1.Problems of Fracture Mechanics and Fatigue

2. Problems of Fracture Mechanics and Fatigue A Solution Guide Edited by E.E. GDOUTOS Democritus University ofThrace, Xanthi, Greece C.A. RODOPOULOS Materials Research Institute, Sheffield Hallam University, Sheffield, United Kingdom J.R. YATES University ofSheffield, Sheffield, United Kingdom SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. 3. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-6491-2 ISBN 978-94-017-2774-7 (eBook) DOI 10.1007/978-94-017-2774-7 Printed on acid-free paper Ali Rights Reserved 2003 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2003 Softcover reprint ofthe hardcover 1st edition 2003 No part of this work rnay be reproduced, stored in a retrieval system, or transrnitted in any form or by any means, electronic, rnechanical, photocopying, rnicrofilrning, recording or otherwise, without written perrnission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 4. A book dedicated to those who can think, observe and imagine 5. Table of Contents Editor's Preface on Fracture Mechanics Editors Preface on Fatigue List of Contributors PART A: FRACTURE MECHANICS 1. Linear Elastic Stress Field Problem 1: Airy Stress Function Method E.E. Gdoutos Problem 2: Westergaard Method for a Crack Under Concentrated Forces E.E. Gdoutos Problem 3: Westergaard Method for a Periodic Array of Cracks Under Concentrated Forces E.E. Gdoutos Problem 4: Westergaard Method for a Periodic Array of Cracks Under xix xxiii XXV 3 11 17 Uniform Stress 21 E.E. Gdoutos Problem 5: Calculation of Stress Intensity Factors by the Westergaard Method 25 E.E. Gdoutos Problem 6: Westergaard Method for a Crack Under Distributed Forces E.E. Gdoutos Problem 7: Westergaard Method for a Crack Under Concentrated Forces E.E. Gdoutos Problem 8: Westergaard Method for a Crack Problem E.E. Gdoutos Problem 9: Westergaard Method for a Crack Subjected to Shear Forces E.E. Gdoutos 31 33 39 41 6. Vlll Table of Contents Problem 10: Calculation of Stress Intensity Factors by Superposition M.S. Konsta-Gdoutos Problem 11: Calculation of Stress Intensity Factors by Integration E.E. Gdoutos Problem 12: Stress Intensity Factors for a Linear Stress Distribution E.E. Gdoutos Problem 13: Mixed-Mode Stress Intensity Factors in Cylindrical Shells E.E. Gdoutos Problem 14: Photoelastic Determination of Stress Intensity Factor K1 E.E. Gdoutos Problem 15: Photoelastic Determination of Mixed-Mode Stress Intensity Factors K1 and Kn M.S. Konsta-Gdoutos Problem 16: Application ofthe Method of Weight Function for the Determination of Stress Intensity Factors L. Banks-Sills 2. Elastic-Plastic Stress Field Problem 17: Approximate Determination of the Crack Tip Plastic Zone for Mode-l and Mode-ll Loading E.E. Gdoutos Problem 18: Approximate Determination of the Crack Tip Plastic Zone for Mixed-Mode Loading E.E. Gdoutos Problem 19: Approximate Determination of the Crack Tip Plastic Zone According to the Tresca Yield Criterion M.S. Konsta-Gdoutos Problem 20: Approximate Determination of the Crack Tip Plastic Zone According to a Pressure Modified Mises Yield Criterion E.E. Gdoutos Problem 21: Crack Tip Plastic Zone According to Irwin's Model E.E. Gdoutos Problem 22: Effective Stress Intensity factor According to Irwin's Model E.E. Gdoutos 45 49 53 57 63 65 69 75 81 83 91 95 99 7. Table of Contents Problem 23: Plastic Zone at the Tip of a Semi-Infinite Crack According to the Dugdale Model E.E. Gdoutos ix 103 Problem 24: Mode-III Crack Tip Plastic Zone According to the Dugdale Model 107 E.E. Gdoutos Problem 25: Plastic Zone at the Tip of a Penny-Shaped Crack According to the Dugdale Model E.E. Gdoutos 3. Strain Energy Release Rate Problem 26: Calculation of Strain Energy Release Rate from Load - Displacement - 113 Crack Area Equation 117 M.S. Konsta-Gdoutos Problem 27: Calculation of Strain Energy Release Rate for Deformation Modes I, II and III E.E. Gdoutos Problem 28: Compliance of a Plate with a Central Crack E.E. Gdoutos 121 127 Problem 29: Strain Energy Release Rate for a Semi-Infinite Plate with a Crack 131 E.E. Gdoutos Problem 30: Strain Energy Release Rate for the Short Rod Specimen E.E. Gdoutos Problem 31: Strain Energy Release Rate for the Blister Test E.E. Gdoutos Problem 32: Calculation of Stress Intensity Factors Based on Strain Energy Release Rate E.E. Gdoutos Problem 33: Critical Strain Energy Release Rate E.E. Gdoutos 4. Critical Stress Intensity Factor Fracture Criterion 135 139 143 147 Problem 34: Experimental Determination of Critical Stress Intensity Factor K1c 155 E.E. Gdoutos 8. X Table of Contents Problem 35: Experimental Determination of K1c E.E. Gdoutos Problem 36: Crack Stability E.E. Gdoutos 161 163 Problem 37: Stable Crack Growth Based on the Resistance Curve Method 169 M.S. Konsta-Gdoutos Problem 38: Three-Point Bending Test in Brittle Materials A. Carpinteri, B. Chiaia and P. Cometti Problem 39: Three-Point Bending Test in Quasi Brittle Materials A. Carpinteri, B. Chiaia and P. Cometti Problem 40: Double-Cantilever Beam Test in Brittle Materials A. Carpinteri, B. Chiaia and P. Cometti Problem 41: Design of a Pressure Vessel E.E. Gdoutos Problem 42: Thermal Loads in a Pipe E.E. Gdoutos 5. J-integral and Crack Opening Displacement Fracture Criteria 173 177 183 189 193 Problem 43: J-integral for an Elastic Beam Partly Bonded to a Half-Plane 197 E.E. Gdoutos Problem 44: J-integral for a Strip with a Semi-Infinite Crack 201 E.E. Gdoutos Problem 45: J-integral for Two Partly Bonded Layers E.E. Gdoutos Problem 46: J-integral for Mode-l E.E. Gdoutos Problem 47: J-integral for Mode III L. Banks-Sills Problem 48: Path Independent Integrals E.E. Gdoutos 207 211 219 223 Problem 49: Stresses Around Notches 229 E.E. Gdoutos Problem 50: Experimental Determination of J1c from J - Crack Growth Curves 233 9. Table of Contents Xl E.E. Gdoutos Problem 51: Experimental Determination of J from Potential Energy - Crack Length Curves 239 E.E. Gdoutos Problem 52: Experimental Determination of J from Load-Displacement Records 243 E.E. Gdoutos Problem 53: Experimental Determination of J from a Compact Tension Specimen 247 E.E. Gdoutos Problem 54: Validity of J1c and K1c Tests E.E. Gdoutos Problem 55: Critical Crack Opening Displacement E.E. Gdoutos Problem 56: Crack Opening Displacement Design Methodology E.E. Gdoutos 6. Strain Energy Density Fracture Criterion and Mixed-Mode Crack Growth Problem 57: Critical Fracture Stress of a Plate with an Inclined Crack M.S. Konsta-Gdoutos Problem 58: Critical Crack Length of a Plate with an Inclined Crack E.E. Gdoutos Problem 59: Failure of a Plate with an Inclined Crack E.E. Gdoutos 251 253 257 263 269 273 Problem 60: Growth of a Plate with an Inclined Crack Under Biaxial Stresses 277 E.E. Gdoutos Problem 61: Crack Growth Under Mode-ll Loading 283 E.E. Gdoutos Problem 62: Growth of a Circular Crack Loaded Perpendicularly to its Cord by Tensile Stress E.E. Gdoutos Problem 63: Growth of a Circular Crack Loaded Perpendicular to its Cord by Compressive Stress E.E. Gdoutos 287 291 10. xu Table of Contents Problem 64: Growth of a Circular Crack Loaded Parallel to its Cord E.E. Gdoutos Problem 65: Growth of Radial Cracks Emanating from a Hole E.E. Gdoutos 293 297 Problem 66: Strain Energy Density in Cuspidal Points of Rigid Inclusions 301 E.E. Gdoutos Problem 67: Failure from Cuspidal Points of Rigid Inclusions 305 E.E. Gdoutos Problem 68: Failure ofa Plate with a Hypocycloidal Inclusion 309 E.E. Gdoutos Problem 69: Crack Growth From Rigid Rectilinear Inclusions 315 E.E. Gdoutos Problem 70: Crack Growth Under Pure Shear 319 E.E. Gdoutos Problem 71: Critical Stress in Mixed Mode Fracture L Banks-Sills Problem 72: Critical Stress for an Interface Crack L Banks-Sills Problem 73: Failure of a Pressure Vessel with an Inclined Crack E.E. Gdoutos Problem 74: Failure ofa Cylindrical bar with a Circular Crack E.E. Gdoutos 327 333 339 343 Problem 75: Failure ofa Pressure Vessel Containing a Crack with Inclined Edges 347 E.E. Gdoutos Problem 76: Failure ofa Cylindrical Bar with a Ring-Shaped Edge Crack 351 G.C. Sih Problem 77: Stable and Unstable Crack Growth 355 E.E. Gdoutos 7. Dynamic Fracture Problem 78: Dynamic Stress Intensity Factor E.E. Gdoutos Problem 79: Crack Speed During Dynamic Crack Propagation 359 365 11. Table of Contents E.E. Gdoutos Problem 80: Rayleigh Wave Speed E.E. Gdoutos Problem 81: Dilatational, Shear and Rayleigh Wave Speeds E.E. Gdoutos Problem 82: Speed and Acceleration of Crack Propagation E.E. Gdoutos 8. Environment-Assisted Fracture xiii 369 373 377 Problem 83: Stress Enhanced Concentration of Hydrogen around Crack Tips 385 D.J. Unger Problem 84: Subcritical Crack Growth due to the Presence of a Deleterious Species 397 D.J. Unger PARTB: FATIGUE 1. Life Estimates Problem 1: Estimating the Lifetime of Aircraft Wing Stringers J.R. Yates Problem 2: Estimating Long Life Fatigue of Components J.R. Yates Problem 3: Strain Life Fatigue Estimation of Automotive Component J.R. Yates Problem 4: Lifetime Estimates Using LEFM J.R. Yates Problem 5: Lifetime of a Gas Pipe A. Afagh and Y.-W. Mai Problem 6: Pipe Failure and Lifetime Using LEFM M.N.James 405 409 413 419 423 427 Problem 7: Strain Life Fatigue Analysis of Automotive Suspension Component 431 J. R. Yates 12. XIV Table of Contents 2. Fatigue Crack Growth Problem 8: Fatigue Crack Growth in a Center-Cracked Thin Aluminium Plate 439 Sp. Pantelakis and P. Papanikos Problem 9: Effect ofCrack Size on Fatigue Life 441 A. Afaghi and Y.-W. Mai Problem 10: Effect of Fatigue Crack Length on Failure Mode of a Center-Cracked Thin Aluminium Plate 445 Sp. Pantelakis and P. Papanikos Problem 11: Crack Propagation Under Combined Tension and Bending 449 J. R. Yates Problem 12: Influence of Mean Stress on Fatigue Crack Growth for Thin and Thick Plates 453 Sp. Pantelakis and P. Papanikos Problem 13: Critical Fatigue Crack Growth in a Rotor Disk Sp. Pantelakis and P. Papanikos Problem 14: Applicability ofLEFM to Fatigue Crack Growth C.A. Rodopoulos 455 457 Problem 15: Fatigue Crack Growth in the Presence of Residual Stress Field 461 Sp. Pantelakis and P. Papanikos 3. Effect of Notches on Fatigue Problem 16: Fatigue Crack Growth in a Plate Containing an Open Hole Sp. Pantelakis and P. Papanikos Problem 17: Infinite Life for a Plate with a Semi-Circular Notch C.A. Rodopoulos Problem 18: Infinite Life for a Plate with a Central Hole C.A. Rodopoulos Problem 19: Crack Initiation in a Sheet Containing a Central Hole C.A. Rodopoulos 467 469 473 477 13. Table of Contents 4. Fatigue and Safety Factors Problem 20: Inspection Scheduling C.A. Rodopoulos Problem 21: Safety Factor of aU-Notched Plate C.A. Rodopoulos Problem 22: Safety Factor and Fatigue Life Estimates C.A. Rodopoulos Problem 23: Design of a Circular Bar for Safe Life Sp. Pantelakis and P. Papanikos Problem 24: Threshold and LEFM C.A. Rodopoulos XV 483 487 491 495 497 Problem 25: Safety Factor and Residual Strength 501 C.A. Rodopoulos Problem 26: Design ofa Rotating Circular Shaft for Safe Life 505 Sp. Pantelakis and P. Papanikos Problem 27: Safety Factor of a Notched Member Containing a Central Crack 509 C.A. Rodopoulos Problem 28: Safety Factor ofa Disk Sander C.A. Rodopoulos S. Short Cracks Problem 29: Short Cracks and LEFM Error C.A. Rodopoulos Problem 30: Stress Ratio effect on the Kitagawa-Takahashi diagram C.A. Rodopoulos Problem 31: Susceptibility of Materials to Short Cracks C.A. Rodopoulos Problem 32: The effect of the Stress Ratio on the Propagation of Short Fatigue Cracks in 2024-T3 C.A. Rodopoulos 519 529 533 539 543 14. xvi Table of Contents 6. Variable Amplitude Loading Problem 33: Crack Growth Rate During Irregular Loading Sp. Pantelakis and P. Papanikos Problem 34: Fatigue Life Under two-stage Block Loading Sp. Pantelakis and P. Papanikos Problem 35: The Application of Wheeler's Model C.A. Rodopoulos Problem 36: Fatigue Life Under Multiple-Stage Block Loading Sp. Pantelakis and P. Papanikos Problem 37: Fatigue Life Under two-stage Block Loading Using Non-Linear Damage Accumulation Sp. Pantelakis and P. Papanikos Problem 38: Fatigue Crack Retardation Following a Single Overload Sp. Pantelakis and P. Papanikos Problem 39: Fatigue Life of a Pipe Under Variable Internal Pressure Sp. Pantelakis and P. Papanikos Problem 40: Fatigue Crack Growth Following a Single Overload Based on Crack Closure Sp. Pantelakis and P. Papanikos Problem 41: Fatigue Crack Growth Following a Single Overload Based on 551 553 555 559 563 565 569 573 Crack-Tip Plasticity 575 Sp. Pantelakis and P. Papanikos Problem 42: Fatigue Crack Growth and Residual Strength of a Double Edge Cracked Panel Under Irregular Fatigue Loading 579 Sp. Pantelakis and P. Papanikos Problem 43: Fatigue Crack Growth Rate Under Irregular Fatigue Loading 583 Sp. Pantelakis and P. Papanikos Problem 44: Fatigue Life of a Pressure Vessel Under Variable Internal Pressure 585 Sp. Pantelakis and P. Papanikos 15. Table of Contents 7. Complex Cases Problem 45: Equibiaxial Low Cycle Fatigue J.R. Yates XVll 589 Problem 46: Mixed Mode Fatigue Crack Growth in a Center-Cracked Panel 593 Sp. Pantelakis and P. Papanikos Problem 47: Collapse Stress and the Dugdale's Model 597 C.A. Rodopoulos Problem 48: Torsional Low Cycle Fatigue 601 J.R. Yates and M. W Brown Problem 49: Fatigue Life Assessment ofa Plate Containing Multiple Cracks 607 Sp. Pantelakis