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)