Platonic localisation: one ring to bind them

The interest in wave propagation and localised waveforms in structured media has led to analysis of defects in crystalline solids, dynamic Green’s kernels in lattices, transmission resonances for grating stacks, and novel designs of structured waveguides. While motivated by practical applications in acoustics and optics, these problems bring new challenges in elasticity: in particular, structured flexural elastic plates incorporate effects attributed to the interaction between evanescent and propagating modes. In turn, this may lead to localised flexural vibration modes within finite structured clusters. Lattice dynamics and analysis of point defects were extensively studied in [1, 2]. Analysis of propagating modes initiated by point defects was published in [3], and stop band Green’s functions representing exponentially localised waveforms were studied in [4]. A scalar problem governed by the Helmholtz operator for a circular cluster of small rigid disc-shaped inclusions was analysed in [5]. Localisation and waveguiding of flexural waves by structured semi-infinite grating stacks were studied in [6, 7], and transmission resonances for flexural waves in an elastic plate containing gratings of rigid pins were analysed in [8].

In the present paper, we develop an asymptotic homogenisation model for evaluation of physical parameters associated with localised flexural waveforms, which occur around circular clusters of point masses or spring–mass resonators embedded in elastic plates. An example of such a localised waveform is shown in Fig. 1. Often, “leaky waves” can be observed near clusters of defects such as point masses, rigid pins or spring–mass resonators. It is also noted that the rate of localisation can be enhanced by using spring–mass resonators with the masses and stiffness chosen according to the procedure discussed in the text below. In particular, Fig. 1 shows the results of computations for a ring of specially designed spring–mass resonators.

The recent papers [9, 10] present an asymptotic analysis of whispering Bloch modes around a circular array of inclusions for acoustic or electromagnetic waves. The results have been obtained for the case of the Helmholtz operator and the Neumann boundary conditions set at the boundaries of the scatterers. The thesis [11] has addressed flexural wave localisation and waveguiding by clusters of point masses inserted in an elastic plate. Clusters of resonators including springs and masses, embedded into an elastic plate, were studied in [6] in the context of wave scattering and trapping by semi-infinite structured gratings.

The asymptotic model developed in the present paper reduces the formulation to a spectral problem for an integral operator. The solution is obtained in a closed analytical form. Special attention is given to localised waveforms in a structured ring associated with spring–mass resonators of negative inertia, a condition we show to be necessary for their occurrence in the finite cluster represented by the structured ring.

The structure of the paper is as follows: a Green’s function formulation for vibrations of point masses in a flexural plate is introduced in Sect. 2. Section 3 includes the homogenisation approximation and reduction to the spectral problem for an integral operator. Normalisation and closed-form evaluation of certain integrals for the special case of a circular cluster are discussed in Sects. 3.2 and 3.3. Localised waveforms for different types of resonators are described in Sects. 4 and 5. Flexural wave localisation in an infinite grating of point masses is discussed in Sect. 6. Finally, Sect. 7 includes concluding remarks and discussion.

The problem is formulated in terms of the dynamic Green function \(G(r;\beta )\), \(r = \left| \varvec{x}\right| \), which satisfies the Kirchhoff–Love equation governing time harmonic flexural waves (see, for example, [12]) associated with a point source at the origin:Here, the spectral parameter \(\beta \) and the radian frequency \(\omega \) are related bywith \(\rho \), h and D being the mass density, the thickness and the flexural rigidity of the plate, respectively. The explicit form for \(G(r;\beta )\) is given in terms of Bessel functions (see, for example, [13, 14])Note that \(G(r; \beta )\) is bounded as \(r\rightarrow 0\), which is important for our model.

For a ring of N points, each with mass m, shown in Fig. 2, we consider a time harmonic flexural displacement \(u(\varvec{x}) e^{-i \omega t}\), where \(u(\varvec{x})\) satisfies the following equation governed by the bi-harmonic operatorwhere \(\varvec{a}^{(j)}, ~ j = 1,\ldots ,N,\) represent the positions of the point masses within the structured ring. Here \(\left| \varvec{a}^{(j)}\right| =R\), where R is the radius of the ring. The solution \(u(\varvec{x})\) satisfies the following relationIn particular, when \(\varvec{x} = \varvec{a}^{(k)}\) we deduceBy looking for a non-trivial solution, incorporating complex \(\beta \) and \(u(\varvec{a}^{(k)}), k = 1,\ldots ,N,\) (see, for example [11]), one will obtain a description of a “leaky wave” around a cluster of point masses, and the imaginary part of \(\beta \) characterises the “leakage” to infinity. When N becomes large, we propose a homogenisation model and find its solution in a simple analytical form. In particular, it is shown that for the case of a “negative inertia” of the structured ring, real roots \(\beta \) are identified for the homogenisation model. It is also shown that the effect of negative inertia is achieved by using spring–mass resonators instead of point masses.

For the circle of radius R centred at the origin, we use the notationsFor a periodic solution \(u(\varvec{x})\) of (6), the displacements of the point masses are \(u\left( \varvec{a}^{(j)}\right) = U \mathrm {e}^{\mathrm {i} k \theta _{j}}\), where k is an integer. The system of Eq. (6) can be rewritten in the form:where \(M = N m\) is the “total mass” of the ring.

