The Application of Metamaterials to Manipulate the Propagation of Elastic Waves
Author:Gao Peng Lin
Supervisor:wu lin zhi
With the rapid development of material science,some basic concepts and theories are undergoing unprecedented challenges in the fields of acoustics and elastodynamics.Phononic crystals and metamaterials are typical examples showing the bright future of novel materials in manipulating the propagation of acoustic and elastic waves.They provide completely new ideas and boost in-depth studies in some interesting fields,typically including low-frequency noise and vibration control,super-resolution imaging,energy redirection,ship stealth and seismic protection,etc.Although remarkable achievements have been reached,there are still some problems that need to be further improved in the existing literatures.First,perfect cloaking is unable to be achieved because of the singular physical parameters(with zero or infinite values)introduced by the transformation-based solutions.Second,for flexural waves propagating in thin plates,the reported metamaterials are usually made of complex structures,and usually open extremely narrow resonant bandgaps.Thrid,it is hard to redirect waves in high efficiency with the Wood’s anomaly mechanism.To solve these typical problems,firstly,the ray tracing method is applied to design a kind of eikonal cloaks without singular parameters.Subsequently,a kind of single-phase resonant structures,whose behavior is well-predicted by the model developed here,are proposed to construct metamaterial plate in low-cost.Finally,the so-called Poisson-like effect is activated by coupling the incident waves with the leaky guided mode,which is useful to redirect flexural waves effectively with platonic crystal slabs.For a given elastic medium,the propagation paths of acoustic and elastic waves are uniquely determined for a specific distribution of physical parameters.However,the reversed statement is not necessarily to be true because the ray trajectories only include parts of the information.Based on this idea,the Hamilton’s equations are employed here to study the inverse problem;that is,for a pre-designed ray equation,can we find some physical profiles without singularity to achieve near perfect cloaking? The typical cases for out-of-plane shear waves(2D)and acoustic waves(3D)are,respectively,considered here to design cylindrical and spherical invisible cloaks.It is demonstrated that the physical profiles obtained by the present method are identical to the transformation-based ones if all equations are fully included in the derivation.If we just take the eikonal equation into consideration,however,a kind of non-singular eikonal cloaks are obtained that accurately control the path of energy flux but with small energy disturbance along it.Therefore,the eikonal cloaks make a good balance between the cloaking performance and the simplicity of material parameters.By analogy to spring-mass oscillators,a kind of single-phase resonant structures(N-beam resonators)are proposed and theoretically studied to manipulate the propagation of flexural waves in thin plates.The N-beam resonator consists of a circular hole containing a smaller plate connected to the background plate by several rectangular beams.Its simple constituent parts facilitate the analytical modeling within the framework of classical plate and beam theories.On this basis,firstly,the multipole expansion method and the impedance matrix method are applied to study the scattering of flexural waves propagating in an infinite thin plate containing a single N-beam resonator.Numerical experiments performed within the framework of the finite element method(FEM)support the accuracy of the model here developed at low frequencies.In a second step,a multiple scattering algorithm is developed to study the properties of flexural waves propagating in a plate with periodically structured N-beam resonators,such as the band structures and transmittance spectra.As an example,the 2-beam resonators are employed to open low-frequency resonant bandgaps and to design gradient index focusing lens,and numerical simulations confirm the accuracy of the model developed here.Based on the semi-analytical modeling,this dissertation further investigates the influence of the number of beams N on the resonant complete bandgap in order to lower and expand it.It is demonstrated that the increasing beams have an adverse effect on the low-frequency resonant bandgaps.First,the increasing beams suppress the omnidirectional excitation of the fundamental mode,and thus hinder the formation of complete bandgaps.In addition,the resonant frequencies raise rapidly with the increasing of the number of beams,which goes against with the goal of lowering bandgaps.On this basis,it is concluded that the 1-beam resonators(N(28)1)are a reasonable choice bacuase the corresponding lattice opens a narrow complete bandgap at low frequencies.In order to broaden the bandgaps,the resonant cavity is restructured by adding multiple number M of 1-beam resonators into the circular hole.This operation successfully opens several low-frequency complete bandgaps with broad bandwidth.The band structures and transmittance spectra show that,for the case M(28)4,three complete bandgaps come into being,and their total bandwidth is enhanced by one order of magnitude when compared with that of 1-beam resonators(M(28)1).Except the resonant bandgaps,platonic crystals can be used to detour the propagation direction of flexural waves as well.It is shown that,for a finite platonic crystal slab,the normal incident waves will be redirected to the perpendicular sides if the lowest order symmetric leaky guided mode(S0 mode)is resonantly excited.This wave coupling effect is similar to the deformation effect in solids and thus is named as Poisson-like effect.For slabs made of weak scatterers,it is a narrow transmittance dip(Wood’s anomaly)that indicates the excitation of a normal S0 mode.On the contrary,the scattering caused by strong scatterers leads to a mixed S0 mode occurring when the incident wave is totally transmitted(transmittance peak).The mixed S0 mode easily couples to the incoming waves and,therefore,the Poisson-like effect excited under this mechanism is much stronger.It can be used to design useful devices for beam splitting and waveguiding.