The propagation of elastic waves in periodic structures is governed by the elastomorphic and material parameters of its unit cell (Kushwaha and Halevi, 1994; Sainidou et al., 2005). Properly conceiving their design in terms of size, shape, and arrangement, as well as choosing their density and elastic properties demonstrated great potential for attaining exceptional dynamic behavior, such as frequency bandgaps (Kushwaha and Halevi, 1994; Mártinez-Sala et al., 1995; Liu et al., 2000; Miniaci et al., 2018a), negative refraction (Morvan et al., 2010; Zhu et al., 2014; Zhu and Semperlotti, 2016), topological protection (Mousavi et al., 2015; Süsstrunk and Huber, 2015; Pal et al., 2016; Miniaci et al., 2018b), etc. This opened up new perspectives in many fields, ranging from microelectromechanical systems to nondestructive evaluation (Pennec et al., 2010; Craster and Guenneau, 2012; Deymier, 2013), including but not limited to wave filters, waveguiding and beam splitting, sensing devices, wave splitters, vibration shielding, subwavelength imaging (Sukhovich et al., 2009; Li et al., 2015; Trainiti et al., 2015; Colombi et al., 2016; Miniaci et al., 2016, 2017; Su et al., 2016; Miniaci et al., 2019).

However, most of the configurations proposed so far are limited by the fact that their unconventional dynamic properties are accessible to specific frequencies only, once the unit cell design is conceived, being its geometrical and mechanical properties constant with respect to time. To overcome this limitation, periodic systems with adaptive elastic properties have received great attention in the recent years. Indeed, introducing an additional degree of freedom, enables the reversible tailoring of the dispersive diagram of the structure, considerably enhancing their functionality and technological applications. To achieve this goal, several approaches have been proposed and explored so far. They include active systems, such as piezoelectric materials (Casadei et al., 2010; Bergamini et al., 2014; Kherraz et al., 2016), temperature (Jim et al., 2009; Airoldi et al., 2011; Cheng et al., 2011; Wu et al., 2018) and magneto-based techniques (Robillard et al., 2009; Matar et al., 2012; Guo and Wei, 2016; Zhang et al., 2017), actuated polymers via electromagnetic waves (Walker et al., 2014, 2017) as well as the application of an external mechanical load. Among them, the latter has shown to be particularly promising to achieve tunable dispersive properties. Bigoni et al. (2008) formulated a theoretical model for an orthotropic, prestressed (compressible) elastic layer vibrating on an elastic half space assuming long-wave asymptotics for the solution. It was found that the influence of the prestress over bandgaps and passbands, so that it could be exploited as a tuning parameter to shift the dispersion curves of the system. A modeling tool for the prediction of controlled bandgap structures responding to flexural vibrations was provided by Gei et al. (2009), also relaxing the hypothesis of perfect periodicity, allowing thus to consider quasi-periodic structures (Gei, 2010). Wang and Bertoldi (2012) showed how mechanical deformations enabled the tuning of the phononic bandgaps in 3D periodic elastomeric structures. The ability of the elastomers to undergo small as well as large strain deformations guaranteed the reversibility and repeatability of the process. Both the linear and nonlinear regimes of elastic deformation were explored, including different geometrical topology for triggering mechanical instability. Galich et al. (2016) showed how local deformations experienced by a composite structure can bring to local stiffening influencing the instability-induced interfaces on elastic wave propagation in finitely deformed layered materials. Finally, Norris and Parnell (2012) tried to exploit the effects of the prestress to theoretically show the feasibility of a hyperelastic cloaking via transformation elasticity.

In this paper, the effects of an applied mechanical load on the dispersive diagram for phononic crystal plates is investigated in the case of elastic deformations, so to have a complete reversibility of the phenomena. The paper is organized as follows: section 2 provides the description of an Updated Lagrangian scheme, in which a representative unit cell of the phononic crystal plate is studied in its static and dynamic deformed configurations. In section 3, the procedure to extract the band diagrams and in particular the energy velocity using the equations described in section 2 is detailed. Finally, section 4 investigates the effect of a reversible compressive and tensile state of prestress on an epoxy plate with cylindrical inclusions.

