Multiple Rayleigh waves guided by the planar surface of a continuously twisted structurally chiral material

The Stroh formalism was adapted for Rayleigh-wave propagation guided by the planar traction-free surface of a continuously twisted structurally chiral material (CTSCM), which is an anisotropic solid that is periodically non-homogeneous in the direction normal to the planar surface. Numerical studies reveal that this surface can support either one or two Rayleigh waves at a fixed frequency, depending on the structural period and orientation of the CTSCM. In the case of two Rayleigh waves, each wave possesses a different wavenumber. The Rayleigh wave with the larger wavenumber is more localized to the surface and has a phase speed that changes less as the angular frequency varies in comparison with the Rayleigh wave with the smaller wavenumber.


Introduction
Elastodynamic surface waves [1,2] have been studied from the 1880s, when Rayleigh [3] hinted at their significance for earthquakes, which was subsequently borne out both experimentally and theoretically [4][5][6]. Seismological applications of surface waves [7] include the detection of subsurface anomalies such as tunnels [8] and mineral deposits [9] as well as investigations of the evolution of planetary crusts [10,11]. Current   Figure 1. The conceptualization of a continuously twisted structurally chiral material (CTSCM), arising from a stack of uniaxial plates. Each plate is constructed by embedding parallel fibres in a homogeneous matrix material, with fibres lying normally to the plate's thickness direction (i.e. the x 3 direction). The orientation of the fibres rotates uniformly from one plate to the plate immediately above. Provided that the fibre diameter is sufficiently small and the plates are sufficiently thin, the stack may be taken to be locally homogeneous. (Online version in colour.)

Theory of Rayleigh-wave propagation (a) Boundary-value problem
The half-space x 3 > 0 is occupied by a CTSCM and the plane x 3 = 0 is traction-free so that it can guide a Rayleigh wave along the x 1 direction. The linearized equation of motion is stated as ∇ •τ (x) = −ρω 2ũ (x), x 3 > 0, (2.1) where ρ is the mass density. The second-rank strain tensor is given bỹ (2. 2) The Rayleigh wave has no variation along the x 2 direction. Therefore, we writẽ where q denotes the wavenumber of the Rayleigh wave. As bothτ (x) andε(x) are symmetric [45], the following column vectors are defined in accordance with the Kelvin notation: As the CTSCM occupying the half-space x 3 > 0 is non-homogeneous along the x 3 direction, Hooke's law may be written in matrix notation [46] as The stiffness matrix of the CTSCM is given by [31] [c( Herein, the Bond matrix [46] denotes a rotation about the x 2 axis by an angle χ towards the x 3 axis in the x 1 x 3 plane, and the Bond matrix [46] denotes a rotation about the x 3 axis by an angle β towards the x 2 axis in the x 1 x 2 plane. The angle γ is an offset from the x 1 axis in the x 1 x 2 plane. The structural handedness parameter h = +1 for right handedness or −1 for left handedness. The theory is general enough to accommodate any symmetric matrix as the reference compliance matrix [c o ref ] . The elastic properties of the CTSCM thus are periodic in the x 3 direction with period 2Ω, and invariant in the x 1 x 2 plane, i.e. c(x 3 ) = c(x 3 + 2Ω) , (2.9) and the CTSCM may be considered to be a one-dimensional phononic crystal [48]. The compliance matrix of the CTSCM is given by (2.14) In effect, there are nine equations in equations (2.13), of which three are algebraic equations and six are ordinary differential equations. The three algebraic equations can be used to eliminate τ 11 (x 3 ), τ 22 (x 3 ) and τ 12 (x 3 ). The remainder of the equations can then be rewritten as the 6 × 6matrix ordinary differential equation d dx 3 [f ( where the column 6-vector Although an expression for [P(x 3 )] is readily derived using a mathematical manipulation package such as Mathematica TM , it is far too cumbersome for reproduction here.

