Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-29T02:37:12.933Z Has data issue: false hasContentIssue false

2 - Linear Elastic Fracture Mechanics

Published online by Cambridge University Press:  05 February 2016

Surjya Kumar Maiti
Affiliation:
Indian Institute of Technology, Bombay
Get access

Summary

Introduction

The foundation for the understanding of brittle fracture originating from a crack in a component was laid by Griffith (1921), who considered the phenomenon to occur within the framework of its global energy balance. He gives the condition for unstable crack extension in terms of a critical strain energy release rate (SERR) per unit crack extension. The next phase of development, which is due to Irwin (1957a and b), is based on the crack-tip local stress–strain field and its characterization in terms of stress intensity factor (SIF). The condition of fracture is given in terms of the SIF reaching a critical value, and the parameter is shown to be related to the critical energy release rate given by Griffith. Later, the scope of the SIF approach was amended to take care of small-scale plastic deformation ahead of the crack-tip. Most of the present applications of the principles of linear elastic fracture mechanics (LEFM) for design or safety analysis have been based on this SIF.

This chapter presents the gradual developments that have taken place to advance the understanding of fracture of brittle materials and other materials that give rise to small-scale plastic deformation before the onset of crack extension. Examples are presented to illustrate the applications of LEFM to design.

Calculation of Theoretical Strength

A fracture occurs at the atomic level when the bonds between atoms are broken across a fracture plane, giving rise to new surfaces. This can occur by breaking the bonds perpendicular to the fracture plane, a process called cleavage, or by shearing bonds along a fracture plane, a process called shear. The theoretical tensile strength of a material will therefore be associated with the cleavage phenomenon (Tetelman and McEvily 1967; Knott 1973).

Generally, atoms of a body at no load will be at a fixed distance apart, that is, the equilibrium spacing a0 (Fig. 2.1). When the external forces are applied to break the atomic bonds, the required force/stress (σ) increases with distance (a or x) till the theoretical strength σ c is reached. Further displacement of the atoms can occur under a decreasing applied stress. The variation can be represented approximately by a sinusoidal variation as follows.

Type
Chapter
Information
Fracture Mechanics
Fundamentals and Applications
, pp. 6 - 64
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

