Issue 43

F. Berto et alii, Frattura ed Integrità Strutturale, 43 (2018) 1-32; DOI: 10.3221/IGF-ESIS.43.01 3 A NALYTICAL BACKGROUND OF THE S TRAIN E NERGY D ENSITY APPROACH he SED approach is based on the idea that under tensile stresses failure occurs when  c W W , where the critical value W c obviously varies from material to material. If the material behaviour is ideally brittle, then W c can be evaluated by using simply the conventional ultimate tensile strength σ t , so that   2 t / 2 c W E . In plane problems, the control volume becomes a circle or a circular sector with a radius R 0 in the case of cracks or pointed V-notches in mode I or mixed, I+II, mode loading (Fig. 1a,b). Under plane strain conditions, a useful expression for R 0 has been provided considering the crack case [49]:               2 IC 0 (1 )(5 8 ) 4π t K R (1) R 0 R 0 2   R 2 =R 0 + r 0   R 0 r 0 (a) (b) (c) 2   2  =0   Figure 1: Critical volume (area) for sharp V-notch (a) , crack (b) and blunt V-notch (c) under mode I loading. R 0 r 0  R 0 r 0  (b) P O’ (a)  Figure 2: Critical volume for U-notch under mode I (a) and mixed mode loading (b) . Distance r 0 =ρ/2 [49]. In the case of blunt notches, the area assumes a crescent shape, with R 0 being its maximum width as measured along the notch bisector line (Fig. 1c) [70, 71]. Under mixed-mode loading, the control area is no longer centered with respect to the notch bisector, but rigidly rotated with respect to it and centered on the point where the maximum principal stress reaches its maximum value [70, 71]. This rotation is shown in Fig. 2 where the control area is drawn for a U-shaped notch both under mode I loading (Fig.2a) and mixed-mode loading (Fig. 2b). It is possible to determine the total strain energy over the area of radius R 0 and then the mean value of the elastic SED referred to the area  . The final relationship is                2 1 1 1 1 1 1 0 4 ( ) I K W E R (2) where λ 1 is Williams’ eigenvalue [77] and K 1 the mode I notch stress intensity factor. The parameter I 1 is different under plane stress and plane strain conditions [7]. In the presence of rounded V-notches it is possible to determine the total strain energy over the area  and then the mean value of the SED. When the area embraces the semicircular edge of the notch (and not its rectilinear flanks), the mean T