Acoustic Invisibility for Elliptic Objects: Theory and Experiments for Underwater Sound (SpringerBriefs in Applied Sciences and Technology) 9783031226021, 9783031226038, 303122602X

Table of contents :
Preface
Contents
1 Introduction and State of the Art
References
2 Wave Propagation in Periodic Media
2.1 Acoustic Waves in Fluids
2.2 Elastic Waves in Homogeneous, Isotropic Solids
2.3 Elastic Waves in Homogenous, Anisotropic Media
2.4 Wave Propagation in Sonic Crystals
2.5 Wave Propagation in Phononic Crystals
2.6 Long-Wavelength Homogenization
2.7 Further Readings
References
3 Transformation Acoustics
3.1 Inertial Cloaking
3.2 Pentamode Materials and Pentamode Cloaking
References
4 Transformation Acoustics in Elliptic Coordinates
4.1 Defining Transformations in Elliptic Coordinates
4.2 Selected Examples of Transformations in Elliptic Coordinates
4.2.1 Spatially Independent Elasticity Tensor
4.2.2 Bulk Moduli Following a Power Law
4.2.3 Spatially Independent Density
4.3 Numerical Test Cases
References
5 Design and Experimental Validation of an Elliptic Cloak
5.1 Problem Setting
5.2 Microstructure Design and Validation
5.3 Experimental Setup
5.4 Results and Discussion
References

Citation preview

SpringerBriefs in Applied Sciences and Technology PoliMI SpringerBriefs Davide Enrico Quadrelli · Francesco Braghin

Acoustic Invisibility for Elliptic Objects Theory and Experiments for Underwater Sound

SpringerBriefs in Applied Sciences and Technology

PoliMI SpringerBriefs Series Editors Barbara Pernici, Politecnico di Milano, Milano, Italy Stefano Della Torre, Politecnico di Milano, Milano, Italy Bianca M. Colosimo, Politecnico di Milano, Milano, Italy Tiziano Faravelli, Politecnico di Milano, Milano, Italy Roberto Paolucci, Politecnico di Milano, Milano, Italy Silvia Piardi, Politecnico di Milano, Milano, Italy

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Davide Enrico Quadrelli · Francesco Braghin

Acoustic Invisibility for Elliptic Objects Theory and Experiments for Underwater Sound

Davide Enrico Quadrelli Dipartimento di Meccanica Politecnico di Milano Milan, Italy

Francesco Braghin Dipartimento di Meccanica Politecnico di Milano Milan, Italy

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISSN 2282-2577 ISSN 2282-2585 (electronic) PoliMI SpringerBriefs ISBN 978-3-031-22602-1 ISBN 978-3-031-22603-8 (eBook) https://doi.org/10.1007/978-3-031-22603-8 © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

As for Bilbo Baggins, even while he was making his speech, he had been fingering the golden ring in his pocket: his magic ring that he had kept secret for so many years. As stepped down he slipped it on his finger, and he was never seen by any hobbit in Hobbiton again. —J. R. R. Tolkien, The Lord of the Rings

Preface

The quest for invisibility devices, unlocked in 2006 by the introduction of Transformation Theory, has pushed the efforts of scientists from the most diverse research fields during the past two decades. Indeed, after the first works dealing with light and optical invisibility, that introduced the “invisibility cloak” as a scientific subject, the idea of a device that is capable of surrounding a body and making it neutral to probing fields has spread to other areas of study, ranging from water waves and sound, to even matter waves and heat conduction. In particular, acoustic cloaking refers to the possibility of shielding an object in such a way that an incident sound field remains undisturbed by the presence of the shielded scatterer, as if the object was not there at all. Such a device could be of great industrial relevance for underwater applications: it is known, indeed, that light does not travel far in salty and dirty water; thus, acoustic signals have been used extensively to communicate and locate things under the surface of the ocean during the last century. Despite its potential industrial relevance, the application of an acoustic cloak in a real-life scenario is still hindered by some practical issues that have to be solved. As an example, Transformation Theory is a very powerful tool but really hard to apply routinely for geometries different than axisymmetrical cylinders and spheres. This book concentrates instead on targets that can be approximated by cylinders having cross-sectional shapes in the form of ellipses and introduces a systematic analytical approach to deal with the design of cloaks that can reduce the acoustic visibility of such objects. Moreover, it has the ambition of guiding the reader through all the design steps, from the mathematical definition of the transformation, to the experimental validation in the water tank, passing through the discussion of pros and cons of design choices related to feasibility and fabrication issues. However, the broad topic of acoustic cloaking is definitely too vast to be thoroughly addressed in a monograph like this one; thus, we apologize the reader for any omission that we made. For example, alternatives to Transformation Methods like active strategies and scattering cancellation cloaks are simply mentioned, and some interesting engineering problems like buoyancy issues related to the weight of the cloak and structural integrity of the microstructure under high pressure acting on vii

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Preface

the exterior of the cloak are left for future studies. This book is primarily intended to reach a broad audience, from graduates to researchers, with the secondary aim to introduce the novice to Transformation Acoustics and to the related technology of wave manipulation with metamaterials. The first chapters, indeed, serve this scope, other than introducing the notions used in the last ones. A monograph is indeed often the outcome of a long journey that no one can obviously travel alone. Many thanks go to all the people that helped us to make this possible. In particular, my gratitude goes to Prof. Francesco Braghin, who supervised and co-authored this work, and to Prof. Gabriele Cazzulani, whose help was crucial during the experimental campaign that is presented in the last section of the book. Mention goes also to Dr. Jacopo Marconi, for the stimulating talks and advice during the way, and to Engr. Matteo Casieri and Engr. Giacomo Ferrari for their precious help. Last but not least, gratitude goes to my family and Sara, for the continuous support. This work is dedicated to them. Milan, Italy October 2022

Davide Enrico Quadrelli

Contents

1 Introduction and State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 7

2 Wave Propagation in Periodic Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Acoustic Waves in Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Elastic Waves in Homogeneous, Isotropic Solids . . . . . . . . . . . . . . . . . 2.3 Elastic Waves in Homogenous, Anisotropic Media . . . . . . . . . . . . . . . 2.4 Wave Propagation in Sonic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Wave Propagation in Phononic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Long-Wavelength Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Further Readings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 14 17 21 28 31 37 37

3 Transformation Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Inertial Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Pentamode Materials and Pentamode Cloaking . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 41 42 47

4 Transformation Acoustics in Elliptic Coordinates . . . . . . . . . . . . . . . . . . 4.1 Defining Transformations in Elliptic Coordinates . . . . . . . . . . . . . . . . 4.2 Selected Examples of Transformations in Elliptic Coordinates . . . . . 4.2.1 Spatially Independent Elasticity Tensor . . . . . . . . . . . . . . . . . . 4.2.2 Bulk Moduli Following a Power Law . . . . . . . . . . . . . . . . . . . . 4.2.3 Spatially Independent Density . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Test Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 56 57 60 61 61 64

5 Design and Experimental Validation of an Elliptic Cloak . . . . . . . . . . . . 5.1 Problem Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Microstructure Design and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 68 81 83 87 ix

Chapter 1

Introduction and State of the Art

Abstract This chapter introduces the notion of cloaking and reviews the main ideas that have been developed during the last fifteen years to take this concept from theory to experimental assessment, underlining issues that remain open challenges for future research.

The idea that invisible things and unseen beings could exist and live among us is a shared cultural heritage between the most diverse populations across the globe. It is not surprising then to realize that also the idea of possessing a tool that can give the unnatural power of not being seen by others is recurrent: Tolkien’s Ring of Power, Harry Potter’s cloak of invisibility and Star Trek’s cloaking device are just a few examples taken from modern pop culture. Achieving this has for long been considered something relegated to fantasy literature and science-fiction, implying some magic involved or the knowledge of some advanced extraterrestrial technology, until recent findings related to the so-called Transformation Theory in optics. Indeed, in 2006 the works of Pendry [1] and Leonhardt [2] introduced a general method to compute the physical properties that a cloak should have to render the wrapped object invisible. This than pushed an explosion of activities aimed at the realization of such a device [3–5]. This method is based on reinterpreting as exotic material properties the metric change coefficients obtained in the wave equation when a map between two domains is considered, exploiting the formal invariance of Maxwell’s equations upon transformations in arbitrary curvilinear coordinate systems [6]. The name of the theory thus comes directly from the transformation that is the core of the method, and it beautifully makes apparent the equivalence between a curved space and an inhomogeneous and anisotropic material distribution, as is implied by Fermat’s principle [7] (rays travel along the shortest-time path), and initially suggested by the pioneering work of Dolin [8]. The use of similar mappings was also already widespread in other contexts, like for solving the fluid flow around bodies with complicated shapes in hydrodynamics [9], or in generating Perfectly Matched Layers to simulate numerically unbounded domains [10, 11]. Nonetheless, it was the rise of the concept of metamaterials [12, 13] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. E. Quadrelli and F. Braghin, Acoustic Invisibility for Elliptic Objects, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-22603-8_1

