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Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

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Abstract

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.

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References

  • Anderson TL (2005) Fracture mechanics. CRC press

  • Bilby BA, Cottrell AH, Swinden KH (1963) The spread of plastic yield from a notch. Proc Roy Soc A 272: 304–314

    Article  Google Scholar 

  • Chen YH, Hasebe N (1997) Explicit formulations of the J-integral considering higher order singular terms in eigenfunction expansion forms. Part I. Analytical treatments. Int J Fract 85: 11–34

    Article  Google Scholar 

  • Chen DH, Ushijima K (2000) Plastic stress singularity near the tip of a V-notch. Int J Fract 106: 117–134

    Article  Google Scholar 

  • Cherepanov GP (1962) An elastic–plastic problem under conditions of antiplanar deformation. J Appl Math Mech 26: 1040–1057

    Article  Google Scholar 

  • Creager M, Paris PC (1967) Elastic field equations for blunt cracks with reference to stress corrosion cracking. Int J Fract Mech 3: 247–252

    CAS  Google Scholar 

  • Davis EA, Tuba IS (1963) Elastic–plastic solutions for notched shafts in torsion. J Appl Mech 33: 79

    Google Scholar 

  • Ellyin F (1997) Fatigue damage, crack growth and life prediction. Chapman & Hall, London

    Google Scholar 

  • Filippi S, Ciavarella M, Lazzarin P (2002) An approximate, analytical approach to the ‘HRR’-solution for sharp V-notches. Int J Fract 117: 269–286

    Article  Google Scholar 

  • Forsyth ARA (1996) Treatise on differential equations. Dover Publications

  • Gdoutos EE (1990) Fracture mechanics criteria and applications. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Glinka G (1985a) Energy density approach to calculation of inelastic strain–stress near notches and cracks. Engng Fract Mech 22: 485–508

    Article  Google Scholar 

  • Glinka G (1985b) Calculation of inelastic notch-tip strain–stress histories under cyclic loading. Engng Fract Mech 22: 839–854

    Article  Google Scholar 

  • Glinka G, Ott W, Nowack H (1988) Elastoplastic plane strain analysis of stresses and strains at the notch root. J Engng Mater Tech 110: 195–204

    Article  Google Scholar 

  • Guo W, Wang CH, Rose LRF (1998) Elastoplastic analysis of notch-tip fields in strain hardening materials. Report DSTO-RR-0137, DSTO Aeronautical and Maritime Research Laboratory, Melbourne, Victoria, Australia

  • Hasebe N, Kutanda Y (1978) Calculation of stress intensity factor from stress concentration factor. Engng Fract Mech 10: 215–221

    Article  Google Scholar 

  • Hui CY, Ruina A (1995) Why K? Higher order singularities and small scale yielding. Int J Fract 72: 97–120

    Google Scholar 

  • Hult JAH (1957) Elastic–plastic torsion of sharply notched bars. J Mech Phys Solids 6: 79–82

    Article  Google Scholar 

  • Hult JAH, McClintock FA (1956) Elastic–plastic stress and strain distribution around sharp notches under repeated shear. 9th Int Cong Appl Mech, 8, Brussels

  • Hutchinson JW (1968a) Singular behavior at the end of a tensile crack in a hardening material. J Mech Phys Solids 16: 13–31

    Article  Google Scholar 

  • Hutchinson JW (1968b) Plastic stress and strain fields at a crack tip. J Mech Phys Solids 16: 337–347

    Article  Google Scholar 

  • Hutchinson JW (1979) Non linear fracture mechanics. Department of Solid Mechanics, Technical University of Denmark

  • Hutchinson JW (1983) Fundamentals of the phenomenological theory of nonlinear fracture mechanics. J Appl Mech 50: 1042–1051

    Article  Google Scholar 

  • Irwin GR (1961) Plastic zone near a crack and fracture toughness, Sagamore Research Conference Proceedings, vol 4. Syracuse University Research Institute, pp 63–78

  • Koskinen MF (1963) Elastic–plastic deformation of a single grooved flat under longitudinal shear. J Basic Engng 85: 585–594

    Google Scholar 

  • Kryven’ VA (2001) Antiplane problem for an elastic perfectly plastic body with biperiodic system of rhombic notches. Mater Sci 37: 866–872

    Article  Google Scholar 

  • Kuang ZB, Xu XP (1987) Stress and strain fields at the tip of a sharp V-notch in a power hardening material. Int J Fract 35: 39–53

    Google Scholar 

  • Larsson SG, Carlsson AJ (1973) Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic–plastic materials. J Mech Phys Solids 21: 263–277

    Article  Google Scholar 

  • Lazzarin P, Zambardi R (2002) The equivalent strain energy density approach re-formulated and applied to sharp V-shaped notches under localized and generalized plasticity. Fatigue Fract Engng Mater Struct 25: 917–928

    Article  Google Scholar 

  • Lazzarin P, Zambardi R, Livieri P (2001) Plastic notch stress intensity factors for large V-shaped notches under mixed load conditions. Int J Fract 107: 361–377

