An edge dislocation interacting with a completely coated semi-infinite crack
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
An edge dislocation interacting with a completely coated semi-infinite crack Xu Wang and Peter Schiavone
Abstract. We derive an analytic solution to the plane problem associated with an edge dislocation interacting with a completely coated semi-infinite crack in the presence of remote in-plane loading. The semi-infinite crack and the parabolic coating–matrix interface are confocal, and the coating and the matrix have the same shear modulus but distinct Poisson’s ratios. Through the introduction of a conformal mapping function and the application of analytic continuation, a closedform representation of a particular auxiliary function can be obtained leading to analytic expressions of the two pairs of analytic functions in the image plane characterizing the elastic fields in the two-phase composite. The local mode I and mode II stress intensity factors at the crack tip and the image force acting on the edge dislocation are then derived. Mathematics Subject Classification. 30B40, 30E25, 35Q74, 74-10, 74B05, 74M25, 74R10, 74S70. Keywords. Edge dislocation, Crack, Coating, Analytic solution, Stress intensity factor, Image force.
1. Introduction The interaction of dislocations with cracks has been extensively and intensively discussed by many researchers (see, for example, [1,5–8,10,12–14,16]). Solutions to the dislocation–crack interaction problem are fundamental to the establishment of dislocation emission and cleavage criteria [5,7,10,13]. The dislocation–crack interaction problem becomes extremely challenging when the crack is entirely surrounded by a coating leading to an elastic mismatch between the coating and the surrounding matrix. Recently, Wang and Schiavone [12] solved the problem of a screw dislocation interacting with a completely coated semi-infinite crack in which the screw dislocation is located either in the matrix or in the coating. Subsequently, Wang and Schiavone [13] established a condition for the emission of a screw dislocation from a completely coated wedge crack. In this study, we continue our studies and consider the more challenging yet more practical problem associated with an edge dislocation interacting with a completely coated semi-infinite crack subjected to far-field in-plane loading. In order to arrive at an analytic solution to the interaction problem, it is assumed that: (i) the focus of the parabolic coating–matrix interface is located only at the tip of the semi-infinite crack; (ii) the coating and the matrix have the same shear modulus but distinct Poisson’s ratios. By means of conformal mapping and analytic continuation, an auxiliary function is constructed. A closed-form expression of the auxiliary function can be obtained by applying the generalized Liouville’s theorem. Consequently, the original two pairs of complex potentials in the image plane can be derived. Furthermore, analytic expressions of the local mode I and mode II stress intensity factors at the crack tip and the image force acting on the edge dislocation are prese
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