INVESTIGATION OF GEOMETRICAL FACTORS FOR
INVESTIGATION OF GEOMETRICAL FACTORS FOR DETERMININGFRACTURE TOUGHNESS WITH THE MODIFIED RING TESTA THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCESOFMIDDLE EAST TECHNICAL UNIVERSITYBYCEYDA ALPAYIN PARTIAL FULFILLMENT OF THE REQUIREMENTSFORTHE DEGREE OF MASTER OF SCIENCEINMINING ENGINEERINGSEPTEMBER 2008
Approval of the thesis:INVESTIGATION OF GEOMETRICAL FACTORS AND SIZE EFFECTFOR DETERMINING FRACTURE TOUGHNESS WITH THEMODIFIED RING TESTsubmitted by CEYDA ALPAY in partial fulfillment of the requirements for thedegree of Master of Science in Mining Engineering Department, MiddleEast Technical University by,Prof. Dr. Canan ÖzgenDean, Graduate School of Natural and Applied SciencesProf. Dr. Celal KarpuzHead of Department, Mining EngineeringAssoc. Prof. Dr. Levent TutluoğluSupervisor, Mining Engineering Dept., METUExamining Committee Members:Prof. Dr. Naci BölükbaşıMining Engineering Dept., METUAssoc. Prof. Dr. Levent TutluoğluMining Engineering Dept., METUProf. Dr. Celal KarpuzMining Engineering Dept., METUAssoc. Prof. Dr. Aydın BilginMining Engineering Dept., METUAssist. Prof. Dr. M. Ali HindistanMining Engineering Dept., Hacettepe UniversityDate:
I hereby declare that all information in this document has been obtainedand presented in accordance with academic rules and ethical conduct. Ialso declare that, as required by these rules and conduct, I have fullycited and referenced all material and results that are not original to thiswork.Name, Last name: Ceyda AlpaySignatureiii:
ABSTRACTINVESTIGATION OF GEOMETRICAL FACTORS FOR DETERMININGFRACTURE TOUGHNESS WITH THE MODIFIED RING TESTAlpay, CeydaM.Sc., Department of Mining EngineeringSupervisor: Assoc. Prof. Dr. Levent TutluoğluSeptember 2008, 120 pagesModified Ring specimens are of the shape of discs having a hole inside andflattened ends. These specimens are used for determination of Mode I fracturetoughness. Finite element program, named ABAQUS, is used for numericalmodeling for finding stress intensity factors. Varying disc geometries were usedfor the experiments and numerical modeling in which size of the flat ends,radius of the hole inside, and external radius of the specimen were varied.Experiments were done by using pink Ankara andesite. Effects of internal holeradius, external disc radius and size of the flat ends on both stress intensityfactor and fracture toughness were studied. In order to compare the results,fracture tests with semi-circular specimens under three point bending (SCB)were also performed. From a similar previous study, fracture toughness valuesof gray andesite were recalculated and compared to the fracture toughnessvalues of pink andesite for varying geometrical factors. Size effect studies wereperformed as well for varying diameter of core specimens.iv
Fracture toughness values of andesite were found to increase with increasingspecimen size. Fracture toughness of 100 mm specimens was determined as1.11 0.07 MPa m, whereas fracture toughness of 75 mm specimens was0.96 0.08 MPa m. 100 mm or larger diameter specimens were suggested forthe fracture toughness determination with the modified ring tests.Keywords: Stress Intensity Factor, Mode I Fracture Toughness, Modified RingTest, Size Effect, Numerical Modelingv
ÖZÇATLAK TOKLUĞU TAYĐNĐ ĐÇĐN GELĐŞTĐRĐLMĐŞ UYARLANMIŞHALKA TESTĐNE GEOMETRĐK FAKTÖRLERĐN ETKĐSĐNĐNARAŞTIRILMASIAlpay, CeydaYüksek Lisans, Maden Mühendisliği BölümüTez Yöneticisi: Doç. Dr. Levent TutluoğluEylül 2008, 120 sayfaÇatlak tokluğu tayini için geliştirilmiş halka numuneleri, içinde bir delikbulunan ve üst ve alt yüzeyleri düzeltilmiş disklerdir ve bu diskler Mod I çatlaktokluğu tayini için kullanılmaktadır. ABAQUS isimli sonlu eleman programıgerilme şiddet faktörünün belirlenmesinde kullanılmıştır. Farklı geometrilerdenumuneler ile deneysel ve sayısal modelleme çalışmaları yapılmıştır. Bunumunelerde, düzeltilmiş kenarların boyutu, içerdeki deliğin yarıçapı,numunenin dış yarıçapı değiştirilmiştir.Deneyler pembe Ankara andesiti kullanılarak yapılmıştır. Düzeltilmişkenarların boyutunun, iç deliğin yarıçapının ve numunenin dış çapının çalışılmıştır.Sonuçlarıkarşılaştırmak amacıyla, çatlak tokluğu testleri, yarım-dairesel örneklerdeeğilme deneyleri ile de yapılmıştır. Daha önceki benzer bir çalışmadan, griAnkara andezitinin de çatlak tokluğu tekrar hesaplanmış ve bu değerler farklıvi