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1 Introduction and State of the Art

and the design freedom unlocked by their employment that ultimately led to recognize that the transformed equations can be reinterpreted as the equations describing the behavior of an unusual material distribution that could be actually implemented, thus providing a design strategy for cloaking devices. Although several definitions have been produced in the last two decades, in this monograph we will refer to metamaterials following the definition provided in [13], i.e. we will consider a metamaterial to be a composite material that was rationally designed in order to obtain equivalent material properties beyond those of the bulk ingredients. Note that this definition stresses the requirement that the design is rational, thus excluding, for example, particulate composites obtained as the result of a random process of suspension of particles in a matrix. As we will see, this requirement often implies the fact that the metamaterial is obtained by an ordered repetition of building blocks. These building blocks can include voids other than one or more solid phases, provided that their arrangement in space is tailored in a rational and deterministic way. Although the use of composites has a long history in engineering and material science [14], the recent advancements of 3D printing technologies have seen a new growth of interest on the matter (and especially on complicated three-dimensional geometries as opposed to random media), leading to the idea that from a very limited set of constituents, a multitude of diverse effective material behaviors can be achieved, thus unlocking the design of devices before considered impossible to realize, as the cloak of invisibility. The success of the idea in electromagnetism pushed the interest in emulating such concept in other fields of classical wave propagation [15, 16]: cloaking is thus now an established research field in acoustics [17, 18], elastodynamics [19–25], surface water waves [26–28] and even in matter waves [29]. By analogy, in these contexts “invisibility” means that the presence of an object does not produce outside the cloak any appreciable distortion of a probing incident radiation, making detection impossible from measurements around the object itself. According to this idea, the concept of cloaking can also be extended to fields not governed by the wave equation: thermal and diffusion cloaks can be designed [30–32], where the goal is to make an inclusion neutral in an otherwise perturbed external field. This monograph will concentrate on acoustic cloaking. From an engineering perspective, this concept finds a natural application in preserving the strategic advantage submarines have, i.e. the difficulty/impossibility of being located when they operate underwater [33]. Indeed, since light does not travel great distances in salty and dirty water, moving underwater proved at the start of the last century to be the quintessential stealth strategy [34]. It is not by chance then that between the first and second World Wars, efforts have been put, especially from the British Navy, into developing the ASDIC system (from the Anti-Submarine Detection Investigation Committee), later on called SONAR (Sound Navigation and Ranging), in order to make “visible” the deadly German U-boats [35]. This system exploits sound instead of light, in such a way that submarines can be detected in the same way as bats, for example, detect their preys by echolocation. In this scenario, and in particular with the development

1 Introduction and State of the Art

3

of the new multi-static SONAR technologies, an acoustic cloak can be a device of invaluable importance. Traditional ways to avoid detection that are nowadays employed, indeed, consist in using sound absorbing tiles directly attached to the outer surface of the hull. These are particularly effective at high frequencies (where the tiles have higher absorption capabilities) and for active SONAR systems where the receiver that is listening the echoes is placed in the same position as the sound source (the tiles suppress backscattering). As opposed to this scenario, called monostatic SONAR, multiple communicating sensors can be placed in different locations, in a multistatic arrangement. This allows detection even if backscattering is reduced: whatever distortion of the expected acoustic field can be a blueprint of the presence of the submarine. In this case, it is easy to understand that only a technology that aims at restoring the acoustic field exactly as it is without the obstacle can preserve the complete invisibility of the vehicle. Acoustic cloaking has also interesting engineering applications in the aeronautical industry [36, 37]: indeed, engine nacelles and wings reflect towards the ground the high intensity sound generated by the airplane during flight. The cloaking technology could make those components neutral to the propagation of sound, allowing designers to cope with the noise mitigation targets that are foreseen for the next decades. Although acoustic cloaking is now an established concept with almost 15 years of related literature, the Technology Readiness Level [38] is low, and actual implementations of the idea in real life scenarios are far from being put in practice. As mentioned, the original idea of an acoustic cloak directly followed from the findings in electromagnetism: due to the formal equivalence of the acoustic wave equation and the equation of the transverse electric and magnetic modes [39], it was initially appealing to convert the requirements on dielectric permittivity and magnetic permeability, that follow from the design of optical cloaks, into requirements on density and bulk modulus, that govern the propagation of pressure waves in fluids [40, 41]. In particular, applying Transformation Theory in this way, it turns out that anisotropic inertial properties are needed. For this reason, Norris [42] later on called this type of acoustic cloaks inertial cloaks. In the same work, Norris also pointed out that the distribution of material properties that realizes the cloak for a given geometry is not unique, as opposed to what happens with the electromagnetic counterpart, establishing acoustic cloaking as a discipline on his own. Indeed, the anisotropy required by non conformal transformations, can be obtained not only assuming a fluid with a tensorial inertial mass, but also employing solids with singular anisotropic elastic properties called pentamode materials [42, 43]. These are a conceptual generalization of fluids that were firstly introduced by Milton and Cherkaev [44]: assuming an elasticity tensor with five out of six null eigenvalues, the constitutive equation implies that only one single state of stress can be generated in the solid, that thus opposes resistance to only one type of applied deformation. As mentioned, this parallels what happens in ideal fluids, that only oppose resistance to volumetric deformations by developing an isotropic state of stress (the so-called isotropic pressure), whereas they are not able to oppose any resistance to changes of shape. The generalization implied in the concept of a pen-

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tamode material is that the eigenvector associated to the non-null eigenvalue is not necessarily the volumetric deformation, but the material can be engineered assigning the desired non-easy way to deform. By consequence, it can be defined a so-called pseudo-pressure (the only non-null component of the stress state when expressed in the basis of the eigenvectors of the elasticity tensor), and the equations of elastodynamics reduce to an equation governing the psuedo-pressure, that can be shown to have the same formal structure the transformed acoustic wave equation has. The most general acoustic cloak thus comprises both anisotropic inertial properties and anisotropic stiffness properties, with the pure inertial cloak and pure pentamode cloak as limiting cases. The implementation of an actual cloak thus implies the capability of realizing a metamaterial that can either behave as a fluid with tensorial inertial mass or behave as a pentamode material (or even the combination of the two). Milton and Cherkaev [44], that firstly introduced the idea of pentamodes, also suggested that it could be realized in practice with an artificial crystal made by a soft matrix (usually air or void) in which a hard structure made by trusses that meet at very tiny joints is embedded. In the limit for the trusses to meet at a single geometrical point, the shear resistance goes to zero, realizing the static fluid-like behavior. Actual practicability of the concept has been firstly shown by Kadic et al. [45], who proved that modern 3D printing technologies allow tiny junctions to be realized in such a way that the shear modulus is lowered down to orders of magnitude below that of the bulk modulus. They also showed how changing the symmetries of such artificial crystal allows to obtain anisotropic versions of the pentamode [46]. The reduction of the static shear modulus in these artificial crystals, implies also some interesting dynamic properties: computing the band distribution of an infinite crystal by Bloch-Floquet analysis, it turns out that the great separation between the equivalent shear and bulk moduli opens a bandgap for the shear mode at frequencies where the wavelength of the longitudinal mode is sufficiently bigger than the size of the cell to allow for long wavelength homogenization [47]. This in turns implies that there is a frequency range where the infinite crystal can only support single mode propagation, exactly as happens in fluids. For what concerns the pure inertial cloak, instead, it has been shown that a distribution of alternating fluids, once homogenized, behaves as an equivalent fluid with tensorial inertial properties [48, 49]. This strategy turns out to be handy when trying to design cloaks for spherical or axisymmetric obstacles: once the cloak is opportunely discretized in the radial direction, concentrical layers of fluids can be adopted to implement the required distribution of material properties (note that, the inhomogeneous continuously varying material properties’ distribution prescribed by Transformation Acoustics has in any case to be discretized in some way to be practically realized, since according to our definition a metamaterial is composed by building blocks of finite size). However, when thinking of practical implementations of the concept, it is not clear how to keep the layers of fluids separated, without introducing a third material in the mix. An alternative strategy was proposed by Fang and coworkers that exploited the analogy between the lumped parameters acoustics and the electrical transmission lines equations to build an underwater cloak made by tiny

