Antiplane elastic wave cloaking using metamaterials,homogenization and hyperelasticity

We consider the problem of how to cloak objects from antiplane elastic waves using two alternative techniques. The first is the use of a layered metamaterial in the spirit of the work of Torrent and Sanchez-Dehesa (2008) who considered acoustic cloaks, motivated by homogenization theories, whilst the second is the use of a hyperelastic cloak in the spirit of the work of Parnell et al. (2012). We extend the hyperelastic cloaking theory to the case of a Mooney–Rivlin material since this is often considered to be a more realistic constitutive model of rubber-like media than the neo-Hookean case studied by Parnell et al. (2012), certainly at the deformations required to produce a significant cloaking effect. Although not perfect, the Mooney–Rivlin material appears to be a reasonable hyperelastic cloak. This is clearly encouraging for applications. We quantify the effectiveness of the various cloaks considered by plotting the scattering cross section as a function of frequency, noting that this would be zero for a perfect cloak.     

Transformational cloaking from seismic surface waves by micropolar metamaterials with finite couple stiffness

Transformational elastodynamics can be used to protect sensitive structures from harmful waves and vibrations. By designing the material properties in a region around the sensitive structure, a cloak, the incident waves can be redirected as to cause minimal or no harmful response on the pertinent structure. In this paper, we consider such transformational cloaking built up by a suitably designed metamaterial exhibiting micropolar properties. First, a theoretically perfect cloak is obtained by designing the properties of an (unphysical) restricted micropolar material within the surrounding medium. Secondly, we investigate the performance of the cloak under more feasible design criteria, relating to finite elastic parameters. In particular, the behavior of a physically realizable cloak built up by unrestricted micropolar elastic media is investigated. Numerical studies are conducted for the case of buried as well as surface breaking structures in 2D subjected to incident Rayleigh waves pertinent to seismic loading. The studies show how the developed cloaking procedure can be utilized to substantially reduce the response of the structure. In particular, the results indicate the performance of the cloak in relation to constraints on the elastic parameters.     

Effect of negative permeability on elastic wave propagation in magnetoelastic multilayered composites

With the advent of left-handed magnetic materials, it is desirable to develop high-performance wave devices based on their novel properties of wave propagation. This letter reports the special properties of elastic wave propagation in magnetoelastic multilayered composites with negative permeability as comparecd to those in counterpart structures with positlve permeability. These novel properties of elastic waves are discerned from the diversified dispersion curves, which represent the propagation and attenuation characteristics of elastic waves. To compute these dispersion curves, the method of reverberation-ray matrix is extended for the analysis of elastic waves in magnctoelastic multilayered composites. Although only the results of a single piezomagnetic and a binary magnetoelastic layers with mechanically free and magnetically short surfaces as well as pelrfect interface are illustrated in the numerical examples, the analysis is applicable lo magnetoelastic multilayered structures with other kinds of boundaries/interfaces.     

Transformation elastodynamics and cloaking for flexural waves

The paper addresses an important issue of cloaking transformations for fourth-order partial differential equations representing flexural waves in thin elastic plates. It is shown that, in contrast with the Helmholtz equation, the general form of the partial differential equation is not invariant with respect to the cloaking transformation. The significant result of this paper is the analysis of the transformed equation and its interpretation in the framework of the linear theory of pre-stressed plates. The paper provides a formal framework for transformation elastodynamics as applied to elastic plates. Furthermore, an algorithm is proposed for designing a broadband square cloak for flexural waves, which employs a regularised push-out transformation. Illustrative numerical examples show high accuracy and efficiency of the proposed cloaking algorithm. In particular, a physical configuration involving a perturbation of an interference pattern generated by two coherent sources is presented. It is demonstrated that the perturbation produced by a cloaked defect is negligibly small even for such a delicate interference pattern.     

