Issue 50

P. Livieri et alii, Frattura ed Integrità Strutturale, 50 (2019) 613-622; DOI: 10.3221/IGF-ESIS.50.52 615 O ORE -B URNS ON A CIRCULAR CRACK et Ω be a fixed set. We can reconstruct the OB integral along the front crack ( )    by its values along  by the equation ( , ', ) (1, '/ , ) I n I K Q K Q       (3) where ( ) n Q     with 0   If we “read” the boundary point Q’ in terms of an angle α that is, for example Ω is star shaped with respect to the origin, (3) takes the simplest form ( , , ) (1, , ) I n I K K        (4) In the particular case when Ω is a disk of radius a centred at the origin of the plane ( , ) x y , we denote by (x,y) the system in dimensionless coordinates x= x a and y= y a (see Fig. 2). Figure 2 : reference circular crack By definition, for the unit disk Ω’, by (1) and (2) it follows:     2 2 ,0 2 2 1 2 ( , ) ( , ) ( ) cos sin I x y a x y h x y K dx dy x y             (5) With ( , ) 1/ ( , ) h x y f x y  (6)     2 2 ( , ) cos sin d f x y x y            (7) By introducing the change of variables       0, , 0, / 2       (1 sin )cos cos sin cos cos (1 sin )sin x r r y r r                  (8) where L