The homogeneous problem in Equation (10) can be solved in the wavenumber k(ϑ, ω) and corresponding Floquet eigenvectors Q~(ϑ,ω) for any fixed direction ϑ and circular frequency ω ∈ ℝ, ω > 0, from which the band structures of the propagation and attenuation constants of a specific mode m are obtained by taking the real and imaginary components of the corresponding wavenumber k_m(ϑ, ω), respectively. For the particular case of a lossless structure (Im(D) = 0) immersed in vacuum, the modes supported by the crystal can be either propagative (Im(k) = 0) or evanescent (Re(k) = 0), the latter belonging to the so called deaf frequency range (bandgap).

In addition to the Bloch wavenumber k(ϑ, ω), from the computed set of solutions (k_m(ϑ, ω), Q_m(ϑ, ω)) it is possible to extract the band structure of the energy velocity, which corresponds to the velocity of propagation of packets of waves having close central frequency (Brillouin, 1953; Mazzotti et al., 2012). The energy velocity of a specific Bloch mode can be found as the ratio between the time-averaged energy flux in the direction ϑ over one period T = 2π/ω and the time-averaged total mechanical energy over the same period, i.e.

where 〈ϕ〉=ω2π∫tt+2π/ωϕdt denotes the time-averaging operation, I(ϑ, ω) denotes the total energy flux along the orientation ϑ of the Bloch wavenumber k, K(ϑ, ω) indicates the total kinetic energy while W(ϑ, ω) and W_σ0(ϑ, ω) represent the total stored elastic energy related to the harmonic motion and prestress, respectively. The time-averaged expressions for these quantities are given by

Equation (18) can be evaluated at any given solution k_m(ϑ, ω) by means of a Gauss quadrature scheme over the finite element mesh of the unit cell in the configuration C, in which the displacement, strain and stress fields can be post-processed from Q_m(ϑ, ω) using nodal interpolations.

The numerical method presented in the previous section is here applied to study a PC plate made of steel cylinders embedded in an epoxy matrix. In the undeformed configuration, the plate is 2.5 mm thick, while the cylindrical inclusions have a radius of 3.0 mm and are arranged in a square lattice of 10.0 mm side length, as shown in Figure 2A. The material properties for the steel and the epoxy are reported in Table 1.

To study how an initial state of stress affects the passbands and bandgaps of the PC plate, two different deformed configuration are considered. In the first configuration, the plate is subjected to a state of compressive stress representative of a normal displacement u₀(x₀) · n₀(x₀) = −0.025 mm applied at each point x₀ belonging to a lateral face of the unit cell. In the second case, a generic tensile state is applied by assuming a normal displacement u₀(x₀) · n₀(x₀) = 0.025 mm for each point x₀ belonging to a lateral face of the unit cell. Figures 2B,C report the von Mises stress consequent to the compression and tension prestress condition, respectively. It is worth noticing that the maximum magnitude of the applied prestretch has been limited to the typical strength at yield of both epoxy and steel in order to ensure the elastic behavior of the materials. This choice enables a full reversibility of the undeformed configuration, and thus of the dispersive behavior of the PC plate, once the load is removed, which is of more practical interest with respect to the case of a permanently induced deformation.

The band diagrams in terms of wavenumber k and energy velocity c_e versus frequency are shown in Figure 3. In these diagrams, the blue and red dots denote the dispersion curves relative to the unit cell subjected to the compressive stress and tensile stress shown in Figures 2B,C, respectively. It should be noted that, since the deformation due to prestress applied to each face of the cell is isotropic in the xy-plane in both cases, the deformation gradient F and its affine component F_L are diagonal and, as a consequence, the orientation of the reciprocal lattice vectors g_j in the deformed configurations does not change with respect to that in the deformed configuration. This implies that the orientation angle θ of the Bloch wavevector also remains unchanged between the undeformed and deformed configurations. The results presented in Figure 3 have been obtained for an angle θ = 0 measured with respect to g₀₁ and g₁. In classical Bloch analysis, this angle corresponds to the Γ − X direction of the irreducible Brillouin zone.