1 Introduction and State of the Art

5

cavities in a disc of aluminum filled by the working fluid [50]. A similar strategy, based on arrangements of fluid-filled cavities [51, 52], allowed to experimentally validate ground cloaks in air [53, 54] and water [55, 56], in both the bidimensional and tridimensional scenarios. Ground cloaks, also known as carpet cloaks, exploit the presence of a reflecting boundary on which the target is placed to hide its presence. This is done by shrinking the equivalent acoustical dimensions of the scatterer to a line: once placed on the reflecting flat surface, it will not produce any appreciable distortion on the reflected waves generated by incident acoustic fields. On the opposite side of the spectrum of employable materials distributions, pure pentamode cloaks, with their intrinsic solid nature and the possibility to avoid working fluids, are generally considered more suitable for practical applications. Note that, in principle, being able to engineer a pentamode material that behaves similarly to an ideal fluid, but with the advantage of not flowing or mixing, could allow to have the alternating layers of fluids replaced by equivalent alternating layers of isotropic pentamodes, making the difference in the implementation effort between inertial cloaks and pentamode ones blurrier. In any case, following the pure pentamode design strategy, Chen et al. [57, 58] produced and experimentally tested a solid underwater cloak for a circular obstacle. All in all, as mentioned before, several engineering challenges are still to be faced to take the acoustic cloaking principle to the readiness level required for actual practical applications. For what concerns the aeronautical applications, for example, the high speed of aircrafts compared with the relatively low sound propagation speed in air (when compared to that of propagation in liquids), does not allow to neglect the effect of the relative motion of the fluid with respect to the cloak. This renders the standard Transformation Acoustic approach not viable for this type of scenarios, since the equations are not form invariant under the previously mentioned transformations, when considering the effects of the moving fluid. To sidestep this problem, Garcia-Meca et al. [59, 60] built on the concept of analogous space-time, previously introduced by Unruh [61], to introduce a development of the standard theory called Analogous Transformation Acoustics. Several works have then followed that address acoustic cloaking with moving fluids [62, 63]. In this work, we will instead focus on underwater applications, where the low Mach numbers allow for neglecting in first approximation the relative movement of the fluid with respect to the target. Another issue is related to the fact that Transformation theories rely on the writing of a suitable mapping which can actually be found easily only for a restricted set of simple geometries. The literature regarding acoustic cloaking is indeed vast if one desires to cloak cylinders or spheres, where transformations are available in cylindrical/spherical coordinates [64]. Instead, the number of works dealing with arbitrary shapes is far less. The problem of the arbitrarity of the shape of the target, has been firstly tackled by Hu and coworkers [65], who proposed a numerical method based on the solution of the Laplace equation to find a mapping between the points in the deformed and undeformed domains. A similar idea was later on introduced by the same group in order to obtain numerical mappings characterized by near zero local rigid rotation [66], i.e. maps allowing the design of pure pentamode cloaks.

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Following a different approach Li et al. [67] proposed a two-step analytical map to design diamond shaped cloaks, in which the material properties are anisotropic but homogeneous inside triangular subregions of the cloak. A similar idea was extended by Vipperman an coworkers [68–70] to arbitrarily shaped targets, dividing both the undeformed and deformed domains into small triangles (in 2D) and polyhedral (in 3D) and defining for each element a transformation that leads to inertially anisotropic but homogeneous properties. A completely different approach to sidestep the difficulties of transformationbased approaches in dealing with arbitrary shapes, is to change the design method. As an example, scattering cancellation [71–75] refers to those cloaks designed as a spatial distribution of scatterers optimized in such a way that, for selected frequencies and incidence directions, the overall scattering of the target and the cloak is close to zero. Clearly, such method does not pose any restriction on the shape of the cloak, but comes with other issues, i.e. the limited working frequency range and incidence directions. Another viable option is to resort to numerical optimization schemes to compute the distribution of material properties that minimizes the scattered field [76–78]. Finally, active cloaking [79–82], refers to the strategy where sources are used to eliminate the scattering of the target, in a similar fashion as in active noise cancellation. In this work we instead stick to Transformation Acoustics and provide analytical quasi-symmetric transformations that can be used to design pure pentamode cloaks for the two-dimensional ellipse in ellipse cloaking problem. This is discussed in Chap. 4, after the next two Chapters, that are used to introduce the setting of wave propagation in periodic media and Transformation Theory. The choice of the ellipse is in part justified by the fact that submarines typically have elongated cross sections that are not perfectly axisymmetric. Compared to the aforementioned numerical method [66], that similarly aims at producing transformations with low local rotations that can be neglected to give material distributions’ prescriptions for pure pentamode cloaks, our approach will be shown to have the advantage to allow infinite analytical solutions for a given geometry, providing the designer with more freedom of choice for metamaterial implementation. This increased freedom is paid, on contrast, with the applicability of the method being restricted to the elliptical shape only. The introduced approximation, implied in the fact that pure pentamode materials are adopted when instead the transformation is not a pure stretch, is quantifiable computing the local rotation angle, and it will be shown to be bounded to low values. In Chap. 5 we then build on such method to design a cloak for experimental validation. It will be shown that, among all the possible transformations in elliptic coordinates, a linear mapping produces a constant anisotropic elasticity tensor with an inhomogeneous scalar density. With this map, the anisotropy of propagation speed required for cloaking is entirely demanded on the elastic properties, while the inhomogeneity is entirely demanded on inertial properties. This in turns reduces the implementation complexity of the cloak. Indeed, it is well known that pentamode metamaterials can be implemented with periodic structures whose topology can be tailored in such a way to obtain independent control on the elastic and inertial properties [83, 84]. Moreover, elliptical coordinates turn out to be handy also in the phase of the accommodation of the metamaterial around the elliptical obstacle. Since the

References

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principal directions of anisotropy prescribed by the transformation are everywhere tangent to coordinate lines, the metamaterial distribution can be built in Cartesian coordinates and then deformed conformally by mapping the rectangular coordinates to the elliptical ones at once. Numerical tests are performed on the whole assembly of the cloak and a comparison with experimental results is provided.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.

Pendry JB, Schurig D, Smith DR (2006) Science 312:1780 Leonhardt U (2006) Science 312:1777 Ergin T, Stenger N, Brenner P, Pendry JB, Wegener M (2010) Science 328:337 Ergin T, Fischer J, Wegener M (2011) Phys Rev Lett 107:173901 Fischer J, Ergin T, Wegener M (2011) Opt Lett 36(11):2059 Milton G, Briane M, Willis J (2006) New J Phys 8:248 Born M, Wolf E (2013) Principles of optics: electromagnetic theory of propagation, interference and diffraction of light. Elsevier Dolin L (1961) Izv Vyssh Uchebn Zaved Radiofizika 4(5):964 Lamb H (1895) Hydrodynamics. University Press Berenger JP (1994) J Comput Phys 114:185 Nicolet A, Remacle JF, Meys B, Genon A, Legros W (1994) J Appl Phys 75:6036 Pendry JB (2000) Phys Rev Lett 85:3966 Kadic M, Milton GW, van Hecke M, Wegener M (2019) Nat Rev Phys 1(3):198 Milton GW (2002) The theory of composites. Cambridge University Press Kadic M, Bückmann T, Schittny R, Wegener M (2015) Philos Trans Roy Soc A Math Phys Eng Sci 373(2049):20140357 Kadic M, Bückmann T, Schittny R, Wegener M (2013) Rep Prog Phys 76(12):126501 Chen H, Chan CT (2010) J Phys D Appl Phys 43(11). https://doi.org/10.1088/0022-3727/43/ 11/113001 Norris AN (2015) Acoust Today 11(1):38. http://acousticstoday.org/acoustic-cloakingandrew-n-norris-2/ Brun M, Guenneau S, Movchan AB (2009) Appl Phys Lett 94(6):061903 Norris AN, Shuvalov AL (2011) Wave Motion 48:525 Farhat M, Guenneau S, Enoch S (2012) Phys Rev B 85(2):020301 Stenger N, Wilhelm M, Wegener M (2012) Phys Rev Lett 108:014301 Norris AN, Parnell WJ (2012) Proc Roy Soc A 468(2146):2881 Xu X, Wang C, Shou W, Du Z, Chen Y, Li B, Matusik W, Hussein N, Huang G (2020) Phys Rev Lett 124:114301 Quadrelli DE, Craster R, Kadic M, Braghin F (2021) Extreme Mech Lett 44:101262 Farhat M, Enoch S, Guenneau S, Movchan AB (2008) Phys Rev Lett 101:134501 Porter R, Newman J (2014) J Fluid Mech 750:124 Dupont G, Guenneau S, Kimmoun O, Molin B, Enoch S (2016) J Fluid Mech 796 Zhang S, Genov DA, Sun C, Zhang X (2008) Phys Rev Lett 100(12):123002 Schittny R, Kadic M, Guenneau S, Wegener M (2013) Phys Rev Lett 110:195901 Guenneau S, Puvirajesinghe T (2013) J Roy Soc Interface 10(83):20130106 Sinigaglia C, Quadrelli DE, Manzoni A, Braghin F (2022) Proc Roy Soc A 478(2258):20210813 Audoly C (2016) Proceedings of the undersea defence technology conference, pp 1–3 Burcher R, Rydill LJ (1995) Concepts in submarine design, vol 2. Cambridge University Press Urick RJ (1975) Principles of underwater sound. McGraw-Hill Book, New York, NY (USA) Iemma U, Carley M, Pellegrini R (2014) INTER-NOISE and NOISE-CON congress and conference proceedings, vol 249. Institute of noise control engineering, 2014, vol 249, pp 274–281

