Issue 36

V. Petrova et alii, Frattura ed Integrità Strutturale, 36 (2016) 8-26; DOI: 10.3221/IGF-ESIS.36.02 9 There is abundant experimental results (e.g., [4-6]) showing that when TBCs are subjected to thermal shock, multiple cracks occur at the ceramic surface. Besides, the crack patterns strongly depend on the microstructure of the materials and on the type of loading. Numerous investigations are devoted to different types of fracture including surface fracture. In previous papers of the authors [7-10] the fracture of FGM/homogeneous bimaterials (an infinite medium) under thermal and mechanical loadings were investigated, besides, in [11] some results for edge cracks in FGM/homogeneous structures (a semi-infinite medium) were obtained in the frame of the approach used in [7-10]. The results show that the fracture of materials (both composites and homogeneous) is significantly affected by a complex crack interaction mechanism, e.g., interacting cracks can enhance or suppress the propagation of each other. During further studying of the fracture of functionally graded coatings on a homogeneous substrate and the preparation of the results for the influence of material non-homogeneity on surface fracture it became clear that a classical problem for edge cracks interaction is still not well examined. Before presenting the results for more complicated cases of non- homogeneous materials (FGMs, bimaterials, and others), modeling of the interaction of edge cracks should be done for a homogeneous medium. From experimental and theoretical investigations it is known that cracks are sensitive to geometry, e.g., to the inclination angle to the load. A small deviation of a crack from the normal direction to a tensile load causes mixed mode conditions near the crack which lead to deviation of the crack from its initial propagation direction. Besides, the presence of other cracks, inhomogeneities, surfaces and their interaction causes additional deformations and stresses which are also influenced on the initiation of the crack propagation and on the direction of this propagation. That is, the picture of the fracture with respect to the crack pattern for a system of arbitrary inclined edge cracks will be different from the picture of regularly distributed cracks, e.g, for periodically distributed equal (and non-equal) cracks, this case was often studied, see [12-14]. The goal of this paper is to show the effects of the interaction of edge cracks on further fracture formation. The main fracture characteristics, such as, stress intensity factors, fracture angles and critical loads are provided for this study. A series of illustrative examples is presented for different geometries of arbitrarily inclined edge cracks. P ROBLEM FORMULATION AND ASSUMPTIONS he geometry of the problem is presented in Fig. 1 a. A homogeneous half-medium contains pre-existing edge cracks inclined arbitrarily on angles β n to the surface. A Cartesian coordinate system ( x, y ) has x -axis along the boundary of the half-plane, and local coordinate systems ( x k , y k ) are attached to each crack. The lengths of the cracks are 2 a n , and the midpoint coordinates are 0 0 0 n n n iy x z   ( 1  i is imaginary unity). The homogeneous medium is subjected to tension p applied parallel to the free surface. (a) (b) Figure 1 : (a) Edge cracks inclined arbitrarily with an angle β n to the surface of the medium. a n – a half length of n -th crack, z n 0 = x n 0 +i y n 0 – the crack midpoint coordinate. (b) The angle ϕ of crack deflection (the fracture angle). The problem is solved by using the method of singular integral equations. The cracks are modeled by displacement jumps on the crack faces and unknown functions in this formulation are the derivatives of displacement jumps   ] [ ] [ )1 ( 2 ) ( n n n vi u x i x g         (1) T