Interfacial acoustic waves in one-dimensional anisotropic phononic bicrystals with a symmetric unit cell

Abstract

The paper is concerned with the interfacial acoustic waves localized at the internal boundary of two different perfectly bonded semi-infinite one-dimensional phononic crystals represented by periodically layered or functionally graded elastic structures. The unit cell is assumed symmetric relative to its midplane, whereas the constituent materials may be of arbitrary anisotropy. The issue of the maximum possible number of interfacial waves per full stop band of a phononic bicrystal is investigated. It is proved that, given a fixed tangential wavenumber, the lowest stop band admits at most one interfacial wave, while an upper stop band admits up to three interfacial waves. The results obtained for the case of generally anisotropic bicrystals are specialized for the case of a symmetric sagittal plane.

Stoneley-type waves in anisotropic periodic superlattices

“…are × 3 3 matrices whose columns A and L are the first three and second three components of = A L ( ) T , respectively. It is at this stage that the case of superlattices with a symmetric unit cell studied in [34] diverges from the present general consideration. By symmetric we mean the unit cells whose arrangement is invariant relative to the midplane.…”

“…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 symmetric we mean the unit cells whose arrangement is invariant relative to the midplane. According to [34], the general identity (8) applied to the transfer matrix through a symmetric unit cell splits into two independent identities = M TM T . This modification does not affect the eigenvalues of M and hence the band structure, but it provides an additional condition on the eigenvectors, namely, that they can be chosen within the full stopbands as = +3 which casts relations (10) and (11) in a more determinative form.…”

“…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].…”

Preface to a special feature dedicated to the memory of Prof. Peter Chadwick FRS

“…The majority of the authors were fortunate to have enjoyed collaborations and interactions with Peter in various capacities. The papers by Barnett [2], Darinskii & Shuvalov [3] and Fu et al . [4] are in the area of Peter's primary research interest related to the most general treatment of surface and interfacial waves.…”