanics based fatigue theory divides the fatigue process into threestages: (1) crack initiation; (2) stable (or subcritical) crack growth; (3) unstable (orcritical) crack growth. It is in the second stage that fracture mechanics based fatigueanalysis concentrates. Linear elastic fracture mechanics essentially assumes that a defectexists originally. Based on the concept of a stress intensity factor, fracture mechanicsbased fatigue theory focuses on the crack growth behaviour (actually the crack growthprocess occupies the major part of life-time for some large components in engineeringpractice) so as to quantify the process from initiation to final failure of a component.In fracture mechanics a crack can be classified into one or a combination of threebasic crack modes, i.e. Mode I (opening), Mode II (sliding) and Mode III (tearing) (seeFig.5.29). Although basic forms of stress intensity factors and crack growth rate laws aresimilar for the three different modes, their crack tip stress field and crack growth behaviorare generally different. Therefore, a crack growth rate is always associated with the crackmode. It can be seen that a tooth gullet crack should be classified as Mode I (openingmode) crack, therefore its growth rate should be obtained using Mode I specimens.Paris (1964) proposed the well-known Paris' law [15]. Based on the concept of stressintensity factor as proposed by Irwin in 1961 [17], Paris introduced a new parameter, the64MODE IOPENING MODEMODE II^MODE IIISUDING MODE^TEARING MODEdadN f(AK,R,,T) (5.16)Chapter 5. Mode I Crack Growth Rate Test^ 65Figure 5.29: Three Modes of cracksstress intensity range (Fig.5.30), to characterize the crack-tip field under fatigue loadingand assumed a general relation:whereda/c/N—Crack extension per cycle.AK—Stress intensity range, defined as shown in Fig.5.30.R—Stress ratio urn /a-m  max •E—Strain rate.T—Current temperature when a material is tested.In the simplest case (no R effect), experimental results show that for most metallicmaterials the relation can be formulated as,Chapter 5. Mode I Crack Growth Rate Test^ 66Figure 5.30: Definition of stress intensity rangeda^ — C(AlObdN(5.17)Here C and b can be seen as material constants which depend on strain rate and envi-ronment.There are many other formulas suggested to account for varying stress ratio andfatigue threshold effect to allow for prediction of component life. Forman's formula [9] istypical of this kind.Through extensive experiments on a vast variety of engineering materials, a generalrange for C and b are suggested in fatigue design handbooks. However, in engineeringpractice, especially in the field where cracking problems are of most concern, experimentaltests have to be carried out on specific materials with specific conditions. ASTM hasissued a standard, E647 [19], for testing specifications and procedures related to fatiguetests.Chapter 5. Mode I Crack Growth Rate Test^ 67As a first attempt to understand the growth behaviour of an gullet crack in a bandsawblade material, and to obtain more complete information for further comparison study,a series of Mode I crack growth rate tests were designed and carried out on specimenscut from a strip of new bandsaw material. The following section discusses the testingprocedure and the results.5.2 Mode I crack growth rate experiment5.2.1 Specimen and equipmentA compact-tension specimen (CT) was chosen with the configuration and dimensionsof the specimen being obtained from the guidelines in ASTM E647 [19]. The crackorientation was perpendicular to the rolling direction (blade length) so as to simulate agullet crack in a bandsaw blade. The crack was measured using Krak-gages connectedto a Fractomat crack monitoring device (see Fig.3.20). The loading was monitored viaa strain gage mounted at a location on the specimen as shown in Fig.5.31. To ensureaccuracy in cycle counting, a separate digital electronic counter was employed.The Sonntag universal fatigue testing machine (Fig.4.27), whose loading system is dis-placement controlled, was used. To maintain a nearly constant load during one crackingstep, load adjustments were needed from time to time during each test. To ensure thatthe specimen was loaded accurately, a calibration relation based on load, crack lengthand strain at a defined location on the specimen (Fig.5.31) was established by usingboth a finite element analysis model and experimental tests. The relationship is showngraphically in Fig.5.32.Two GripsSpecimenKra k—gageStrain GaugeChapter 5. Mode I Crack Growth Rate Test^ 68Figure 5.31: The specimen and fixtures for crack growth rate testChapter 5. Mode I Crack Growth Rate Test^ 69COMPARISON OF CALIBRATIONFEM vs experiment3.532:7 2.50 2• 1.50<• 10.5 * 002 03^04^05^06^07RELATIVE CRACK LENGTH (a/W)expt.^FEMFigure 5.32: The calibration relation for CT specimen5.2.2 Tests and data analysisExperimental testingIn order to investigate stress ratio effects, the tests were divided into three groupshaving different stress ratios (R) corresponding to the range of commonly employed pre-strain force levels used in actual saw blade operations. Each group of three specimenswas tested to obtain the data needed for establishment of a daldN curve.In order to cover a wide stress intensity range AK, as well as provide a reasonabledensity of points, three specimens from each group were tested, while each specimen wastested using a different method. One of them was tested with the so-called load sheddingmethod, one with the constant load and one with the load increasing method.• load shedding—the load is reduced stepwise (R ratio constant) so that AK de-creases. When a desired low AK is reached, the load is increased stepwise. In thisChapter 5. Mode I Crack Growth Rate Test^ 70method low as well as high values of AK can be covered. The data from the loadincrease and decrease process overlap each other so as to gain statistical certaintyfor each case.• constant load—the load is kept constant so that a reasonable density of points aswell as the medium and high range of AK can be covered.• load increasing—the load is increased stepwise(R constant) so that AK increasesquickly to reach a high level of AK before the crack grows beyond the Krak-gagelimit length.To minimize crack retardation effects [19] which may be introduced by a large loaddrop, 8% shedding and an increasing rate was used. An interval of 0.02-0.03 in.(0.5-0.8mm) was used for each step [19]. The tests were conducted on the Sonntag universalfatigue testing machine. The tests were manually controlled with each load step.Data and analysisUsing the discussed approach, a total of twelve specimens were tested to cover thefour different stress ratios. The parameters measured included load, crack length andcycles elapsed (Fig.5.33). The data was correlated into typical daldN vs AK form bymeans of the secant method [19] and were formulated by using the least-squares approach.Fig.5.33 is a typical record of a — N curve. The log —log plots of the dal dN — AK curvesare shown in Fig.5.34—Fig.5.38.The least-squares fitted expression for the range of lOksi.in1/2 < AK < 50ksi.in112follows as (see Fig.5.38) (note: in eqn.(5.18), (5.19) and (5.20), the unit for AK isksi.in112 instead of MPa.m112),dadN^1.68(AK)2-86 (10-8mmIcycle)^(5.18)Chapter 5. Mode I Crack Growth Rate Test^ 71Figure 5.33: A typical crack length versus cycle recordFigure 5.34: The crack growth rate: R = 0.4Chapter 5. Mode I Crack Growth Rate TestFigure 5.35: The crack growth rate: R-=-  0.295Figure 5.36: The crack growth rate: R = 0.24372Chapter 5. Mode I Crack Growth Rate Test^ 73Chapter 5. Mode I Crack Growth Rate Test^ 74with an upper bound:da2.22(AK)2.86^(10-8mm/cycle)^(5.19)(10-8mmicyc/e)^(5.20)The upper and lower bounds are produced by three standard deviations of da/dNdata which, according to statistics theory, can cover the statistical behaviour with a 95%confidence level.Based on the experimental results the following observations can be made:Observations1. There is little stress ratio effect on the propagation rate within AK = 10 —50ksi.in1/2(11 — 55MPa.m1/2). To see the stress ratio (R) effect, all of the datawas plotted on the same graph (Fig.5.38). It can be clearly seen that in the rangeof stress intensity range AK = 10 —50ksi.in1 1'2(11 _ 55mpa.mv2) all data clustersclosely to the fitted line. This means that within this range of AK, stress ratiohas no significant influence on the propagation of a mode I crack in this material.Based on this data, it is reasonable to quantify the propagation rate by a singlecurve when operating within this AK range.2. Stress ratio effects show up in the region of low AK ,i.e., AK < 10ksi.ini/2(11MPa.m1/2). A clear deviation begins when AK falls below 10 ksi.in1/2 (11mpa.m1/2\.) The two deviating curves belong to the higher R ratio tested, i.e.R = 0.295 and R = 0.4 groups respectively. The different extent of crack closure,which is closely related to stress ratio, is believed to be responsible for this deviation.dNa lower bound:dadN = 1.28(AK)286Chapter 5. Mode I Crack Growth Rate Test^ 75While closure itself is caused by the permanent plastic deformation