International Journal of Fracture

, Volume 146, Issue 1–2, pp 33–51 | Cite as

J-integral evaluation for U- and V-blunt notches under Mode I loading and materials obeying a power hardening law

  • F. Berto
  • P. Lazzarin
  • Yu. G. MatvienkoEmail author
Original Paper


The paper deals with calculations of the J-integral for a plate weakened by U- and V-blunt notches under mode I loading in the case of a linear and nonlinear elastic material. The main aim of the study is to suggest simple equations suitable for rapid calculations of the J-integral. The semicircular arc of the notch, which is traction free, is assumed as integration path and the J-integral is given as a function of the strain energy over the notch edge. For a numerical investigation of the strain energy density distribution on the notch edge the equation W(θ)=Wmax cosδ(θ) has been assumed, where δ has been determined from finite element analyses. In particular, the following values of the notch acuity a/ρ and the opening angle 2α have been analyzed: 4  ≤  a/ρ   ≤   400 and 0  ≤   2α  ≤   3π /4. Considering plates weakened by lateral and central notches under symmetric mode I loading, the approximate relationships for the strain energy density, which require the presence of a non zero notch radius for their application, and the J-integral are discussed firstly considering a linear elastic material and then a material obeying a power hardening law during the loading phase. The predicted results of the J-integral are consistent with those directly obtained from finite element analyses.


Strain energy density distribution J-integral U- and V-notches Finite element results 



Notch depth (lateral notch) or notch semi-depth (central notch)


Width of the specimen


Young’s modulus


J-integral (as due to the notch arc)


J-integral under linear elastic conditions


Constant in the Ramberg-Osgood law


Theoretical stress concentration factor


Hardening exponent in the Ramberg-Osgood law (1 ≤  n  <  ∞)


Distance from the notch tip


Strain energy density on the semicircular edge of the notch

V-notch opening angle (2α = 0 in a U-notch)


Exponent in the equation of the strain energy density


Poisson’s ratio


Angular coordinate


Notch radius


Yield stress (0.2% offset stress)


Remotely applied tensile stress


Beltrami equivalent stress at the notch tip


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Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Management and EngineeringUniversity of PadovaVicenzaItaly
  2. 2.Mechanical Engineering Research InstituteRussian Academy of SciencesMoscowRussia

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