On an effective two-phase model of a medium with cracks of finite length

Abstract

To investigate wave propagation in cracked media in which the cracks are filled with liquid, an effective two-phase model was suggested in [1][2][3]. This model allowed us to elucidate a number of important peculiarities of wave propagation in cracked media [4][5][6] and has received support in experiments [7,8]. In deriving this model several assumptions on the form of cracks are made. An infinite length of the cracks is the main assumption.The purpose of the present paper is to develop the model mentioned above and to generalize it to the case of cracks of finite length. Within the framework of the model obtained, the investigation of wave propagation in a number of cracked media is carried out. The results of these considerations allow us to elucidate the influence of the mean length of cracks on wave propagation and also to show a good agreement of the theoretical conclusions for the new model with the experimental data. w 1. THE TWO-PHASE MODEL OF A MEDIUM CONTAINING INFINITE CRACKS FILLED WITH LIQUIDSuppose that a cracked medium containing infinitely long periodic cracks with plane-paxallel interfaces is given. These cracks are filled with a homogeneous liquid medium characterized by the density px, the velocity of propagation Vl, and the thickness hi.Homogeneous isotropic elastic media, situated between the cracks, are determined by the Lam6 parameters ,k2, #2, the density p2, and the thickness h2.Such an elastic-liquid medium, situated in the region 0 < z < H and consisting of n periods along the z-axis with thickness h = H/n = hi +h2, was averaged under the conditions h --~ 0, n ~ oo, and H = const in [1][2][3]. After averaging, the medium is described by an effective two-phase model of cracked media [3]. The displacements u (a). , u (2), and u,, and also the stresses t (~)** = t,,, t(~ satisfy the equations of continuous media az =P at 2 ' 0x -pl 0t 2 , G0 x --P2 at 2(1.1) and of Hooke's lawhere the superscripts (1) and (2) represent the first (liquid) and the second (elastic) phases, respectively. The porosity ~, density p, and parameters s a, and b are expressed by the formulas