of the materialin the wake of the crack [16]. The ultimate result of this kind of deviation fromeach other produces quite different values of the so-called threshold stress intensityrange, denoted as AKth, which is defined as a value of AK below which da I dN isso low that a crack does not practically grows. The AKth can be defined accordingto engineering circumstances. For instance, ASTM E647 recommends that AK =AKth when daldN = 10-1°m/cycle [19].3. The maximum threshold value of the bandsaw steel under investigation is about7.9ksi.in1/2(8.69MPa.m1/2) in the case where R= 0.4. The minimum AK is below3.16ksi.in1/2(3.48MPa.m1/2) in the case where R= 0.2). The threshold values areobtained by the extrapolation method recommended in ASTM E647.Chapter 6Crack Propagation Test for Out-of-Plane Bending6.1 IntroductionIn the previous tests, crack propagation can be considered as one dimensional, i.e. thestress intensity factor is the same along the whole crack front. Therefore the wholestress field along crack front can be characterized by one single stress intensity factor.However, in actual service a bandsaw blade experiences both tensile and out-of-planebending during one revolution. According to the simple beam theory [11], the out-of-plane bending produces a triangle stress distribution across the blade thickness, thereforethe stress intensity factor varies along the crack front, which results in different crackextension rates along the crack front. Furthermore, out-of-plane bending may also maketwo crack surfaces contact each other (Fig.6.39). The contact force between the two cracksurfaces introduces another stress field onto the existing field, which also may changesthe crack extension rates. It can be expected that the pre-strain force has a strong effecton the stress intensity and crack growth rate. In order to find more evidence regardingthe crack growth pattern under a out-of-plane bending load, a series of crack growthtests using on out-of-plane bending load were conducted. These tests were also used toinvestigate the effect of stress ratio. Description and discussion of these tests is presentedin following sections.76CRACKEXTENSIONI CONTACT AREAA—A CRACK FRONT( AFTER EXTENSIONCRACK FRONTBEFORE EXTENSIONB—BChapter 6. Crack Propagation Test for Out-of-Plane Bending^ 77CRACK FRONTFigure 6.39: Schematic relation of crack surfaces interacting: saw blade and crackedgullet positionChapter 6. Crack Propagation Test for Out-of-Plane Bending^ 786.2 Experiment6.2.1 Basic considerationsIn order to assess the effect of the stress ratio, the tests were conducted at six differentvalues of stress ratio. stress ratios. The stress ratios were selected such that the loadconditions in both the inner and outer edges could be simulated to some extent, i.e. boththe maximum and minimum stress in the outer and inner edges of the test specimenapproximated those of a working blade, while at the same time maintaining the nominalstress intensity range AK at the tip of the crack in the outer edge.6.2.2 Specimens, equipment and devicesSpecimenThe test specimens were cut from new blade material in its 'as received' conditionand were machined to the geometry shown in Fig.6.40. The specimens were pre-crackedunder a cyclic tensile loading on the Sonntag fatigue testing machine (see Fig.4.28 inChapter 4). The crack was oriented perpendicular to the rolling direction to simulate agullet crack in an actual blade. The starting crack length was about 20-35% of the widthof the specimen.Equipment and devicesTo achieve a pure bending condition, a four point bending fixture (Fig.6.41) attachedto the Sonntag fatigue testing machine was employed. The fixture allowed a degree offreedom in the axial direction so that there was little or no axial force in the specimenwhile a desired bending load was imposed on the specimen. Its working mechanism isshown in Fig.6.41. The loading mechanism is the same as that discussed in Chapter4. Because different crack lengths on the upper and lower surfaces were expected, oneKrak-gage was mounted to each side of the specimens. The Krak-gages were then wiredji___ KRAK —gageOMBstrain gaugeChapter 6. Crack Propagation Test for Out-of-Plane Bending^ 79^5"(127mm)^^1 2" (305mm)^Figure 6.40: Geometry of the specimen for out-of-plan bending loadto the Fractomat crack monitoring device (see Fig.3.21). Two strain gages mounted onthe each side of the specimen were used to monitor the current nominal applied stresslevel. The extension of the crack for each step was about 0.8mm. The data collection (crack length and cycles) was performed manually stepwise. The machine had to be shutoff for each step for data collecting and load adjusting.6.2.3 Test results and analysisThe recorded data included the crack lengths on each side of the specimen, the appliedstress and the number of cycles. The relationship between crack length (on tension side)and cycles under different stress ratios are presented in Fig.6.43-6.48. Fig.6.49 showsall six curves (least-square fitted) superimposed on the same daldN — a/W graph forcomparison. Observing Fig.6.49, the following observations can be made:specimen\\k"-LstationaryplatformmovingplatformFigure 6.41: A four-point bending fixtureelectoniccycle counterimamFractomatcrack monotoringdevicespecimenstrain indicatorChapter 6. Crack Propagation Test for Out-of-Plane Bending^ 80Figure 6.42: A test set-up for out-of-plane bending test32.5---,^2•..........•0.50Chapter 6. Crack Propagation Test for Out-of-Plane Bending^ 81comparison of a—Nmeasure. vs predict. (stress ratio=0.313)32.5.--..^2•_0.500^200^400^600^800^1000^1200Thousandscycles^ measured _ predictedFigure 6.43: Crack growth: R=0.3131400comparison of a—Nmeasure. vs predict (stress ratio=0.227)0^200^400^600^800^1000^1200^1400Thousandscycles^ measured _ predictedFigure 6.44: Crack growth: R=0.227Chapter 6. Crack Propagation Test for Out-of-Plane Bending^ 82comparison of a—Nmeasure. vs predict. (stress ratio=0) 500 1000 1500Thousandscycles^ measured • predictedFigure 6.45: Crack growth: R=0comparison of a—Nmeasure. vs predict.(stress ratio= —1)32.50.500 500^1000^1500ThousandscyclesmeasuredFigure 6.46: Crack growth: R=-12000t 4,5 01.5tuCs10.5032.5—Chapter 6. Crack Propagation Test for Out-of-Plane Bending^ 83comparison of a—Nmeasure. vs predict. (stress ratio= —0.373)0 250 30050^100^150^200Thousandscycles^ measured • predictedFigure 6.47: Crack growth: R=-0.373comparison of a—Nmeasure. vs predict. (stress ratio=—O.49)32.50.500 100 200^300^400^500^600Thousandscycles^measured^• predictedFigure 6.48: Crack growth: R-=-0.49700Chapter 6. Crack Propagation Test for Out-of-Plane Bending^ 84.Sv-)03503025rg•020.15te)0.10.05dK unit= ksi.in ^ 1/2ieetrio--0444-444-41-fri-"4"4+-+R = -0.373 dK=16.9—_dyi,j0:12fi . _^1.... ,^•00,„,I.A,„40.4.4.*:—,..<','4;4.:<::' ',,,,,,II 11111,1111i.WU 11 11 11 11 11 MI DIMaoa _ _R —0.313 dK=,.,..1....40.,,,,,,e,,,,,,AN.,...,a/A/A/Ap.A"= —1 dK=N/A -..._____ R=0 dK=10.30.2^ 03^ 0 4^ 05^0 6^0 7relative crack length a/W0 R=0.313^O 0.227^6 R=0x R=-1^• R= —0.373^x R=-0.49Figure 6.49: Comparison of different crack growth rate1. The crack growth rate is only slightly dependent on crack length. In Fig.6.43—6.48 one common tendency is shared by all specimens—crack extension is almostproportional to numbers of cycles, in other words, the crack growth rates werealmost constant. This implies that the stress intensity factor range AK does notchange very much in pure out-plane-bending load condition.2. Stress ratio effect is distinguishable (Fig.6.49).Fractographical analysisIn order to have a better understanding of the above phenomena, all of the testedspecimens were cut open and the crack surfaces examined using the Scanning ElectronicMicroscope (SEM). The final crack profile of each specimen was measured using an opticalmicroscope and the data were used for the calculations of crack growth.uncrackedsectioncrack front profileChapter 6. Crack Propagation Test for Out-of-Plane Bending^ 85Figure 6.50: A typical crack front profileFig.6.50 shows a typical final crack front profile--a smooth, elliptical curve front,which was shared by all six specimens.In order to obtain a record of the crack propagation process using discrete points atinting technique, a method to mark the crack front by heating the specimen to form anoxide film on the newly created crack surfaces, was used. The specimen was tinted fourtimes by using four different temperatures to mark the four different crack fronts withfour different tints of colour. Fig.6.51 shows a photo of the crack front traces.In Fig.6.51 the process of crack advance can be seen, i.e. the crack growth starts fromthe corner on the tensile side, grows radially to some distance, then starts to accelerateat a point on the tensile surface—the circular crack front grows into an elliptically curvedfront, finally the lower part of the front reaches the compressive surface. After this pointthe crack front advances while maintaining its shape relatively unchanged. This evidenceis very similar to that found by Ingema on a sample from a failed blade [1 ].1st crack front3rd crack front4th crack frontZuncracked