geometrilerde pembe andezitinkilerle karşılaştırılmıştır. Numune boyutununçatlak tokluğuna etkisi de incelenmiştir.Andezitin çatlak tokluğu değerinin artan numune çapıyla arttığı bulunmuştur.100 mm’lik numunelerin çatlak tokluğu değeri 1.11 0.07 MPa m olarakbulunurken, 75 mm’lik numunelerin çatlak tokluğu değeri 0.96 0.08 MPa molarak bulunmuştur. Uyarlanmış halka testlerinde 100 mm’lik numunelerinkullanılması uygun görülmüştür.Anahtar Kelimeler: Gerilme Şiddet Faktörü, Mod I Çatlak Tokluğu, UyarlanmışHalka Testi, Boyut Etkisi, Sayısal Modellemevii
TO MY FAMILYviii
ACKNOWLEDGEMENTDuring the writing of this thesis I was fortunate enough to obtain substantialguidance from my supervisor, Assoc. Prof. Dr. Levent Tutluoğlu, for hishelpful comments and advice. I would like to take this opportunity to give himmy deepest appreciation.I want to thank to the examining committee the members, Prof. Dr. CelalKarpuz, Prof. Dr. Naci Bölükbaşı, Assoc. Prof. Dr. Aydın Bilgin andAssist. Prof. Dr. Mehmet Ali Hindistan, for being interested in this thesis.For their help in my laboratory works, I’m grateful to Tahsin Işıksal, HakanUysal and Arman Koçal.I have to give a lot of credit for what Çiğdem Alkılıçgil has done for me. Shewas there for me whenever I need help.I’d like to show my gratitude to all my friends. They are always with me, in mybest and worst times.I truly appreciate to all my family, for their unconditional support during mythesis and all my life.I want to express my deepest appreciation to Berk Alpay for his patience andlove.ix
TABLE OF CONTENTSABSTRACT . ivÖZ . viACKNOWLEDGEMENT. ixTABLE OF CONTENTS . xINTRODUCTION . 11.1 General. 11.2 Fracture Toughness Applications . 11.3 Statement of the Problem . 31.4 Objective of the Thesis . 51.5 Methodology. 51.6 Sign Convention . 61.7 Outline of the Thesis. 6FRACTURE MECHANICS. 72.1 History . 72.2 Linear Elastic Fracture Mechanics . 92.3 Fracture Modes . 102.4 Plane Stress and Plane Strain. 112.5 Crack Tip Stress Fields in an Isotropic, Linear Elastic Solid . 122 .6 Stress Intensity Factor . 142.8 Fracture Toughness. 172.7.1 Fracture Toughness Values of Some Rock Types. 18PREVIOUS STUDIES . 233.1ISRM Suggested Methods . 243.1.1 Chevron Bend (CB) Specimen Tests . 243.1.2 Short Rod (SR) Specimen Tests . 263.1.3 Cracked Chevron Notched Brazilian Disc (CCNBD) Specimen . 273.2Semicircular Core in Three Point Bending (SCB) . 29x
3.3 Modified Ring Test Studies . 304.1 Finite Element Programs . 344.1.1 ABAQUS CAE . 354.1.2 Verification Problem . 384.1.3 FRANC2D/L . 454.2 Numerical Modeling of the Modified Ring Specimens. 484.2.1 Geometry of the Models . 534.2.2 Stress Intensity Factor Computation. 534.3 Stress Distributions at the Crack Front . 554.4 Variation of Stress Intensity Factor with Geometrical Parameters of MRSpecimens . 57EXPERIMENTAL STUDIES . 625.1 Physical and Mechanical Properties of Pink Ankara Andesite . 625.1.1 Uniaxial Compressive Strength (UCS) Test . 635.1.2 Indirect Tensile Strength (Brazilian) Test . 655.2Fracture Toughness Tests . 675.2.1 SCB Specimen Preparation . 685.2.2 Modified Ring Specimen Preparation . 695.2.3 SCB Specimen Geometries . 715.2.4 MR specimen Geometries . 725.3Rock Fracture Testing . 745.3.1 SCB Tests . 745.3.2 SCB Test Results . 745.3.3 Modified Ring Tests . 775.3.4 MR Method Testing Procedure . 785.3.5 MR Test Results . 795.3.6 Computation of Fracture Toughness Values for MR Tests . 815.4Effects of Geometric Parameters on Fracture Toughness . 825. 5 Analysis with the Results of a Previous Study . 885.6Size Effect Studies. 90xi