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37. Iemma U (2016) Aerospace 3(2):15 38. Héder M (2017) Innov J 22(2):1 39. Banerjee B (2011) An introduction to metamaterials and waves in composites. Taylor and Francis 40. Cummer SA, Schurig D (2007) New J Phys 9. https://doi.org/10.1088/1367-2630/9/3/045 41. Chen H, Chan C (2007) Appl Phys Lett 91(18):183518 42. Norris AN (2008) Proc Roy Soc A 464:2411 43. Norris AN (2009) J Acoust Soc Am 125(2):839 44. Milton GW, Cherkaev A (1995) J Eng Mater T 117:483 45. Kadic M, Bückmann T, Stenger N, Thiel M, Wegener M (2012) Appl Phys Lett 100:191901 46. Kadic M, Schittny R, Bückmann T, Wegener M (2013) New J Phys 15:023029 47. Martin A, Kadic M, Schittny R, Bückmann T, Wegener M (2012) Phys Rev B 86:155116 48. Cheng Y, Yang F, Xu JY, Liu XJ (2008) Appl Phys Lett 92(15):151913 49. Torrent D Søanchez-Dehesa J (2008) New J Phys 10(6):063015 50. Zhang S, Xia C, Fang N (2011) Phys Rev Lett 106(2):024301 51. Pendry JB, Li J (2008) New J Phys 10(11):115032 52. Popa BI, Wang W, Konneker A, Cummer SA, Rohde CA, Martin TP, Orris GJ, Guild MD (2016) J Acoust Soc Am 139(6):3325 53. Popa BI, Zigoneanu L, Cummer SA (2011) Phys Rev Lett 106:253901 54. Zigoneanu L, Popa BI, Cummer SA (2014) Nat Mater 13(4):352. https://doi.org/10.1038/ nmat3901 55. Bi Y, Jia H, Lu W, Ji P, Yang J (2017) Sci Rep 7(1):1. https://doi.org/10.1038/s41598-01700779-4 56. Bi Y, Jia H, Sun Z, Yang Y, Zhao H, Yang J (2018) Appl Phys Lett 112(22):1. https://doi.org/ 10.1063/1.5026199 57. Chen Y, Liu X, Hu G (2015) Sci Rep 5:15745 58. Chen Y, Zheng M, Liu X, Bi Y, Sun Z, Xiang P, Yang J, Hu G (2017) Phys Rev B 95(18):180104 59. García-Meca C, Carloni S, Barceló C, Jannes G, Sánchez-Dehesa J, Martínez A (2013) Sci Rep 3(1):1 60. García-Meca C, Carloni S, Barceló C, Jannes G, Sánchez-Dehesa J, Martínez A (2014) Photonics and nanostructures-fundamentals and applications, vol 12, no 4, p 312 61. Unruh WG (1995) Phys Rev D 51(6):2827 62. Iemma U, Palma G (2017) Math Prob Eng 2017 63. Iemma U, Palma G (2020) Sci Rep 10(1):1 64. Gokhale NH, Cipolla JL, Norris AN (2012) J Acoust Soc Am 132:2932 65. Hu J, Zhou X, Hu G (2009) Comput Mater Sci 46(3):708 66. Chen Y, Liu X, Hu G (2016) J Acoust Soc Am 140(5):EL405. https://doi.org/10.1121/1. 4967347 67. Li T, Huang M, Yang J, Lan Y, Sun J (2012) J Vib Acoust Trans ASME 134(5). https://doi. org/10.1115/1.4006633 68. Li Q, Vipperman JS (2014) Appl Phys Lett 105(10). https://doi.org/10.1063/1.4895765 69. Li Q, Vipperman JS (2018) J Appl Phys 124(3). https://doi.org/10.1063/1.5028136 70. Li Q, Vipperman JS (2019) J Vib Acoust Trans ASME 141(2):1. https://doi.org/10.1115/1. 4041897 71. García-Chocano VM, Sanchis L, Díaz-Rubio A, Martínez-Pastor J, Cervera F, LlopisPontiveros R, Sánchez-Dehesa J (2011) Appl Phys Lett 99(7):074102 72. Amirkulova F, Norris A (2017) J Acoust Soc Am 142(4):2578 73. Lu Z, Sanchis L, Wen J, Cai L, Bi Y, Sánchez-Dehesa J (2018) Sci Rep 8(1):1 74. Andkjær J, Sigmund O (2013) J Vib Acoust 135(4) 75. Sanchis L, García-Chocano VM, Llopis-Pontiveros R, Climente A, Martínez-Pastor J, Cervera F, Sánchez-Dehesa J (2013) Phys Rev Lett 110(12):124301 76. Ahmed WW, Farhat M, Zhang X, Wu Y (2021) Phys Rev Res 3(1):013142 77. Chen P, Haberman MR, Ghattas O (2021) J Comput Phys 431:110114

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78. Cominelli S, Quadrelli DE, Sinigaglia C, Braghin F (2022) Proc Roy Soc A 478(2257):20210750 79. Vasquez FG, Milton GW, Onofrei D (2009) Phys Rev Lett 103(7):073901 80. Vasquez FG, Milton GW, Onofrei D (2009) Opt Express 17(17):14800 81. Vasquez FG, Milton GW, Onofrei D (2011) Wave Motion 48(6):515 82. Lin C, Liu D, Eggler D, Kessissoglou N (2021) J Acoust Soc Am 149(3):1803 83. Kadic M, Bückmann T, Schittny R, Gumbsch P, Wegener M (2014) Phys Rev Appl 2(5):054007 84. Layman CN, Naify CJ, Martin TP, Calvo DC, Orris GJ (2013) Phys Rev Lett 111(2):024302

Chapter 2

Wave Propagation in Periodic Media

Abstract This chapter is devoted to the review of some useful results related to wave propagation both in homogeneous and in periodic media and is intended to recall the fundamental knowledge on which the following of this work builds, other than to set the general notation used.

2.1 Acoustic Waves in Fluids Acoustic phenomena involve small perturbations of the total pressure p [Pa] around a reference pressure state p0 , assumed here constant in space and time. According to the constitutive behavior of the fluid, the pressure fluctuation p  = ( p − p0 ) is accompanied by fluctuations of other physical quantities such as the local density ρ  = (ρ − ρ0 ) [kg/m3 ], the particle velocity v [m/s] (the fluid is assumed at rest, such that no particle velocity is considered in the reference state), and temperature T  = (T − T0 ) [K] (although early works by Newton on acoustics assumed the process to be isothermal, it has been indeed verified experimentally that assuming the process to be adiabatic leads to more accurate results). Such constitutive behavior can thus be represented by an equation that relates pressure to density and entropy [1]: p = p(ρ, s).

(2.1)

Applying the total time derivative one gets: ∂ p dρ ∂ p ds dp = + . dt ∂ρ dt ∂s dt

(2.2)

Since, as mentioned, the process is assumed to be so fast that there is no time for heat to be exchanged (hypothesis of adiabiaticity), the total time derivative of entropy ds/dt is null. The following is thus obtained: dp ∂p = dt ∂ρ



 ∂ρ + v · ∇ρ , ∂t

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. E. Quadrelli and F. Braghin, Acoustic Invisibility for Elliptic Objects, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-22603-8_2

(2.3)

11

12

2 Wave Propagation in Periodic Media

where the material time derivative has been expressed according to the spatial representation d/dt = (∂/∂t + v · ∇). From Eq. 2.3, a linear relationship between the variables of interest can be obtained invoking the assumption that fluctuations are infinitesimal. Dropping the higher order terms: B0 ∂ p = ∂t ρ0



∂ρ  + v · ∇ρ0 ∂t

 (2.4)

is obtained. Note that the reference state of density is not in general considered homogeneous, thus the term related to its spatial gradient cannot be neglected. The following has been also adopted:  ∂p B0 ∂ p  = ≈  ∂ρ ∂ρ 0 ρ0

(2.5)

where the definition of the isoentropic bulk modulus B0 [Pa], that relates at first order volumetric strains to variations of pressure, has been introduced. Equation 2.4 represents the constitutive behavior that is adopted in the linear modeling of acoustic wave propagation. Given an infinitesimal fluid volume element, the conservation of linear momentum reads: d (2.6) (ρv) = −∇ p dt where absence of body forces and null viscosity has been assumed. Considering the assumed infinitesimal character of the acoustic fluctuations of variables, the conservation of momentum can be simplified into: ρ0