and P. Papanikos Problem 50: Fatigue Crack Growth and Residual Strength in a Simple MSD Problem 611 Sp. Pantelakis and P. Papanikos INDEX 615 16. Editor's Preface On Fracture Mechanics A major objective of engineering design is the determination of the geometry and dimensions of machine or structural elements and the selection of material in such a way that the elements perform their operating function in an efficient, safe and economic manner. For this reason the results of stress analysis are coupled with an appropriate failure criterion. Traditional failure criteria based on maximum stress, strain or energy density cannot adequately explain many structural failures that occurred at stress levels considerably lower than the ultimate strength of the material. On the other hand, experiments performed by Griffith in 1921 on glass fibers led to the conclusion that the strength of real materials is much smaller, typically by two orders of magnitude, than the theoretical strength. The discipline of fracture mechanics has been created in an effort to explain these phenomena. It is based on the realistic assumption that all materials contain crack-like defects from which failure initiates. Defects can exist in a material due to its composition, as second-phase particles, debonds in composites, etc., they can be introduced into a structure during fabrication, as welds, or can be created during the service life of a component like fatigue, environment-assisted or creep cracks. Fracture mechanics studies the loading-bearing capacity of structures in the presence of initial defects. A dominant crack is usually assumed to exist. The safe design of structures proceeds along two lines: either the safe operating load is determined when a crack of a prescribed size exists in the structure, or given the operating load, the size of the crack that is created in the structure is determined. Design by fracture mechanics necessitates knowledge of a parameter that characterizes the propensity of a crack to extend. Such a parameter should be able to relate laboratory test results to structural performance, so that the response of a structure with cracks can be predicted from laboratory test data. This is determined as function of material behavior, crack size, structural geometry and loading conditions. On the other l}.and, the critical value of this parameter, known as fracture toughness, is a property of the material and is determined from laboratory tests. Fracture toughness is the ability of the material to resist fracture in the presence of cracks. By equating this parameter to its critical value we obtain a relation between applied load, crack size and structure geometry, which gives the necessary information for structural design. Fracture mechanics is used to rank the ability of a material to resist fracture within the framework of fracture mechanics, in the same way that yield or ultimate strength is used to rank the resistance of the material to yield or fracture in the conventional design criteria. In selecting materials for structural applications we must choose between materials with high yield strength, but comparatively low fracture toughness, or those with a lower yield strength but higher fracture toughness. 17. XX Editor's Preface The theory of fracture mechanics has been presented in many excellent books, like those written by the editor of the first part of the book devoted to fracture mechanics entitled: "Problems of Mixed Mode Crack Propagation," "Fracture Mechanics Criteria and Applications," and "Fracture Mechanics-An Introduction." However, students, scholars and practicing engineers are still reluctant to implement and exploit the potential of fracture mechanics in their work. This is because fracture is characterized by complexity, empiricism and conflicting viewpoints. It is the objective of this book to build and increase engineering confidence through worked exercises. The first part of the book referred to fracture mechanics contains 84 solved problems. They cover the following areas: The Westergaard method for crack problems Stress intensity factors Mixed-mode crack problems Elastic-plastic crack problems Determination of strain energy release rate Determination of the compliance of crack problems The critical strain energy release rate criterion The critical stress intensity factor criterion Experimental determination of critical stress intensity factor. The !-integral and its experimental determination The crack opening displacement criterion Strain energy density criterion Dynamic fracture problems Environment assisted crack growth problems Photoelastic determination of stress intensity factors Crack growth from rigid inclusions Design of plates, bars and pressure vessels The problems are divided into three groups: novice (for undergraduate students), intermediate (for graduate students and practicing engineers) and advanced (for researchers and professional engineers). They are marked by one, two and three asterisks, respectively. At the beginning of each problem there is a part of "useful information," in which the basic theory for the solution of the problem is briefly outlined. For more information on the theory the reader is referred to the books of the editor: "Fracture Mechanics Criteria and Applications," "Fracture Mechanics-An Introduction," "Problems of Mixed-Mode Crack Propagation." The solution of each problem is divided into several easy to follow steps. At the end of each problem the relevant bibliography is given. 18. Editor's Preface XXl I wish to express my sincere gratitude and thanks to the leading experts in fracture mechanics and good friends and colleagues who accepted my proposal and contributed to this part of the book referred to fracture mechanics: Professor L. Banks-Sills of the Tel Aviv University, Professor A. Carpinteri, Professor B. Chiaia and Professor P. Cometti of the Politecnico di Torino, Dr. M. S. Konsta-Gdoutos of the Democritus University of Thrace, Professor G. C. Sib of Lehigh University and Professor D. J. Unger of the University of Evansville. My deep appreciation and thanks go to Mrs Litsa Adamidou for her help in typing the manuscript. Finally, a special word of thanks goes to Ms Nathalie Jacobs of Kluwer Academic Publishers for her kind collaboration and support during the preparation of the book. April, 2003 Xanthi, Greece Emmanuel E. Gdoutos Editor 19. Editor's Preface On Fatigue The second part of this book is devoted to fatigue. The word refers to the damage caused by the cyclic duty imposed on an engineering component. In most cases, fatigue will result into the development of a crack which