section2nd crack frontChapter 6. Crack Propagation Test for Out-of-Plane Bending^ 86crack advance directionFigure 6.51: The trace of crack advanceAs noted in Fig.6.49, the rate of crack growth (da/dN) as a function of crack size isnearly constant. Given the above propagation pattern, the following can be one of theexplanations for this behaviour:According to Paris' assumption [151, crack growth rate is controlled by stress intensityfactor range. Therefore, the small change in da/dN means little change in the effectivestress intensity factor range. From Fig.6.51 it can be seen that as the lower part of thecrack front reaches the compressive surface of the specimen, the crack advances withalmost the same crack front shape. On the other hand, the bending load also remainsconstant, therefore it is reasonable to expect little difference in the stress field aroundthe crack-tip, thus little change in the stress intensity factor range.Stress intensity factor for out-of-plane bendingIn the above analysis, the crack front advances with a curved profile under the out-of-plane bending load. To determine its stress intensity factor, the crack front can beChapter 6. Crack Propagation Test for Out-of-Plane Bending^ 87approximated as a quarter elliptical crack.In 1981, J.C. Newman and his co-worker [23] published their work on stress inten-sity factor for a surface crack. They correlated their previous three dimensional finite-element analysis results which covered a wide range of configuration parameters for asemi-elliptical surface crack under tension and out-of-plane bending. Being multipliedby free edge correction factor [9], the formula were used to calculate the stress intensityfactor for the crack in the out-of-plane bending tests. The details of the formula arequoted in Appendix A.The stress intensity factors for the final profiles of crack front of all specimens werecalculated. The results proved that the higher SIF range is, the faster the crack grows(see Fig.6.49).Calculation of the crack growth rateTo show the validity of the crack growth rate experimentally obtained (see Chapter5), the crack front profile data and the crack growth rate described by equation (5.17)(in Chapter 5) were used to calculate the crack growth in the out-of-plane bendingtests. These results were presented in Fig.6.43-6.48. The maximum difference for allsix specimens is 4.56% (for details see Appendix A). The predictions are encouraging,which proves that the stress intensity factor formula obtained by C.J. Newman andthe crack propagation rate experimentally obtained for the bandsaw material are useful.Therefore, they were both used in the failure analysis of a bandsaw blade. It is worthmentioning that without the crack front profile data the calculations would be basedon some assumptions of the crack initiation shape which needs further investigation infuture study.Chapter 7Fatigue-Life Prediction of a Bandsaw Blade7.1 IntroductionLife predictions of bandsaw blades have been made by previous investigators [10, 5].The predictions are based on two approaches: (1) the traditional smooth body designapproach [10, 5] and (2) the fracture mechanics based fatigue theory approach [5].Smooth body design approachSmooth body design assumes that there are no cracks present in the structure. Itdoes, however, include surface roughness and stress concentration effects. The smoothbody design approach also qualitatively ignores the distinction between the initiation,propagation and failure phases and only estimates the time to failure. Allen [10] madeuse of the modified Goodman Diagram to calculate blade life and the allowable stressesin the blade. Allen's approach, however, is based upon the assumption of achieving aninfinite life of one million cycles, which is not achieved in practice. In addition, likeother previous investigators, Allen did not distinguish the SCF under tension loadingfrom those under out-of-plane bending. The stress concentration factor he used was farlarger than expected in service. Lehmann [5] performed finite life calculations based onan S-N diagram employing the material fatigue limit and yield strength. However theseresults could not explain the existence of the cracks in the blades at low lives as seen inservice. Unfortunately, Lehmann's assessment lacked first-hand experiments to providedata on the material studied, therefore, the results could not satisfactorily explain the88Chapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 89blade life-span of typically less than four hours.Fracture mechanics approachThe fracture mechanics approach allows for a quantitative differentiation betweeninitiation, propagation and failure stages of crack growth. Specifically,• crack initiation—quantified by the numbers of cycles consumed for a crack to benurtured and grow to some arbitrarily defined length a,.• crack propagation—(1) propagation rate: quantified by experimentally obtainedrelations [15], e.g. the Paris' law: daldN = C(AK)b (C and b are material con-stants). (2) propagation time : quantified by integrating the above relation withan upper limit of ac defined in the following item and a lower limit of a,.• failure—quantified by critical crack length ac which is calculated from the fracturetoughness IC1c or K.As noted previously, Lehmann applied fracture mechanics concepts to explain crackingproblems in saw blades [5]. However most of his results were based on assumed materialproperties, such as fracture toughness and crack propagation rates. In the followingsections the assessment of fatigue life for a bandsaw blade in this research program arediscussed. In comparison to the previous investigators, a much better assessment ispossible as more first-hand information about the fracture aspects of bandsaw blades hasbeen obtained. In this analysis, it is assumed that the pre-strain stresses and the bendingstress fields as created by the wheel provide the dominant driving force for cracks formingin the blade (see Chapter 2).Chapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 90criticalcracklengthloadcasesetatIcload(In)dynamicload(In)difference(in)el.E-60EWe04---t-.g .±.)rEta.oo0.. .26c\I2.825 1.24 1.04occi; x).5 --43.00 1.34 1.660otri,--,3.12 1.44 1.68---6, 7-03t "eroiE.0,._ —.....RIIEa.oco0 -0 2.48 0.94 1.5407=1-co,__,2.52 0.99 1.5400o -c'It)-,,2.62 1.09 1.53Table 7.4: The assessment of critical crack length7.2 Critical crack length of a bandsaw bladeThe criteria for component fracture is typically assumed to be the critical crack size,beyond which the crack growth may become and catastrophically fast. The critical cracksizes for commonly used blade loading conditions have been estimated. For comparison,two cases were considered: the blade (1) without residual stress and (2) with an assumedresidual stress of 10 ksi.(69 MPa) [5]. The model assumed the an edge crack in a plate offinite width subjected to tension and bending. Table 7.4 lists the results of this analysis(See Appendix A for the details of the calculations).According to the results listed in Table 7.4 it is clear that both pre-strain force levelsand residual stresses have a significant influence on the critical crack length. As far asthe pre-strain force is concerned, the difference in critical crack length can be up to 16%.If a residual stress of 10 ksi (69 MPa) is involved, the difference jumps to as high as 50%.Chapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 91The difference in crack length between the dynamic and static load condition can be asmuch as 1.68 in.(42.7mm). The range covered by the dynamic critical crack length underthe condition considered is 0.94-1.24 in.(25.1mm-31.5mm).In practice [5], when a cracked blade is removed from the machine and examined,the length of the crack(s) are typically on the order of 1/16 in.(1.6mm), i.e. just visible,to a maximum of about 1.5 in. (38mm). The Worker's Compensation Board of BritishColumbia defines (see [12] article 66.36 (1)) the maximum safe length of a crack to be1/10 of the blade width (i.e. about 0.95 in.(23mm) for the blade studied in this program).This limit is believed to be on the conservative side of practical experience. Generallyspeaking, a crack in a blade can grow in a stable manner up to about 0.95 in.(23mm)in length. At this point, it is below or very close to the estimated non-residual stresscritical length range from 1.24-1.44 in.