∂v = −∇ p  . ∂t

(2.7)

This is the Euler equation for inviscid fluids, that is used to relate the acoustic pressure p  with the acoustic velocity. The conservation of mass instead reads: ∂ρ + ∇ · (ρv) = 0 ∂t

(2.8)

that, by virtue of the previous assumptions can be linearized in: ∂ρ  + ∇ρ0 · v + ρ0 ∇ · v = 0. ∂t

(2.9)

Substituting the linearized constitutive Eq. 2.4 into the conservation of mass the following is obtained: 1 ∂ p + ∇ · v = 0. (2.10) B0 ∂t

2.1 Acoustic Waves in Fluids

13

Taking the time derivative of 2.10 and the divergence of 2.7: ∂v 1 ∂ 2 p =∇· − 2 B0 ∂t ∂t

∂v ∇· = −∇ · ∂t



1 ∇ p ρ0

 (2.11)

the two can be combined into:  ∇·

1 ∇ p ρ0

 =

1 ∂ 2 p B0 ∂t 2

(2.12)

that is the wave equation with inhomogeneous coefficients. If the fluid is homogeneous, i.e. the assumption of ρ0 and B0 being constant in space can be invoked, than the following is obtained: 1 ∂ 2 p (2.13) ∇ 2 p = 2 2 c ∂t √ that is the scalar wave equation, with c = B0 /ρ0 [m/s] the speed of propagation. The symbol ∇ 2 is the Laplacian operator and stands for (∇ · ∇). In the following, the prime symbol that stands for fluctuation of variables will be dropped to ease the notation, when there is no possibility of confusion. Assuming time harmonic ˆ iωt ), with i the imaginary unit and ω [rad/s] the dependency of variables, p  = Re( pe angular frequency, the Helmholtz equation is obtained: ∇ 2 pˆ +

ω2 pˆ = 0 c2

(2.14)

for the complex amplitude p. ˆ Solutions satisfying this equation in the form of: pˆ = Ae±iκ·x κ = |κ| =

ω2 c2

(2.15)

represent plane harmonic waves p = A cos(ωt ± κ · x)

(2.16)

with amplitude A [Pa], traveling with speed c along the direction identified by the wavevector κ [rad/m]. The modulus of the wavevector is called wavenumber κ and is linearly related to the circular frequency ω. This relationship is called dispersion relation, and can be plotted in a linear graph as ω = cκ that represent the locus of all the acceptable (ω, κ) pairs for plane waves to travel in the homogeneous, isotropic fluid. The constant slope is c, the speed of propagation. Typical boundary conditions arise at pressure release surfaces  D (Dirichlet Boundary condition): pˆ = 0

x ∈ D

(2.17)

14

2 Wave Propagation in Periodic Media

or at acoustically rigid surfaces  N (Neumann boundary condition): vˆ · n = −

1 ∇ pˆ · n = 0 iωρ

x ∈ N ,

(2.18)

n being the surface normal. The characteristic acoustic impedance: Z=

1 [Pa · s/m3 ] ρc

(2.19)

represents the ratio between the amplitudes of pressure and the associated acoustic velocity for plane waves (not necessarily harmonic) traveling in the medium.

2.2 Elastic Waves in Homogeneous, Isotropic Solids The constitutive equation of linear elastic solids reads: σ =C:ε

(2.20)

where ε = 1/2(∇u + ∇u T ) is the dimensionless second order strain tensor, obtained as the symmetric part of the gradient of the displacement field u [m]. The symmetric second order stress tensor σ [Pa] instead allows the computation of traction vectors t [N/m2 ] over the infinitesimal surface with normal n, according to t = σ · n.

(2.21)

The fourth order elasticity tensor C [Pa] linearly maps strain into stress, and has in principle 81 independent components. By virtue of the symmetries of stress and strains it follows that (minor symmetries): Ci jkl = C jikl = Ci jlk

(2.22)

thus the independent components are reduced to 36. The assumption of pathindependent work, i.e. the material recovers its initial state with no energy dissipation upon a complete loading/unloading cycle, implies the existence of an elastic strain energy density which is related to C by: Ci jlk =

∂ 2 Wel ∂εi j ∂εlk

(2.23)

which in turns implies the major symmetries Ci jlk = Clki j , that reduce the independent coefficients to 21. For isotropic solids, the independent coefficients can be further reduced to 2 [2]. A popular choice is to express the constitutive equation for

2.2 Elastic Waves in Homogeneous, Isotropic Solids

15

isotropic solids as a function of the two Lamè parameters λ [Pa] and μ [Pa]: σ = λtr(ε)I + 2με

(2.24)

where tr(·) is the trace operator, and I is the second order identity tensor. Another popular choice is to express the constitutive equation in terms of the bulk modulus B [Pa] and the shear modulus G = μ: σ = 3B

tr(ε) tr(ε) I + 2G(ε − I) = 3Bεiso + 2Gε dev 3 3

(2.25)

where the isotropic and deviatoric part of the deformation tensors are introduced. The conservation of momentum in absence of body forces for an infinitesimal volume element reads: ∂2u (2.26) ∇ ·σ =ρ 2 ∂t Note that, considering the definition of pressure as p = −tr(σ )/3, the conservation of momentum 2.7 for an ideal fluid can be seen as a particular case of Eq. 2.26, where the material has shear modulus equal to zero. The stress/strain relationship of an ideal fluid can be thus written as: σ = − p I = 3Bεiso = Bδ I

(2.27)

where δ = tr(ε) is the volumetric strain. Substituting the constitutive equation in the conservation of momentum, a vector wave equation is found for the displacements: ∇ · (C : ∇u) = ρ

∂2u ∂t 2

(2.28)

where the identity C : ε = C : ∇u that follows from the minor symmetries of C has been used. Considering the constitutive equation of an isotropic medium, the following is obtained: ∂2u ∇λ(∇ · u) + ∇μ · [∇u + (∇u)T ] + (λ + 2μ)∇(∇ · u) − μ∇ × ∇ × u = ρ 2 ∂t (2.29) which, under the hypothesis of homogeneous material properties’ distribution, reduces to: ∂2u (2.30) (λ + 2μ)∇(∇ · u) − μ∇ × ∇ × u = ρ 2 . ∂t This equation can be further simplified applying the Helmholtz decomposition to the displacement field: u = ∇φ + ∇ × ψ (2.31)

16

2 Wave Propagation in Periodic Media

where φ is a scalar potential and ψ is a divergence free vector potential. Upon substitution of the decomposition, and rearranging terms, the following is obtained:     ∂ 2φ ∂ 2ψ 2 2 ∇ (λ + 2μ)∇ φ − ρ 2 = ∇ × μ∇ ψ − ρ 2 , ∂t ∂t

(2.32)

a solution of which is found nullifying each term in the square brackets separately: (λ + 2μ)∇ 2 φ = ρ μ∇ 2 ψ = ρ

∂ 2ψ . ∂t 2

∂ 2φ ∂t 2

(2.33)

Two wave equations are thus obtained for the scalar and vector potentials. The introduction of the potentials allows to separate two contributions to the displacement field that have a clear physical meaning: considering that, since ∇ × ∇φ = 0

(2.34)

displacements associated to the scalar potential do not involve local rotations (∇ × u = 0), while since ∇ · (∇ × ψ) = 0 (2.35) displacements associated to the vector potential do not involve volume changes (∇ · u = δ = 0). The solution of the scalar wave equation that governs φ thus represents so-called compressional waves that travel at a speed:  cp =

λ + 2μ . ρ

(2.36)

ˆ iωt , a Helmholtz equation is obtained Assuming time harmonic dependency φ = φe that can be solved for plane harmonic waves: φ = A cos(ωt ± κ p · x) u = ∓κ p A sin(ωt ± κ p · x) ω |κ p | = κ p = . cp

(2.37)

Note that the displacement is purely in the direction of the wavevector, thus compressional waves are longitudinal waves. They are also called P waves from “primary”, since they are the faster and thus the first to be measured during heartquakes. On contrast, the solution of the vector wave equation for ψ represents the so called shear waves (or S-waves from “secondary waves”), that travel at the speed:

2.3 Elastic Waves in Homogenous, Anisotropic Media

 cs =

17

μ . ρ

(2.38)

Assuming harmonic dependency, the obtained Helmholtz equation provides the following solution: ψ = A cos(ωt ± κ s · x) u = ∓κ s × A sin(ωt ± κ s · x) (2.39) ω |κ p | = κ p = cp which shows that the associated displacements are orthogonal to the propagation direction indicated by κ s and thus are purely transverse waves. We underline the fact that the two types of waves have different speeds and thus when considering the same frequency, P-waves have longer wavelengths than S-waves. In unbounded domains the propagation of P and S waves is completely independent, while boundary conditions can couple them, and both types of waves are in general reflected when one hits a reflecting boundary [3]. Typical boundary conditions are fixed surfaces  D (Dirichlet boundary condition): u=0

x ∈ D

(2.40)

or traction free surfaces  N (Neumann boundary condition): t = (C : ∇u) · n = 0

x ∈ N

(2.41)

