The equations of the effective two-phase cracked model with finite length cracks

Abstract

For the investigation of wave propagation in a cracked media a number of assumptions about the shape of cracks is made. Infinite length of cracks is one of the main assumptions. On the base of similar assumptions and by means of the matrix method and the method of layer averaging, a two-phase effective model of the cracked medium was derived specifically in the case where the cracks are filled with liquid [1,2]. The study of this model showed a good coincidence of results of model theory with experimental data [3][4][5]. For luther development of effective models, it is useful to get rid of the assumptions made. In this paper cracks filled with liquid are considered in the case of finite length. For media with such cracks, the equations of the effective model are derived. w 1. THE CONDITIONS ON THE INTERFACE BETWEEN TWO CRACKED HALF-SPACESLet us consider two cracked half-spaces 0 (x < 0) and 1 (x > 0). The interface between them, x = 0, is perpendicular to cracks in both half-spaces. For simplicity we assume that the layers of the media x > 0 and x < 0 are periodic along the z-axis and that the fields of displacements contain no components along the y-axis and are independent of the coordinate y. Such media are represented schematically in Fig. 1.The cracks filled with liquid are shaded. After averaging both half-spaces are described by the two:phase effective model of the cracked media [1]. The displacements u (1) -(2) and Uz as well as the stresses t (1) and t (2) satisfy the equations of continuous medium Ox --P2 0t 2 (1.1)