(31.5-36.6mm). The above predictions of criticalcrack length suggests that depending on the initial assumptions, a crack may grow aslong as 0.94-1.44 in (23.9-36.6mm) prior to reaching the critical length.7.3 Propagation life of a bandsaw blade7.3.1 Fatigue loads experienced by a bladeAccording to the experimental evidence obtained from the crack growth tests (see Chap-ter 6), a crack starts at the corner of the outer surface (Fig.6.50), then propagates radiallyas a corner crack until the crack front reaches the inner surface. The crack then propa-gates as a through-thickness crack. In this process, both bending and tensile stresses areinvolved in driving the crack. As discussed in Chapter 6, in half of a revolution, theblade experiences two different load periods: 1) pure tension; and 2) tension plus bend-ing. Thus, the outer surface of the blade experiences 1) tension produced by pre-strainforce, 2) pre-strain tension plus the tension induced by bending, while the inner surfaceChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 92experiences 1) pre-strain tension, 2) pre-strain tension plus the compression induced bybending (Fig.1.2). Therefore, both the inner and the outer surfaces experience two dif-ferent alternating stress fields. The changes in crack configuration and stress field requiredifferent methods to evaluate the SIF along the crack front.Before the corner crack becomes a through thickness crack the stress field changesalong the whole crack front can be calculated using the formulation developed by New-man [23] (see Appendix A for details), while after the crack penetrates the thickness,the through thickness feature must be considered in the calculation of the SIF alongthe curved front. During this stage, the SIF under tension as expressed by eqn. (3.11)(Chapter 3), and the SIF under out-of-bending loads correlated by Wilson and Thomp-son [24] (see Appendix A for details) were used to evaluate the stress intensity factorsat both the outer and the inner tips of the crack front.An assumption for the initial crack shapeThe actual initial flaw shape in the blade is typically unknown, therefor, an assump-tion was made that the initial crack front has a specific symmetric geometric profile, e.g.a quarter circle. Certainly, this assumption effects the final prediction of life-time of thecomponent studied. In the case of a handsaw blade, as discussed in Chapter 6, becauseof the out-of-plane bending load, a crack initiates as a corner crack, and extends in twodimensions. If the dimension of the initial crack is sufficiently small, the assumed profileof a quarter circle should be a close approximation. In the following prediction, a quartercircle profile with a lmm, radius was assumed for the profile of initial crack front in theblade.An assumption for the transition from a corner crack to through-thicknesscrackIn a blade, crack growth from a corner crack to the critical crack length is not asingle-phase process in terms of crack configuration, i.e. the crack grows, firstly, asChapter 7. Fatigue-Life Prediction of a Ban dsaw Blade^ 93a corner crack, then in a transition phase, and finally as a through-thickness crack.Theoretically, there is a critical point through which the crack transforms itself froma part-through or corner crack into a through-thickness crack. Crack growth behaviorbefore and after this point are different because of the different crack configurations,therefore, the corresponding SIF's must be evaluated in different ways. How a crackbehaves during this transition phase is still unknown. An assumption must be madeabout this transition (see Fig.7.52)In the following assessment, it is assumed that as soon as the side tip of the crackfront reaches the inner surface of the blade (point A), the crack growth at point A startsto propagate towards the center of the blade. At the two tips of the crack front, crackgrowth occurs at different rates as they precede into the blade until the final critical cracklength is reached.Notch field dimensionAs discussed in Chapter 2, each individual tooth of the blade acts as a notch whichsharply raises the stress level when the blade is loaded. The contributions to the stressincrease are different for the pre-strain loading component and for the out-of-plane bend-ing component. In the crack propagation calculations the stress concentration effects areaccounted for in the form of the so-called 'notch field' [25]. The notch field defines thedistance to which the stress concentration effect extends. In 1981, Smith [25] developeda simple equation to quantify the dimension of the notch field which is dependent onboth the notch depth and its radius of curvature (see Fig.7.53). For a notch with a depthof ari and radius of curvature of pin the dimension of the notch field can be expressedas 0.13Vanpn. For the bandsaw tooth studied having a,, = 0.55in, pi, = 1.08in, thenotch field is evaluated as 0.13Vanpn = 0.13V0.54 x 1.08 = 0.1in = 2.54mm, which isclose to the thickness of the blade (0.073 in), i.e. the stress concentration factors for thetension and the bending have effect only within this distance. It can be seen that a crackChapter 7. Fatigue-Life Prediction of a Ban dsaw Blade^ 94Figure 7.52: Transition of crack growthChapter 7. Fatigue-Life Prediction of a Ban dsaw Blade^ 95a.illiitififfiffii i iiiiicrFigure 7.53: Definition of a notch fieldexperiences different stages, i.e. initiation, growth through the notch field, and growthup to the critical crack length. During the whole propagation period, the crack drivingforce, and the SIF along the curved crack front varies from stage to stage, thus the crackgrowth rate varies in these different stages. Therefore, any prediction must evaluate thevarying growth rates in these individual stages.7.3.2 Analysis of growth life for a bladeBased on the critical crack length, assumed initial crack shape and notch field dimen-sion described, and using the assumptions discussed previously, assessments of the cycliclife of a blade were made using the experimentally obtained daldN—AK relationship(eqn.(5.18) in Chapter 5). Table 7.5., 7.6 and Fig.7.55-7.63 show the results (see Ap-pendix A for the details of calculations). The crack initiation life as experimentallyChapter 7. Fatigue-Life Prediction of a Ban dsaw Blade^ 96obtained from the small specimen tests were used (see Chapter 4). The specific geom-etry and load parameters used in the prediction are listed in Appendix A.How a crack grows in the blade—qualitative analysisFig.7.54 shows a schematic illustration of the crack growth process in a blade of abandsaw machine having a driving wheel of 5 ft.(1.5m) diameter working with a pre-strainforce of 20,000 lb.(89,000 N). To assist in the description of the crack growth process,the entire growth period is divided into five 'snap-shots'. Each of these 'snap-shots'represents the crack front profile at one or more typical points during its growth period.Snap-shot (a)—Crack initiation. The crack initiates one way or another as a quartercrack with a radius of lmm.Snap-shot (b)—The crack grows as a corner crack. In this period, the SIF along theentire crack front can be evaluated from Newman's equation (eqn.(A.21) in AppendixA). However, in this period there are two stages. In the early stage, it is clear fromNewman's equation that the SW range along the crack front are determined only by thebending load (AK -= (o-B HaT)F — Ho-TF aBF, where H, F are geometry factorsaccounting for the shape of the crack front). Therefore, the crack growth in this stageis controlled by the bending load. Since the stress field produced by the bending is alinear distribution across the thickness, the SIF range of the side tip of the crack frontbecomes zero at some point A. From point A down, the second stage starts, at the sidetip, the SIF produced by the bending is either equal or less than zero, while the SIFgenerated by the tension stays constant, thus the SIF range at the side tip of the crackfront is evaluated by the pre-strain force. Therefore, crack growth in this second stageis controlled by the pre-strain force. In the entire period, the outer tip (point C of thecrack front grows faster because of its higher SIF range, while the side tip of the crackfront grows slower and slower because its stress intensity becomes lower and lower asthe tip approaches point A. Therefore, the crack front is stretched into a quarter of an(c) thru—thk crack(c) Reaches transition pointcriticalcracklength111."-1outer tip of crackextends into bladeouter tip of crackextends into blade/ /Ainner tip of crackextends into bladeinitialcrackrt}(a) Crack initiatesAK=0(b) Grows as corner crackChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 97inner tip of crackextends into blade(e) Inner tip catchs up and surpass outer tipFigure 7.54: Schematics of crack growth in a bladeChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 98elliptic profile.Snap-shot (c)—Transition into a through-thickness crack. As soon as the side tipof the crack front reaches the inner surface, the tip starts to precede into the centerof the blade. The extension in each of the first several thousand cycles are relativelylarge because the SIF at point B is driven up by both its sharp corner and the stressconcentration of the tooth 'notch field'. After the crack becomes through-thickness, theSIF generated by the bending can be evaluated using Wilson's equation (eqn.(A.36) inAppendix A), while the SIF produced by the pre-strain force was calculated usingeqn.