Some new practical equations for rapid calculation of J-integral in plates weakened by U-notches under bending

The paper deals with calculations of the J-integral for a plate weakened by U-notches under Mode I bending loading in the case of a material obeying a linear elastic law. The main aim of the study is to suggest some new equations suitable for calculations of the J-integral under bending loading. Although some closed-form solutions exist in the literature based on elliptic integrals, some expressions, simple to be implemented, are useful as a rapid engineering tool.The semi-circular arc of the notch, which is traction-free, is assumed as the integration path and the J-integral is given as a function of the strain-energy density. For a numerical investigation of the strain-energy density distribution on the notch edge, the equation W(θ) = Wmax cos δ(θ) has been assumed, where the exponent δ has been determined from finite element analyses. The influence on δ of the notch acuity of the specimen width to the notch depth ratio w/a has been investigated by means of several parametric finite element analyses. In particular, the following values of the notch acuity (a/ρ) and notch depth ratio (w/a  ) have been analyzed: 4⩽a/ρ⩽2004⩽a/ρ⩽200 and 2⩽w/a⩽1002⩽w/a⩽100.