Some experimental studies that Poisson's ratio of fissured media can be small and even negative. A possible explanation for this circumstance is thus ...

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UDC 550.344

Some experimental studies that Poisson's ratio of fissured media can be small and even negative. A possible explanation for this circumstance is thus proposed on the basis of the application of the matrix method to fissured media.

Poisson's ~atio v, which is the ratio of the lateral compression to the longitudinal elongation under uniaxial tension, is of great importance in studies on the physicomechanical properties of rocks and the stressed state of rock massifs. In practice, v is found from measurements of the velocities v s and Vp of the S (transverse) and P (longitudinal) waves, since ~ is uniquely related to the ratio T = Vs/V p by the functional dependence --

~ -J~

9

(I)

From the theory of elasticity for an isotropic solid we know that v can vary over the interval (0, 0.5), which corresponds to the variation of 7 in the range (0, I/V~). Moreover, the literature reports a number of experimental data on the study of fissured media, in which the values of T, determined from the ratio of the velocities of the slow and fast waves (identified as the longitudinal and transverse waves), are 0.7-0.9. Such values of T correspond to very small, even negative, values of ~ [i]. In this communication we propose one variant of a possible explanation of this effect, which follows from a theoretical investigation of the effective model of a cracked massif constructed in [2-4]. Suppose that in the cylindrical coordinate system r, 8, z we have a homogeneous isotropic elastic medium 0 < z < H = nh, which is cut infinitesimally thin, infinitely long cracks z = h, z = 2h ..... z = (n - l)h, where the condition for contact with slippage is satisfied, i.e.,

(2) the "square brackets" symbol here expresses a jump, upon passage through the crack, by the quantity inside the brackets. This elastic medium is characterized by the [email protected] constants and ~ and the density p. In the range of wavelengths exceeding H/v~, according to [2-4], the specified elastic medium can be replaced by an effective model with the aid of the matrix method. Such a substitution is also possible when the distances between parallel cracks are different. The effective model is described by the equations of a continuous medium

~e

~te '

~e

,

(3)

=

and Hooke's law

=

~-~ +

~e + ~e /

~ z -~

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 143-147, 1986.

0090-4104/90/5004-1771512.50

9 1990 Plenum Publishing Corporation

1771

~ZZ

~"

~O !

~Z- '

(4)

Equations (3) and (4) differ from the equations of a transversally isotropic medium with isotropy planes z = const (see, e.g., [5]) in that I' = X, p' = ~, and ~" = 0. Since the condition ~" > 0 is assumed in transversally isotropic media, the effective model under consideration is a special particular case of a transversally isotropic medium. In this model waves of the SH type cannot propagate and the fronts of waves of the P-SV type differ markedly from the corresponding fronts in transversally isotropic media. In the case of point axisymmetric sources (center of expansion and force acting along the z axis), which form a field of displacements, according to [4] are represented by the equations o=

~+~

0

~-'~oo

(s) 2~'~ 0

]" z

+

e

,

~-'~=~

in which

i ii~

j

(6) ,

=

.~, + ~,~

The explicit expressions for the functions US(k, h) and U~(k, h) in the cases of an unbounded medium and sources at the boundary of the half-space z > 0 are written here on the basis of the monograph [3]. The position of the fronts for these sources was found in [4]. Figure I shows the fronts inside the right angle z e 0, r e 0, when a point source acts at the origin. The leading front pertains to a quasilongitudinal wave. Along the axes this front propagates w i t h the velocity of longitudinal waves, Vp = /(X + 2p)/p. The trailing front has an unusual concave shape. In the case of a transversally isotropic medium, in which ~" > 0, the shape of the second front is completely different. In particular, when p" ~ %, ~" ~ p, I' = %, and p' = ~ the position of the corresponding fronts is shown in Fig. 2, where the second front is shown with a loop. As the shear modulus ~" decreases, this loop increases and at ~" = 0 the ends of the concave part of the loop approach the coordinate axes and the other parts of the second front shrink. Let us return to our analysis of the concave front in Fig. i. This front touches the z axis and the plane z = 0 and the tangency point moves with the longitudinal plastic velocity. The existence of a wave along the plane z = const with such a velocity is attributed to the fact that a longitudinal plastic wave propagates in the elastic layer which is in contact, with slippage, with the surrounding medium [6]. This wave is conserved after averaging. The existence of a wave with velocity Vps along the z axis is explained by the condition trz = 0 that no shear stress exists. The wave pattern in Fig. 1 has a symmetry about the bisector of the right angle under consideration. On this straight line, as shown by the analysis carried out in [4], are points of strongs which propagate with the lowest velocities /~/p and /(% + ~)/p among the points of the trailing and leading fro___nts, respectively. At the same time, the velocities along the coordinate axes, 4~a'/p and /a/p, are the maximum values for the trailing and leading fronts, respectively. Using the values given for the velocities, we form two functions 1772