(A.37).Snap-shot (d)—The inner side of the crack front catches up. After the transition intoa through-thickness crack, the SIF range at the inner crack tip of the crack front is higherthan that at the outer crack tip, therefore the inner crack tip advances faster and catchesup to the outer crack tip.Snap-shot (e)—The inner-side tip surpasses the outer-side tip. After the inner-sidetip catches up the outer-side tip, for a period of time, it leaves the outer-side tip behindfurther and further. At a point, the difference between the two tips is large enough sothat the outer tip develops a sharp corner which raises its SIF range, thus advancingfaster than before for a period of time. Then the distance between the two tips becomessmaller and smaller until SIF range of the outer tip becomes lower than that of the innertip, then the distance starts to wider again. This process repeats itself again and againuntil the critical crack length is reached. In this period, the inner-side crack tip is alwaysahead of the outer tip after it catches up the outer crack tip in the case of 20,000lb.pre-strain force. It should be stressed that although the inner tip is always ahead of theouter one, the final failure is still controlled by the outer crack tip because no matterwhat the pre-strain force is, the maximum SW (not SIF range) at the outer tip is alwaysfar higher than that of the inner-side tip.Chapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 99How a crack grows in the blade—quantitative analysisTable 7.5 and 7.6 present the prediction of the crack propagation life for a bandsawblade. The tables also include the crack initiation lives of the specimens with differentsurface roughness (see Chapter 4). A tensile residual stress of 10 ksi. was also consideredas different cases in the calculations. The residual stress was estimated in reference [5]accounting for the stress induced by the roll tensioning process. In this calculation,the residual stress was treated as if an extra pre-strain stress field was imposed on theblade. To visualize the prediction, the information contained in tables is also presentedin graphical form in Fig.7.55-7.63. The crack growth lives are all presented in units oftime (i.e. minutes) which is converted from number of cycles based on the assumed speedof the saw blade. The loads and geometry parameters used in the calculations are alllisted in Appendix A.Crack growth lifeAccording to Table.7.5-7.6, without the assumed residual stress, the crack growth lifefor each of the three pre-strain levels exceeds 240 minutes (4 hours). The crack growthlife for a 20,000lb pre-strain force level is just over 240 minutes (4 hours) However, withthe assumed residual stress, the growth life for all cases are reduced dramatically, eventhe longest does not reach 120 minutes (2 hours). The residual stress reduces the crackgrowth life by at least 70%, at the worst by 90%. This observation has great practicalsignificance as in actual sawmills, it is typical that a new blade is rolled [5] so that thecenter strip is slightly longer than the tooth edge (called 'back crown') to maintain astraight cutting edge. This rolling introduces a compressive stress in the middle stripand tensile stresses at the tooth and back edges. Therefore it is expected that the crackpropagation life may be much shorter than 240 minutes (4 hours). For instance, in thecase of a 10 ksi. residual stress, the crack growth life can be as short as 76-113 minutes(1.2-1.6 hrs). Even if a crack initiation life of 44-47 minutes (for a rough gullet surface)Chapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 100A. without residual stress unit: minutesPre—strainForce Level(1000 lb)crack initiation crack •rowth life total fatigue lifefinesurfaceroughsurfaceas cornercrackas thru—thk.crackfor finesurfacefor roughsurface20 427 44 49 194 670.2 287.418 455 44 55 268 778.4 367.415 600 47 70 461 1131 577.8B. with residual tensile stress of 10 ksi. unit: minutes)Pre—strainForce Level(1000 lb)crack initiation (assumed) crack growth life total fatigue lifefinesurfaceroughsurfaceas cornercrackas thru—thk.crackfor finesurfacefor roughsurface20 427 44 36 40 503.2 120.418 455 44 37 48 540.4 129.415 600 47 39 74 713 159.8Table 7.5: Crack-growth life prediction in a bandsaw blade (A)Chapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 101A. without residual stressPre—strainForce Level(1000 lb)crack initiation as cornercrackas thru—thk.crack% of total fatigue lifefine surface rough surface % of total growth life20 64 15 20 8018 59 12 17 8315 53 8 13 87B. with residual tensile stress of 10 ksi.Pre—strainForce Level(1000 lb)crack initiation as cornercrackas thru—thk.crack% of total fatigue lifefine surface rough surface % of total growth life20 85 9 47 5318 84 8 44 5615 84 7 35 65Table 7.6: Crack-growth life prediction in a bandsaw blade (B)Chapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 102Figure 7.55: Crack-growth life: as a corner crackChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 103Figure 7.56: Crack-growth life: as a through-thickness crackChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 104Figure 7.57: Total crack-growth life: no residual stressChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 105Figure 7.58: Total crack-growth life: 10 ksi. residual stressChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 106Figure 7.59: Crack-growth life distribution (%): no residual stressChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 107Figure 7.60: Crack-growth life distribution (%): 10 ksi. residual stressChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 108Figure 7.61: Total fatigue life of a blade: fine surfaceChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 109Figure 7.62: Total fatigue life of a blade: rough surfaceChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 11012001000a)5 8006004002000total fatigue life of a bandsaw bladetotal fatigue life (no residual stress)20^18^15pre—strain force level (1000 lb.)1111 fine surface ^ rough surfaceFigure 7.63: Total fatigue life of a blade: surface roughness effectChapter 7. Fatigue-Life Prediction of a Ban dsaw Blade^ 111are added to the crack propagation life, the blade is still predicted to fail prior to oneshift of 240 minutes (4 hours). This explains why it is possible that failure can happento a freshly ground saw blade (although visible cracks are expected to be removed in thegrinding process).Residual stress effect can also be used in favor of a blade fatigue life. If a compressiveresidual stress field is introduced in the gullet region and its immediate neighbor area,crack growth can be slowed down dramatically and even stopped. The widely used'hammering' technique is based on this idea.Distribution of crack growth life and the role of pre-strain forceIn Table 7.5-7.6 (or Fig.7.55-7.63) it can be seen that in the case of a 20,000 lb.pre-strain force, most of the crack growth life is consumed as a through-thickness crack,i.e. the propagation life is controlled mainly by pre-strain force since the pre-strain forcecontrols the SIF during the through-thickness propagation (see the discussion in thesnap-shots (b) and (e)). On the other hand, as the pre-strain level increases, includingtensile residual stress, the percentage of the propagation life occupied by corner crackincreases up to 47%. This means that the bending load has relatively more contributionin promoting crack advance.Residual stress effectFrom Fig.7.56 it can be seen that in the process of through-thickness growth a tensileresidual stress of 10 ksi can reduce the growth life upto 80%. This is a very stronginfluence. If a compressive residual stress can be introduced into the blade, the crackgrowth life can be prolonged in the same scale.Role of crack-initiation life—surface roughness effectThe crack initiation lives obtained empirically are used in calculating the total fatiguelife for the assumed blade configuration. For the same pre-strain force, the crack initiationlives are different for varying surface roughness. Fig.7.61-7.63 show the influence of theChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 112surface roughness on the total fatigue life of the blade (here 10 ksi. residual stress isconsidered).A fine surface finish can prolong the fatigue life by at least 300 more minutes (5 hours),while the severe residual stress and high pre-strain force can reduce the propagation lifeto about 60 minutes (one hour), ideally, a fine surface finish on the tooth gullet still caneliminate the possibility of failure during a single 240-minute (4-hour) shift.The efficient method to improve fatigue life of a bladeThis detailed study of each phase of the cracking process of the blade leads one torecommend more realistic methods to improve the total fatigue life of a blade. Generallyspeaking, there are a number of ways to improve fatigue life of a bandsaw blade, theyare:1. To improve fracture properties such as increasing critical stress intensity factor,increasing the fatigue limit and decreasing the propagation rate.2. To decrease pre-strain force level to as low as is practical.3. To redesign the gullet region tooth shape.4. To introduce compressive residual stress on the gullet region.5. To improve the surface finish of the tooth gullet.To improve the fracture properties of a material is usually a long term and expen-sive proposition. Almost every stage of the manufacturing process has to be involved.Furthermore, to increase the critical stress intensity