Localised modes in the ring of masses are obtained as follows:Note that there are real solutions \({\tilde{\beta }}\) only if \(M < 0\). For positive total mass, the system exhibits only leaky modes.

Let us consider a plate where the masses are replaced by spring–mass resonators of stiffness c and mass m, as shown in Fig. 5. The dynamic response of gratings consisting of spring–mass resonators has been analysed in [6]. Compared to the case of point masses, for the case of spring–mass resonators it has been shown that in the equations of motion the inertia term \(m \omega ^{2}\) should be replaced by \(c {{m}} \omega ^{2} / (c - {{m}} \omega ^{2})\). Consequently, Eq. (8) becomesLet \({\mathscr {M}} = N {{m}}\) be the total mass of resonators and \({\mathscr {C}} = N c\) be the total stiffness of resonators. Then Eq. (27) is given byThe fraction in the last term of (28) can be chosen to be negative. In the limit as \(N \rightarrow \infty \), we have that \(\Lambda \) in Eq. (28) is given by

Using the normalisation, we obtainAfter determining \({\tilde{\beta }}\) from (22), R, \({\mathscr {C}}\) and \({\mathscr {M}}\) are chosen so thatWhen \({\tilde{\beta }} \gg 1\), we haveSpring–mass resonators can deliver the required negative inertia, which provides a larger range of admissible parameters to simulate exponentially localised vibration modes along the structured ring. Figure 7 shows how to choose the stiffness and the mass of the spring–mass resonator system to achieve the required negative inertia M. Namely,where \({\mathscr {C}}\) and \({\mathscr {M}}\) represent the total stiffness and the total mass of the cluster of resonators; the other parameters are the same as in the previous sections. Given the required negative inertia M and the value of the spectral parameter \({\tilde{\beta }}\), Fig. 7 shows \({\mathscr {C}}\) versus \({\mathscr {M}}\) according to the formula (33).

We show that the homogenised model can also be developed to identify trapped “waveguide modes” for an infinite grating of point masses m embedded in an infinite elastic plate, as shown in Fig. 8. These are often referred to as Rayleigh–Bloch waves [13]. Although it is tempting to think of this grating as the limit case of a circular ring as \(R \rightarrow \infty \), we show that the trapped waves observed here cannot be obtained as the limit of the localised waveforms studied in Sect. 3.

The system of equations is now given bywhere d denotes the distance between the masses. We note that \(\left| \varvec{a}^{(k)}-\varvec{a}^{(p)}\right| = \left| k-p\right| d\). We seek the solution u in the form \(u\left( \varvec{a}^{(k)}\right) = U \mathrm {e}^{\mathrm {i} \alpha k d}\), where \(\alpha \) is the Bloch parameter. Then, Eq. (34) becomeswhere the mass per unit length \(\mu = m/d\) has been introduced. The sum on the left-hand side of (35) can be approximated by an integral, such thatCompared to Sect. 3, this now is no longer the limit as \(N \rightarrow \infty \), as N has no meaning for the infinite array, but is instead the limit as \(d \rightarrow 0\) and \(m/d = O(1)\).

For real and positive values of \(\alpha \) and \(\beta \), Eq. (36) takes the formAssuming that \(\alpha> \beta > 0\) and using (2), we obtainor, equivalently,This equation can be compared to (6.3) and (4.9) of [13]. We observe that the homogenisation approximation extracts the \(n=0\) term from (4.9), and then, Eqs. (6.3) of [13] and (39) are consistent with each other.

At low frequencies, the dispersion relation (39) leads to the following asymptotic approximation for the radian frequency \(\omega \) as a function of the wavenumber \(\alpha \):when \(\alpha > 4 \rho h/\mu \).

An example of a trapped flexural waveform for the infinite grating in the elastic plate is shown in Fig. 9, where quasi-periodic boundary conditions have been imposed on the vertical sides of the square domain. In this example, the wavenumber is chosen to be \(\alpha = 2\) 1/m, which corresponds to the numerical value of the frequency \(f = 5.3\) Hz. We note that this value of the frequency is close to the analytically predicted value, that is \(f = 5.5\) Hz.

We point out that Eqs. (36) and (37) do not have real and positive solutions \(\alpha , \omega (\alpha )\) such that \(\omega '(\alpha )=0\), i.e. the homogenisation approximation does not capture localised standing waves supported by the infinite grating of point masses in the elastic plate. This statement also holds true if point masses are replaced by spring–mass resonators. We also note that localised standing waves can be observed when the wavelength is comparable with d, which is outside the framework of the homogenisation approximation. This is described in Sect. 6 of [13] and illustrated in Fig. 5a of the same paper.

We have presented a homogenisation model, which leads to closed-form solutions that correspond to exponentially localised waveforms in structured plates. The cases considered here include flexural plates containing structured rings as well as infinite gratings.

Compared to the leaky waves, studied in detail in [11], we have obtained exponentially localised ring-shaped waveforms by identifying the regime of negative inertia, which can be achieved by implementing specially tuned spring–mass resonators, as discussed in Sect. 5. It has also been derived that a non-unique choice of resonator parameters [as shown in formula (33)] can be used to achieve the same value of the effective negative inertia.

The work has interesting extensions to the design of energy absorbers in flexural plates and construction of flexural elastic waveguides.