-x

Z

Fig. 2

Fig. 1 TABLE 1

O.Ob 0.I 0.15 0.2 0.25 0.3 0.35 0.4 0.4b O.b 0.55 0.6 0.65 0.7 0.75 0.8 0.85

0.9 0.95

0.I0 0.20 0.30 0.39 0.48 0.57 0.66 0.73 0.80 0.87 6.92 0.96 0.99 1.00 0.99 0.96 0.90 0.78 0.89

0.05 0.!0 O. I5 0.20 0.26 0.31 0.37 0.44 0. 0.~8 0.66 0.75 0.86 0.98 ~ I 9 I >I ;'i >I

(7) which describe the ratio of the two velocities along the coordinate axes and at a 45 ~ angle to them. If we take into account the ratio

(8) of the transverse and longitudinal velocities in an isotropic medium, which is located within each period, then function (7) becomes

The ratios ~0 and ~i of the velocities in the model under discussion along cracks and at a 45 ~ angle as a function of the ratio u of the transverse and longitudinal velocities in the initial medium are shown in Table i. For most isotropic media the ratio y of the velocities of the longitudinal and transverse waves lies in the interval (0.5, 0.7). Ratios ~0 and 71 in the interval (0.58, 1.00) correspond to these values, according to Table I. As we see from Table i, the ratio of velocities in a fissured medium is markedly higher than the corresponding ratios in a medium without fissures. This circumstance, which has been detected in experimental studies, is confirmed by theoretical studies on fissured media. The velocities recorded in fissured media, however, cannot be considered as the velocities of longitudinal, and transverse waves and, therefore, Poisson's ratio cannot be found from formula (I) for these media. The model of a fissured medium is a particular case of a transversely isotropic medium. The polarization of waves in such a medium is much more complex than in an isotropic medium. LITERATURE CITED i.

B. P. Sibiryakov, M. A. Tatarnikov, and L. A. Maksimov, Propagation of Elastic Waves in Microinhomogeneous Media Containing Fluids (a Review) [in Russian], Novosibirsk (1978). 1773

2. 3. 4. 5. 6.

L . A . Molotkov, "On the equivalence of layered-periodic and transversally isotropic media," media," J. Soy. Math., 19, No. 4 (1982). L . A . Molotkov, The Matrix Method in the Theory of Wave Propagation in Layered Elastic and Liquid Media [in Russian], Leningrad (1984). L . A . Molotkov and A. E. Khilo, "Study of single-phase and multiphase effective models describing periodic media," J. Sov. Math., 32, No. 2 (1986). L . A . Molotkov and U. Baimagambetov, "On the study of wave propagation in layered transversally isotropic media," J. Sov. Math., 22, No. 1 (1983). P . V . Krauklis and L. A. Molotkov, "On low-frequency vibrations of a plate in an elastic half-space," Prikl. Mat. Mekh., 27, No. 5, 947 (1963).

EFFECTIVE MODELS OF LAYERED ELASTIC MEDIA WITH LINEAR CONTACTS OF THE GENERAL TYPE L. A. Molotkov and A. E. Khilo

UDC 550.34

Wave propagation in a layered medium with contacts of the general type at some boundaries is considered. An effective model, which in the general case is a medium with elastic aftereffect, is found for the medium under discussion.

It is useful to employ the matrix method when constructing effective models of layered media and liquid media [i, 2]. This method has been used, in particular, to carry out averaging in the case of nonideally elastic media (viscous with elastic aftereffect, and thermoelastic) [3]. In this paper we construct effective models for elastic media with linear contacts of the general type at some boundaries. For simplicity in the discussion, the medium is assumed to be periodic, but the studies carried out here can also be made in the absence of periodicity. I.

Construction of Effective Models

Given an elastic periodic medium 0 < z < H = nh, consisting of n periods. Each period is either a homogeneous layer, a layer that is inhomogeneous along the z axis, or a packet of homogeneous layers which are in tight contact with each other. At the boundaries z = ih (i = i, 2 ..... n - I) between the periods the conditions

[~] = O,

(1.i)

are satisfied; here Am(~/3t) and B~(8/St) are operator polynomials with real degrees m and and the "square brackets" symbol expresses a jump in the quantity inside the brackets. For example, [u z] = Uz(ih + 0) - Uz(ih - 0). The contact specified by conditions (i.i) gives a break in the energy flux vector at the boundary. Conditions which ensure absorption of energy at the boundary between periods and rules our self-excitation, therefore, must be imposed on the polynomials A m and B~. Such conditions will be given and examined in Sec. 2. Contact (i.i) is a generalization of the contacts postulated, e.g., in [4]. Two-dimensional waves of the P - S V and SH type can propagate in the given medium. Since both types of waves are considered in similar fashion, we shall study only waves of the P - S V type. In this case the field of displacements does not depend on they coordinate and does not contain a component Uy. The displacements Ux, u z and the stresses txz, tzz are represented by

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 148-157, 1986.

1774

0090-4104/90/5004-1774512.50

9 1990 Plenum Publishing Corporation