Issue 41

A. Carpinteri et alii, Frattura ed Integrità Strutturale, 41 (2017) 175-182; DOI: 10.3221/IGF-ESIS.41.24 177 For the considered geometry, the internal wedge angle is 2 ߙ = 3/2 ߨ and the angle corresponding to the line of the crack is ߴ = ߨ /4. Therefore we have:             0 0 0.5445 , 0.9085 0.543 , 0.219 0.7304 , 1.087 I II I II r r I I II II g g K K K K (2) where K I 0 , K II 0 are simply scaled versions of the Mode I and Mode II notch stress intensity factors K I , K II , obtained from a calibration with the finite element method. Let us consider the crack to be of length c ≪ a and lying along the x -axis. A distribution of edge dislocations of densities B x , B y is added, so that the resulting integral equations which govern the problem are the following:                                0 0 ( 1 ( ) ( , ) ( ) ( , ) ) ( 1 ( ) ) 2 c c x xxy yxy t y B F x d B F x d x x x G (3)                                0 0 ( 1 ( ) ( , ) ( ) ( , ) ( ) ( ) 2 1) x xyy y y y c n y c B F x d B F x d x x x G (4) where G is the shear modulus, ߢ = 3 - 4 ߥ is the Kolosov constant for plane strain, and ߥ is the Poisson ratio. The influence function F kij (x, ξ ) , connecting the stress component ߪ ij (x) to a dislocation b k ( ξ ) , can be found in [3]. The right- side terms of Eqs. (3)-(4) contain the far-field stresses ߬ ∞ (x) and ߪ ∞ (x) , obtained from the Williams solution, and the shear and normal stresses applied to the crack surfaces, ߪ t (x) and ߪ n (x) , respectively. Introducing the normalized variables t,s , with crack extremes ± 1, instead of x and  respectively, we can write the singular integral equations in the usual form, and express the unknown dislocation densities B x , B y as follows:         1 ( ) ( ) ( ) ( ) , , 1 j j j s B s s s s j x y s (5) where we have chosen the fundamental form of the solution ߱ (s) to be singular at the crack tip s = 1 and bounded at s = - 1. A numerical solution of the integral equations is needed, and this can be achieved by means of the Gauss-Chebyshev quadrature described in [10]. The resulting 2N equations in the 2N unknown ߶ j (s i ) are the following:                                   1 1) ( 1 ( ) ( ) ( , ) ( ) ( , ) ( ) ( ) 2 i x i xxy k i i yxy k i k t i k k i N y W s s F t s s F t s t t t s G (6)                                   1 ( 1 ( ) ( ) ( , ) ( ) ( , ) ( ) 1 ( ) 2 ) N i x i xyy y k i i yyy k i k n k i k i W s s F t s s F t s t t t s G (7) where W(i) are the weight functions, s i are the integration points at which the unknown functions and the displacements are computed, whereas t k are the collocation points at which we evaluate the stresses. Stress intensity factors at the crack tip are directly related to the value of ߶ j (s) for s = +1:         2 2 ( ( 1) 1), , , , i j G K i I II j x c y (8) Values of ߶ j at end-points are not included in this quadrature method, thus Krenk’s interpolation is used [11]. Interface constitutive law In Eqs. (6)-(7), the stresses on the crack surface ߪ t and ߪ n are related to the relative displacements by means of a constitutive law which describes friction and roughness. According to this law, the crack surfaces are assumed to be