    Article  Google Scholar 

  • Lazzarin P, Sonsino CM, Zambardi R (2004) A Notch Stress Intensity approach to predict the fatigue behaviour of T butt welds between tube and flange when subjected to in-phase bending and torsion loading. Fract Engng Mater Struct 27: 127–141

    Article  CAS  Google Scholar 

  • Lazzarin P, Zappalorto M, Yates J (2007) Analytical study of stress distributions due to semi-elliptic notches in shafts under torsion loading. Int J Engng Sci 45: 308–328

    Article  CAS  Google Scholar 

  • Molski K, Glinka G (1981) A method of elastic–plastic stress and strain calculation at a notch root. Mater Sci Engng 50: 93–100

    Article  CAS  Google Scholar 

  • Neuber H (1958) Theory of notch stresses. Splinger-Verlag, Berlin

    Google Scholar 

  • Neuber H (1961) Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress–strain law. J Appl Mech 28: 544–550

    Google Scholar 

  • Radaj D, Sonsino CM, Fricke W (2006) Fatigue assessment of welded joints by local approaches. 2nd edn. Woodhead Publishing, Cambridge

    Google Scholar 

  • Radi E (2007) Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials. Int J Plasticity 23: 1439–1456

    Article  CAS  Google Scholar 

  • Rice JR (1966) Contained plastic deformation near cracks and notches under longitudinal shear. Int J Fract Mech 2: 426–447

    CAS  Google Scholar 

  • Rice JR (1967a) Stresses due to a sharp notch in a work-hardening elastic–plastic material loaded by longitudinal shear. J Appl Mech 34: 287–298

    Google Scholar 

  • Rice JR (1967b) Mechanics of crack tip deformation and extension by fatigue. In: Fatigue crack propagation. ASTM STP 415, American Society for Testing and Materials, Philadelphia, PA, pp 247–311

  • Rice JR (1974) Limitations to the small scale yielding approximation for crack tip plasticity. J Mech Phys Solids 22: 17–26

    Article  Google Scholar 

  • Rice JR, Rosengren GF (1968) Plane strain deformation near a crack tip in a power-law hardening material. J Mech Phys Solids 16: 1–12

    Article  Google Scholar 

  • Sanford RJ (2003) Principles of fracture mechanics. Prentice Hall

  • Saxena A (1998) Nonlinear fracture mechanics. CRC press

  • Sharma SM, Aravas N (1991) Determination of higher-order terms in asymptotic elastoplastic crack tip solutions. J Mech Phys Solids 39: 1043–1072

    Article  Google Scholar 

  • Shin CS (1994) Fatigue crack growth from stress concentrations and fatigue life predictions in notched components. In: Carpinteri An(eds) Handbook of fatigue crack propagation in metallic structures. Elsevier Science Publishers BV, Amsterdam, pp 613–652

    Google Scholar 

  • Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York

    Google Scholar 

  • Tuba IS (1966) Elastic–plastic torsion of shafts with hyperbolic notches. Int J Mech Sci 8: 683–701

    Article  Google Scholar 

  • Unger DJ (2001) Analytical fracture mechanics. Dover publication

  • Valiron G (1984) The geometric theory of ordinary differential equations and algebraic functions. Math Sci Pr

  • Wang TJ, Kuang ZB (1999) Higher order asymptotic solutions of V-notch tip fields for damaged nonlinear materials under antiplane shear loading. Int J Fract 96: 303–329

    Article  Google Scholar 

  • Xia L, Wang T (1993) Singular behavior near the tip of a sharp V-notch in a power-law hardening material. Int J Fract 59: 83–93

    Google Scholar 

  • Xia L, Wang T, Shih CF (1993) Higher-order analysis of crack tip fields in elastic power-law hardening materials. J Mech Phys Solids 41: 665–687

    Article  Google Scholar 

  • Yang S, Yuan FG, Cai X (1996) Higher order asymptotic elastic–plastic crack-tip fields under antiplane shear. Eng Fract Mech 54: 405–422

    Article  Google Scholar 

  • Yuan FG, Yang S (1994) Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear. Int J Fract 69: 1–26

    Article  Google Scholar 

  • Zappalorto M, Lazzarin P, Yates J (2008) Elastic stress distributions for hyperbolic and parabolic notches in round shafts under torsion and uniform antiplane shear loadings (submitted)

  • Zhang N, Joseph PF (1998) A nonlinear finite element eigenanalysis of singular plane stress fields in bimaterial wedges including complex eigenvalues. Int J Fract 90: 175–207

    Article  Google Scholar 

  • Zhang L, Huang Y, Chen JY, Hwang KC (1998) The mode III full-field solution in elastic materials with strain gradient effects. Int J Fract 92: 325–348

    Article  CAS  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Management and Engineering, University of Padova, Stradella. S. Nicola 3, Vicenza, 44100, Italy

    M. Zappalorto & P. Lazzarin

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  1. M. Zappalorto
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  2. P. Lazzarin
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Correspondence to P. Lazzarin.

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Zappalorto, M., Lazzarin, P. Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading. Int J Fract 148, 139–154 (2007). https://doi.org/10.1007/s10704-008-9185-7

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  • DOI: https://doi.org/10.1007/s10704-008-9185-7

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