Stoneley-type waves in anisotropic periodic superlattices
Introduction
Surface and interfacial acoustic wave solutions normally come about within such ranges of frequency and wave vector which preclude existence of the propagating partial modes, so that the boundary condition can be satisfied by a wave packet that wholly consists of modes decaying into the depth of surrounding media and thus is localised near the guiding boundary. Localised waves in homogeneous halfspaces, e.g. the Rayleigh and Stoneley waves, are non-dispersive and typically exist in the subsonic velocity interval extending from zero up to a certain threshold velocity [1]. By contrast, localised waves in periodic halfspaces are dispersive and may exist inside the infinite sequence of subdomains of the space called forbidden bands or stopbands [2], [3], [4]. Thus the realm of localised waves occurring in periodic media is certainly much more diverse than that in homogeneous materials. Much work has been done in this direction. The largest amount of results has been obtained on surface [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and interfacial [17], [18], [19], [20], [21] waves in one-dimensionally periodic structures, or superlattices, in which case analytical developments can be advanced further and the numerical calculations are not that costly as compared to the structures with two- and three-dimensional periodicity. At the same time, no general and rigorous knowledge on the existence of localised waves in anisotropic superlattices can be gained by way of explicit derivations, which do not provide a closed-form dependence on the involved parameters even in the simplest case of isotropic materials; neither can it be extrapolated from numerical results, which are certainly incapable of embracing infinite variety of possible input data. In this regard, it is clear that an appropriate approach to the question of existence of localised waves should capture the essence of the problem without a need to find these wave solutions themselves.
An efficient approach to surface and interfacial waves in homogeneous anisotropic half-spaces has been developed and applied in the seminal papers [22], [23], [24], [25], [26], [27], [28], [29]. It is based on the Stroh formalism and the concept of surface impedance matrix, whose powerful properties follow from its link to energetic quantities. Recently we have applied an extension of this approach coupled with the Floquet-Bloch concepts to the problem of existence of surface waves in elastic and piezoelectric superlattices (1D phononic crystals) of general anisotropy [30], [31], [32], [33]. In particular, the analysis reveals that the maximum possible number of surface waves per stopband essentially depends on whether the unit cell of a superlattice is asymmetric or symmetric relative to its midplane (note that the symmetry in this context concerns the ordering of constituent layers and has nothing to do with their crystallographic symmetry which may be as low as triclinic). Existence of interfacial waves in 1D phononic bicrystals formed by two superlattices with a symmetric unit cell each was studied in [34].
The present paper is concerned with the most general case of interfacial waves in 1D phononic bicrystals, i.e. of the waves localised at the interface between two perfectly bonded half-infinite superlattices. Each superlattice is periodically layered, possibly functionally graded, medium of arbitrary elastic anisotropy with an arbitrary piecewise constant or continuous variation of material properties within the unit cell (period). It appears that asymmetry of unit cells much ramifies possible scenarios of the occurrence of interfacial waves and eventually augments their possible maximum number allowed at fixed tangential wavenumber per stopband frequency range. It is found that a bicrystal with any asymmetric unit cells admits up to three interfacial waves in the lowest and up to six waves in any upper stopband. This is in contrast to at most one and three waves which may exist in the lowest and upper stopbands of a bicrystal with both halves having a symmetric unit cell [34]. Moreover, the aforementioned bound, which is three or six waves per stopband, is shown to actually be the maximum for the total number of waves occurring per the same stopband in a given bicrystal and in the “complementary” bicrystal, which is obtained by swapping upper and lower superlattices of the initial one. By way of appropriate examples, it is confirmed that the upper bound of the number of interfacial waves per stopband is attainable, i.e., that the number of waves is less or equal (not just less) than the established bound and so this upper bound is a maximum in the formal meaning of it.
The paper has the following structure. The properties of the transfer and impedance matrices are outlined in Section 2. The existence and number of interfacial waves is analysed in Section 3. Numerical examples are presented in Section 4 and the results obtained are summarized in Section 5.
Section snippets
Transfer matrix
Consider a solid multilayered medium whose density and stiffness tensor vary along the stratification axis Y normal to the perfectly bonded layer interfaces. The layers may be homogeneous and/or functionally graded. Assume a displacement wave of the formwhere k and are real wave number and frequency, and with and being unit vectors parallel and orthogonal to the plane of interfaces, respectively. The Stroh formalism [35], [36] casts the
Dispersion equation
Consider a 1D bicrystal consisting of two perfectly bonded half-infinite periodic superlattices. Each, or at least one of them, is supposed to have an asymmetric unit cell (otherwise see [34]). Let the bonding interface be the plane and the superlattices occupying the half-spaces and be labeled by indices and , respectively. Alongside a given bicrystal, we will consider its counterpart consisting of the superlattice at and of the superlattice at . These two
Maximum number of IAWs: Examples
It was found above that the number of IAWs per stopband cannot be greater than the certain quantity. The purpose of the present section is to prove by evidence that this upper bound is really attainable (i.e., in formal terms, that this bound is a maximum rather than supremum). With this in mind, we present below a number of examples where the maximum possible number of IAWs does occur in a canonical bilayered structure involving the materials of practical use and the model ones.
Consider an
Conclusions
We have studied interfacial acoustic waves (IAWs) guided by the perfectly bonded interface between two half-infinite periodic superlattices. Each superlattice may consist of homogeneous or functionally graded layers of general anisotropy and may have an arbitrary assembling of its unit cell. The waves under study may be seen as an analogue of the Stoneley wave at an interface between homogeneous media, except that they are dispersive and their frequency and wavenumber form multiple branches
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors thank V. I. Alshits for helpful discussions. The work of A.D. was supported by the Ministry of Science and Higher Education of the Russian Federation within the State assignment FSRC “Crystallography and Photonics” RAS.
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