ON ELASTIC WAVES IN A THINLY-LAYERED LAMINATED MEDIUM WITH STRESS COUPLES UNDER INITIAL STRESS

The present work is concerned with a simple transformation rule in finding out the composite elastic coefficients of a thinly layered laminated medium whose bulk properties are strongly anisotropic with a microelastic bending rigidity. These elastic coefficients which were not known completely for a layered laminated struc- ture, are obtained suitably in terms of initial stress components and Lame's constants xi' ui of initially isotropic solids. The explicit solutions of the dynamical equations for a prestressed thinly layered laminated medium under horizontal compression in a gravity field are derived. The results are discussed specifying the effects of hydrostatic, deviatoric and couple stresses upon the characteristic propa- gation velocities of shear and compression wave modes.


INTRODUCTION
Biot [1] has indicated that some of the basic properties of anisotropic elastic media are provided by analyzing a laminated medium superposed of thin adhering layers which are alternately hard and soft.He also suggested that an equivalent continuous anisotropic elastic medium can approximately be used to provide useful insight into some of the basic features of the statical or dynamical problem of elasticity.The validity of such an approximation is based upon the fact that rigidity contrast of the layers is not too large, and that the thickness of the layers remains sufficiently small with respect to the wavelength of the displacement field.Such approximation depends also on the type of problem considered and in many cases it requires some additional refinements.
In a classic work on the theory of deformation of anisotropic elastic solids, Biot  [2] suggested that a skin effect is associated with anisotropy.He showed that near a free surface or a surface of discontinuity certain components of the stress field vary rapidly from zero to a maximum value within a thin skin.The concentration of stress also occurs where certain stress components are amplified with the skin thickness.In order to make an additional refinement of his theory, Biot [3] has proposed the effect of couple stresses in elasticity and viscoelasticity of an initially stressed anisotropic solids.He showed that this theory is intended to provide an approximate continuous model valid over a wide range for the mechanics of layered laminated media.This theory when applied to a wide variety of geological structures provides remarkably simple and useful results, and hence can be used for a better understanding of the involved physical features.
When a laminated medium of compressible material is replaced by a continuous medium of anisotropic properties, Biot's [1] analysis gives the composite elastic coefficients.This study is found to be somewhat similar to that used by Postma [4], and also to that considered by Helbig [5] in the theory of acoustic propagation in a laminated medium composed of layers of isotropic materials and is initially stress free.The laminated medium is obtained by superposition of thin adherent layers which are assumed to be alternately hard and soft.In these studies, Biot [1] obtained elastic coefficients in terms of the unknown individual material coefficients after deformation.Recently, Tolstoy [6] has made some slight modifications of Biot's formulation and then obtained simple explicit solutions of the dynamical equations for a prestressed isotropic homogeneous solid under horizontal compression in a gravita- tional field.
In spite of the above progress, several problems concerning the propagation of elastic waves in a thinly-layered laminated medium with stress couples under initial stresses remain fully or partially understood.This paper is intended to address some of these problems.
The main purpose of this paper is to discuss how to obtain the composite elastic coefficients after deformation in terms of Lame's constants and prestress components of the individual layers before deformation.After determination of such coeffi- cients, an attempt is made to find the explicit solutions of the dynamical equations for a pre stressed thinly-layered laminated medium under horizontal compression in a gravitational field.The bending rigidity of the laminated medium is also taken into account by introducing stress couples.

BASIC ASSUMPTIONS AND STRESS-STRAIN RELATIONS
We consider a thinly laminated medium composed of thin adhering layers which are alternately hard and soft.These hard and soft materials occupy, respectively, fractions 1 and 2 of the total thickness.If the rigidity contrast of the layers is not too large, and if the layer thickness remains sufficiently small with respect to the wavelength of the deformation field, then such a layered medium behaves approxi- mately like an elastic continuum with anisotropic properties even though the individual layers may be isotropic.
We consider the case of a compressible material.The stress-strain relations for he composite medium are where L is the composite slide modulus.Due to compressibility the strain component eyy is not the same in each layer.However, exx and t22 remain the same in both materials.The normal stresses in the hard material and (Biot [1], page 190), t a I exx eyy (2.7) The average stress e is YY eyy 1 e(1) + 2 (2)   yy eyy (2.8) The average strain tll is (2) tll a t + a 2 tll (2.9) The composite elastic coefficients Cij

THE HARD AND SOFT INITIALLY ISOTROPIC MATERIALS
We assume that the acceleration due to gravity points downward and the medium lies in the space y < O.The external gravitational field has the components (0, -g, 0).

