Propagation of edge waves in a thinly layered laminated medium with stress couples under initial stresses

The propagation of edge waves in a thinly layered laminated medium with stress couples under initial stresses is examined. Based upon an approximate representation of a laminated medium by an equivalent anisotropic continuum with average initial and couple stresses, an explicit form of frequency equation is obtained to derive the phase velocity of edge waves. Edge waves exist under certain conditions. The inclusion of couple stresses increases the velocity of wave propagation. For a specific compression, the presence of couple stresses increases the velocity of wave propagation with the increase of wave number, whereas the reverse is the case when there is no couple stress. Numerical computation is performed with graphical representations. Several special cases are also examined.


INTRODUCTION
Considerable attention has been given to the specific features and propagation characteristic of edge waves in an elastic medium. These studies were recorded in several monographs including Love [1] and Ewing et al. [2] Subsequently, Kumar [3] "made an elaborate study of the propagation of edge waves in homogeneous isotropic plates. Das and Dey [4] have extended Kumar's work by incorporating the effect of initial stresses in the governing equations. In spite of these studies, the propagation of edge waves in composite structures of thinly laminated materials has received little attention. However, such structures are found to have many geophysical applications. They are also used extensively in the study of buckling and vibrations. Biot [5] provided an approximate representation of a laminated medium by a continuous structure with anisotropic properties. The assumptions required for the validity of such an approximate representation imply that the rigidity contrast of the layers is not too large, and that layer thicknesses must remain sufficiently small with respect to the wavelength of the deformation field. Based upon the averaging process, Biot [6] studied the folding problems of the first and the second kind, and showed that the same thing can be applied to the problem of the folding of a single anisotropic layer. Biot [7,8] also offered the static and dynamic analyses of multilayered orthotropic elastic and viscoelastic plates which include the case in which anisotropy of the individual layer is due to a thinly laminated structure.
In a series of papers, Pal Roy [9], Pal Roy and Sinha [10], and Pal Roy and Debnath [11] have investigated the propagation characteristics of elastic waves in a layered laminated medium under initial and couple stresses. These studies also consider the dynamics of a laminated medium of Maxwell type solids and surface instability of a laminated material. This paper is concerned with the propagation of edge waves in a thinly layered laminated medium with stress couples under initial stresses. This work is based on an approximate representation of a laminated medium by an equivalent anisotropic continuum with average initial and couple stresses. The explicit form of the frequency equation is determined so that the phase velocity of the edge waves can be calculated. Certain conditions among the couple stress coefficient, initial stresses, and the wave numbers ensure the existence of edge waves in the layered medium. The inclusion of couple stresses increases the velocity of wave propagation. For a specific compression, the presence of couple stresses increases the velocity of wave propagation with the increase of wave numbers, whereas the reverse is the case when there is no couple stress. Several special cases are also considered. Numerical computation is performed for a specific laminated structure of ten thin adhering layers with different elastic moduli. The graphical representation is given to show the importance and effects of couple stresses, initial stresses, and wave numbers. We consider a laminated medium that is made up of n thin, alternating hard and soft layers. The ith layer occupies a fraction a of the total thickness H. If the layers are sufficiently thin and the rigidity contrast of the layers is not too large, such a meditim behaves like an elastic continuum with anisotropic properties, although the individual layers may be isotropic see Biot [5]). The ith layer is specified by the elastic coefficients N and Q 1, 2, 3 n). We suppose that the y-axis is normal to the plane of the laminations. The stress-strain relations of the composite medium are given by (Biot [5], p. 186): We now consider the influence of initial stresses in the composite structure. The (i+1) principal directions are oriented along the x-axis and the y-axis. If (i) and S are the '-' 11 11 principal initial stresses along the laminations in the ith and (i + 1)st layers respectively, then the average initial stress in the x-direction is given by The y-component of the initial stresses $22 is constant throughout. In particular, when $22 0, the quantities p(i) -S ( i ) l l (i 2, n) represent compressive stresses in a direction parallel to the layers. The average compression is then

COUPLE STRESS ANALOGY
It follows from the theory of Biot [7] that the use of an equivalent anisotropic continuum with equivalent elastic coefficients N and Q is not sufficient for good predictions. A better representation can be made by considering the bending rigidity of the laminations. According to Biot's analysis [7] the bending moment C. of the ith layer of thickness h. is given by Propagation of Edge Waves: Debnath and Pal Roy 275 The average bending moment C of the equivalent anisotropic medium is obtained by extending the result (3.1) in each layer so that We shall restrict our discussion to the plane strain analysis only. The modified dynamical equations of equilibrium of the composite anisotropic medium are given by 0S12 00) O72U (4.1) )Sla + -P -p ckx

Oy
Oy The condition of incompressibility is

BOUNDARY CONDITIONS
The boundary conditions of the problem are concerned with the bounding planes of the composite anisotropic medium and are supposed to be free from tractions. The boundary conditions as suggested by Biot [5] are Afx S12 + P exy O dispersive in nature. The phase velocity depends on 6 and hence on the average initial compression P. The relation between V and can be established for particular values of the parameters. The velocity depends on N and Q, which in turn depend on the initial stress. The velocity also depends on b, the average couple-stress coefficient of the equivalent continuum.
7. SOME PARTICULAR CASES Let us consider as a first case the situation wherein the total thickness H of the composite structure is itself small. In this situation the case of a homogeneous isotropic elastic medium. When condition (ii)or (iii) is satisfied, surface instability ensues in the medium as soon as small compression is applied so that waves do not propagate under either of the two conditions. Thus the existence of edge waves is solely due to couple stresses involved in (i).

NUMERICAL CALCULATIONS AND DISCUSSIONS
We now consider a laminated plate constituted by an alteration of ten thin adhering layers with different elastic moduli, in which the first and sixth layers are isotropic in nature. The following dimensions are taken into account: In the first case of Section 7, it is found when b 0 that edge waves exist for P 0; and instability will occur at once if a certain amount of compression begins. When # 0, there is no wave propagation in the compression-free case; and if a certain amount of compression is applied, instability will occur at once.
In the second case of Section 7, the values of V 2 / Q are calculated from equation .800 4" vZ/6 as a function of s/obi//'fy parameter played for lhree vlues of X with oad without takiny sfress effect. The following conclusions can be drawn from Fig. 4.
The increase in (hence the wave number) brings about a decrease in V when be 0. While increases, the velocity (v) increases when b 0. The inclusion of couple stress (b) increases the velocity of wave propagation. When there is a couple stress, then for a specific compression, the velocity increases as 2 increases; but the converse is the case when there is no couple stress.