Exam (elaborations) TEST BANK FOR Problems of Fracture Mechanics

Exam (elaborations) TEST BANK FOR Problems of Fracture Mechanics 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 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 of the 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 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 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 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 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 of a 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 of a Cylindrical bar with a Circular Crack E.E. Gdoutos 327 333 339 343 Problem 75: Failure of a Pressure Vessel Containing a Crack with Inclined Edges 347 E.E. Gdoutos Problem 76: Failure of a 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 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 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 of Crack 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 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 of a 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 of a 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 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 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 of a 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 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. 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. 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 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 of a 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 of Steel 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 of Fracture 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.