2.1 Abdelshehid, M., K., Mohmodieh, K., Mori, L., Chen, P., Stoyanov, D., Davlantes, J., Foyos, J., Orgen, R., Clark Jr. and O.S., Es-Said. 2007. ‘On the correlation between fracture toughness and precipitation hardening heat treatments in 15-5PH stainless steel.Engineering Failure Analysis 14: 626–31.CrossRefGoogle Scholar
2.2 Anderson, T. L. 2005. Fracture Mechanics: Fundamentals and Applications. Boston: CRC Press.Google Scholar
2.3 ASM Handbook. 1996a. ‘Fatigue & Fracture’, Vol. 19. Paper by S., Lapman, Fatigue and Fracture Properties of Stainless Steels, 621–24. Ohio: ASM International.Google Scholar
2.4 1996b. ‘Fatigue & Fracture’, Vol. 19, Paper by R.J., Bucci, G., Nordmark and E.A., Starkes Jr. Selecting Aluminum Alloys to Resist Failure by Fracture Mechanisms, 771–812 [refer p. 722]. Ohio: ASM International.Google Scholar
2.5 1996c. ‘Fatigue & Fracture’, Vol. 19. Paper by R.J., Bucci, G., Nordmark and E.A., Starkes Jr. Selecting Aluminum Alloys to Resist Failure by fracture Mechanisms, 771–812 [refer p. 776]. Ohio: ASM International.Google Scholar
2.6 1996d. ‘Fatigue & Fracture’, Vol. 19. Paper by R.J., Bucci, G., Nordmark and E.A., Starkes Jr. Selecting Aluminum Alloys to Resist Failure by fracture Mechanisms, 771–812 [refer p. 777]. Ohio: ASM International.Google Scholar
2.7 1996e. ‘Fatigue & Fracture’, Vol. 19. Paper by R.J., Bucci, G., Nordmark and E.A., Starkes Jr. Selecting Aluminum Alloys to Resist Failure by fracture Mechanisms, 771–812 [refer p. 779]. Ohio: ASM International.Google Scholar
2.8 ASTM E399-90. 2000 (Reapproved 1997). ‘Standard Test Method for Plane- Strain Fracture Toughness of Metallic Materials.’ In Annual Book of Standards, Section 3, Vol. 03. 01, Metals Test Methods and Analytical Procedures, 431–61. American Society for Testing and Materials.
2.9 Barenblatt, G.I. 1962. ‘The Mathematical Theory of Equilibrium Cracks in Brittle Fracture.’ In Advances in Applied Mechanics, Vol. VII, 55–129. New York: Academic Press.Google Scholar
2.10 Bluhm, J.I. 1961. ‘A Model for the Effect of Thickness on Fracture Toughness’. ASTM Proceedings 61: 1324–31.Google Scholar
2.11 Barbagallo, S. and E., Cerri. 2004. ‘Evaluation of KIC and JIC fracture parameters in a sand cast AZ91 Magnesium alloy.Engineering Failure Analysis 11(1) : 127–40.CrossRefGoogle Scholar
2.12 Broek, D. 1986. Elementary Engineering Fracture Mechanics, 4th revised edn. The Netherlands: Noordhoff.CrossRefGoogle Scholar
2.13 Broek, D. and H., Vlieger. 1974. The Thickness Effect in Plane Stress Fracture Toughness. Amsterdam: National Aerospace Institute [Rept. TR 74032].Google Scholar
2.14 Brown Jr., W.F. and J.E., Srawley. 1965. ‘Fracture Toughness Testing Methods’. In Fracture Toughness Testing and its Applications, 133–195. Philadelphia: American Society for Testing and Materials [ASTM STP 381].Google Scholar
2.15 Brown Jr., W.F. and J.E., Srawley. 1966. Plane Strain Crack Toughness Testing of High Strength Metallic Materials. Philadelphia: American Society for Testing and Materials [ASTM STP 410].CrossRefGoogle Scholar
2.16 Crews Jr., J.H. and J.R., Reeder. 1988. A Mixed-mode Bending Apparatus for Delamination Testing, NASA Technical Memorandum 100662.
2.17 Ducept, F., P., Davies and D., Gamby. 2000. ‘Mixed Mode Failure Criteria for a Glass/Epoxy Composite and an Adhesively Bonded Composite/Composite Joint.International Journal of Adhesion and Adhesives 20: 233–44.CrossRefGoogle Scholar
2.18 Dugdale, D.S. 1960. ‘Yielding of Steel Sheets Containing Slits.Journal of Mechanics and Physics of Solids 8: 100–04.CrossRefGoogle Scholar
2.19 Gdoutos, E.E. 1993. Fracture Mechanics – An Introduction. Kluwer. Dordrecht/Boston/ London: Kluwer Academic Publishers.CrossRefGoogle Scholar
