Theoretical background

Lazzarin and Zambardi developed the Strain Energy Density (SED) criterion, according to which, under tensile stresses, failure occurs when the average value of the strain energy density over a given control volume is equal to a critical value \(W_{c}\) 18. According to Beltrami’s hypothesis 28, we have:
\(W_{c}=\frac{\sigma_{c}^{2}}{E}\) (1)
where \(\sigma_{c}\) is the ultimate tensile strength and \(E\) is the Young’s modulus. The critical value \(W_{c}\) varies from material to material but it does not depend on the notch geometry and sharpness22,29. The control volume is thought of as dependent on the ultimate tensile strength, on the Poisson’s ratio \(v\) and on the fracture toughness \(K_{\text{Ic}}\) 30. Such a method was first formalized and applied to sharp, zero radius, V-notches under mode I and mixed mode I/II loading and later extended to blunt U and V-notches and to real components 21,31–36. When dealing with sharp V-notches, the critical volume is a circle of radius\(R_{c}\) centered at the tip (Fig. 2a).
Under plane strain conditions, the critical length \(R_{c}\) can be evaluated according to the following expression37–40:
\(R_{c}=\frac{\left(1+v\right)\left(5-8v\right)}{4\pi}\left(\frac{K_{\text{Ic}}}{\sigma_{c}}\right)^{2}\)(2)
In the case of blunt V-notches and U-notches (Figs. 2b and c, respectively), the volume assumes a crescent shape, where \(R_{c}\) is the depth measured along the notch bisector line. The outer radius of the crescent shape is equal to \(R_{c}+r_{0}\) where \(r_{0}\) depends on the notch opening angle \(\left(2\alpha\right)\) according to the following expression:
\(r_{0}=\frac{q-1}{q}\rho\) (3)
with \(q\) defined as
\(q=\frac{2\pi-2\alpha}{\pi}\) (4)