will propagate until either the component is retired or the component experiences catastrophic failure. Even though fatigue research dates back to the nineteenth century (A. Wohler1860, H. Gerber 1874 and J. Goodman 1899), it is within the last five decades that has emerged as a major area of research. This was because of major developments in materials science and fracture mechanics which help researchers to better understand the complicated mechanisms of crack growth. Fatigue in its current form wouldn't have happened if it wasn't for a handful of inspired people. The gold medal should be undoubtedly given to G. Irwin for his 1957 paper Analysis of Stresses and Strains Near the End ofa Crack Traversing a Plate. The silver medal should go to Paris, Gomez and Anderson for their 1961 paper A Rational Analytic Theory of Fatigue. There are a few candidates for the bronze which makes the selection a bit more difficult. In our opinion the medal should be shared by D.S. Dugdale for his 1960 paper Yielding ofSteel Sheets Containing Slits, W. Biber for the 1960 paper Fatigue Crack Closure under Cyclic Tension and K. Kitagawa and S. Takahashi for their 1976 paper Applicability ofFracture Mechanics to Very Small Cracks or the Cracks in the Early Stage. Unquestionably, if there was a fourth place, we would have to put a list of hundreds of names and exceptionally good works. To write and editor a book about solved problems in fatigue it is more difficult than it seems. Due to ongoing research and scientific disputes we are compelled to present solutions which are well established and generally accepted. This is especially the case for those problems designated for novice and intermediate level. In the advanced level, there are some solutions based on the author's own research. In this second part, there are 50 solved problems. They cover the following areas: Life estimates Fatigue crack growth Effect of Notches on Fatigue Fatigue and Safety factors Short cracks Variable amplitude loading Complex cases As before, the problems are divided into three groups: novice (for undergraduate students), intermediate (for graduate students and practicing engineers) and advanced (for researchers and professional engineers). Both the editors have been privileged to scientifically mature in an department with a long tradition in fatigue research. Our minds have been shaped by people including Bruce Bilby, Keith Miller, Mike Brown, Rod Smith and Eduardo de los Rios. We thank them. We wish to express our appreciation to the leading experts in the field of fatigue who contributed to this second part of the book: Professor M. W. Brown from the University of Sheffield, Professor M. N. James from the University of Plymouth, Professor Y-M. 20. xxiv Editor's Preface Mai from the University of Sydney, Dr. P. Papanikos from the Institute of Structures and Advanced Materials, Dr. A. Afaghi-Khatibi from the University of Melbourne and Professor Sp. Pantelakis from the University of Patras. Finally, we are indebted to Ms. Nathalie Jacobs for immense patience that she showed during the preparation of this manuscript. April, 2003 Sheffield, United Kingdom Chris A. Rodopoulos John R. Yates Editors 21. List of Contributors Afaghi-Khatibi, A., Department of Mechanical and Manufacturing Engineering. The University of Melbourne, Victoria 3010, Australia. Banks-Sills, L., Department of Solid Mechanics, Materials and Systems, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Brown, M. W., Department of Mechanical Engineering, The University of Sheffield, Sheffield, S1 3JD, UK. Carpinteri, A., Department of Structural Engineering and Geotechnics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Chiaia, B., Department of Structural Engineering and Geotechnics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Cometti, P., Department of Structural Engineering and Geotechnics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Gdoutos, E. E., School of Engineering, Democritus University ofThrace, GR-671 00 Xanthi, Greece. James, M. N., Department of Mechanical and Marine Engineering, University of Plymouth, Drake Circus, Plymouth, Devon PL4 8AA, UK. Konsta-Gdoutos, M., School of Engineering, Democritus University of Thrace, GR-671 00 Xanthi, Greece. Mai, Yiu-Wing, Centre for Advanced Materials Technology, School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia. Pantelakis, Sp., Department of Mechanical Engineering and Aeronautics, University of Patras, GR 26500, Patras, Greece. Papanikos, P., ISTRAM, Institute of Structures & Advanced Materials, Patron-Athinon 57, Patras, 26441, Greece. Rodopoulos, C. A., Structural Integrity Research Institute of the University of Sheffield, Department of Mechanical Engineering, The University of Sheffield, Sheffield, S1 3JD, UK. Unger, D. J., Department of Mechanical and Civil Engineering, University of Evansville, 1800 Lincoln Avenue, Evansville, IN 47722, USA. Yates, J. R, Department of Mechanical Engineering, The University of Sheffield, Sheffield, S1 3JD, UK. 22. PART A: FRACTURE MECHANICS 23. 1. Linear Elastic Stress Field 24. Problem 1: Airy Stress Function Method *** E.E. Gdoutos 1. Problem In William's eigenfunction expansion method [I] the Airy stress function for a semi- infinite crack in an infinite plate subjected to general loading is assumed in the form (1) where r, 9 are polar coordinates centered at the crack tip and). is real. Using the boundary conditions along the crack faces, determine the function U and find the expressions for the singular stress and displacement components for opening mode and sliding mode loading. Observe that negative values of A. are ignored since they produce infinite displacements at the crack tip. Furthermore, use the result that the total strain energy contained in any circular region surrounding the crack tip is bounded to show that the value ). = 0 should also be excluded from the solution. 2. Useful Information In the Airy stress function method the solution of a plane elasticity problem in polar coordinates is reduced to finding a function U = U(r, 9) (Airy function) which satisfies the biharmonic equation in polar coordinates and the appropriate boundary conditions [2]. The stress components are given by (3) 25. 4 E.E. Gdoutos 3. Solution 3.I GENERAL REMARKS From Equation (I) we obtain (4) Thus Equation (2) becomes (5) The solution ofthis equation is (6) (7) f 2 = C2 sin (A. -I) e+C4 sin (A.+ I) e where the symmetric part f1 corresponds to opening-mode and the anti-symmetric part f2 corresponds to sliding-mode. The boundary conditions are 0. As in Problem 2 we obtain that this force is equal toP. K1 is calculated from [I] (8) We have I (9) . 