Skip to main content Accessibility help
×
Home
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 1
  • Print publication year: 2015
  • Online publication date: February 2016

2 - Linear Elastic Fracture Mechanics

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.

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.
2.2 Anderson, T. L. 2005. Fracture Mechanics: Fundamentals and Applications. Boston: CRC Press.
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.
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.
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.
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.
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.
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.
2.10 Bluhm, J.I. 1961. ‘A Model for the Effect of Thickness on Fracture Toughness’. ASTM Proceedings 61: 1324–31.
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.
2.12 Broek, D. 1986. Elementary Engineering Fracture Mechanics, 4th revised edn. The Netherlands: Noordhoff.
2.13 Broek, D. and H., Vlieger. 1974. The Thickness Effect in Plane Stress Fracture Toughness. Amsterdam: National Aerospace Institute [Rept. TR 74032].
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].
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].
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.
2.18 Dugdale, D.S. 1960. ‘Yielding of Steel Sheets Containing Slits.Journal of Mechanics and Physics of Solids 8: 100–04.
2.19 Gdoutos, E.E. 1993. Fracture Mechanics – An Introduction. Kluwer. Dordrecht/Boston/ London: Kluwer Academic Publishers.
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.
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.
2.22 Hahn, G.T. and A.R., Rosenfield. 1965. ‘Local Yielding and Extension of Crack Under Plane Stress.Acta Metallurgica 13: 293–306.
2.23 Hellan, K. 1985. Introduction to Fracture Mechanics. New York: McGraw-Hill.
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.
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. 2
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.
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.
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.
2.29 Irwin, G.R. 1948. ‘Fracture Dynamics.’ In Fracturing of Metals, 147–66. Cleveland: ASM Publication.
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.
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.
2.32 Irwin, G.R. 1958. ‘Fracture.’ In Hanbuch der Physik, Vol. VI, ed. Flugge, S., 551–90. Berlin: Springer-Verlag.
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.
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.
2.36 Kanninen, M.F. and C.H., Popelar. 1985. Advanced Fracture Mechanics. New York: Oxford University Press.
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.
2.38 Knott, J.F. 1973. Fundamentals of Fracture Mechanics. London: Butterworths.
2.39 Knott, J.F. 1975. ‘The Fracture Toughness of Metals.Journal of Strain Analysis 10: 201–06.
2.40 Kumar, P. 2009. Elements of Fracture Mechanics. New Delhi: Tata McGraw-Hill.
2.41 Liebowitz, H., ed. 1968. Fracture – and Advanced Treatise, Vol. II, Mathematical Fundamentals. New York: Academic Press.
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].
2.43 Meguid, S.A. 1989. Engineering Fracture Mechanics. London: Elsevier Applied Science.
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.
2.50 Murakami, Y. (Editor-in-Chief). 1987. Stress Intensity Factors Handbook, Vols I and II. Oxford: Pergamon Press.
2.51 Orowan, E. 1949. ‘Fracture and Strength of Solids’, Report on Progress in Physics, Vol. 12, 185–232.
2.52 Orowan, E. 1955. ‘Energy Criteria of Fracture.Welding Journal Research Supplement 20: 157S–160S.
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].
2.54 Parton, V.Z. and E. M., Morozov. 1978. Elastic Plastic Fracture Mechanics. Moscow: Mir Publishers.
2.55 Peters, S.T., ed. 1998. Handbook of Composites, 2nd edn, 325–27. London: Chapman and Hall.
2.56 Putatunda, S.K. 2001. ‘Fracture Toughness of a High Carbon and High Silicon Steel.Material Science and Engineering A 297: 31–43.
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.
2.59 Sih, G.C. 1973a. Handbook of Stress Intensity Factors. Pennsylvania: Lehigh University.
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.
2.61 Somekawa, H. and T., Sakai. 2006. ‘Fracture Toughness in an Extruded ZK60 Magnesium Alloy.Material Transactions 47: 995–98.
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.
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].
2.64 Tada, H., P.C., Paris and G.R., Irwin. 2000. The Stress Analysis of Cracks Handbook, 3rd edn. New York: ASME Press.
2.65 Tetelman, A.S. and A., McEvily, Jr. 1967. Fracture of Structural Material. New York: John Wiley.
2.66 Timoshenko, S.P. and J.N., Goodier. 1970. Theory of Elasticity, International Student Edition. New York: McGraw-Hill.