(c) Dispersion equation
In order to find the stress and displacement vectors of the Rayleigh wave, as well as the corresponding surface wavenumber q, equation ( ], the solution of equation (2.15) would be very simple [50]: For the CTSCM, [P(x 3 )] can be written as a non-terminating matrix polynomial series with respect to x 3 , which allows the solution of equation (2.15) also in terms of a non-terminating matrix polynomial series with respect to x 3 [49]. A compact solution for which Floquet theory [51] can be invoked is desirable because [P(x 3 )] varies periodically with x 3 [52]. According to Floquet theory, a compact solution of the form be the eigenvector corresponding to the nth eigenvalue σ n of [Q]; then, the corresponding eigenvalue α n of [A] is given by After labelling the eigenvalues of [A] such that Im{α 1 } > 0, Im{α 2 } > 0 and Im{α 3 } > 0, we set  [17,21]; therefore, such instances need not be considered here. The existence of exactly three eigenvalues (i.e. α 1 , α 2 and α 3 ) with positive imaginary parts is an assumption which is not proven mathematically. However, it is a physically reasonable assumption because propagation and attenuation along the +x 3 axis must have the same characteristics as propagation and attenuation along the −x 3 axis since the CTSCM is a reciprocal medium [53], and this assumption holds true for all numerical results presented in §3.
The piecewise-uniform-approximation method [5,17] is used to calculate [Q], and thereby [f (x 3 )] for all x 3 > 0, as follows. We introduce for all integers n ∈ [0, ∞). The half-space x 3 > 0 is partitioned into slices of equal thickness, with each cut occurring at the plane x 3 = x (n) 3 for n > 0. Thus, the integer N > 0 is the number of slices per period along the +x 3 axis. The matrices are defined. As propagation from the plane for n > 0 is characterized approximately by the matrix [W] (n) , it follows that [54] [ (1) . (2.28) The integer N should be sufficiently large that the piecewise-uniform approximation captures well the continuous variation of [P(x 3 )]. The piecewise-uniform approximation to [f (x 3 )] for arbitrary x 3 > 0 is accordingly given by Now, let us enforce the traction-free boundary conditions As a result, we get wherein the 3×3 matrix

Numerical studies
We begin numerical studies by assuming that the CTSCM has the same local symmetry as that of a hexagonal crystal with its symmetry axis parallel to the x 1 axis [55].     The morphology of the single Rayleigh wave found for the large-Ω approximation is quite different. Although the maximum magnitudes of the nine stress and displacement components in figure 5 lie in the vicinity of, but generally not on, the surface x 3 = 0, all components decay monotonically as x 3 increases thereafter. The morphology of the single Rayleigh wave found for the small-Ω approximation is also quite different. In this case, the profiles of |u 2 |, |τ 23 | and |τ 12 | are null valued, while the profiles for all other components of the displacement and stress vectors resemble those for the large-Ω approximation.
Lastly, we turn to the influence of the angular frequency ω on the phase speed defined as ω/q of Rayleigh waves. In figure 6, ω/q is plotted against ω. As in figure 5, for these calculations χ = 60 • , γ = 20 • and Ω = 0.1 mm. The two solutions found for every value of ω ∈ (π , 3π ) × 10 7 rad s −1 can be organized in two branches: the phase speed on the small-wavenumber branch decreases dramatically, but the phase speed on the large-wavenumber branch increases very slowly, as ω increases.
The phase speeds for the large-Ω and the small-Ω approximations are also displayed in figure 6 for comparison. Both phase speeds are independent of ω. Also, both phase speeds are lower than the phase speeds of the two Rayleigh waves for the CTSCM with finite Ω.

Closing remarks
In this paper, we developed the theory for Rayleigh waves guided by the planar traction-free surface of a CTSCM, which is an anisotropic material whose stiffness tensor rotates at a uniform rate along the direction normal to the planar surface. Application of the Kelvin notation and the Stroh formalism yielded a 6×6-matrix ordinary differential equation that can be solved using the piecewise-uniform-approximation method. Imposition of the traction-free boundary conditions on the solution of the 6×6-matrix ordinary differential equation led to the dispersion equation for Rayleigh-wave propagation. As the dispersion equation is analytically intractable, its solutions had to be extracted by graphical means.
Our numerical studies revealed that either one or two Rayleigh waves can exist at a fixed frequency, depending on the structural period and orientation of the CTSCM, for the chosen constitutive parameters. In fact, in many instances, the dispersion equation (2.33) yielded more than two solutions for q, but the additional solutions were rejected because they corresponded to null stress and displacement fields. Each of the two Rayleigh waves possesses a distinct wavenumber. The Rayleigh wave with the larger wavenumber is more localized to the tractionfree surface than the Rayleigh wave with the smaller wavenumber. In addition, the phase speed of the Rayleigh wave with the smaller wavenumber varies strongly with angular frequency whereas the phase speed of the Rayleigh wave with the larger wavenumber does not. This multiplicity of Rayleigh waves contrasts with the single Rayleigh wave that exists for the homogeneous elastic solid obtained by making the CTSCM's period either infinitely large or very small. Parenthetically, the observed multiplicity of Rayleigh waves guided by the tractionfree surface of a periodically non-homogeneous half-space mirrors findings for electromagnetic surface waves wherein periodic non-homogeneity delivers a multiplicity of Tamm waves [57] and Dyakonov-Tamm waves [58] at a fixed frequency.
Lastly, we note that the process of finding solutions of equation (2.15) is numerically very challenging, especially given the spatially non-homogeneous nature of [P(x 3 )]. At most, we found two Rayleigh-wave solutions at a fixed frequency, but we cannot definitively rule out the existence of further solutions that escaped our detection.
Data accessibility. This article has no additional data.