2[x(a+~)J . 2 (xa)SID -SID - w w and sin [ x (~+~)J = sin ( ~) + ~ cos ( ~) (IO) . 2 [x(a+~)J . 2 (xa) x2 ~2 2 (xa) 2x~ . (xa) (xa)SID -sm - =--cos - +--sm - cos - w w w2 w w w w (11) Thus . Psin ( ~) K1 =hmJ2x~ Z1 =----''----'- i~l-+0 w [ x2 ~2 2 (xa) 2x~ . (xa) (xa)]w2 cos w +-wsm w cos w (I2) We have for K1 39. 20 E.E. Gdoutos p (13) W. (2xa}-Stn -~ 2 w Note that for W/a ~ oc the above solution reduces to the case of a single crack (K1 = PI ,J;;, Equation (2) ofProblem 2 with b = 0). 4. References [l] E.E. Gdoutos (1993) Fracture Mechanics - An Introduction, Kluwer Academic Publishers, Dordrecht, Boston, London. 40. Problem 4: Westergaard Method for a Periodic Array of Cracks Under Uniform Stress** E.E. Gdoutos 1. Problem Consider an infinite periodic array of equally spaced cracks along the x-axis in an infi- nite plate subjected to equal uniform stresses u along the x- and y-axes at infinity (Fig- ure l). VerifY that the Westergaard function is 0 (n:z)USID W (l) Then show that the stress intensity factor is given by ( ) 1/2 112 W n:aK 1 =u(n:a) -tan- n:a W (2) r-------------1~------------,I I I I I Yf I I W W I a 1 1 a ...._.. - .--...I X I I ~~~ ~~~ ~~~ I I I I I I I L-------------l~------------J Figure lo An infinite periodic array of equally spaced cracks in an infinite plate subjected to equal unifunn stresses a at infinityo 41. 22 E.E. Gdoutos 2. Useful Information See Problem 2. 3. Solution From Equation (I) we have For y = 0, lx-WI< awe have z=x Thus, . (1tx) . (1ta)sm W tan W =W the above solution reduces to the case of a single crack (K 1 = a/ii). 4. References [1] E.E. Gdoutos (1993) Fracture Mechanics- An Introduction, Kluwer Academic Publishers, Dordrecht, Boston, London. 43. Problem 5: Calculation of Stress Intensity Factors by the Westergaard Method** E.E. Gdoutos 1. Problem The Westergaard function Z for the concentrated forces P and Qapplied at the point x = b(b o 21t JC+l ~ (l;+a)z -az b-(a+ I;) (a+ Q2 -a2 r .j21tf,(Q+iP)[(JC-l) 1 1 [ b2 -a2 1]] =:~1~!1J 1t ~ K+l ~1;(1;+2a) + b-(a+ I;) 1;(1;+2a) + = (Q+iP) [(~) ~+-I~ x (b 2 -a2 ) l21t IC+I v-; b-a a (5) = (Q +iP) [(~) + r;;+b l21t IC+I V~ The mode-l and mode-II stress intensity factors K1 and Krr are given by (6a) p (IC-1) Q ra+b Ku =- 2~ K+I + 2~ v;=t; (6b) 3.2. STRESS INTENSITY FACTOR FOR FIGURE Ib Letting P = cry (x, 0) dx and Q= r,y (x, 0) dx integrating from x = -a to x = a the K1 and Kn expressions for a crack subjected to arbitrary loads on the upper crack surface are I af Jg[+x I (IC-1) 3 K 1 = r-- cry(x,O) --dx+ r-- -- Jrxy(x,O)dx 2-yxa a-x 2-vxa K+I-a -a (7a) I (IC-I) 3 1 a J+xK 11 =- r-- -- Jay (x, 0) dx + r-- Jrxy (x, 0) - - dx 2-vxa K+l 2-vxa a-x-a -a (7b) For equal and opposite forces on the upper and lower crack faces using the symmetry equations (8a) 46. 28 E.E. Gdoutos (8b) we obtain a~I a+x KI = ~ Jay(x, 0) - - dx -vna a-x -a (9a) a ~I a+x K 11 = ,...- Jtxy(x, 0) - - dx vna a-x -a (9b) 3.3. STRESS INTENSITY FACTOR FOR FIGURE 2a For uniform stress distribution Cfy (x, 0) = cs0 we obtain from equation (9a) a~I a+x KI = ,...- Ja0 - - dx vna a-x -a (10) Putting u = x/a we have a~ I~ Ia +x l+u . -I 2 J - dx =a J - du =a [sin u- ~]a-x I-u -1 -a -1 (11) and cso ,-- KI = ,...-an= cs0 -vna vna (I2) which is identical to the value of K1 for a crack of length 2a subjected to a uniform re- mote stress cs0 3.4. STRESS INTENSITY FACTOR FOR FIGURE 2b For the triangular stress distribution of Figure 2b, Cfy = x cs0/a. We obtain from Equa- tion (9a) 47. Calculation of Stress Intensity Factors by the Westergaard Method 29 K, =_I_ aJ 0. We have 52. Westergaard Method for a Crack Under Concentrated Forces 35 ] Introducing this value of tm into the photoelastic law expressed by Equation (2) we obtain for the distance of a point on the isochromatic fringe of order N from the crack tip (6) where (7) This equation can be solved in a computer to give the polar distance r as a function of the polar angle e withy, a, a and~ as parameters. Physically accepted solutions ofthis equation should give real values of r, such that r/a < I. The four parameters y, a, a and ~ can be adjusted so that the analytical isochromatics match the experimental ones. When a close fit is achieved the stress intensity factor K1 is determined. This analysis permits the fringe loops to tilt, stretch and become unsymmetrical, so that they can be used to determine K1 for a wide variety of specimen geometries and loading conditions [I]. 4. References (I] J.W. Dally and W.F. Riley (1991) Experimental Stress Analysis, Third Ed., McGraw-Hill, New York. 78. Problem 15: Photoelastic Determination of Mixed-Mode Stress Intensity Factors K1 and K11 ** MS. Konsta-Gdoutos 1. Problem Consider a crack in a mixed-mode stress field governed by the values of the opening- mode K1 and sliding-mode Krr stress intensity factors. Obtain the singular stress com- ponents and subtract the constant term O"ox from the stress O"x to account for distant field stresses. Determine the isochromatic fringe order N from equation [I] Nf 2T =- m t (l) where Tm is the maximum in-plane shear stress, f is the stress-optical constant and t is the plate thickness. Obtain an expression for N. Consider the opening-mode. Ifrm and 9m are the polar coordinates of the point on an isochromatic loop, furthest from the crack tip (Figure I), show that [2] _ Nf~[ ( 2 ) 2 ] 112 ( 2tan(39m /2)K 1 - I + 1 + ---'---"::._- tsin9m 3tan9m 3tan9m (2) (3) For the problem of a mixed-mode stress field if only the singular stresses are consid- ered, show that the maximum in-plane shear stress Tm is given by I r. 2 2 2 2 f'2 Tm = ,;;--!_SID 9K1 + 2sm 29K 1Kn + (4-3sm 9)Kn . 2v2xr (4) Then show that the polar angle 9m ofthe point furthest from the crack tip on the curve Tmax = constant (Figure I) satisfies the following equation 79. 66 M.S. Konsta-Gdoutos (5) y TaConstant X Figure 1. A crack-tip isochromatic fringe loop. 2. Useful Information See Problem 13. 3. Solution The stress field in the vicinity ofthe crack tip for mixed-mode conditions is given by 1 [ 6 (I . 6 . 36} K . 6 ( 2 6 36 }]crx = r;:;-- K1 cos- -SID- sm- - II sm- +cos- cos- - crox -v2xr 2 2 2 2 2 2 1 [K 6 ( 1 . 6 . 36} K . 6 9 39]cry= ~ 1cos- +sm -sm- + IIsm- cos -cos- -v2u 2 2 2 2 2 2 (6) I [K . 6 6 36 K 6 ( 1 . 6 . 36 }]t xy = r;:;-- 1 sm - cos - cos - + II cos - - sm - SID - -v2xr 2 2 2 2 2 2 80. Photoelastic Determination ofMixed-Mode Stress Intensity Factors K1 and Kn 67 where K1 and Kn are the mode-l and mode-II stress intensity factors and r, 9 the polar coordinates referred to the crack tip. The maximum in-plane shear stress 'tm is given by We obtain from Equations (6) and (7) (2mY = - 1-[(K1 sinO+ 2Ku cos of +(Ku sin of] 21tr (7) + ~sin! [K1sin0 (1 + 2cos0 )+ Ku(1 + 2cos2 0 +coso)]+ cr~x (8) v21tr 2 For opening-mode (Kn = 0) we obtain (9) The position ofthe farthest point on a given loop is dictated by (IO) which gives -K1 sin9m cos Om cro = X~( .39 3. 39)V. JL lm COS e Slfl __tJI_ + - Slfl 0 COS ___IJI_ m 2 2 m 2 (II) From Equation (9) and (II) using the photoelastic law ofEquation (I) we obtain for K1 and O"ox Nf ~21trm 2 2 2 tan 2 t sin9m (3tanem) 3tan9m [ l-1/2 [ 30m ] K 1 = - 1+ 1+ (I2) 81. 68 M.S. Konsta-Gdoutos Nf cosem O"ox =- ----------=------ t (36 ) ( 9 )112 cos 2m cos 2 6m+ 4sin 2 em (13) From Equation (8) we obtain for Oox = 0 1 r. 2 2 2 2 ] 112 m= r.:;--I_Sm 6K1 +2sm26K1Ku+(4-3sm 6)Ku 2v2 n:r (14) From Equations (5) and (14) we obtain (15) 4. References [1) J.W. Dally and W.F. Riley(l99l)Experimental Stress Analysis, Third Ed., McGraw-Hill, New York. [2) G.R. Irwin (1958) Discussion ofpaper ''The Dynamic Stress Distribution Surrounding a Running Crack-A Photoelastic Analysis", by A. Wells and D. Post, Proc. SESA, Vol. XVI, pp. 69-92, Proc. SESA, Vol. XVI, pp. 93-96. 82. Problem 16: Application of the Method of Weight Function for the Determination of Stress Intensity Factors ** L. Banks-Sills 1. Problem (a) By means of the weight function, determine an integral expression for the stress intensity factor of the geometry and loading shown in Fig. I. Assume plane strain conditions. (b) Carry out the integration to obtain an explicit expression for K1 - 2 a - Figure. I A crack in an infinite plate subjected to a triangular stress distribution along the crack surfaces 2. Useful Information The stress intensity factor K1 may be written as (l) 83. 70 L. Banks-Sills where Ti is the traction vector on the boundary Sr along which the tractions are applied, ds is differential arc length, and m; is the Bueckner-Rice weight function given by [1, 2) ( n)- H au~(x,y,l) mi x,y, cr2 are given by (10) or Kn . e Kn cr1 2 =- r;;::: sm- r;;::: ...;2Jtr 2 ...;2Jtr . z e 2 el-3sm -cos - 2 2 (II) For conditions ofgeneralized plane stress (cr3 = 0) the Mises yield criterion becomes (I2) or 2 Kn (6 + 2sin 2 ~- 2.sin 2 e)= 2crt 21tr 2 2 (13) 0.4 r/(K,/a;r 0.6 Figure 2. Approximate estimation of the crack-tip plastic zones for mode-II loading under plane stress and plane strain. v=I/3. The radius ofthe plastic zone is given by 91. Crack Tip Plastic Zone for Mode-l and Mode-ll Loading rp(9) =r =-1-(~)2 (14- 2cos9- 9sin 2 9) 87t Oy For conditions ofplane strain (aJ = v(a1 + a2)) the Mises yield criterion yields 79 (14) rP (9) =r =-1-(~)2 [12 + 2(1 - 2v)2 (1-cos 9)- 9sin 2 9] (15) 87t Oy Figure 2 shows the shapes ofthe plastic zones for plane stress and plane strain with v = 1/3. 4. References [I] A. Nadai (1950) Theory ofFlow and Fracture ofSolids, McGraw-Hill, New York. [2] E.E. Gdoutos (1993) Fracture Mechanics- An Introduction, Kluwer Academic Publishers, Dordrecht, Boston, London. 92. Problem 18: Approximate Determination of the Crack Tip Plastic Zone for Mixed-Mode Loading * E.E. Gdoutos 1. Problem Determine the radius of the plastic zone accompanying the crack tip for mixed-mode (opening-mode and sliding-mode) loading under plane strain conditions according to the Mises Yield criterion. Plot the resulting elastic-plastic boundary for a crack of length 2a in an infinite plate subtending an angle p= 30 with the direction ofapplied uniaxial stress at infinity. v =0.3. 2. Useful Information See Problem 17 3. Solution By superimposing the stresses for opening-mode and sliding-mode loading and omit- ting the constant term we obtain, after some algebra, for plane strain conditions (cr.= v (crx+ cry), see Equations (7) of Problem 13) for the radius r ofthe plastic zone r =- 1- 2-[Kf cos2 ~[ r2 for 9 < 38.94. Thus, for 9 < 38.94 the elastic- plastic boundary is represented by Equation (8), while for 9 > 38.94 the elastic-plastic boundary is represented by Equation (9). The elastic-plastic boundary for v = 113 according to the Tresca yield criterion for con- ditions of plane stress and plane strain is plotted in Figure I. Comparing Figure I with Figure I ofProblem 17 we observe that the elastic-plastic boundaries according to Tre- sca yield criterion are slightly different than the elastic-plastic boundaries according to Mises yield criterion. ..0.4 Figure 1. Plastic zones around the crack tip for mode-l under plane stress and plane strain conditions accord- ing to the Tresca yield criterion. 3.2 MODE II 3.2.1. Plane stress_(a3 = 0): 97. 86 M.S. Konsta-Gdoutos We have (11) The Tresca yield criterion is expressed by (12) Since (13) we have for the Trasca criterion (14) and the elastic-plastic boundary is determined by 2KnW2r:;::: 1- -SID 9 = Oy -v27rr 4 (15) or Kn ( . e ~1 3 2 e)r;;=: sm-+ - -sm =cry -v2nr 2 4 (16) The radius ofplastic zone is the larger of (17) and 98. Crack Tip Plastic Zone According to the Tresca Yield Criterion 87 Note that r2 > r1 for 1e1 >76.7. Thus the elastic-plastic boundary is represented by equation (17) for 1e1 < 76.7 and by equation (18) for lei > 76.7. 3.2.2. Plane strain: (cr3 = v (cri + crz)): We have cr1-cr3 = -~[(1- 2v) sin!!__~I- ~sin2 e]~r 2 4 (19) The Tresca yield criterion is expressed by (20) Since (21) we have for the Tresca criterion (22) and the elastic-plastic boundary is determined by 2Ku ~ 3 . 2 r;;::::: 1- -Sin 9 = Oy v21tr 4 (23) or 99. 88 M.S. Konsta-Gdoutos (24) The radius ofplastic zone is the larger of (25) and rp(9) =r3 =-1-(Ku ) 2 [(1-2v) sin!+~1- ~sin2e]2 2n oy 2 4 (26) Since r1 > r3 the elastic-plastic boundary is represented by Equation (25) and coincides with the elastic-plastic boundary for conditions ofplane stress for 1e1 < 76.7". The elastic-plastic boundary for v = 1/3 according to Tresca yield criterion for condi- tions of plane stress and plane strain is plotted in Figure 2. Comparing Figure 2 with Figure 2 of Problem 17 we observe that the elastic-plastic boundaries according to Tre- sca yield criterion are slightly different than the elastic-plastic boundaries according to Mises yield criterion. Figure 2. Plastic zones around the crack tip for mode-11 under plane stress and plane strain conditions ac- cording to the Tresca yield criterion. 100. Crack Tip Plastic Zone According to the Tresca Yield Criterion 89 4. References [I] E.E. Gdoutos (1993) Fracture Mechanics- An Introduction, Kluwer Academic Publishers, Dordrecht, Boston, London. [2] A. Nadai (1950) Theory ofFlow and Fracture ofSolids, McGraw-Hill, New York. 101. Problem 20: Approximate Determination of the Crack Tip Plastic Zone According to a Pressure Modified Mises Yield Criterion** E.E. Gdoutos 1. Problem Determine the crack tip plastic zone for opening-mode loading for a pressure modified von Mises yield criterion expressed by where R = crcfcr, and cr, and crc are the yield stress of the material in tension and com- pression, respectively. Plot the resulting elastic-plastic boundaries for plane stress and plane strain conditions when R = 1.2 and 1.5. Compare the results with those obtained by the von Mises criterion [I]. 