FORMULATION OF THE PROBLEM FOR THE COMPOSITE MATERIAL
The average initial stresses in the x and y directions are B22 C22, Q L-P + 1 / 2 pgy where C's are given by (4.1abcde). 6.

COUPLE STRESS ANALOGY
If the hard layer is sufficiently stiff a couple stress of moment M per unit area is produced in the plane normal to the x axis.In this case t12 # t21.Equilibrium of moments for an element of material requires the condition aM (6.1)

T12-t21 @x
If we now introduce the effect of couple stress, the equilibrium equations are obtained as (Biot [3]) @t12 2 u at11 + @ (6.2) @x By p at22 a2v @2v B2M 3) The value of the moment M in terms of the deformation is obtained as (Biot [3]) 4(},i+2i)(1 + (6.6) When one layer is much more rigid than the other the couple stress coefficient has the simple value b 1 / 2 h 2 M with M M 1 1 (6.7) 7.

GOVERNING DYNAMICAL EQUATIONS AND THEIR SOLUTIONS
In view of (5.3abc)-( @2u 82v @2u a2u Cll ax-'+ B''+ L a y -y -y -P a-t a 2v a2v a2v a2v a4v C22 + B--+ (L-P) ax---p + b a x -where B C12 + L and L Q + -1/2 gpy ( It is noted that the effect of the buoyancy terms in equations (7.1)-(7.2) is appreciably smaller than that of pre-stressed values for a period of half a minute or less.In dealing with seismic wave problems, we can neglect the buoyancy terms in (7.1)-(7.2) compared with the pre-stressed terms.Using now (5.5abc) and neglecting the buoyancy terms in the above equations, it turns out that a2u a2v a2u a2u Bll ax--+ (B12-P+Q) + (Q + P) ay--p (7.4) a2v a2v a2u a2v a4v For some interesting solutions we take the trial functions as (alX+BlY+Clt (7.9) (a2x+b2Y+c2t (7.10) Assuming now a pure shear wave travelling in the x-direction, we set Thus when the functional form of is known then (7.13) corresponds to a shear mode of the SV type (u O) propagating in the $11 i.e. x-direction, with velocity c 2.
In general, for plane harmonic waves along the x-direction, we write exp[i(x+c2t)] (7.14) Then, from equation (7.13), we obtain 2 Csx (Q+b-P)/p (7.15) Evidently, the shear wave velocity, Csx depends on the initial stress, couple stress and hydrostatic stress.

Csy
In this case of shear waves travelling in the y-direction, there is no effect of couple stress.
We also note In the case of harmonic waves the velocity also remains the same.Cpy px p As in the case of a single isotropic layer (Tolstoy [6]) there are however, apart from the principal axes of pre-stress, two further conjugate directions for which uncoupled i.e. pure modes of shear of the pre-stress field occur.Based upon} the equilibrium equations for the composite elastic material and using an analysis similar to Tolstoy [6], the velocities of the P and SV wave modes are obtained explicitly.
The effects of couple stresses and initial stresses are found to be large for values approaching the instability value.The effect on the propagation velocity can then be quite large.As for the effects of gravity, these seem to be sensible for a very weak (low rigidity) sediments and for a very low frequency at which buoyancy effects are significant.This point deserves further attention and will be discussed in a subsequent paper.9.

CONCLUDING REMARKS
Although the present theory is developed in the context of plane strain deforma- tion, it can be extended to three-dimensional problems of transverse isotropic materials with rectangular, triangular and circular plane form.Also, further extension of results is possible for a thermoclastic laminated medium.Such problems will be treated in a subsequent paper.