Issue 48

T. Profant et alii, Frattura ed Integrità Strutturale, 48 (2019) 503-512; DOI: 10.3221/IGF-ESIS.48.48 504 the idealization of the problem geometry, the analytical mathematical tools can propose an energetic balance at the crack tip initiated in this stress field. The energy consideration is a basic concept widely used not only in the fracture mechanics. The energy release rate (ERR) associated with a finite small and arbitrary oriented crack initiation at the interface between the interfacial zone of the circular inclusion and the matrix is considered to assess the direction of the crack path influenced by the presence of the inclusion and its interfacial zone. The topological derivative field indicates the variation of a response functional when an infinitesimal hole is introduced into the body, where the response functional is the total potential energy of the fractured matrix. As the first, it is introduced the fundamental solution in the form of the unit point force or edge dislocation interacting with the inclusion and the interfacial zone, [4]. The fundamental solution is found in the form of the Laurent series whose coefficients are evaluated from the compatibility conditions between the inclusion, interfacial zone and infinity matrix. The fundamental solution is applied to the boundary integral method and the continuously distributed dislocation method [5]. Under the assumption of a relatively short crack with respect to the inclusion and matrix dimensions, the asymptotic analysis [6] introduces the outer and inner solution of the problem represented by the uncracked finite matrix and cracked infinity matrix, respectively. The asymptotic analysis also demonstrates that the mismatch between the stress values along the outer boundary of the uncracked matrix and the stresses at the same points of the unbounded cracked matrix is proportional to the crack length. The energy momentum tensor [7], [8] and the approximation of the energy release rate for any crack size and orientation by means of a topological derivative can be evaluated from the inner solution and the corresponding stress intensity factors at the crack tip lying in the matrix. Figure 1 : The problem formulation. The cracked matrix under the statically equilibrated external load and containing the inclusion with thin zone. The length of the crack is small with respect to the dimension of the inclusion. Figure 2 : The point force f and the dislocation with Burgers vector b at the circular inclusion with the interfacial zone. F UNDAMENTAL SOLUTION he derivation of the fundamental solution, see Fig. 2, describing the interaction between the unit point force f or dislocation with Burgers vector b and the circular inclusion with an interfacial zone is based on [4], but contrary to its results it is supplemented by a rigid displacement of the body, because the fundamental solution is also to be T