Characteristics And Control Of Elastic Wave Propagation In Nonlinear Phononic Crystals And Cloaking Structures

The development of modern equipment requires the less-vibration and lowness-noise environment.Large vibration and noise seriously affect the working performance,reliability and efficiency of the equipment.The nature of vibration and noise in structure/material is in the form of elastic waves.Therefore,the manipulation of elastic wave behavior in structure/material is an effective means to control vibration and acoustic characteristics in an equipment.Recently,acoustic metamaterials such as phonon crystals(PCs)and cloaking structures are proposed to guide and manupulate elastic waves in structures/materials.Some basic research and concepts in this field are being transformed into engineering applications,which is a valuable topic.However,there are some problems and deficiencies as follows:At present,calculations for dispersion relations of PCs are mainly based on linear wave dynamic models,while linear dispersion relations cannot predict band-gap variations as the wave intensity increases.Wave propagation in structures usually exhibits nonlinear wave dynamic response when new materials,high-intensity wave environment and high-precision analysis requirements are considered.Nonlinear PCs can control wave propagation direction and have amplitude-dependent band gap applications.Recent studies in the literature mainly focus on discrete systems;most studies only consider weakly nonlinear regimes and cannot accurately obtain some relations between wave propagation characteristics and nonlinearities.Therefore,constructing an accurate nonlinear wave dynamic model is the key to study effects of nonlinearities on wave propagation in PCs for accurate guiding of elastic wave and energy.At present,linear cloaking structures mainly designed by linear coordinate transformation theory,which have drawbacks of being narrowly banded and even negative cloaking efficiency that results from singular material parameters on internal boundaries of the cloaks.Elastic wave propagate in traditional linear cloaking structures along fixed trajectories,which cannot be flexibly controlled and tuned.Moreover,the displacement field near internal boundary of linear cloak is distorted and discontinuous,which leads to more elastic wave energy flowing into the region inside the cloak and brings wave field instability.Aiming at the above problems,a nonlinear wave propagation dynamic model is developed in this dissertation to study influences of nonlinearities on wave propagation characteristics in PC based on wavelet finite element method(WFEM).B-spline wavelet on the interval(BSWI)interpolation functions are used to construct wavelet-based elements,which have analytical expressions at all levels and sufficient continuity compared with piecewise continuous polynomials of the traditional finite element method(TFEM).Wave propagation dynamics models of in-plane and out-of-plane Timoshenko curved beam are constructed by using the WFEM.Finally,nonlinear ray trjectry equations are used to construct structural dynamic analysis models of nonlinear elastic wave cloaks for accurately guiding and tuning shear waves,bending waves and acoustic waves.Research contents and conclusions are shown as follows:Propagation characteristics of longitudinal elastic waves in a thin rod and coupled longitudinal and transverse waves in an Euler-Bernoulli beam are presented by using exact Green-Lagrange strain relations.Band structure relations for a periodic rod and beam are derived to predict their nonlinear wave propagation characteristics using the WFEM.Influences of nonlinearities on wave propagation characteristics are discussed.Numerical examples show that the proposed method is more effective for nonlinear static and band structure problems than the TFEM and illustrate that nonlinearities can cause band-gap width and location changes,which is similar to results reported in the literature for discrete systems.The proposed methodology is not restricted to weakly nonlinear systems and can be used to accurately predict wave propagation characteristics of nonlinear structures.This study can provide good support for engineering applications,such as sound and vibration control using tunable band gaps of nonlinear PCs.An efficient formulation of a curved Timoshenko beam is developed for static,vibration,and wave propagation problems.This dissertation derives static equilibrium,vibration,and wave propagation WFE models of in-plane and outof-plane motions of curved beams according to the Hamilton’s principle.Wave propagation characteristics of in-plane and out-of-plane waves in a PC curved beam are analyzed using the WFEM in combination with the Bloch theorem.Effects of geomertic and material parameters on in-plane and out-of-plane wave propagation characteristics of the PC curved beam are discussed.Numerical simulations show that the proposed method is more effective for static,vibration,and band structure problems than the TFEM.The proposed method can achieve the same accuracy as the TFEM with much fewer numbers of elements and degrees of freedom.It is shown that both in-plane and out-of-plane elastic wave band gaps exist in the PC curved beam,which exhibits some interesting phenomena due to coupling effects.This study can provide good support for wave filtering and vibration control of PC curved beam structures.A means is proposed to design cylindrical cloaks for shear waves and flexural waves based on a nonlinear transformation.This dissertation uses the nonlinear mapping,but from a ray trajectory perspective to construct acoustic cloaks with tunable non-singular material properties.Use of the ray trajectory equation is a straightforward and alternate way to study propagation characteristics for different types of waves,whichthereby allows us more flexibility in controlling waves.Nonlinear ray trajectory equations for out-of-plane shear waves and flexural waves are derived and two parameters to adjust the efficiency of cylindrical cloaks are introduced.Qualities of the nonlinear invisibility cloak are discussed by comparison with those of a cloak with the linear transformation.Numerical examples show that the nonlinear cloak is more effective for shielding out-of-plane shear waves and flexural waves from outside the cloak than the linear cloak and illustrate that the nonlinear cloaks for shear waves and flexural waves remain highly efficient in a broad frequency range.A new theory of nonlinear-transformation-based acoustics is proposed and a nonlinear ray-tracing equation for acoustic waves is derived.A broadband cylindrical cloak for acoustic waves in an inviscid fluid is realized with layered non-singular,homogeneous,and isotropic materials based on a nonlinear transformation.Some advantages and improvements of the invisibility nonlinear cloak over a traditional linear cloak are analyzed.The invisibility capability of the nonlinear cloak can be tuned by adjusting a tuning parameter that is shown to have influence on the acoustic wave energy flowing into the region inside the cloak.Numerical examples show that the nonlinear cloak is more effective for making a domain undetectable by acoustic waves in an inviscid fluid and shielding acoustic waves from outside the cloak than the linear cloak in a broad frequency range.