factor and decrease the propagationrate are not a positive method because these properties show their influence only after acrack occurs. If a crack comes into being, the performance of the blade will be affected,Chapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 113particularly when the crack grows longer. Therefore, in this sense, to improve theseproperties does not help to improve the performance of the blade significantly.To decrease the pre-strain force level is always limited by production requirements.The room for decrease is always limited in actual service.Improvement on the design of the tooth profile is very effective method to decreasethe SCF at the tooth gullet region. Its effect can be seen in Fig.2.5. The 'flattening'and 'shaving' proposed by Lee and co-worker was also this kind of effort. However,modification of the gullet region profile is always balanced by the production rate (sincethe tooth profile effects the tooth capacity), tooth strength and its stability.Introducing a compressive residual stress distribution in the gullet region can eitherincrease crack initiation time and slow crack propagating, or may even prevent a crackfrom initiating if the residual stresses are sufficenttly high. This is effectively equiva-lent to a decrease in pre-strain force level locally and will improve the fatigue life, whilestill maintaining the globe pre-strain force level unchanged, therefore retaining the de-sired blade stiffness. An example of imposing a compressive residual stress field is the'hammering' or 'pressing' method used currently on some bandsaw blades. It is recalledthat tensile residual stresses are produced through blade 'rolling' employed to assist inmaintaining blade position on the wheel and cutting edge stability. As this analysis indi-cates that the presence of tensile stresses can significantly reduce crack propagation time,blades free of rolling will ultimnatelly have a longer life. These alternative methods to'rolling' which maintain blade position and stability should be considered.As the previous discussion shows, surface roughness in the gullet region of a blade isthe most sensitive factor affecting its initiation life, and therefore the total fatigue lifeof a blade. Accordingly, in addition to introduction of compressive residual stress field,improving the surface finish appear to be the most economical, efficient and practicalway to prolong the fatigue life of a blade. In the above example, it can be seen that if theChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 114tooth gullet of a blade is carefully ground, the blade life could be several times longer.What is more important is that the initiation life could be extended beyond one shift, sothat cracks initiated could be ground away before the next shift starts.Future workAlthough this author believes that the prediction is more realistic than those done bythe previous researchers since several very important fracture properties of blade materialare now available, the following factors may still make it difficulty to provide accuratepredictions of fatigue life for actual bandsaw blades:1. The nature of crack initiation is random. A test of the specific blade steel witha large sample population is required to gain a high confidence level. Even so, aspecific life-time, e.g. its mathematical expectation, is meaningful only in the senseof probability. Also the initial shape of the corner crack has a strong effect on howthe crack will extend. To quantify the initial crack shape requires further theoreticaland experimental study. However, up to date predictions for 2-dimension crackpropagation have to be based on assumed initial crack shapes.2. The actual residual stresses induced by the manufacturing process are different foreach individual blade. The information about the residual stress distribution foreach individual blade is usually not available. Also each individual residual stressdistribution has a different effect on initiation and propagation.3. Other loads such as those induced by cutting, guiding, tilting, crowning etc. maybecome more influential when a crack appears, and may become even more so asthe crack grows.4. The global behavior of a blade may change when a crack becomes long and bladevibration becomes involved. For example, cracking may substantially release theChapter 7. Fatigue-Life Prediction of a Bandsaw Blade^ 115strain on one part of the blade, the consequent unbalance of strain plus the bladevibration may, in the worst case, derail the blade. In practice it is highly possiblethat subsequently the stress field may change dramatically so that the current stressmodelling becomes invalid.The above factors should be the focus of future study.Chapter 8ConclusionsIn this research program, fracture mechanics theory was employed as a guide to conduct aseries of experimental tests to obtain the fracture properties of bandsaw material. Basedon the data obtained, failure prediction of a bandsaw blade was performed to explaingullet cracking in handsaw blades, and a method to improve fatigue-life of bandsaw bladewas recommended. The following are the major conclusions of this research work:Material properties---tensile strengthThe bandsaw material tested has a very high strength, i.e its 2% offset yield strengthis 184 ksi. (1270 MPa) and the ultimate strength is 207 ksi. (1421 MPa). These resultsare in very good agreement with those obtained by a previous researcher [2].Material Properties—R-curveThe crack-line-wedge-loaded specimen technique was employed and the related load-ing devices were designed and constructed. A more reliable strain based double-calibrationmethod was developed and used in correlating the experimental records into a crackingresistance curve of the bandsaw material.The cracking resistance curve for the bandsaw material was obtained. The criticalstress intensity factor assessed using the R-curve is 220ksi.in1/2(242MPa.m1/2) whichreflects the material ability to handle a crack of a specific length when it is subjectedto a static load. When the load is cyclic, as in a bandsaw blade, the toughness of70ksi.in112(77MPa.m1/2), which is the turning point on the R-curve, should be used .Therefore, the toughness under dynamic loading is only about 30% of that under static116Chapter 8. Conclusions^ 117loading.Crack initiation lifeBoth the stress level and surface roughness of a blade have effects on the crack initi-ation life. However, within the range of the pre-strain forces used in an actual sawmill,the stress level, or in other words, the pre-strain level has a far weaker influence on thecrack initiation life than does the gullet surface roughness. A carefully ground gullet canprolong crack initiation life by as much as nine times longer than that produced withthe same sharpening wheel but without the extra care. What is more important is thata fine surface finish on a gullet can help a blade survive one shift of four hours so thatthe cracks initiated can be removed in the upcoming resharpening process.What is alarming is that for those specimens with a surface roughness which is consid-ered to be very close to an actual blade, crack initiation take place only after 44 minutesunder a 20,000 lb. per-strain force! Microscopic examination on a sample from a failedblade shows that the actual surface of a tooth gullet can be much rougher. It is easy toaccept that if the blade is in high-strain system, crack initiation life can be expected tobe even shorter.Crack growth rate for mode I crack in the handsaw materialThe mode I crack growth rate tests of the bandsaw material shows that withinAK > lOksi.in112(11MPa.m112) the stress ratio R has no noticeable effect on the crackpropagation rate.When AK > 46ksi.ini/2(50.6MPa.m1/2, the crack starts to advance with stable, butmuch larger growth increments, which shows the dynamic feature of the R-curve.Crack growth behavior under out-of-plane bendingWhen subjected to an out-of-plane bending load, crack growth starts from the corneron the tensile side, grows radially to some distance, then starts to accelerate at a pointon the tensile surface—the circular crack front grows into an elliptically curved front,Chapter 8. Conclusions^ 118and finally the lower part of the front reaches the compressive surface. After this pointthe crack front advances while maintaining its shape relatively unchanged.The crack growth life predicted using the empirical crack-growth rates were in agree-ment with the actual testing data, which proved that both the crack-growth rate andNewman's stress intensity factor equation are close to reality and useful.Failure analysis—critical crack lengthWith the critical stress intensity factor under both static and dynamic loads, thecritical crack lengths under different pre-strain forces have been assessed. The maximumpossible net critical crack lengths under dynamic loads range from 1.04 to 1.24 in (26.4-31.5mm) if residual stress is not considered. This explains why the existence of 1 in.