Issue 50

P. Livieri et alii, Frattura ed Integrità Strutturale, 50 (2019) 613-622; DOI: 10.3221/IGF-ESIS.50.52 616 2 2sin sin r    (9) by some simple calculations 4 sin sin cos dx dy r d d       (10) 2 1 2sin r r      (11) By using Eqn. (8–11), we have the following expression for the stress intensity factors on the unitary disk 2 ,0 4 ( ) ( , ) sin cos I K x y d d           (12) where the integral is computed on the “longitude”   0,    and the “latitude”   0, / 2    . Moreover, the pressure σ(x,y) is “read” in the new coordinates (  , φ) for any fixed α, in the sense that x and y are given by (8), with r being defined by (9). If the crack has a radius equal to a the stress intensity factor becomes 2 ,0 4 ( ) ( , ) sin cos I a K x y d d           (13) When   1   , from (13) we obtain the well-known result: ,0 2 ( ) 1.12837 I a K a     (14) We may test the efficiency of Eqn. (13) by comparison with the special cases of nominal stress distribution considered in the literature [18]. Furthermore, many other new examples have been obtained in reference [19] by changing the shape of the nominal stress σ . SIF FOR AN IRREGULAR CRACK SHAPE LIKE A STAR DOMAIN or a crack like a star domain as reported in Fig. 3, the Oore-Burns integral can be evaluated without a particular numerical procedure. The Oore-Burns integral will be approximated by means of Riemann sums. Let us use the Cartesian reference system x,y, Q’ is a point of coordinate (R,α) on  . Now we consider a new orthogonal reference system u,v with origin in Q’ with n tangent to  (see Fig. 3). A mesh of size δ on  can be considered, where δ divides the length of  . Q jk of coordinate (kδ, jδ) in the u,v plain, and also Q jk = Rδ(cos(jδ), sin(jδ)). P m in Fig. 3 is a point of coordinate mδ with respect to the initial point P O . The Riemann sums K I (δ, Q’) is given by: ( ) 2 ( , ') jk ij I Q A K Q k       (15) where   1 2 2 ij jk m A Q P      (16) The sum (15) is made on 2 0 1, jk m Q       . F