直角平面内弹性圆夹杂对入射平面SH波的散射

利用复变函数法、多极坐标移动技术及傅立叶级数展开求解二维直角平面内圆形弹性夹杂对稳态入射平面SH波的散射问题。首先写出直角平面内不含夹杂时的入射波场和反射波场;其次建立直角平面内含夹杂时夹杂外的散射波解和夹杂内的驻波解,并利用叠加原理写出问题的总波场,借助夹杂边界处应力和位移的连续条件建立求解散射波解和驻波解中未知系数的无穷代数方程组并求解,通过算例具体讨论了直角平面水平边界点的位移幅度比和夹杂边界处径向应力集中系数随不同无量纲波数、入射角及圆孔位置的变化情况,结果表明了算法的有效实用性。     

Theory of scattering from multilayered bodies of arbitrary shape

Green 's functions and boundary integral equation methods are used to derive a matrix set of equations for scattering from a multilayered homogeneous elastic body embedded in an infinite elastic material. The surfaces separating the layers have arbitrary shape. The formalism for the three-layer material is derived in detail and generalized to N-layers. A matrix factorization method (MFM) is shown to considerably simplify the computational problem. The relation to the problems of acoustic waves in fluids and electromagnetic waves in a dielectric material is briefly indicated.     

Dynamics of structural interfaces: Filtering and focussing effects for elastic waves

Dynamics of thick interfaces separating different regions of elastic materials is investigated. The interfaces are made up of elastic layers or inertial truss structures. The study of evanescent mode propagation and transmission properties reveals that the discrete nature of structural interfaces introduces unusual filtering characteristics in the system, which cannot be obtained with multilayered interfaces. An example of metamaterial is presented, namely, a planar structural interface, which acts as a flat lens, therefore evidencing the negative refraction and focussing of elastic waves.     

Elastodynamic behavior of mechanical cloaks designed by direct lattice transformations

Steering waves in elastic solids is more demanding than steering waves in electromagnetism or acoustics. As a result, designing material distributions which are the counterpart of optical invisibility cloaks in elasticity poses a major challenge. Waves of all polarizations should be guided around an obstacle to emerge on the downstream side as though no obstacle were there. Recently, we have introduced the direct-lattice-transformation approach. This simple and explicit construction procedure led to extremely good cloaking results in the static case. Here, we transfer this approach to the dynamic case, i.e., to elastic waves or phonons. We demonstrate broadband reduction of scattering, with best suppressions exceeding a factor of five when using cubic coordinate transformations instead of linear ones. To reliably and quantitatively test these cloaks efficiency, we use an effective-medium approach.     

Numerical analysis of three-dimensional acoustic cloaks and carpets

We start by a review of the chronology of mathematical results on the Dirichlet-to-Neumann map which paved the way toward the physics of transformational acoustics. We then rederive the expression for the (anisotropic) density and bulk modulus appearing in the pressure wave equation written in the transformed coordinates. A spherical acoustic cloak consisting of an alternation of homogeneous isotropic concentric layers is further proposed based on the effective medium theory. This cloak is characterized by a low reflection and good efficiency over a large bandwidth for both near and far fields, which approximates the ideal cloak with an inhomogeneous and anisotropic distribution of material parameters. The latter suffers from singular material parameters on its inner surface. This singularity depends upon the sharpness of corners, if the cloak has an irregular boundary, e.g. a polyhedron cloak becomes more and more singular when the number of vertices increases if it is star shaped. We thus analyze the acoustic response of a non-singular spherical cloak designed by blowing up a small ball instead of a point, as proposed in [Kohn, Shen, Vogelius, Weinstein, Inverse Problems 24, 015016, 2008]. The multilayered approximation of this cloak requires less extreme densities (especially for the lowest bound). Finally, we investigate another type of non-singular cloaks, known as invisibility carpets [Li and Pendry, Phys. Rev. Lett. 101, 203901, 2008], which mimic the reflection by a flat ground.     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