CONCLUSIONS AND RECOMMENDATIONS . 96REFERENCES . 99A. LOAD – DISPLACEMENT GRAPHS OF SPECIMENS. 104B. SPECIMEN PHOTOS AFTER EXPERIMENTS . 118xii
LIST OF TABLESTable 2.1 Fracture toughness values of some rock types with related testingmethod . 19Table 4.1 Comparison of analytical solution with numerical modelingsolutions. 43Table 5.1 UCS Test Data and Results . 64Table 5.2 Brazilian test results . 66Table 5.3 Fracture data for SCB specimens . 76Table 5.4 Fracture data for MR specimens with 54 mm diameter . 83Table 5.5 Fracture data for MR specimens with 75 mm diameter . 85Table 5.6 Mechanical properties of gray Ankara andesite . 88Table 5.7 Corrected fracture toughness values of gray Ankara andesite . 89Table 5.8 KIC values of pink Ankara andesite with internal hole radiusof 8 mm (ri 8 mm, diameter 16 mm) . 91Table 5.9 KIC values of gray Ankara andesite with internal hole radiusof 8 mm (ri 8 mm, diameter 16 mm) . 92xiii
LIST OF FIGURESFigure 2.1 Fracture modes . 11Figure 2.2 Plane stress and plane strain . 11Figure 2.3 Local stresses near a crack tip . 12Figure 2.4 KI expressions for some common loading conditions . 15Figure 3.1 CB specimen geometry . 24Figure 3.2 Geometry of SR specimen . 26Figure 3.3 Geometry of CCNBD Specimen . 27Figure 3.4 Valid geometrical ranges for CCNBD . 28Figure 3.5 SCB specimen . 29Figure 3.6 Specimen geometry for diametral compression test . 31Figure 3.7 Modified ring test geometry. 33Figure 3.8 Modified Brazilian test with machined flat contact surfaces . 33Figure 4.1 Crack tip coordinate system and typical line integral contour . 36Figure 4.2 Geometry of the verification example . 38Figure 4.3 Brazilian disc stress field around the crack tip . 39Figure 4.4 Boundary conditions and crack region of the verification example 40Figure 4.5 Geometry of MR specimen . 41Figure 4.6 Stress Intensity Factor vs crack length. 42Figure 4.7 Plot of comparison of KI results; for re 38.5mm,ri 5mm, 2L 14mm. 44Figure 4.8 Deformed and undeformed model in FRANC2D/L . 47Figure 4.9 Mesh, boundary conditions, and the deformed shape of ABAQUSmodified ring specimen model . 48Figure 4.10 Crack front, crack extension direction and boundary conditions . 50Figure 4.11 Symmetry conditions for uncracked and cracked model . 51Figure 4.12 Mesh of a specimen with re 37.5 mm, ri 5 mm, β 10 . 52Figure 4.13 Crack propagation and contour region . 53xiv
Figure 4.14 Stress intensity factor versus a/re . 54Figure 4.15 Distribution of σyy at the crack front for different re/ri ratios . 56Figure 4.16 Distribution of σxx at the crack front for different re/ri ratios . 57Figure 4.17 Variation of KImax with respect to L/re ratio . 58Figure 4.18 3D Graph of KImax for different L/re and ri/re for 54 mmspecimens . 59Figure 4.19 3D Graph of KImax for different L/re and ri/re for 75 mmspecimens . 60Figure 4.20 Stress intensity factor change with respect to re/ri ratio . 61Figure 5.1 Uniaxial compressive test . 63Figure 5.2 Stress-Strain graph of UCS1 . 64Figure 5.3 Indirect tensile strength test . 65Figure 5.4 Stress–displacement graph of one of the Brazilian tests . 66Figure 5.5 Coring machine . 67Figure 5.6 Grinding machine. 68Figure 5.7 Opening of notches to SCB specimen . 69Figure 5.8 Lathe and drilling frame. 70Figure 5.9 Flattening of the ends of the specimen in the grinding machine . 71Figure 5.10 Geometry of SCB specimen. 72Figure 5.11 Geometry of the MR specimen . 73Figure 5.12 Load-Displacement graph of SCB-14 . 75Figure 5.13 Load-displacement graph of S5410A16 . 80Figure 5.15 Fracture toughness versus L/re for 75 mm specimens . 86Figure 5.16 Fracture Toughness versus ri/re . 87Figure 5.17 Fracture Toughness vs re/ri for ri 8 mm . 93Figure 5.18 Average fracture toughness of all specimens with varying ri and Lvs diameter of the specimen . 94xv