2.20 Ghosh, S., V., Kain, A., Ray, H., Roy, S., Sivaprasad, S., Tarafdar and K.K., Ray. 2009. ‘Deterioration in Fracture Toughness of 304LN Austenitic Stainless Steel due to Sensitization.Metallurgical and Materials Transactions A 40(12): 2938–49.CrossRefGoogle Scholar
2.21 Griffith, A.A. 1921. ‘The Phenomenon of Rupture and Flow in Solid.Philosophical Transactions, Royal Society of London, Series A 221: 163–69.CrossRefGoogle Scholar
2.22 Hahn, G.T. and A.R., Rosenfield. 1965. ‘Local Yielding and Extension of Crack Under Plane Stress.Acta Metallurgica 13: 293–306.CrossRefGoogle Scholar
2.23 Hellan, K. 1985. Introduction to Fracture Mechanics. New York: McGraw-Hill.Google Scholar
2.24 Hudson, C.M. and S.K., Seward. 1978. ‘A Compendium of sources of Fracture Toughness and Fatigue-crack Growth Data for Metallic Alloys.International Journal of Fracture 14: R151–84.CrossRefGoogle Scholar
2.25 Hudson, C.M. and S.K., Seward. 1982. ‘A Compendium of sources of Fracture Toughness and Fatigue-crack Growth Data for Metallic Alloys – Part II.International Journal of Fracture 20: R59–117. 2CrossRefGoogle Scholar
2.26 Hudson, C.M. and S.K., Seward. 1989. ‘A Compendium of sources of Fracture Toughness and Fatigue-crack Growth Data for Metallic Alloys – Part III.International Journal of Fracture 39: R43–63.CrossRefGoogle Scholar
2.27 Hudson, C.M. and J.J., Ferrainlo. 1991. ‘A Compendium of sources of Fracture Toughness and Fatigue-crack Growth Data for Metallic Alloys – Part IV.International Journal of Fracture 48: R19–43.CrossRefGoogle Scholar
2.28 Inglis, C.E. 1913. ‘Stresses in a Plate due to the Presence of Cracks and Sharp Corners.Transaction of the Institute of Naval Architects 55: 219–41.Google Scholar
2.29 Irwin, G.R. 1948. ‘Fracture Dynamics.’ In Fracturing of Metals, 147–66. Cleveland: ASM Publication.Google Scholar
2.30 Irwin, G.R. 1957a. ‘Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate.Journal of Applied Mechanics, Transactions of ASME 24: 361–64.Google Scholar
2.31 Irwin, G.R. 1957b. ‘Relation of stresses near a crack to the crack extension force’. Proceedings of Ninth International Congress of Applied Mechanics, Brussels.Google Scholar
2.32 Irwin, G.R. 1958. ‘Fracture.’ In Hanbuch der Physik, Vol. VI, ed. Flugge, S., 551–90. Berlin: Springer-Verlag.Google Scholar
2.33 Irwin, G.R. 1960. ‘Plastic Zone Near a Crack and Fracture Toughness’, 4–63. Proceedings of Seventh Sagamore Conference.
2.34 Irwin, G.R. and J.A., Kies. 1952. ‘Fracture and Fracture Dynamics.Welding Journal Research Supplement 31: 95S–100S.Google Scholar
2.35 Isherwood, D.P. and J.G., Williams. 1970. ‘The Effect of Stress–Strain Properties on Notched Tensile Fracture in Plane Stress.Engineering Fracture Mechanics 2: 19–22.CrossRefGoogle Scholar
2.36 Kanninen, M.F. and C.H., Popelar. 1985. Advanced Fracture Mechanics. New York: Oxford University Press.Google Scholar
2.37 Kim, B., J., Do, S., Lee and I., Park. 2010. ‘In Situ Fracture Observation and Fracture Toughness Analysis for Squeeze Cast AZ31-xSn Magnesium Alloys.Materials Science and Engineering A 527(24–25) : 6745–757.CrossRefGoogle Scholar
2.38 Knott, J.F. 1973. Fundamentals of Fracture Mechanics. London: Butterworths.Google Scholar
2.39 Knott, J.F. 1975. ‘The Fracture Toughness of Metals.Journal of Strain Analysis 10: 201–06.CrossRefGoogle Scholar
2.40 Kumar, P. 2009. Elements of Fracture Mechanics. New Delhi: Tata McGraw-Hill.Google Scholar
2.41 Liebowitz, H., ed. 1968. Fracture – and Advanced Treatise, Vol. II, Mathematical Fundamentals. New York: Academic Press.Google Scholar
2.42 McClintock, F.A. and G.R., Irwin. 1965. ‘Plasticity Aspects of Fracture Mechanics.’ In Fracture Toughness Testing and its Applications, 84–113. Philadelphia: American Society for Testing and Materials [ASTM STP 381].Google Scholar