Elastic field due to an edge dislocation in a multilayered composite

A numerical procedure is presented for the analysis of the elastic field due to an edge dislocation in a multilayered composite. The multilayered composite consists of n perfectly bonded layers having different material properties and thickness, and two half-planes adhere to the top and bottom layers. The stiffness matrices for each layer and the half-planes are first derived in the Fourier transform domain, then a set of global stiffness equations is assembled to solve for the transformed components of the elastic field. Since the singular part of the elastic field corresponding to the dislocation in the full-plane has been extracted from the transformed components, regular numerical integration is needed only to evaluate the inverse Fourier transform. Numerical results for the elastic field due to an edge dislocation in a bimaterial medium are shown in fairly good agreement with analytical solutions. The elastic field and the Peach–Kohler image force are also presented for an edge dislocation in a single layered half-plane, a two-layered half-plane and a multilayered composite made of alternating layers of two different materials.     

Design and analysis of nonlinear-transformation-based broadband cloaking for acoustic wave propagation

A coordinate-transformation method can be used to design invisibility cloaks for many types of waves, including acoustic waves. The traditional method for designing a cloak depends on a transformation from a virtual space to a physical space. Previous acoustic cloaks that are mainly designed with linear-transformation-based acoustics have drawbacks that acoustic wave trajectories in the cloaks cannot be controlled and tuned. This work uses a nonlinear mapping from a ray trajectory perspective to construct acoustic cloaks with tunable non-singular material properties. Use of a ray trajectory equation is a straightforward and alternate way to study propagation characteristics of different types of waves, which allows more flexibility in controlling the waves. 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-transformation cloak over a traditional linear-transformation cloak are analyzed. The invisibility capability of the nonlinear-transformation cloak can be tuned by adjusting a design 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-transformation 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-transformation cloak in a broad frequency range. The methodology developed here can be used to design nonlinear-transformation cloaks for other types of waves.     

Scattering of elastic waves in an elastic matrix containing an inclusion with interfaces

Based on the theory of elastic dynamics, the scattering of elastic waves and dynamic stress concentration in fiber-reinforced composite with interfaces are studied. Analytical expressions of elastic waves in different medium areas are presented and an analytic method of solving this problem is established. The mode coefficients are determined by means of the continuous conditions of displacement and stress on the boundary of the interfaces. The influence of material properties and structural size on the dynamic stress concentration factors near the interfaces is analyzed. It indicates that they have a great influence on the dynamic properties of fiber-reinforced composite. As examples, numerical results of dynamic stress concentration factors near the interfaces are presented and discussed. This paper provides reliable theoretical evidence for the study of dynamic properties in fiber-reinforced composite. Project supported by the National Natural Science Foundation of China (No. 19972018).     

A boundary element method for calculating elastic waves scattered by axisymmetric inclusion

A Boundary Element Method (BEM) is described to compute the scattering of elastic waves by an axisymmetric inclusion in an infinite elastic medium. The boundary loads applied to the inclusion is expanded in terms of Fourier series in an infinite space. The boundary integral equation is solved in the general direction of the axisymmetric inclusion by BEM. The problem of the 3-D scattering of elastic waves is reduced to a 1-Done. According to the geometric features of the axisymmetric in clusion the ring shell elements are adopted in this method. A comparison is made with other BEM methods. The numerical results show this method can reduce the amount of calculation and enhance the speed of convergence. Supported by Foundation of Ph. D Program of State Education Commission of China     

Identification of an ellipsoidal defect in an elastic solid using boundary measurements

Elastostatic problem of identification of an ellipsoidal cavity or inclusion (rigid or linear elastic) in an isotropic, linear elastic solid is considered. The reciprocity gap functional method is used for solving the problem. It is shown that the parameters of the ellipsoidal defect (coordinates of its center, the directions and magnitudes of the semiaxes and elastic moduli in the case of isotropic, linear elastic inclusion), located in an infinite elastic solid are expressed by means of the values of the reciprocity gap functional. The values of the reciprocity gap functional can be calculated if the loads and displacements corresponding to uniaxial tension (compression) of an infinite solid are known on the closed surface containing the defect inside. Applications of the results to the problem of ellipsoidal defect identification in a bounded body are discussed. A number of numerical examples showing the efficiency of the developed identification method are considered.     