2.3 Elastic Waves in Homogenous, Anisotropic Media Several natural materials exhibit a mechanical response that depends on the considered direction of loading: a typical example is wood, which due to the orientation of natural fibers shows anisotropic properties. In the same way, single crystals as quartz (a material widely used, for example, in the watchmaking industry) that are made by the ordered repetition of unit cells at lattice points, show different levels of anisotropy depending on the distinct symmetries shown by the crystal structure [2]. In the picture of continuum mechanics, where infinitesimal material elements are assumed to be sufficiently small to be considered geometrical points, but sufficiently big to homogenize the atomic scale (or the scale of the biological wood cells), this behavior is encoded in the fact that the elasticity tensor has in general more than 2 independent coefficients. As mentioned, the symmetry elements of the point group of the crystal are the key to understand the associated structure of the elasticity tensor: indeed, if the crystal is mapped onto itself by a symmetry operation, the elasticity tensor should be mapped onto itself too upon the same symmetry operation (Neu-

18

2 Wave Propagation in Periodic Media

mann’s principle [2]). The set of symmetry operations characterizing a given crystal thus produces a set of constraints on the 21 independent coefficients of C, whose number is thus reduced. Let us consider again the vector wave equation of elastodynamics with homogeneous material properties: ∇ · (C:∇u) = ρ

∂2u ; ∂t 2

or

Ci jkl u k,l j = ρu i,tt

(2.42)

where in the component form summation is implied for repeated indices and comma is used to indicate partial derivation. Without making any assumption on the structure of C other than the minor and major symmetries, a plane harmonic wave solution in the form u = Re( Aeiωt−κ·x ) is substituted to obtain: κ · Cκ A = ω2 A; ρ

κi Ci jkl κl Ak = ω2 A j ρ

(2.43)

identifying the wavevector with κ = κn, the following can be written n · Cn A = c2 A; ρ

n i Ci jkl n l A j = c2 Ai ρ

(2.44)

where c = ω/κ is the propagation speed. This constitutes an eigenvalue problem in the form:  A = c2 A (2.45) ρ where the positive definite, symmetric second order tensor  is called acoustical or Christoffel tensor [4], whose components read: i j = n i Ci jkl n l ,

(2.46)

and allows computation of three wave speeds for assigned propagation direction n. For each eigenvalue speed, a corresponding eigenvector A is obtained, that is called polarization vector. It indicates the direction of the displacement vector associated to that particular wave. The locus of all wave speeds as a function of the three components of n is called the velocity surface, while the locus of the reciprocal of the speed is called slowness surface. Note that the velocity surface corresponds also to the dispersion diagram ω = ω(κ) for unitary wavevectors, and that rescaling by κ = |κ| the velocity surface the whole dispersion can be obtained. The slowness surface, instead, corresponds to the locus of all the possible wavevectors κ in the wavevector space when a unitary angular frequency is selected. Scaling the slowness surface by ω thus provides the isofrequency surfaces (contours in a two dimensional setting) in the wavevector space. Note thus that since the propagation speed only depends on n and not on ω or κ, all the information regarding the dispersion relation are contained in the velocity and slowness surfaces only.

2.3 Elastic Waves in Homogenous, Anisotropic Media

19

Example 2.1 Christoffel Tensor and Wave Speeds for Orthotropic Solids The case of orthotropic symmetry is of particular interest for this work, thus in the following the computation of the associated wave speeds and polarization vectors for selected propagation directions is detailed. Adopting Voigt’s notation, i.e. applying the following indices substitutions: 11 → 1; 22 → 2; 33 → 3 23 → 4; 13 → 5; 12 → 6

(2.47)

the constitutive equation of orthotropic solids can be written in compact matrix notation as: ⎡ ⎤ ⎡ ⎤⎡ ⎤ C11 C12 C13 0 0 0 ε1 σ1 ⎢σ2 ⎥ ⎢C12 C22 C23 0 0 0 ⎥ ⎢ ε2 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢σ3 ⎥ ⎢C13 C23 C33 0 0 0 ⎥ ⎢ ε3 ⎥ ⎢ ⎥=⎢ ⎥⎢ ⎥ (2.48) ⎢σ4 ⎥ ⎢ 0 0 0 C44 0 0 ⎥ ⎢2ε4 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣σ5 ⎦ ⎣ 0 0 0 0 C55 0 ⎦ ⎣2ε5 ⎦ σ6 2ε6 0 0 0 0 0 C66  T Propagation along x1 direction: for n = 1 0 0 , the Christoffel tensor reads: ⎡ ⎤ C11 0 0  x1 = ⎣ 0 C66 0 ⎦ (2.49) 0 0 C55 the velocities and associated polarizations are  c1 =

 C11 ρ

c2 =

A1 = [1, 0, 0]T

 C66 ρ

c3 =

A2 = [0, 1, 0]T

C55 ρ

(2.50)

A3 = [0, 0, 1]T

Thus a longitudinal wave is obtained and two transverse waves with different speeds.  T Propagation along x2 direction: for n = 0 1 0 :  x2

⎡ ⎤ C66 0 0 = ⎣ 0 C22 0 ⎦ 0 0 C44

(2.51)

20

2 Wave Propagation in Periodic Media

the velocities and associated polarizations are  c1 =

 C66 ρ

c2 =

A1 = [1, 0, 0]T

 C22 ρ

c3 =

A2 = [0, 1, 0]T

C44 ρ

(2.52)

A3 = [0, 0, 1]T

Also in this case a purely longitudinal wave is obtained and two purely transverse waves.  √ √ T Propagation at 45◦ : for n = 1/ 2 1/ 2 0 :  45

⎤ ⎡ C + C66 C12 + C66 0 1 ⎣ 11 ⎦ 0 C12 + C66 C22 + C66 = 2 0 0 C44 + C55

(2.53)

For the evaluation of the eigenvectors consider a generic 3 × 3 matrix in the form: ⎡ ⎤ m 11 m 12 0 [M] = ⎣m 12 m 22 0 ⎦ (2.54) 0 0 m 33 the eigenvalues are:    1 m 11 + m 22 − (m 11 − m 22 )2 + 4m 212 2    1 m 11 + m 22 + (m 11 − m 22 )2 + 4m 212 ρc22 = 2

ρc12 =

(2.55)

ρc32 = m 33 while the associated eigenvectors read:   T   1 −m 11 + m 22 + (m 11 − m 22 )2 + 4m 212 , 1, 0 A1 = − 2m 12      1 −m 11 + m 22 − (m 11 − m 22 )2 + 4m 212 , 1, 0 A2 = − 2m 12 A3 = [0, 0, 1] (2.56) Thus one wave is purely transverse with out of plane polarization, while the two wave propagating with in plane displacements are not completely longitudinal neither transverse. The faster is referred to as the quasi-P, the slower as the quasi-S.

2.4 Wave Propagation in Sonic Crystals

21

Note that a cubic crystal can be seen as a special case where C11 = C22 = C33 , and C44 = C55 = C66 , while for the hexagonal symmetry C11 = C22 = C33 , C44 = C55 = C66 , C66 = (C11 − C12 )/2.

2.4 Wave Propagation in Sonic Crystals As opposed to natural crystals, we refer to artificial crystals to indicate those periodic arrangements of materials that are man-made by repeating unit blocks whose size is usually orders of magnitude bigger than the atomic scale. Among these crystals, those that are designed to work for pressure waves in fluids are referred to as sonic crystals [5]. This definition would in principle imply all material constituents of a sonic crystal to be fluid (a typical example could be a periodic arrangement of air bubbles in water). Nonetheless, due to the practical difficulty of producing such periodic arrangements, a sonic crystal is defined by extension also in cases where solid inclusions are considered in a fluid matrix. In that case, no shear waves can propagate through the hosting medium, that thus effectively carries only longitudinal waves. By consequence, the modeling of sonic crystals implies, in the most accurate picture, the use of coupled acousto-elastic equations. If the coupling of longitudinal acoustic waves in the fluid matrix with the shear waves in the solid can be neglected, the solid inclusion can be substituted with an equivalent acoustic fluid exhibiting the same longitudinal wave behavior. More than that, when the impedance of the longitudinal mode in the solid is so high that the inclusion can be considered infinitely rigid, the sonic crystal can be simply modeled with a periodic repetition of hard-walled boundary conditions. Assuming harmonic time dependence, the governing equation that will be considered is thus the Helmholtz equation with periodic coefficients:  B0 (x)∇ ·

 1 ∇ pˆ = −ω2 pˆ ρ0 (x)

(2.57)

where ρ0 (x + d) = ρ0 (x) and B0 (x + d) = B0 (x) for arbitrary vectors d in the set of lattice vectors (see Example 2.2 below). Note that this constitutes an eigenvalue problem: L( p) ˆ = λ pˆ (2.58) for the operator L = −B0 (x)∇ · (ρ0 (x)−1 ∇), with eigenvalue λ = ω2 . Due to the properties of the L operator owing to the periodicity of the crystal [6], the BlochFloquet theorem implies that acceptable solutions are in the form of a Bloch waves [7]: −iκ·x (2.59) pˆ = p(x)e ˜ where p(x) ˜ is a periodic function with the same periodicity of the lattice, i.e.