LIST OF SYMBOLS AND ABBREVIATIONSa: Crack lengthBDT: Uncracked Brazilian Disk TestCB: Chevron BendCCBD: Central Cracked Brazilian Disc under diametralcompression testCCP: Centre Cracked PanelCMOD: Crack Mouth Opening DisplacementCNBD: Chevron-Notched Brazilian DiscCTOD: Crack Tip Opening DisplacementD: Specimen diameterE: Young’s ModulusF: Applied loadFcr: Critical loadFImax: First maximum load in the modified ring testFmax: Maximum loadG: Strain energy release rateISRM: International Society for Rock MechanicsJ: J-integralK: Stress intensity factorKIc: Fracture toughnessKI: Stress intensity factor in Mode IKII: Stress intensity factor in Mode IIKIII: Stress intensity factor in Mode IIIKCB: Fracture toughness of chevron bend specimenKSR: Fracture toughness of short rod specimenxvi
L: Half length of flattened end of MRspecimenLEFM: Linear Elastic Fracture MechanicsLVDT: Linear Variable Differential TransducerMR: Modified Ring testMTS: Material Testing Systemr: Specimen Radiusre: External radius of the specimenri: Inner hole radiusS: Support SpanSCB: Semi-Circular specimen under three-point bendingSR: Short Rodt: Thickness of the specimenT: Traction vectorT0: Tensile strengthu: Displacement vectorUCS: Uniaxial Compressive StrengthW: Strain energy densityµ: Shear modulusν: Poisson’s Ratioε: Strainσ: Applied stressσcr: Critical stressσ11: Stress in horizontal (x) directionσ22: Stress in vertical (y) directionτ: Shear stress2γ: Surface energyг: Arbitrary path around crack tipxvii
CHAPTER 1INTRODUCTION1.1GeneralCrack, in its dictionary meaning, is to break or burst with or without entireseparation of the parts. Fracture mechanics is the science which examines thecracked bodies. Formation of cracks is important for many sciences andengineering uses.Fracture toughness is a parameter to determine the stress required to drive apreexisting crack. It is an important parameter; because materials generallyhave a preexisting crack. Although, the ways to determine fracture toughnesshave been studied by many researchers, still there is no standard method. Butsome suggested methods are present.Suggested methods for determination of fracture toughness are not very easy toperform. Modified ring test is not a suggested method; but this method usespecimens which are easily prepared. Since there are not many studies onmodified ring test, this method has to be improved.1.2Fracture Toughness ApplicationsEngineering materials are full of cracks. For civil engineering applications;although structures can be safely built with these materials, this fact is relevantforthe designof a wide class of structures. The presence of stress1
concentrations in notches, around holes, in connections, etc. requiresophisticated design procedures. With the development of Fracture Mechanics,whether the cracks are stable or not becomes more important than if it exists ornot. The high end in Fracture Mechanics applied to concrete indicates a greatvariety of models to simulate concrete behavior. These models require thatparameters be obtained from concrete samples to characterize, basically,resistance to crack propagation. (Prado and Mier, 2003)Medical use of fracture mechanics shows its importance in biomaterials such asstents. Stents are small tubes that are inserted into the body for several reasons.Inside the body, stents are exposed to many cycles of loading and sometimessome overloads which may result in fracture. After this point, fracturemechanics is the science which should be used.Many researchers used strength or fracture toughness to find the toughness ofbones. In 2006, Yan et al. stated in their paper, that they applied