Wave momentum and scattering of elastic waves by two-dimensional thin objects

The elastic wave momentum equation is applied to scattering of dilatational and shear waves by two-dimensional thin objects. It is shown that the sources of wave momentum are located at the edges of these objects. For a stress-zero crack or for a rigid inclusion there are two sources at each edge, for a fluid-filled crack there is just one. The scattered wave is expressed in terms of these sources. This reduces the number of independent variables by a factor two. An application to inverse scattering problems is also given.     

Propagation behavior of elastic waves in one-dimensional piezoelectric/piezomagnetic phononic crystal with line defects

Using a stiffness matrix method, we in- vestigate the propagation behaviors of elastic waves in one-dimensional （1D） piezoelectric/piezomagnetic （PE/PM） phononic crystals （PCs） with line defects by calculating energy reflection/transmittion coefficients of quasi-pressure and quasi-shear waves. Line defects are created by the re- placement of PE or PM constituent layer. The defect modes existing in the first gap are considered and the influences on defect modes of the material properties and volume fraction of the defect layers, the type of incident waves, the location of defect layer and the number of structural layers are discussed in detail. Numerical results indicate that defect modes are the most obvious when the defect layers are inserted in the middle of the perfect PCs; the types of incidence wave and material properties of the defect layers have important effects on the numbers, the location of frequencies and the peaks of defect modes, and the defect modes are strongly de- pendent on volume fraction of the defect layers. We hope this paper will be found useful for the design of PE/PM acoustic filters or acoustic transducer with PCs structures.     

Analysis of scattering waves in an elastic layered medium by means of the complete eigenfunction expansion form of the Green’s function

A scattering problem due to an object and a plane incident wave in an elastic layered half space is presented in this paper. The complete eigenfunction expansion form of the Green’s function developed by the author and the boundary integral equation method are introduced into the analysis. First, the complete eigenfunction expansion form of the Green’s function is investigated for its application to the scattering problem. A comprehensive explanation is also given for the fact that the complex Rayleigh wave modes exhibit standing waves. Next, a method for the analysis of scattering waves by means of the Green’s function is presented. The advantage of the present method is that the formulation itself is independent of the number of layers and the scattering waves can be decomposed into the modes for the spectra defined for the layered medium. Several numerical calculations are performed to examine the efficiency of the present method as well as the properties of the scattering waves. According to the numerical results, the complete eigenfunction expansion form of the Green’s function provides accurate values for application to a boundary element analysis. The spectral structure and radiation patterns of the scattering wave are presented and investigated. The differences in directionality can be found from the radiation patterns of the scattering waves decomposed into the modes for the spectra.     

缺陷岩体中弹性波传播的多次散射效应分析

弹性波在岩体中传播时与岩体缺陷相互作用形成复杂的传播图案。为研究缺陷对弹性波多次散射作用的影响,建立了双椭圆缺陷模型,基于Green函数基本解,采用边界积分的计算方法,得到了反映缺陷界面条件的刚度矩阵,分析了弹性波在双椭圆缺陷间的多次散射效应。结果表明：与单椭圆缺陷模型相比,双缺陷的相互作用使得弹性波频散和衰减效应增强,定量给出了缺陷的影响区域,从而明确了多次散射效应的尺度界限。进一步探讨了弹性波传播的多尺度效应,结果表明频散的Rayleigh峰、Mie峰和衰减的峰值频率同椭圆长轴和入射波波长两个尺度密切相关,存在明确的定量关系。相应的数值模拟结果表明,弹性波和缺陷相互作用在缺陷界面上诱发界面波,该界面波也存在频率相关性,影响了弹性波宏观传播的频散和衰减特征。