From the dispersion analysis of the undeformed unit cell, it can be found that, at frequencies below 110 kHz, only one bandgap is present along the direction ϑ = 0, which is located in the [68.3–88.6] kHz frequency range (see Figure A1 in Appendix B). This result, which has not been reported in the Figure 3 for the sake of clarity, can be readily obtained by carrying out the analysis of sections 2.3 and 3 on the undeformed geometry and by setting σ₀ = 0 in Equation (15). However, when the cell is subjected to a compressive stress as shown in Figure 2B, the bandgap experiences a shift towards higher frequencies, being it located, in this case, in the [70.7 − 87.3] kHz frequency range. On the other hand, when the cell is subjected to traction, the lower bound of the bandgap is observed at 68.3 kHz while its upper bound at 84.3 kHz, which corresponds to a frequency downshift of approximately 3.5% of the bandgap with respect to the case of the PC plate under compression. These results suggest that, even in the elastic regime, a deformation due to prestress can lead to significant changes in the passband and bandgap behavior.

Furthermore, looking at the behavior of the energy velocity for the two deformed configurations, it is possible to infer that, similarly to the case of homogeneous plates (Pau and Lanza di Scalea, 2015), a generic state of compression leads to an increase of the energy velocity, whereas a state of traction leads, in general, to its decrease. This situation can be readily deduced from the inspection of the energy velocity band diagrams in Figure 3, from which it can be noted that the largest variations take place in proximity of the cutoff frequencies. Moreover, in the case of prestressed homogeneous plates (Dubuc et al., 2017), characteristic spikes in the energy velocity dispersion curves appear due to mode coupling. This behavior can also be observed for the third fundamental mode at around 36 kHz when the PC plate is subjected to a compression.

Finally, in the case of the PC plate under compression, it is possible to observe the presence of a cutoff wavenumber for the first fundamental mode at about 108 rad/m and null frequency. This value can be derived by solving the eigenvalue problem reported in Equation (10) in k(ϑ) by imposing ω = 0. In this case, the eigenvalue problem corresponds to that of the buckling load and, consequently, the lowest eigenvalue (the cutoff wavenumber) corresponds to a bifurcation point that indicates the onset of buckling for the infinite PC plate. It should be noted that the specific value of the cutoff wavenumber depends on the magnitude and distribution of the compressive stress in the PC.

An Updated Lagrangian computational scheme has been presented for the calculation of the band diagrams of phononic crystal plates subjected to a generic state of prestress. The scheme involves the solution of a geometrically nonlinear static problem for a representative unit cell and the application of the Floquet-Bloch theorem to the linearized equations of acousto-elasticity on the deformed configuration of the unit cell undergoing the static load. For the case of an epoxy plate with cylindrical steel inclusions, it has been demonstrated that the existence of a prestress state of compression or tension can lead to significant changes in the passbands and bandgaps of a phononic crystal plate, even in the case of prestress inducing full reversible deformation state, which is of more practical interest with respect to the case of a permanently induced deformation. It has been observed that when the cell is subjected to a compressive stress, the bandgap experiences a shift toward higher frequencies, whereas when the cell is subjected to traction, we observe a frequency downshift of approximately the 3.5% of the bandgap with respect to the case of the phononic plate under compression. Similarly to the case of homogeneous plates, a generic state of compression leads to an increase of the energy velocity, whereas a state of traction tends to lower its value. The largest variations have been observed in proximity of the cutoff frequencies. Characteristic spikes in the energy velocity dispersion curves appear due to mode coupling for the third fundamental mode at around 36 kHz when the phononic plate is subjected to a compression.

The results presented herein provide insights in the behavior of phononic crystal plates with tunable dispersive properties, and suggest new leverages for wave manipulation valuable in many application fields such as wave filters, waveguiding and beam splitting, sensing devices, and vibration shielding.

MMa and MMi conceived the idea and wrote the first draft of the paper. MMa performed numerical calculations. All the authors discussed the manuscript.