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2 Wave Propagation in Periodic Media

p(x ˜ + d) = p(x). ˜

(2.60)

Substituting the Bloch wave solution in the differential governing equation, all the gradient operators are substituted by: ∇ pˆ = (∇ − iκ) pe ˜ −iκ·x 

to obtain: − B0 (x)(∇ − iκ) ·

 1 (∇ − iκ) p˜ = ω2 p˜ ρ0 (x)

(2.61)

(2.62)

which now represents an eigenvalue problem for the operator L (κ) = −B0 (x)(∇ − iκ) · (ρ0 (x)−1 (∇ − iκ)), that depends on κ. Solving the eigenvalue problem for assigned values of κ provides all the possible (ω, κ) pairs that are admissible for Bloch waves to exist in the unbounded infinite crystal. Different methods can be used for the solution of this problem, that are reviewed, for example, in [5, 8]. In this work, the Finite Element Method will be extensively used to solve for the dispersion relation of artificial crystals, thus it will be briefly recalled in the following. Starting again by the Helmholtz equation  ∇·

 1 1 ∇ pˆ = − ω2 pˆ ρ0 (x) B0 (x)

(2.63)

we multiply both sides by a test function qˆ and integrate over the domain of a unit cell :        1 1 ∇· ∇ pˆ qd ˆ = − ω2 pˆ q ˆ (2.64) ρ0 (x) B0 (x)



using the divergence theorem    qˆ

      1 1 ∇ pˆ · d S − ∇ pˆ · ∇ qˆ d ρ0 (x) ρ0 (x) ∂    1 pˆ qd , ˆ = − ω2 B0 (x)

(2.65)



recalling that pˆ = pe ˜ iκ·x and choosing a test function in the form qˆ = qe ˜ iκ·x , with q(x ˜ + d) = q(x): ˜    q˜

      1 1 (∇ − iκ) p˜ · d S − (∇ − iκ) p˜ · (∇ − iκ)q˜ d ρ0 (x) ρ0 (x) ∂    1 = − ω2 p˜ qd . ˜ B0 (x)

(2.66)

2.4 Wave Propagation in Sonic Crystals

23

The surface integral can be split into a contribution on the external borders of the unit cell plus a contribution on the rigid boundaries adopted in those cases where solid inclusions are modeled as infinitely rigid obstacles. Both of these contributions reduce to zero: the latter is null by virtue of the Neumann boundary condition on rigid surfaces, while the former is because of the periodicity of p, ˜ q. ˜ This leads to:  

     1 1 2 (∇ − iκ) p˜ · (∇ − iκ)q˜ d − ω p˜ qd ˜ = 0 (2.67) ρ0 (x) B0 (x)

which is the weak formulation of Eq. 2.62. Dividing the domain of the unit cell in finite elements, and choosing appropriate finite element basis functions Ni (x) such that the test function and the trial solution are written as:   q˜ = Ni (x)Q i (2.68) p˜ = Ni (x)Pi after invoking the arbitrarity of the test function, a linear eigenvalue problem is obtained in the form:   (2.69) [K (κ)] − ω2 [M] P = 0 where P is a vector collecting the coefficients Pi . The characteristic equation:   det [K (κ)] − ω2 [M] = 0

(2.70)

allows to plot the dispersion relation, by imposing real valued κ and finding the associated ω values. Note that, from Eqs. 2.59 and 2.60 it follows that: −iκ·d , p(x ˆ + d) = p(x)e ˆ

(2.71)

from which it can be seen that κ represents a phase difference between points separated by a lattice vector. It is also implied that the quantity κ · d is only defined modulo 2π , which means that replacing κ with κ + i n i bi , with n i arbitrary integer numbers and bi · d j = 2π δi j ;

δi j = 1 if i = j, 0 otherwise

(2.72)

will not change the result of Eq. 2.70, and will lead to the same pressure field [7]. The dispersion relation ω = ω(κ) computed with Eq. 2.70 is thus periodic in the wavevector space with lattice points defined by vectors bi , which are thus called inverse lattice vectors. From this fact it also follows that it is sufficient to compute the eigenvalues of Eq. 2.70 only in one unit of the reciprocal lattice, to get all the required information regarding the ω = ω(κ) relation. Such unit cell is taken to be the set of wavevectors closer to the null wavevector than to any other lattice point in the wavevector space. This is called the First Brillouin zone [7], and corresponds to the Wigner-Seitz unit cell of the inverse lattice (see Example 2.2 below). Further

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2 Wave Propagation in Periodic Media

considerations on the symmetry of the problem can further reduce the computational burden, e.g. realizing that substituting κ with −κ produces the same solution for the eigenvalue problem, and represents the same Bloch wave with wavefronts propagating in the direction identified by −κ instead of κ. The smallest set in the wavevector space that allows the computation of the whole dispersion exploiting symmetries is called the Irreducible Brillouin Zone. Example 2.2 Inverse Lattice and First Brillouin Zone for the Centered Rectangular Lattice Figure 2.1 represents a centered rectangular lattice of points. Such arrangement of points can be obtained tessellating the space with elemental elements called unit cells, that repeat equal to itself at locations that can be found by linear combinations of a set of vectors d i that are called direct lattice vectors (i.e. the lattice is invariant upon the translation operations represented by such vectors). As Fig. 2.1 shows, the choice of the unit cell and the associated direct lattice vectors is not unique. The Wigner-Seitz unit cell, that is represented in Fig. 2.1b, is the one that comprises all the geometrical points that are closer to one lattice point than any other lattice point, and can be obtained as the volume (area in 2D) enclosed by the bisector planes of the segments joining one lattice point to the neighbors. The Wigner-Seitz unit cells is also a primitive unit cell, in that it has the smallest possible volume a unit cell can have for the considered lattice. The set of wavevectors that are included in the Fourier expansion of the material properties of an artificial crystal form a lattice of points in the wavevector space, which is called the inverse lattice. The Wigner-Seitz unit cell of such lattice is called First Brillouin Zone.

Fig. 2.1 Different choices for the unit cell of a centered rectangular lattice

As a matter of example, let us build the first Brillouin zone of the centered rectangular lattice, considering the primitive unit cell depicted in Fig. 2.1b. The inverse lattice vectors are found from Eq. 2.72. In practice, if the direct lattice vectors are expressed with respect to a Cartesian basis, Eq. 2.72 can be rewritten as: (2.73) [B]T = 2π [D]−1 where [D] is a matrix collecting the direct lattice vectors components as columns, and [B] contains the inverse lattice vectors as columns. In the case at hand (Fig. 2.1b):

2.4 Wave Propagation in Sonic Crystals

25

Fig. 2.2 First Brillouin Zone of the centered rectangular lattice as shown in Fig. 2.1b. The Irreducible Brillouin Zone is highlighted in yellow



d cos(γ ) d cos γ [D] = −d sin γ d sin γ

 (2.74)

where d is the length of direct lattice vectors. This leads to [B]T = and finally:

  π d sin(γ ) −d cos γ d 2 cos γ sin γ d sin γ d cos γ

  1 π 1 ex − ey d cos γ sin γ   1 π 1 ex + ey b2 = d cos γ sin γ

(2.75)

b1 =

(2.76)

with ex and e y the basis unit vectors. From the knowledge of the bi the location of the inverse lattice points in the wavevector space can be retrieved, and the First Brillouin Zone can be obtained graphically as depicted in Fig. 2.2. As a matter of example, in Fig. 2.3a a square lattice of rigid inclusions separated by a distance a is depicted. In Fig. 2.3b it is shown a simple mesh discretizing the fluid domain, which is then used to compute the real branches of the dispersion relation with the Finite Element Method, implemented in the commercial software Comsol Multiphysics. The result is depicted in Fig. 2.3c, for wavevectors pointing in the x direction as depicted in Fig. 2.3b. Black dots represent the numerical results, expressed as normalized frequency f (a/c0 ), with f = ω/(2π ) and c0 indicating the wave speed of propagation in the hosting fluid. Note that, exploiting the mirror symmetry with respect to the origin of the wavevector space, the computation is carried for a set of wavevectors from the null wavevector up to κ = π/a only. Comparing