2. Useful Information See Problem I7. 3. Solution 3.1. PLASTIC ZONE ACCORDING TO EQUATION (I) Introducing the values of crh cr2(Problem 17) into the modified von Mises yield crite- rion expressed by Equation (I) we obtain for the radius r of the elastic-plastic boundary (3) 102. 92 E.E. Gdoutos for plane stress, and { 2;~~)=(~;1J[~[[2(l-2v)2 +3(1-cos e)](l+cos e)]112 ] 2 R-t e + 2 (l+v)(--) cos- R+l 2 (4) for plane strain. The elastic-plastic boundaries for conditions of plane stress (v =0) and plane strain for v = 0.3 and v = 0.5 when R = l (von Mises yield criterion), R = 1.2 and R = 1.5 are shown by the upper halves ofFigures l, 2 and 3. Figure 1. Plastic zones for conditions of plane stress (v =0) and plane strain for v = 0, R = I, 1.2 and 1.5 according to the modified Mises yield criteria expressed by Equations I (upper half curves) and Equation 2 (lower halfcurves). 103. Crack Tip Plastic Zone According to a Pressure Modified Mises Yield Criterion 93 Figure 2. As in Figure I for v = 0.3 Figure 3. As in Figure I for v =0.5 104. 94 E.E. Gdoutos 3.2. PLASTIC ZONE ACCORDING TO EQUATION (2) Working as in the previous case we obtain: for plane stress and r (21t~~) =~[2(1+v)(R -1)cos ~+[4(1+v)2 (R -1)2 cos2 ~K1 4R 2 2 +R(1+cos9) [2(1- 2v)2 + 3(1-cos 9)]]112 ] 2 (6) for plane strain. The elastic-plastic boundaries for conditions of plane stress (v = 0) and plane strain for v = 0.3 and v = 0.5 when R = 1 (von Mises yield criterion), R = 1.2 and R = 1.5 are shown by the lower halves ofFigures 1, 2 and 3. 4. References [I] A. Nadai ( 1950) Theory ofFlow and Fracture ofSolids, McGraw-Hill, New York. 105. Problem 21: Crack Tip Plastic Zone According to Irwin's Model * E.E. Gdoutos 1. Problem Consider a central crack of length 2a in an infinite plate subjected to uniaxial stress a at infinity perpendicular to the crack plane. According to the Irwin model, the effective crack is larger than the actual crack by the length of plastic zone. Show that the stress intensity factor corresponding to the effective crack, called effective stress intensity factor Keffi for conditions ofplane stress, is given by (1) Then, consider a large plate of steel that contains a crack of length 20 mm and is sub- jected to a stress a = 500 MPa normal to the crack plane. Plot the ay stress distribution directly ahead of the crack according to the Irwin model. The yield stress of the mate- rial is 2000 MPa. 2. Useful Information Irwin [I, 2] presented a simplified model for the determination of the plastic zone at- tending the crack tip under small-scale yielding. He focused attention only on the ex- tent along the crack axis and not on the shape of the plastic zone, for an elastic- perfectly plastic material. The model is based on the notion that as a result of crack tip plasticity the stiffness of the plate is lower than in the elastic case. The length of the plastic zone c in front of the crack is given by c = _!_(.!S..)2 1t ay (2) for plane stress, and 106. 96 E.E. Gdoutos c =_I(.!S_)2 31t CJy for plane strain, where K1 is the stress intensity factor and CJyis the yield stress. 3. Solution The effective crack has a length 2(a+c/2) where for plane stress c/2 is (Equation (2)) ~=-1(KerrJ2 2 21t CJy (3) (4) The stress intensity factor Ketr for a crack of length 2(a+c/2) in an infinite plate sub- jected to the stress cr is (5) or (6) This Equation leads to Equation (I). Since the plate is large the effective stress intensity factor Keffis computed from Equa- tion (1). We have [ ] 1/2 K = cr 1t (O.Ol) =90MP r err 1/2 a-vm [~-0.5 ( ;oooooyl The length ofthe plastic zone cis c =_!_ (__2Q_)2 = 0.64mm 1t 2000 (7) (8) 107. Crack Tip Plastic Zone According to Irwin's Model 97 The cry stress is constant along the length of plastic zone, while in the elastic region it varies according to Keff cr =---- y (21tx)I/2 where xis measured from the tip of the effective crack (x > 0.32 mm). The cry stress distribution is shown in Figure 1. original crock fictitious crock }-- o=20 mm ----.i T2000UPo 1--x (2-nx)l'l Figure 1. Original and fictitious crack and cr, stress distribution according to the Irwin modeL 4. References (9) (I] G.R. Irwin (1960) Plastic Zone Near a Crack Tip and Fracture Toughness. Sagamore Ordnance Mate- rial Conference. pp. IV63-N78. (2] G.R. Irwin (1968) Linear Fracture Mechanics, Fracture Transition, and Fracture Control, Engineering. Fracture Mechanics., 1, 241-257. 108. Problem 22: Effective Stress intensity Factor According to Irwin' Model ** E.E. Gdoutos 1. Problem Consider a crack in a finite width plate subjected to opening-mode loading. Establish an iterative process for determining the effective stress intensity factor Keffaccording to the Irwin model. Then consider a thin steel plate of width 2b = 40 mm with a central crack of length 2a = 20 mm that is subjected to a stress a = 500 MPa normal to the crack plane. Plot the ay stress distribution directly ahead of the crack according to the Irwin model. The yield stress ofthe material is 2000 MPa. 2. Useful Information See Problem 21and references I and 2. 3. Solution The effective crack has a length 2(a + c/2), where for conditions ofplane stress c/2 is (Equation (2) ofProblem 21) ~=-1 (~)2 2 2n ay (I) and for conditions ofplane strain c/2 is (Equation (3) ofProblem 21) ~=-1 (~)2 2 6n ay (2) Kefffor a crack oflength 2(a + c/2) in a finite width plate is [ ] 1/2 KelT =f((a+c/2)/b)a n(a+i) (3) 109. 100 E.E. Gdoutos where the function t((a + c/2)/b) depends on the ratio (a+ c/2)/b, where b is the plate thickness. A flow chart of a computer program for the solution ofequations (I) and (3) or (2) and (3) is shown below: START I Aaoume SlJIISs Intensity factor K. F1 Calculate length of Plane srrain ptasUc zone c I c -~[~f -~r~r I I K, roJn(a i> K,foJn(ail T ff (ABS(K.-KJJ < T NO I YES I I K.K, K,,=K. -T I Print K,, I END From the computer program based on the above flow chart it is found Kerr = 109.48 MParrn The length ofplastic zone calculated from Equation (I) is c= 0.954 mm The cry stress distribution directly ahead ofthe crack is calculated from Kerf 0 =--- y .J21tX (4) (5) (6) 110. Effective Stress intensity Factor According to the Irwin Model where xis measured from the tip ofthe fictitious crack. It is shown in Figure 1. 