2.43 Meguid, S.A. 1989. Engineering Fracture Mechanics. London: Elsevier Applied Science.Google Scholar
2.44 Metals Handbook. 1990a. Properties & Selection: Nonferrous Alloys and Special- Purpose Materials, Vol. 2. 10th edn, p. 54 and p. 77. USA: ASM International.
2.45 Metals Handbook. 1990b. Properties & Selection: Nonferrous Alloys and Special-Purpose Materials, Vol. 2, 10th edn, 120–21. USA: ASM International.
2.46 Metals Handbook. 1990c. Properties & Selection: Nonferrous Alloys and Special-Purpose Materials, Vol. 2, 10th edn, 117–18. USA: ASM International.
2.47 Metals Handbook. 1990d. Properties & Selection: Nonferrous Alloys and Special-Purpose Materials, Vol. 2, 10th edn, 115–17. USA: ASM International.
2.48 Metals Handbook. 1990e. Properties & Selection: Nonferrous Alloys and Special-Purpose Materials, Vol. 2, 10th edn, p. 112–13. USA: ASM International.
2.49 Minnay, D.P. 1998. Fracture Mechanics. New York: Springer-Verlag.CrossRefGoogle Scholar
2.50 Murakami, Y. (Editor-in-Chief). 1987. Stress Intensity Factors Handbook, Vols I and II. Oxford: Pergamon Press.Google Scholar
2.51 Orowan, E. 1949. ‘Fracture and Strength of Solids’, Report on Progress in Physics, Vol. 12, 185–232.CrossRefGoogle Scholar
2.52 Orowan, E. 1955. ‘Energy Criteria of Fracture.Welding Journal Research Supplement 20: 157S–160S.Google Scholar
2.53 Paris, P.C. and G. C., Sih. 1965. ‘Stress Analysis of Cracks.’ In Fracture Toughness Testing and its Applications, 30–81. Philadelphia: American Society for Testing and Materials [ASTM STP 381].Google Scholar
2.54 Parton, V.Z. and E. M., Morozov. 1978. Elastic Plastic Fracture Mechanics. Moscow: Mir Publishers.Google Scholar
2.55 Peters, S.T., ed. 1998. Handbook of Composites, 2nd edn, 325–27. London: Chapman and Hall.CrossRefGoogle Scholar
2.56 Putatunda, S.K. 2001. ‘Fracture Toughness of a High Carbon and High Silicon Steel.Material Science and Engineering A 297: 31–43.CrossRefGoogle Scholar
2.57 Rooke, D.P. and D. J., Cartwright. 1975. Compendium of Stress Intensity Factors. Her Majesty's Stationery Office.
2.58 Sasaki, T., H., Somekawa, A., Takara, Y., Nishikawa and K., Higashi. 2003. ‘Plane-strain Fracture Toughness on Thin AZ31 Wrought Magnesium Alloy Sheets.Materials Transactions 44(5): 986–90.CrossRefGoogle Scholar
2.59 Sih, G.C. 1973a. Handbook of Stress Intensity Factors. Pennsylvania: Lehigh University.Google Scholar
2.60 Sih, G.C., ed. 1973b. ‘Methods of Analysis and Solution of Crack Problems.’ In Mechanics of Fracture, Vol. 1. Leyden: Noordhoff International Publishing.CrossRefGoogle Scholar
2.61 Somekawa, H. and T., Sakai. 2006. ‘Fracture Toughness in an Extruded ZK60 Magnesium Alloy.Material Transactions 47: 995–98.CrossRefGoogle Scholar
2.62 Srawley, J.E. 1976. ‘Wide Range Stress Intensity Factor Expressions for ASTM E399 Standard Fracture Toughness Specimens.International Journal of Fracture 12: 475–76.Google Scholar
2.63 Suresh, S., G.F., Zamiski and R.O., Ritchie. 1982. ‘Fatigue Crack Propagation Behavior of 21 4 Cr-1Mo Steels for Thick Walled Pressure Vessels.’ In Application of 214 Cr-1Mo Steels for Thick Walled Pressure Vessels, eds. Sangdahl, G. S. and M., Semchyshen, 49–67. Philadelphia: American Society for Testing and Materials [ASTM STP 755].Google Scholar
2.64 Tada, H., P.C., Paris and G.R., Irwin. 2000. The Stress Analysis of Cracks Handbook, 3rd edn. New York: ASME Press.CrossRefGoogle Scholar
2.65 Tetelman, A.S. and A., McEvily, Jr. 1967. Fracture of Structural Material. New York: John Wiley.Google Scholar
2.66 Timoshenko, S.P. and J.N., Goodier. 1970. Theory of Elasticity, International Student Edition. New York: McGraw-Hill.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×