Issue34

F. Berto, Frattura ed Integrità Strutturale, 34 (2015) 11-26; DOI: 10.3221/IGF-ESIS.34.02 17   1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1 )cos(1 ) cos(1 ) 1 (3 )cos(1 ) (1 ) cos(1 ) 1 1 (1 )sin(1 ) sin(1 ) rr b b r f f f                                                                               (2)     1 1 1 1 1 1 1 1 1 1 1 1 1 1 (1 )cos(1 ) cos(1 ) (3 )cos(1 ) cos(1 ) 4 1 1 1 (1 )sin(1 ) sin(1 ) rr d c b r g q g q g                                                                                   (3) The eigenfunctions f ij depend only on Williams’ eigenvalue,    which controls the sharp solution for zero notch radius [66]. The eigenfunctions g ij mainly depend on eigenvalue   , but are not independent from   . Since   <   , the contribution of  -based terms in Eq. (1) rapidly decreases with the increase of the distance from the notch tip. All parameters in Eqs.(2,3) have closed form expressions but here, for the sake of brevity, only some values for the most common angles are reported in [65]. Under the plane strain conditions, the eigenfunctions f ij and g ij satisfy the following expressions:   ( ) ( ) ( ) zz rr f f f          ( ) ( ) ( ) zz rr g g g        (4) whereas     zz zz 0 f g     under plane stress conditions. Figure 2 : Notch geometry and coordinate system The 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  t , so that 2 t / 2 c W E   . Often un-notched specimens exhibit a non-linear behaviour whereas the behaviour of notched specimens remains linear. Under these circumstances the stress  t should be substituted by “ the maximum normal stress existing at the edge at the moment preceding the cracking ”, as underlined in Ref. [21] where it is also recommended to use tensile specimens with semicircular notches. 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. 3a,b). Under plane strain conditions, a useful expression for R 0 has been provided considering the crack case [41]: 2 IC 0 (1 )(5 8 ) 4π t K R             (5) If the critical value of the NSIF is determined by means of specimens with 2   0, the critical radius can be estimated by means of the expression: