Time transformations, anisotropy and analogue transformation elasticity

The transformational paradigm for metamaterials was recently generalized by the introduction of an “analogue transformation” method. In this project, we will extend this method to include anisotropy and general spacetime transformations in acoustics, as well as towards the construction of an “analogue transformation elasticity”, with possible applications for vibration control in solids.

Introduction

The transformational paradigm in the field of electromagnetism has been one of the most interesting recent developments in material sciences. Using the straightforward generalization of the Maxwell equations to curved spacetime, and the fact that in Relativity a curved spacetime bends light in the same way as a refractive medium, one is able to connect vacuum solutions of the Maxwell equations in a curved spacetime with solutions of the Maxwell equations in materials with a given permeability and permittivity [1–3]. Together with the increasing availability of novel artificial materials or metamaterials, in which the propagation of electromagnetic radiation can be controlled to the point of obtaining an effective negative refraction index, the transformational paradigm has opened the door to the design of exotic devices such as invisibility cloaks and superlenses.

The great success of this “transformation optics” approach has led the research community to look for ways in which this transformational approach could be extended to other fields such as acoustics. Unfortunately such attempts have been undermined by the deep structural differences between the Maxwell theory with its underlying relativistic geometry on the one hand, and the Galilean character of (fluid) mechanics on the other hand, which drastically reduces the power of the traditional transformation approaches when applied to acoustics [4,5].

In the recent paper [6], the problem of acoustic metamaterial design was approached via another angle. Instead of using the direct symmetries in spacetime to bridge between different solutions for the propagation of acoustic waves, one constructs a complementary relativistic theory, an “analogue theory”. Such analogue theory is defined in an auxiliary relativistic spacetime or “virtual spacetime” such that the coordinate transformations in this virtual spacetime correspond to symmetry transformations of the underlying real model, which can in principle be implemented by an adequate choice of the material parameters. In [6] this was achieved by recasting the acoustic wave equation in terms of a velocity potential (or phase), and by exploiting the analogy with the propagation of a scalar wave in a relativistic curved spacetime [7].

Using this “analogue transformation acoustics” scheme, one is able to overcome many of the limitations of the first attempts to construct a transformation acoustics. For example, one obtains results which are easier to implement technologically, and one is able to consider a much wider range of fluid backgrounds and of transformations. However, some important issues still remain to be clarified. In particular, in [6] only the isotropic case was considered. Moreover, the method described in [6] allows us to consider, for the first time, transformations that mix space and time, but the possibilities and potential applications of this set of transformations remain largely unexplored.

Initial Study Objectives

The aim of this project is twofold. The first part will be dedicated to the extension of the work of [6] with the goal of making the theory able to treat anisotropic materials and to analyze the introduction of time transformations in acoustics

The inclusion of anisotropy is important because: (i) real fluids (as opposed to idealized ones) typically display at least a small amount of anisotropy; (ii) metafluids are often based on composite structures which definitely display anisotropy [8]; (iii) an arbitrary geometrical configuration (e.g., a cloaking geometry) will in general correspond to an anisotropic transformational spacetime structure [9]. Analogue transformational acoustics also allows to explore time transformations in acoustics, which so far were only possible in electromagnetism. Such extension will permit to look into the possibility of designing time-dependent metafluids, i.e. metafluids whose properties are able to change in time.

Such devices could have some interesting uses such as, for example, time-dependent acoustic beam aperture modifiers, the cloaking of space-time events [10] or the frequency conversion of acoustic waves [11].

The second part of the project is devoted to the construction of an “analogue transformation elasticity” in the same sense as the “analogue transformation acoustics” summarized above. The results of [6] have taught us that, regardless of the spacetime transformation properties of a given physical theory, if it is possible (in any coordinate system) to construct an analogue version of this theory which is relativistic, then the spacetime transformations of this analogue theory can be linked to the symmetries of the original theory, and this scheme can be used to perform the central task of a transformation approach, i.e. to connect different types of solutions and thereby to design different types of geometrical devices (cloaks, lenses,...).

The possibility to extend this method to the theory of elasticity in solids and the limitations of this analogue transformational approach in this case are the central topics of this second part of the project. It is clear that the definition of a transformational elasticity would have a substantial technological impact not only on the space industry but in many sectors of engineering. Possible applications include noise attenuation based on vibration control of window panels, or vibration isolation by elasticity cloaks.

Study Results

The study started in March 2013 together with the Nanophotonics Technology Center, Universitat Politècnica de València, involving also researchers from the Modelling & Numerical Simulation Group, Universidad Carlos III de Madrid, the Instituto de Astrofísica de Andalucía (CSIC), and the Wave Phenomena Group, Universitat Politècnica de València.

The results of the first part of the report were achieved using transformational techniques combined with the tools of analogue gravity and homogenization techniques. The first one was used to be able to implement time transformations in the transformation acoustics whilst the second one considered non-isotropic transformations in both static and moving backgrounds. The methods of elasticity theory in anisotropic media, seismology, and more specific recent research on the elastic properties of composite materials have been used in the second part of the report. They helped to uncover some (so-far unknown) aspects of elasticity that need to be further clarified in order to construct a meaningful transformation approach for elastic waves.

The detailed results are described in the final report (final report, executive summary). In summary the study achieved the following results:

1. Development of a novel route (Analogue Transformation Acoustics, ATA) for transformations mixing space and time in acoustics. This kind of transformation is not allowed in Standard Transformation Acoustics (STA).
2. Design of novel acoustic devices based on ATA: a frequency shifter, a spacetime cloak, and a spacetime compressor.
3. Development of a homogenization process for the acoustic velocity potential wave equation. Our finding reveals a different way of tailoring acoustic properties through gradients of the static pressure. In contrast to standard metafluids based on isobaric composites, this alternative kind of metafluids is suitable for the implementation of transformational devices designed via the velocity potential equation.
4. Identification of the key issues and current roadblocks in the development of a general theory for transformational elasticity, and definition of several possible future strategies.

Selected Highlights

We have clarified fundamental differences between the newly developed ATA techniques and the STA method. This has allowed us to develop spacetime transformations that can only be performed with ATA. We have applied a homogenization process to the velocity potential acoustic wave equation, which has allowed us to design a multilayer structure able to cloak the acoustic velocity potential. About transformation elasticity, we have concluded that the scope of transformational elasticity is vastly broader than originally expected. For instance, the implementation of spacetime transformations would require the temporal control of some of the properties of the metamaterial system so as to lead to (apparent) flows in the metamaterial.

References

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Outcome