elastic-plasticfracture mechanics to find the fracture toughness of the bone. They used the Jintegral, which is a parameter used to calculate energy per unit fracture surfacearea in a material, to quantify the total energy spent before bone fracture.Fracture mechanics is also used for the determination of fracture properties ofadhesively bonded aerospace material systems.In the paper of Choupani(2008), finite-element analyses of bonded joints and also mixed-mode fracturetoughness tests of the specimens consisting of several combinations ofadhesive, composite and metallic adherends were studied for the determinationof interfacial mixed-mode fracture properties.2
Fracture mechanics have several usages in material sciences. Application offracture mechanics to adhesive joints is one of them. The effect on fracturetoughness of joint geometry, section size, strain rate and fracture mode arediscussed by Ripling et al. in 1963.Fracture mechanics is used in analyzing problems such as rock slope stabilityand well bore stability. Also, the applications of fracture mechanics fordesigning operations like rock blasting, hydraulic fracturing are increasing. Themost essential parameter in these studies is fracture toughness of rock material.Fracture toughness studies are also used to study the energy requirement tobreak coal during crushing. Cutting results in dynamic breakage of rocks wheredrilling and blasting operations break the rock partly in tension along preexisting or newly formed cracks which produces dust. Karl & Bieniawski(1986) studied effect of fracture toughness of coal to the formation of coal dust.One of the most important use fields of fracture mechanics is undoubtedlyhydraulic fracturing. Thiercelin (1989) showed the importance of the exactvalue for fracture toughness in hydraulic fracture. Abou-Sayed et. al. (1978)introduced a fracture mechanics approach to the hydraulic fracturing criterionassuming the existence of arbitrarily oriented cracks in rock. Rummel (1987)suggest that when the opening mode (Mode I) stress intensity factor KI at thetip of the crack reaches a critical value, KIC (fracture toughness), hydraulicfracturing will occur.1.3Statement of the ProblemSince rock fracture mechanics has been widely applied to many branches ofcivil and mining engineering, researchers are still studying to find a simple and3
accurate standard method to determine fracture toughness. It is difficult toachieve accurate fracture toughness values in the case of soft and heterogeneousrocks. In the case of subsized specimens, the existence of a large process zoneahead of the crack tip makes the determination of the apparent fracturetoughness crack-length dependent.Most of the existing test techniques for fracture toughness determinationrequire specimens with preliminary notches or cracks of different shapes.Preparation of specimens with these sophisticated notch shapes and fatiguepre-cracking for some test techniques, take a lot of effort and time. With itssimplicity in specimen preparation and with its compressive loading procedurethe modified ring testing method is a future candidate as one of the suggestedmethods of ISRM for fracture toughness testing on core-based specimens.The Modified Ring Test uses the concept of effective crack length andconsiders an adequately long crack to be able to neglect subcritical crackpropagation. This test also prevents development of a large process zone aheadof the crack tip (Thiercelin and Roegiers, 1986). It is easy to prepare modifiedring test specimens compared to the other specimen geometries since there is ahole inside and crack initiates from this hole, there is no need to open a newcrack. So crack forms by itself.In order to become one of the suggested or standard methods for fracture testingextensive studies including effects of geometrical factors and specimen size areto be conducted. For this testing method, stress intensity factors for differentspecimen geometries and sizes should be computed and served for the universalapplications of this specimen geometry.4
