Some Solutions That Are Useful in Fracture Prediction

  • C. K. H. Dharan
  • B. S. Kang
  • Iain Finnie


Our primary objective is to use solutions that have been developed for elastic, elastic–plastic, and fully plastic behavior to predict fracture. The detailed development of these solutions and a discussion of techniques, analytical and numerical, used to obtain them are beyond the scope of this work. However, for completeness, some of the analytical techniques that have been used to obtain stress fields for sharp cracks and other configurations will be described. We consider first the elastic solutions for sharp cracks, circular holes, and elliptical holes in flat plates. Also, the elastic solutions for cylindrical inclusions, for spherical holes, and inclusions will be summarized. Then, approximate and more exact solutions, based on elastic–plastic behavior for sharp and blunted cracks, will be described. Finally, solutions for notched members will be presented, both for the initial elastic stress distribution and for the case in which yielding has occurred over the entire cross section. A few solutions have been omitted from this chapter because it seems more appropriate to treat them at a later stage. These are the Dugdale model for the plastic zone ahead of a crack for plane stress (Chap.  5), the solutions for stresses and strains (in terms) of the J integral (Chap.  6), and the solutions for void growth in ductile fracture (Chap.  7).


Stress Intensity Factor Plane Strain Plastic Zone Stress Function Stress Concentration Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supplementary material


  1. 1.
    Westergaard HM. Bearing Pressure and Cracks. J Appl Mech Trans ASME. 1939(6);A49–53.Google Scholar
  2. 2.
    Muskhelishvili NI. Some basic problems of the mathematical theory of elasticity. Groningen, The Netherlands: P. Noordhoff Ltd; 1953. published originally in Russian in 1933.MATHGoogle Scholar
  3. 3.
    Williams ML. On the stress distribution at the base of a stationary crack. J Appl Mech. 1957;24:109–14. Trans. ASME, Vol. 79.MathSciNetMATHGoogle Scholar
  4. 4.
    The solution is originally due to G. Kirsch, VDI, Vol. 42, 1898. It is described in detail in Timoshenko and Goodier’s “Theory of Elasticity.”Google Scholar
  5. 5.
    Greaves RH, Jones JA. The effect of temperature on the behaviour of iron and steel in the notched-bar impact test. J Iron and Steel Inst. 1925;112:123–65.Google Scholar
  6. 6.
    Hult JA, McClintock FA. Proc. 9th Int. Con. App. Mech., Brussels. 8; p. 51 (1956)Google Scholar
  7. 7.
    Rice JR, Contained plastic deformation near cracks and notches under longitudinal shear. Int. J. Fract. Mech., 1966;2:426–447.Google Scholar
  8. 8.
    Rice JR and Rosengren GF, Plane Strain Deformation Near a Crack in a Power Law Hardening Material. Journal of the Mechanics and Physics of Solids, 16, 1968, pp. 1–12.CrossRefMATHGoogle Scholar
  9. 9.
    Hutchison AA, J Mech Phy Solids. Singular behavior at the end of a tensile crack in a hardening material. 1968;(16)13;337Google Scholar
  10. 10.
    Tuba IS. A method of elastic–plastic plane stress and strain analysis. J Strain Anal. 1966;(1) 2:115.Google Scholar
  11. 11.
    Marcal PV, King IP. Elastic–plastic analysis of two-dimensional stress systems by the finite element method. Int J Mech Sci. 1967; 9:143.CrossRefGoogle Scholar
  12. 12.
    Rice JR, Johnson MA. The Role of Large Crack Tip Geometry Changes in Plane Strain Fracture, in Inelastic Behavior of Solids (eds. Kanninen MF et al.), McGraw-Hill, N.Y., 1970, pp. 641–672.Google Scholar
  13. 13.
    Levy N, Marcal PV, Ostergren WJ and Rice JR. Small Scale Yielding Near a Crack in Plane Strain: A Finite Element Analysis, International Journal of Fracture Mechanics, 7, 1971, pp. 143–156.CrossRefGoogle Scholar
  14. 14.
    Green AP. The plastic yielding of notched bars due to bending. Quart. J. Mech. Appl. Math., Vol. 6, Pt. 2, p. 233 (1953).Google Scholar
  15. 15.
    Ewing DJF. A series method for constructing slip line fields. J. Mech. Phys. Solids, Vol. 16, p. 205 (1968).CrossRefGoogle Scholar
  16. 16.
    Alexander JM, Komoly TJ. On the yielding of a rigid plastic bar with Izod notch. Mech. Phys. Solids, Vol. 10, p. 265 (1962).CrossRefGoogle Scholar
  17. 17.
    Johnson W. Sowerby R. On the collapse of some simple structures. Int. J. Mech. Sci., Vol. 9, p. 433 (1967).CrossRefGoogle Scholar
  18. 18.
    Green AP, Hundy BB. Initial plastic yielding in notch bend tests. J Mech Phys Solids 1956;16:128–44.CrossRefGoogle Scholar
  19. 19.
    Owen DRJ, Nayak GC, Kfouri AP, Griffiths JR. Stresses in a partly yielded notched bar—An assessment of three alternative programs. Int. Journ. for Numerical Methods in Eng., Vol. 6, p. 63 (1973).Google Scholar
  20. 20.
    Bridgmen PW. Studies in large plastic flow and fracture. New York: McGraw-Hill; 1952.Google Scholar
  21. 21.
    Alpaugh HE. Investigation of the mechanisms of failure in the ductile fracture of mild steel. S.B. Thesis, Dept. of Mech. Eng., M.I.T. (1965). Results reported by McClintock, F.A. In: Ductility, ASM, Metals Park, OH, p. 255 (1968).Google Scholar
  22. 22.
    Clausing, DP. Effect of plane-strain sensitivity on the Charpy toughness of structural steels. Journal of Materials, Vol. 4, p. 566 (1968).Google Scholar
  23. 23.
    Morrison HL, Richmond O. Large deformation of notched perfectly plastic tensile bars. J Appl Mech. 1972;39(4):971.CrossRefGoogle Scholar
  24. 24.
    Neimark, J. E., The fully plastic plane strain tension or a notched bar. J. Appl. Mech., Vol. 35, p. 111 (1968).CrossRefGoogle Scholar
  25. 25.
    Hill R, The plastic yielding of notched bars under tension. Quart. Journ. Mech. and Appl. Math., II, Pt. I, p. 40 (1949).Google Scholar
  26. 26.
    McClintock FA. A criterion for ductile fracture by the growth of holes. Int. Jour. Fracture Mechanics, 4, 2, p. 101 (1968).Google Scholar
  27. 27.
    McClintock FA, Liebowitz HA (eds.) Treatise on fracture. Academic Press: New York; 19713:191.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • C. K. H. Dharan
    • 1
  • B. S. Kang
    • 2
  • Iain Finnie
    • 1
  1. 1.University of California at BerkeleyBerkeleyUSA
  2. 2.School of EngineeringPusan National University Beon-gilBusanKorea, Republic of

Personalised recommendations