Elastic Wave Cloak Research Articles

Symmetric elastic wave cloak design for underground protective structures based on multi-center coordinate transformation

Symmetric elastic wave cloak design for underground protective structures based on multi-center coordinate transformation

Rational design of hyperelastic semi-linear material and its application to elastic wave control

Incremental elastic wave in a hyperelastic semi-linear (SL) material can be purposely controlled by pre-deformation with help of hyperelastic transformation (HT) theory. However, this material model, assuming a linear stress-strain relation for large deformation, has neither been realized with naturally occurring materials nor through microstructure design. By balancing the stiffness of axial and rotational springs in a hexagonal mass-spring lattice, we propose a lattice prototype to realize the SL material. The performance of the model is systematically verified from static strain energy to wave property on different finitely deformed configurations. Based on this SL lattice, an elastic wave cloak is designed and validated through numerical simulation. This work provides the first two-dimensional (2D) microstructure model for realizing SL material and paves the way for elastic wave control with HT method.

Elastic wave cloak and invisibility of piezoelectric/piezomagnetic mechanical metamaterials.

In this paper, a piezoelectric cloaking mechanism is proposed, which makes the enclosed piezomagnetic cylinder invisible to elastic shear horizontal (SH) waves. Based on the scattering cancellation technique, the piezoelectric cloaking mechanism and dynamic stress concentration factor (DSCF) is obtained by the plane wave expansion method. A nonlinear ray trajectory equation for SH waves is derived based on the nonlinear transformation. Furthermore, piezoelectric effects on both cloaking mechanism and dynamic stress concentration are analyzed. The numerical results show that the scattering cancellation can be attributed to the cloak density, and the piezoelectric property can enhance the object's invisibility. The piezoelectric cloaking design can be applied to reduce the DSCF in some frequency regions, which means that it can change the stress distribution. It means that piezoelectric scattering cancellation can enhance both the cloaking results and structural strength of the mechanical metamaterials. This study is expected to have significance for the development and design of elastic wave metamaterials.

An asymmetric elastic metamaterial model for elastic wave cloaking

Elastic material with its elastic tensor losing minor symmetry is considered impossible without introducing artificially body torque. Here we demonstrate the feasibility of such material by introducing rotational resonance, the amplified rotational inertia of the microstructure during dynamical loading breaks naturally the shear stress symmetry, without resorting to external body torque or any other active means. This concept is illustrated through a realistic mass-spring model together with analytical homogenization technique and band structure analysis. It is also proven that this metamaterial model can be deliberately tuned to meet the material requirement defined by transformation method for full control of elastic wave, and the relation bridging the microstructure and the desired wave functionality is explicitly given. Application of this asymmetric metamaterial to design elastic wave cloak is demonstrated and validated by numerical simulation. The study paves the way for material design used to construct the transformation media for controlling elastic wave and related devices.

Nonlinear transformation-based broadband cloaking for flexural waves in elastic thin plates

Controlling and guiding elastic waves in solids is more complicated compared with electromagnetic and acoustic waves; design and fabrication of an elastic wave cloak with non-singular, homogeneous, and isotropic material parameters is a challenging task. Recent studies in the literature that focus on manipulating flexural waves in elastic thin plates mainly use a linear transformation with a linear radial-dependent mapping function, which has drawbacks of being narrowly banded or even having negative cloaking efficiency that results from singular material parameters on the internal boundary of the cloak. This paper presents a theory of nonlinear transformation-based flexural waves and derives the nonlinear ray-tracing equation for flexural waves. A broadband cylindrical cloak for flexural waves in an elastic thin plate is realized based on a nonlinear transformation, whose materials can be simplified as layered non-singular, homogeneous, and isotropic materials using an effective medium theory. Some advantages and improvements of the invisibility nonlinear-transformation cloak are analyzed by comparison with the linear-transformation cloak. The invisibility capability of the nonlinear-transformation cloak can be tuned by adjusting an impact parameter that is shown to have influence on flexural wave energy emitting into the region inside the cloak. Numerical simulations show that the nonlinear-transformation cloak is more effective for guiding flexural waves that propagate in the region outside the cloak than the linear-transformation cloak in a broad frequency range. They also show that the nonlinear-transformation cloak can accurately control ray tracing of different types of flexural waves under disturbances outside the cloak. The methodology developed here can be used to construct nonlinear-transformation cloaks for other types of waves.

Seismic invisibility: elastic wave cloaking via symmetrized transformation media

Transformation media theory, which steers waves in solids via an effective geometry induced by a refractive material (Fermat’s principle of least action), provides a means of controlling vibrations and elastic waves beyond the traditional dissipative structures regime. In particular, it could be used to create an elastic wave cloak, shielding an interior region against elastic waves while simultaneously preventing scattering in the outside domain. However, as a true elastic wave cloak would generally require materials with stiffness tensors lacking the minor symmetry (implying asymmetric stress), the utility of such an elastic wave cloak has thus far been limited by the challenge of fabricating these materials. Here we develop a means of overcoming this limitation via the development of a symmetrized elastic cloak (SEC), sacrificing some of the performance of the perfect cloak for the sake of restoring the minor symmetry. We test the performance of the SEC for shielding a tunnel against seismic waves, showing that it can be used to reduce the average displacement within the tunnel by an order of magnitude (and reduce energy by two orders of magnitude) for waves above a critical frequency of the cloak. This critical frequency, which corresponds to the generation of surface waves at the cloak-interior interface, can be used to develop a simple heuristic model of the SEC’s performance for a generic problem.

Controlling Out-of-Plane Elastic Shear Wave Propagation With Broadband Cloaking

A major challenge in designing a perfect invisibility cloak for elastic waves is that the mass density and elasticity tensor need to be independent functions of its radius with a linear transformation medium. The traditional cloak for out-of-plane shear waves in elastic membranes exhibits material properties with inhomogeneous and anisotropic shear moduli and densities, which yields a poor or even negative cloaking efficiency. This paper presents the design of a cylindrical cloak for elastic shear waves based on a nonlinear transformation. This excellent broadband nonlinear cloak only requires variation of its shear modulus, while the density in the cloak region remains unchanged. A nonlinear ray trajectory equation for out-of-plane shear waves is derived and a parameter to adjust the efficiency of the cylindrical cloak is 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 elastic shear waves from outside the cloak than the linear cloak and illustrate that the nonlinear cloak for shear waves remains highly efficient in a broad frequency range. The proposed nonlinear transformation in conjunction with the ray trajectory equation can also be used to design nonlinear cloaks for other elastic waves.

The form-invariance of wave equations without requiring a priori relations between field variables

According to the principle of relativity, the equations describing the laws of physics should have the same forms in all admissible frames of reference, i.e., form-invariance is an intrinsic property of correct wave equations. However, so far in the design of metamaterials by transformation methods, the form-invariance is always proved by using certain relations between field variables before and after coordinate transformation. The main contribution of this paper is to give general proofs of form-invariance of electromagnetic, sound and elastic wave equations in the global Cartesian coordinate system without using any assumption of the relation between field variables. The results show that electromagnetic wave equations and sound wave equations are intrinsically form-invariant, but traditional elastodynamic equations are not. As a by-product, one can naturally obtain new elastodynamic equations in the time domain that are locally accurate to describe the elastic wave propagation in inhomogeneous media. The validity of these new equations is demonstrated by some numerical simulations of a perfect elastic wave rotator and an approximate elastic wave cloak. These findings are important for solving inverse scattering problems in many fields such as seismology, nondestructive evaluation and metamaterials.