1.4Objective of the ThesisThe aim of this study is to investigate the effects of geometrical factors onMode I fracture toughness of rocks. Effects of inner hole radius, size of loadingends, and specimen size, that is, the diameter of core specimens will beinvestigated. This way it is hoped here that contributions is made in efforts todevelop another core-based testing technique, and in efforts to include thistechnique as one of the suggested methods of ISRM. The modified ring testingtechnique is expected to find wide applicability, since it does not require anypreliminary notch or crack, provided that appropriate geometries and specimensizes, and accurate stress intensity factors in general are determined and servedfor general use.1.5MethodologyExperimental studies for development of modified ring test start with cuttingthe rock blocks into smaller pieces so that cores can be taken. After the coresare taken, these cores are cut into discs with desired width. Then these discs aredrilled with drill bits having different diameters. Following that the surfaces ofthe discs are flattened and polished, and then two ends of the specimens areflattened. After preparing the specimens, experiments start under compressiveloading, the data are collected, and load - displacement graphs are drawn.Tests are carried out with specimens having 54, 75, 100 and 125 mm diameterspecimens. Like the size of the specimen, inner hole diameter and the size ofthe flat ends are changed.ABAQUS program was used for numerical modeling. After drawing thegeometry of the specimens in the program and giving their properties, load and5
boundary conditions, stress intensity factor versus crack length graphs weredrawn. From these graphs maximum stress intensity factor was computed.Using this maximum stress intensity factor together with the maximum loadtaken from the load-displacement graph of the experiments, fracture toughnesswas calculated. The same procedure was repeated for varying specimendiameters, inner hole diameters and flat end sizes.Half-disc specimens were prepared for the semicircular bending tests underthree point bending (SCB). Fracture toughness values obtained from modifiedring tests are compared with the results of SCB tests.1.6Sign ConventionIn rock mechanics, compressive stresses are accepted to be positive and tensilestresses are accepted to be negative. But in this study, since general linearelastic fracture mechanics and a finite element program; ABAQUS is used,stresses of compression and tension are negative and positive respectively.1.7Outline of the ThesisThis thesis begins with a brief introduction in Chapter 1 and continues withhistory and some basic principles of fracture mechanics. Chapter 3 covers someprevious studies on fracture toughness determination. Explanation of the finiteelement program used and numerical modeling studies are given in Chapter 4.Laboratory work with experimental setup is elucidated in Chapter 5 and finallyChapter 6 covers the conclusion of this study and recommendations for furtherstudies.6
CHAPTER 2FRACTURE MECHANICS2.1HistoryIn 1913, C. E. Inglis noticed that, an elliptical hole at a thin plate of glassbecomes greater when he pulled the plate at both ends. He found the points thatare feeling the most stress. He then looked at other plates having non-ellipticalholes and he found that the shape of the crack does not really matters incracking. The important thing in cracking is the length of the crack in
investigation of geometrical factors for determining fracture toughness with the modified ring test a thesis su
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