(25.4mm) cracks in a blade is possible.Failure analysis crack growth modelA crack grows from the assumed initial corner crack of lmm radius, then the crackfront gradually becomes elliptical as the crack grows into the blade. When the side tipof the crack front reaches the inner surface of the blade, the crack becomes a through-thickness crack and extends as a through-thickness crack until the critical crack lengthis reached.The predictions based on the above model indicate that there are some cases that abandsaw blade can only last about two or even just over one hour. This explains why amid-service failure is possible. These predictions also indicate a significant effect of thepresence of residual stress on blade fatigue life. For the pre-strain force levels consideredin this study, a reduction factor of 4 on propagation life of a crack in the blade waspredicted as result of the presence of a 10 ksi tensile residual stress. Since tensile residualstress effects crack initiation life in the same manner as pre-strain force level does, totalfatigue life of blade are expected being reduced even further. This indicates strongly that'rolling' will have a detrimental effect on blade service life. In contrast, the introductionChapter 8. Conclusions^ 119of compressive residual stress should act to increase life. Thus methods which imposethese compressive stress (e.g. hammering or pressing) should be considered as a meansof extending blade service life.Improvement of bandsaw blade fatigue lifeIn addition to introduction of compressive residual stress field in gullet region, themost efficient, economical and practical way to improve bandsaw fatigue life is to grindit with extra care. In this way, the crack initiation life can be prolonged many times aslong as that without extra care.Bibliography[1] Ingema, E., 1977, "Reports on Gullet cracking in Bandsaw Blades", Uddeholm SteelResearch[2] Porter, A.W., 1971, "Some Engineering Considerations of High-Strain Band Saws",Forest Products Journal, vol. 21, No. 4, pp. 24-32[3] Lee, C., Robinovitch, S., Romilly, D.P., 1988, "Stress Analysis of a Bandsaw Bladefor Improved Fatigue Life", Final Report for Mech 455/456, Department of Me-chanical Engineering, University of British Columbia, Vancouver, B.C., Canada.[4] Jones, D.S. 1965 "Gullet Cracking Saws" Australian Timber J, 31 (7): pp 22-25[5] Lehmann, B.F., 1988 "Factors in the Cracking of Bandsaw Blades", Report forMech 550, Department of Mechanical Engineering, University of British Columbia,Vancouver, B.C., Canada.[6] Eschler, A., 1982 "Stress And Vibrations In bandsaw Blade", M.A.Sc. Thesis,Department of Mechanical Engineering, University of British Columbia, Vancouver,B.C., Canada.[7] Targard, C., 1989, "A Test Report: the Composition of Bandsaw Blade steel ",Stress Analysis Lab., Department of Mechanical Engineering, University of BritishColumbia, Vancouver, B.C., Canada.[8] Osgood, C.C., 1970, "Fatigue Design", Wiley-Interscience pub.[9] Broek, D., 1986, "Elementary Engineering Fracture Mechanics", 4th edition, Mar-tinus Nijhoff Publishers[10] Allen, F.E., 1973, "High-Strain/Thin Kerr', proceedings of the Modern SawmillTechniques, Portland, Oregon, Feb.1973[11] Timoshenko, S., 1955, "Strength of Materials" 3rd edition, McGRAW-Hill BookCompany.[12] 1980, "Industrial Health and Safety Regulations", Workers' Compensation Boardof British Columbia120Bibliography^ 121[13] Krafft, J.M., Sullivan, A.M., and Boyle, R.W, 1961, "Effect of Dimensions onFast Fracture Instability of Nothch Sheets", Proceedings of the Crack PropagationSymposium, College of Aeronautics, Cranfied, England, Vol. 1, pp. 8-26.[14] ASTM Designation: E561-81, "Recommended practice for R-curve Determina-tion", Am. Soc. Testing Mats., 1981.[15] Paris, P.C., 1964, "The Fracture Mechanics Approach to Fatigue", Proceedingsof the Tenth Sagamore Army Material Research Conference, Syracuse UniversityPress, 1964, pp 107-132.[16] Elber,W.,1971, "The Significance of Fatigue Crack Closure", ASTM STP 486, 1971,pp 230-242.[17] Irwin, G.R., 1957, "Analysis of Stresses and Strains Near the End of Crack Trav-elling a Plate", J. of Applied Mech., 24, pp 361-364, 1957.[18] Griffith, A.A., 1920, "The Phenomena of Rupture and Flow in Solids", Phil. Trans.Roy. Soc. of London. A221, pp 163-198,1920.[19] ASTM Designation: E647-86a, "Standard Test Method for Measurement of FatigueCrack Growth Rates", Am. Soc. Testing Mats., 1986[20] Cook, R.D., 1988 "Concepts and Applications of Finite Element Analysis", 3rdedition, John Wiley & Sons.[21] Savin, G.N., 1961, "Stress Concentration Around Holes", Pergamon Press.[22] Heyer, R.H., and McCabe, D.E., 1972 "Plane-Stress Fracture Toughness TestingUsing a Crack-Line-Loaded Specimen", Engineering Fracture Mechanics, EFMEA,Vol. 4, pp. 393-412[23] Newman, Jr. J.C., and Raju, I.S. 1981 "An Empirical stress-Intensity Factor Equa-tion for the Surface crack", Engineering Fracture Mechanics, Vol.15, No.1-2, pp.185-192[24] Wilson, W.K., and Thompson, D.C., 1971 Engineering Fracture Mechenics, Vol. 3,PP 97-[25] Smith, R.A., and Miller, K.J., 1977 International Journal of Mechanical Science,Vol 19, pp 11-[26] ANSYS FEM Software, Version 4.3, Swanson Analysis System Inc, 1992.[27] Roberts, R., and Rich, T., 1967 "Stress-Intensity Factor for Plate Bending", J. ofApplied Mech., 34, pp777-779, 1967.Bibliography^ 122[28] Alkins, A.G., and Mai, Y-M., 1988 "Elastic and Plastic Fracture Mechanics", EllisHorwood Ltd.Appendix ACalculations in the Failure AnalysisA.1 Bandsaw parameters used in the calculationsThe following assumptions are made in the following calculations:Sawmill machine and blade parameters:1. The diameter of the wheel: 5 ft.(1524mm).2. The distance between the axis of the wheels: 5 ft.(1524mm).3. The travelling speed of the bandsaw blade: 147.6 ft./sec. (45m sec4. Effective width of the blade: 9.5 in. (241mm).5. The thickness of the blade: 0.073 in.(1.85mm).Simplification of the loads1. Only the bending load caused by the wheel and the tensile load caused by thepre-stain force are counted because of their magnitude and cyclic nature.2. Simple beam theory [11] is used for the bending stress and one dimension tensiletheory [11] for stress caused by pre-strain force.3. Three pre-strain force levels are used: 20,000 lb (90,000 N), 18,000 lb.(72,000 N)and 13,000 lb.(58,500 N).See the Table A.7 for the results.123Appendix A. Calculations in the Failure Analysis^ 124stresslevelloadcasesstaticstress(ksi)bendingstress(ksi)V)rn05-o-I-' 4-30 ul0:9 74.-. 0..C/3(I)a.)004-1.--1Oa)74!i"'^.112,4-4I(I)F-1oCo dccp-CV14.6 350op,0 --cd,--I11.68 35or-,6 41.6-19.49 35gi"-' a)rng 2-;o '''-1 73..0 --.., -0. R' . FAa.)a)0&-.0ef -I5 TuI•FJ.^t- ....4-,rnIa )04C00.4oC•224.6 35gcl. 'L,co'21.68 35000 4in19.49 35Table A.7: The calculated loadsAppendix A. Calculations in the Failure Analysis^ 125A.2 K-formula used in the calculationsAccording to the load-crack configuration of the blade, the actual stress field at the cracktip is produced either by pure tensile load or by combination of bending and tensile loads.A crack goes through two different stages, i.e. a corner crack, and as through-thicknesscrack. The formula used to evaluate the stress intensity factor depends on the load typeas well as the crack configuration. Therefore, in the case of a gullet crack in a bandsawblade subjected to pre-load and out-of-plane bending, three formula were used in thecalculations.Newman's equation for a corner crack under tension and bendingNewman and co-worker (23] developed an empirical stress-intensity factor equationfor a surface crack. The stress-intensity factors used to develop the equation were ob-tained from their previous three-dimensional, finite element analysis of a semi-ellipticalsurface crack in finite elastic plates subjected to tension or bending loads. A wide rangeof configuration parameters were included in the equation. The ratios of crack length toplate thickness and the ratio of crack depth to crack length ranged from 0 to 0.1. The ef-fects of plate width on stress-intensity variations along the crack front were also included.In order to facilitate readers in refering to his original paper, the original symbols werepreserved in the following citation.Notations used in Newman's equationFor the following definitions of the notations, refer to Fig.A.64 and Fig.A.64.a -,- depth of surface crack.b^half-width of cracked plate.c = half-length of surface crack.F = stress-intensity boundary-correction factor.h = half-length of cracked plate.Appendix A. Calculations in the Failure Analysis^ 126= mode I stress-intensity factor.K. = elastic fracture toughness.Al = applied bending moment.Q = shape factor for elliptical crackSb = remote bending stress on outer fiber, 3M/be.St = remote uniform-tension stress.t = plate thickness.= parametric angle of the ellipse.Stress-intensity factor equation for the surface crackThe empirical equation for the stress-intensity factor for a surface crack in a finiteplate subjected to tension and bending loads were fitted as follows,K1= (St+ HSb) 7r rt—F(c±,--,,43)^ (A.21)Q t c bThe equation is valid for 0 < alc <= 1.0,0 <= alt <1.0,c1b < 0.5 and <= cP <=A useful approximation for geometric factor Q isQ = 1 + 1.464()1.65^(c± < 1)^(A.22)The function F and H are defined so that the boundary-correction factor for tensionis equal to F and boundary-correction factor for bending is equal to the product of Hand F. The function F was taken to beF =1M1+ M2(7)2 + M3(c-:)41.49f.