4. References 4000 I I I a _ 109.48 ~-72RX 2 4 6 x(mm) Figure/. Stress distribution ahead of the cmck tip 8 10 101 [I] G.R. Irwin (1960) Plastic Zone Near a Cmck Tip and Fmcture Toughness, Sagamore Ordnance Mate- rial Conference, pp. IV63-IV78. [2] G.R. Irwin (1968) Linear Fmcture Mechanics, Fmcture Transition, and Fracture Control, Engineering Fracture. Mech., l, 241-257. 111. Problem 23: Plastic Zone at the Tip of a Semi-Infinite Crack According to the Dugdale Model * E.E. Gdoutos 1. Problem The stress intensity factor for an infinite plate with a semi-infinite crack subjected to concentrated loads Pat distance L from the crack tip (Figure I) is given by [I) K _ 2P I- (2nL)l/2 (I) For this situation determine the length of the plastic zone according to the Dugdale model. p p t-- L--t (a) t r llllll::p -x--1 1-- L ..,.. C----1 Figure 1. (a) A semi-infinite crack subjected to concentrated loads P and (b) calculation of the length of plastic zone according to the Dugdale model. 112. 104 E.E. Gdoutos 2. Useful Information Calculation ofthe plastic state of stress around the crack tip and the extent of the plas- tic zone is a difficult task. A simplified model for plane stress yielding which avoids the complexities ofthe true elastic-plastic solution was introduced by Dugdale [2]. The model applies to very thin plates in which plane stress conditions dominate, and to materials with elastic-perfectly plastic behavior which obey the Tresca yield criterion. According to the Dugdale model there is a fictitious crack equal to the real crack plus the length of plastic zone (Figure 1b). This crack is loaded by the applied loads P and an additional uniform compressive stress equal to the yield stress of the material, av. along the plastic zone. The length of plastic zone c is determined from the condition that the stresses should remain bounded at the tip ofthe fictitious crack. This condition is expressed by zeroing the stress intensity factor. 3. Solution The stress intensity factor at the tip of the fictitious crack is obtained by adding the stress intensity factors due to the applied loads P and the uniform compression stress av along the plastic zone. 3.1. STRESS INTENSITY FACTOR DUE TO APPLIED LOADS P The stress intensity factor K ~P) at the tip ofthe fictitious crack due to applied loads P is according to Eq. (I) (2) 3 2. STRESS INTENSITY FACTOR DUE TO THE STRESS ov The stress intensity factor Kov l at the tip of the fictitious crack due to the uniform compressive stress av along the length of plastic zone is calculated by integrating the expression for the stress intensity factor due to a pair of concentrated loads along the length ofthe plastic zone (Eq. (1)). We have (3) or 113. Plastic Zone at the Tip ofa Semi-Infinite Crack According to the Dugdale 105 Model K (Gy)- I - 3.3. SUPERPOSITION OF STRESS INTENSITY FACTORS (4) The stress intensity factor at the tip of the fictitious crack is obtained from Equations (2) and (4) as (5) 2P 40"yC112 [21t (c+L)]112 (21t)112 3.4. CONDITION OF ZEROING THE STRESS INTENSITY FACTOR The condition that the stress intensity factor be zero at the tip of the fictitious crack expressed as (6) leads to (7) Equation (7) expresses the length of plastic zone ahead of the crack tip according to the Dugdale model. 4. References [I] G.C. Sih (1973) Handbook of Stress Intensity Factors, Institute of Fracture and Solid Mechanics, Lehigh University. [2] D.S. Dugdale (1960) Yielding ofSteel Sheets Containing Slits, Journal. ofthe Mechanics and Physics of. Solids, 8, 100-104. 114. Problem 24: Mode-III Crack Tip Plastic Zone According to the Dugdale Model E.E. Gdoutos 1. Problem The stress intensity factor for an edge crack of length a in a semi-infinite solid sub- jected to a pair of concentrated shear forces S applied to the crack at a distance b from the solid edge (Figure I) is [I] (I) Determine the length of plastic zone according to the Dugdale model [2], and plot the variation of cia versus SlaTy for different values of b/a, where Tv is the yield stress in shear. -------------.., -------------~ Figure I. A crack of length a in a semi-infinite solid subjected to a pair of shear forces S. 2. Useful Information Look in Problem 23. 115. 108 E.E. Gdoutos 3. Solution The stress intensity factor at the tip of the fictitious crack is obtained by adding the stress intensity factors due to the applied shear forces Sand the uniform shear stress tv along the plastic zone (2]. 3.1. STRESS INTENSITY FACTOR DUE TO APPLIED SHEAR FORCES S The mode-III stress intensity factor K~> at the tip of the fictitious crack of length (a+c), where cis the length ofthe plastic zone, due to applied shear forces S, is accord- ing to Equation (I) K (S) _ 2S~7t(a+c) III- 1t~ (a+c)2 -b2 3.2. STRESS INTENSITY FACTOR DUE TO THE STRESS tv (2) The stress intensity factor K j;t> at the tip of the fictitious crack due to the uniform shear stress tv along the length of plastic zone is calculated by integrating the expres- sion for the stress intensity factor due to the concentrated shear forces along the length ofthe plastic zone. We have: 0) Equation (I) gives, or d 2 (u)..!_ dR > dA2 P R dA- __ dA2 R dA- dC dA while for stability in hard testing machines (du/u > 0) Equation (I) gives d2 (p) ..!_ dR > dA 2 u R dA - _!_(_!'_) . dA u or (2) (3) (4) 168. Crack Stability 165 d2 ( l)_!_ dR > dA C R dA- ~(_!_) dA C (5) where C = u/P is the compliance ofthe cracked plate. Equations (2), (3) and (4), (5) define the stability conditions in soft and hard testing machines. The right-hand size ofEquations (2), (3) and (4), (5) depends on the geome- try ofthe specimen and is called the geometry stability factor ofthe specimen. 3. Solution To determine the stability condition for the two cases under consideration, we first cal- culate the compliance of the system and then apply Equation (3) for load-controlled or Equation (5) for displacement-controlled conditions. We study the two cases sepa- rately. 3.1. CANTILEVER BEAM (DCB) 3.1.1. Compliance The two arms of the DCB may be considered to a first approximation as cantilevers with zero rotation at their ends. According to elementary beam theory the deflection of each cantilever at its end is. Bh 3 1=- 12 where 11 = l or 11 = 1 - v2 for generalized plane stress or plane strain, respectively. The relative displacement u ofthe points ofapplication ofthe loads Pis The compliance ofthe DCB is From Equation (8) we obtain u = 2u' = 211Pal. 3EI C=~=211aJ P 3EI (6) (7) (8) 169. 166 E.E.Gdoutos dC 3Tta2 d2C 3Tta dA = 8EB2 h3 ' dA2 = 4EB3 h3 3.1.2 ii. Soft machine (Load-Controlled) For stability in a soft (load-controlled) testing machine Equation (3) becomes 1 dR 2 --