^ (A.23)where= 1.13 — 0.09(:)^ (A.24)2hAppendix A. Calculations in the Failure Analysis^ 127Figure A.64: Surface crack in a finite plateI^ I2C2b^••••2hIAppendix A. Calculations in the Failure Analysis^ 128S tM/0"."••••\-t\.-......"ki3MSt =btaSt(a) Tension^(b) BendingFigure A.65: Surface-cracked plate subjected to tension or bending loadsAppendix A. Calculations in the Failure Analysis^ 129(A.25)(A.26)(A.27)1 . 0M3 = 0.5 — ^ + 14(1.0 — c±d240.65 + (a/c)g =1 +10.1 + 0.35(-t )2111 — sin4))2The angular function ff. issin2c1/4= [(a)2)24,The finite-width correction function fu, is[sec'71 C a 112tThe function H is0.89M2 = —0.54 + 0.2 + (a/c)(A.28)(A.29)H = H1+ (H2 — Hi)sinP(D^ (A.30)whereIn the equation for H2p= 0.2 + (c±c)H1 = 1 — 0.30t) — 0.11(:)(7)H2 = 1 + C1() G2(t-t1-)(A.31)(A.32)(A.33)Appendix A. Calculations in the Failure Analysis^ 130G1 = —1.22— 0.12(-2i)^(A.34)G2 = 0.55 - 1.05( (±)° 75 + O.47()15^(A.35)For a corner crack as in the case of bandsaw blade, eqn.(A.21) should be correctedby factor 1.12 [9] to account for the free-edge effect.In the prediction calculation, eqn.(A.21) was used in snap-shot (a), and (b) (seeChapter 7).Wilson's equation for a through-thickness crack under bendingWilson and co-worker's work [24] concerns a long plate of width 2b and thickness twhich contains a crack of length 2a located symmetrically between the edges of the sheetand perpendicular to them (Fig.A.66). The plate is subjected, remote from the crack,to a uniform moment M about an axis parallel to the crack line. Using a finite elementmethod, they studied this configuration. The opening mode stress intensity factor K1 atPoint A was calculated and fitted to a curve K1/K0 vs alb (see ref.[24]. Ka is the openingmode stress intensity factor at point A for an isolated (b = oo) crack and is given byKa = (1+ v) 6M Ora^ (A.36)(3 + v) t2For a edge crack as in the case of bandsaw blade, eqn.(A.36) should be corrected byfactor 1.12 [9] to account for the free-edge effect.It should be noted that in this equation, 6M/t2 is the maximum tensile stress in theouter fiber of the plate.It is worth mentioning that within alb = 0.19 the maximum ratio K1/K0 is less than1.02. In the case of bandsaw blade, the critical crack length is less than 19% of its width,A•^1 AA2bAppendix A. Calculations in the Failure Analysis^ 131Figure A.66: A long plate with a crack subjected to bendingtherefore eqn.( A.36) times a free edge factor (1.12 for tension, 1 for bending) can beused directly.In the prediction calculation, eqn.(A.36) was used in snap-shot (c), (d),(e) and thecritical crack length calculation (see Chapter 7).K-formula for edge crack in a plateA gullet crack in a blade under a pre-strain load can be classified as a typical openingmode (mode I) edge crack in a plate subjected to pure tensile stress. Its stress intensityfactor formula is documented in reference [9] and re-presented as follows,aK =^— 0.41() + 18.7(-a )2w+ 53.85( wa )41 (A.37)Appendix A. Calculations in the Failure Analysis^ 132A.3 Calculations of critical crack lengthThe critical conditionK = Ke^ (A.38)WhereK—the stress intensity factor at the tip of the crack in the blade produced both bythe pre-strain force and the bending.Kc —critical stress intensity factor for the blade material (obtained in Chapter 3).To obtain the critical crack length ac, substitute the equation (A.36) and equation(A.37) into equation (A.38), use static stress in the Table.A.7 for a- in equation (A.37)and bending stress in equation (A.36) and solve equation (A.38) for a numerically.Critical crack length under static loading conditionAs discussed in Chapter 3, in the case of 'absolutely' static load, the critical stressintensity factor is as high as 220 ksi.in112, with an upper limit of 246 ksi.in1/2 and alower limit of 210 ksi.in112 (see Chapter 3). Therefore, in the calculations of criticalcrack length under static load, the above toughness were used.In the prediction calculation, equation (A.37) was also used in snap-shot (c), (d) and(e) (see Chapter 7).Critical crack length under a dynamic loadFrom engineering design point of view, even a static load should be control belowsuch a level that there is no more crack extension. Therefore, for a alternating load, itsamplitude should be always restricted to a level so that there is no more crack extension.To this end, a alternative 'dynamic critical stress intensity factor' is defined as the pointon R-curve where the R-curve starts to bend [28]. for the bandsaw blade this is 71ksi.in V'. The 95% confidence upper limit and lower limit are 120 ksi.inii2 and 42Appendix A. Calculations in the Failure Analysis^ 133ksi.inli 2 respectively. The effect on the critical crack lengths can be seen in Table 7.5.A.4 Residual stress considerationThe estimated 10 ksi (68.89 MPa) tensile residual stress induced by the manufacturingprocess is assumed [5] and superimposed upon the static stress.A.5 Calculations of crack growth lifeCrack propagation rateIn the calculation of fatigue crack propagation, the following empirical formula wasused (see Chapter 5).dadN 0.66(AK)2.86 (in/cycle)^(A.39)^ _ The unit for AK is ksi.in1/2.Crack growth model for the out-of-plane bendingIn Fig.6.51 (Chapter 6) the process of crack advance can be seen, i.e. the crackgrowth starts from the corner on the tensile side, grows radially to some distance, thenstarts to accelerate at a point on the tensile surface—the circular crack front grows intoan elliptically curved front, finally the lower part of the front reaches the compressivesurface. After this point the crack front advances while maintaining its shape relativelyunchanged. Based on the above analysis and the crack profile data measured using aoptical microscope, prediction of the crack growth for the six specimens tested weremade. The following are the parameters and data used in the prediction.Different crack growth rate in the two directionsAccording to the research done by J.C. Newman and co-worker [23], a surface crackhas a different crack growth rate in the thickness and surface directions (see Fig.A.64)da—dN CA(AKA) (A.40)dadN CB(AKB) (A.41)Appendix A. Calculations in the Failure Analysis^ 134even if the stress intensity factor ranges are the same. The crack-growth rates at pointA and point B along the crack front can be described independently as follows,At point A,At point B,For the same material, Newman and co-worker found that for their test data, arelationship GB^0.9nCA exists to corelate the empirical data.In the prediction for the out-of-plane bending testing, the result produced by usingGB = 0.91nCA was found to be closer to the actual crack-growth data.The crack closure/crack surface contact co-efficientsSince the specimens were subjected to out-of-plane bending, it is possible that crackclosure or crack surface contact occurs during crack growth, especially in the case of neg-ative stress ratios. The deviations in the predictions from the actual data were attributedto this closure or surface contact. If a coefficient U is introduced to account for theseeffects to match the actual data, the effective stress intensity factor can be correlated tothe calculated stress intensity factor using the geometry of the crack, i.e. AK, = UAK.In the prediction, the data (the crack extension and number of cycles) recorded for thelast step of the crack growth was employed to develop the coefficient U and the AK, wasused to calculate the da/dN. Table A.8 shows the results. The values of AK explain thedifference in growth rate shown in Fig.6.49 (Chapter 6).The calculation procedure for the fatigue-life predictionWith the initial crack length assumed and the crack growth rate rule available, theAppendix A. Calculations in the Failure Analysis^ 135Final Profilestress ratioat outersurfacehand meassured final • ofile 3stimated from dN =final step dNa(In)c(In)Thal stepdcfinal stepdNC_c =0.91" n•C_ade a/calternatingstress(i)meanstres8(ks0dK at Cksi.in ^1/2 Umax din %in daJdNR=^0.313 0.073 0.224 0.039 57906 0.039 0.32 16 30.6 12.38 0.93 2.95R.^0.227 0.071 0.266 0.039 55344 0.039 0.27 16 25.4 12.55 0.89 4.65R=^0.000 0.056 0.189 0.039 109891 0.043 0.30 16 18 1028 0.96 0.49R= —1.000 N/A N/A 0.015 97536 N/A N/A 32 0 N/A N/A N/AR= —0.373 0.068 0.177 0.039 16462 0.027 0.38 32 14.6 16.88 1.10 0.95R= —0.490 0.057 0.162 0.039 40461 0.038 0.35 32 10.95 13.94 0.96 12Table A.8: The crack growth calculation for the out-of-plane bending testingcrack growth life was calculated using a accumulation or cycle-by-cycle procedure, thatis,• input the initial crack geometry and stress intensity factor range, obtain da and deafter one cycle.• add the crack increment da and dc obtained in the previous step to the initial cracklengths to obtain new crack lengths.• input the new crack lengths and current stress intensity factor, obtain da and dcafter one cycle.• repeat ... until the desired crack length is reached.


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