26

2 Wave Propagation in Periodic Media

the obtained dispersion relation with that of a homogeneous fluid, it can be seen how there are frequency ranges in which no solution is found, i.e. those frequency values are not associated to any real wavevector. These frequency ranges are called bandgaps or stop bands: it can be shown that in those frequency ranges a Bloch wave solution exists only if the wavevector is a complex number [9]. A purely imaginaly wavenumber κ = −iα corresponds to so-called evanescent waves, i.e. Bloch solutions that cannot propagate freely inside the artificial crystal, but are characterized by attenuation in space according to: ˜ −αx pˆ = pe ˜ −iκ x = pe

(2.77)

Although there can be other phenomena leading to the opening of a bandgap [10], the most common ones are: • Bragg Scattering: bandgaps appear every time the distance between the inclusions is a multiple of half the wavelength of the Bloch wave. Intuitively, in this case the scattering occurring at each row of inclusions produces a destructive interference with the propagating wave, preventing the field to propagate freely along the crystal [11]. • Local Resonance: If the unit cell comprises local resonators, at the resonance frequency the badgap is opened by the interaction between the resonance and the propagating mode [12, 13]. The results obtained for the sonic crystal depicted in Fig. 2.3 show bandgaps occurring for both the mechanisms. Indeed, the C-shape of the inclusion creates a cavity connected with the hosting fluid by a tiny neck, i.e. the typical geometry of a Helmholtz resonator [1]. The first bandgap opens thus slightly above f a/c0 = 0.2 at the resonance frequency of the Helmholtz resonator. On contrast, the gap opening with normalized central frequency ≈ 0.5 is related to Bragg scattering. This geometry is well known as a prototype of sonic crystal (more can be found on this crystal in [14]) and is directly inspired from the split ring resonator, that was originally used in electromagnetism [6, 15]. In Fig. 2.4 it is represented the response of a finite arrangement of 10 unit cells in the direction of propagation, when a plane wave traveling in a semi-infinite homogeneous fluid domain impinges on the surface separating it from the sonic crystal. In Fig. 2.4a it is shown what happens when the frequency of the incoming wave corresponds to f a/c0 = 0.26, i.e. inside the locally resonant bandgap. It can be seen that the pressure field is mainly localized inside the resonators and decreases exponentially from the interface between the crystal and the homogeneous fluid domain to which is coupled. In Fig. 2.4b, instead, it is depicted the response for f a/c0 = 0.51, i.e. inside the Bragg scattering bandgap. The field inside the crystal shows a wavelength equal to two times the length of a cell, and shows the typical spatial exponential decay. One last interesting feature of the dispersion graph depicted in Fig. 2.4 is that the phase velocity c ph (κ) = ω(κ)/κ is not constant. In other words, the relationship between ω and κ is not linear as happens for homogeneous materials. This implies that for a wave packet traveling inside a sonic crystal, each Bloch wave will propagate

2.4 Wave Propagation in Sonic Crystals

27

Fig. 2.3 a Scheme of a square lattice of Helmholtz resonators. The length of the side of the unit cell is indicated by a. b Simple example of discretization of the unit cell. Solid inclusions are modeled as acoustically rigid boundaries. c Graph of the real dispersion branches, for wavevector κ aligned with the side of the square unit cell, f [Hz] indicates the frequency, while c0 the wave propagation speed in the background fluid

Fig. 2.4 a A plane wave incident on the sonic crystal represented in Fig. 2.3 at f a/c0 = 0.26. b Same plane wave but now characterized by f a/c0 = 0.51

with a different speed. The resulting field will change its shape as it travels along the crystal, according to the superposition of the individual Bloch modes. This is called dispersive behavior, as opposed to the non-dispersive behavior characterizing homogeneous isotropic media, where the shape of the field does not change as it propagates, by virtue of the fact that individual monocromatic waves all have the same propagation speed. The group velocity vector: cg =

∂ω(κ) ei ∂κi

(2.78)

with κi the components of the wavevector according to the Cartesian basis defined by the unit vectors ei , indicates the speed of propagation of the wavepacket with central frequency ω(κ), and is aligned with the intensity vector I = pv [W/m2 ] (Poynting vector) [5, 7]. This means that, while the wavefronts propagate at the phase speed in the direction identified by the wavevector, the energy is propagated along the direction of the group velocity (which can be, in general, not aligned with the wavevector).

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2.5 Wave Propagation in Phononic Crystals In analogy with what previously introduced for sonic crystals, phononic crystals are artificial crystals that work for elastic waves in solids, or elastic waves propagating in a solid matrix with fluid inclusions (note that not all authors recognize the difference between phononic crystals and sonic crystal, and sometimes the term phononic crystal is used loosely to indicate both). Similarly to what done for the case of sonic crystals, we can consider at first approximation a fluid inclusion to be a solid inclusion with zero shear (i.e. a solid without transverse waves), or even a simple traction free boundary condition, when the impedance mismatch is such high that the fluid inclusion can be replaced by void. The governing equations that are thus considered are the elastodynamic equations with inhomogeneous periodic coefficients, i.e.: 1 ˆ = −ω2 uˆ ∇ · (C(x) : ∇ u) ρ(x)

(2.79)

ˆ iωt , and: with u = ue ρ(x + d) = ρ(x);

C(x + d) = C(x)

(2.80)

As done in the previous chapter, this can be recasted as an eigenvalue problem: ˆ = λuˆ L(u)

(2.81)

with λ = −ω2 and L = ρ −1 (x)(∇ · (C(x) : ∇)), that admits solutions in the form of Bloch waves: −iκ·x ˜ , (2.82) uˆ = u(x)e that once substituted into the differential equation produce the following: 1 ˜ = −ω2 u. ˜ (∇ − iκ⊗) · (C(x) : (∇ − iκ⊗) u) ρ(x)

(2.83)

The weak formulation is obtained integrating Eq. 2.79 over the domain of a unit ˆ cell, after scalar multiplication by a test function q: 

  ˆ · qˆ d = − ∇ · (C(x) : ∇ u)





  ρ(x)ω2 uˆ · qˆ d

(2.84)



and adopting the divergence theorem:  ∂

  ˆ · dS = qˆ · (C(x) : ∇ u)



  ˆ d − ∇ qˆ : (C(x) : ∇ u)



  ρ(x)ω2 uˆ · qˆ d . (2.85)

2.5 Wave Propagation in Phononic Crystals

29

If qˆ is interpreted as a virtual displacement, the left hand side is the virtual work of external forces on the boundary of the domain, while the right hand side represents the virtual work of internal elastic forces and of the inertia forces. Choosing the test ˜ + d) = q(x), ˜ ˜ iκ·x , with q(x the left hand side reduces to zero function to be qˆ = qe by virtue either of the periodicity of the solution on the outer boundary of the unit cell, or on the Neumann boundary condition on free traction surfaces. The following is thus obtained:       2 ˜ d − ω (∇ − iκ⊗)q˜ : (C(x) : (∇ − iκ⊗)u) ρ(x)u˜ · q˜ d = 0.



(2.86) Upon discretization of the domain in Finite Elements and expressing both u˜ and q˜ as a combination of finite element’s basis functions:   q˜ = Ni (x) Q i (2.87) u˜ = Ni (x)U i the following is obtained:   [K (κ)] − ω2 [M] U¯ = 0

(2.88)

with the vector U¯ collecting all the vectors U i of coefficients. For illustrative purposes, the dispersion relations of acoustic/elastic waves in isotropic homogeneous fluids/solids are depicted in Fig. 2.5, along with a schematic representation of the dispersion relation of sonic and phononic crystals. This allows to summarize and compare the characteristic features of each different case. In Fig. 2.5a the dispersion relation of acoustic waves in homogeneous isotropic fluids is reported for comparison with Fig. 2.5b, where the real dispersion branches of a prototypical sonic crystal are schematically depicted along a propagation direction. As mentioned in the previous section, while the dispersion of the homogeneous fluid is linear (nondispersive) with slope equal to the wave speed, the branches of the sonic crystal are non-linear (dispersive) and periodic, with Bragg bandgaps opening each time the size of the unit cell is a multiple of half the wavelength. Note that the branch that emanates from the origin has a linear behavior for κ → 0, i.e. when the wavelength is much bigger than the size of the unit cell (long-wavelength limit) the material behaves at first order as an equivalent homogeneous material. In that range, the phase velocity corresponds to the slope of the linear part of the dispersion of the sonic crystal. From the knowledge of such propagation speed, equivalent material properties can be found for a fictitious homogeneous material showing the same dynamic behavior of the inhomogeneous material distribution that constitutes the sonic crystal. This process is known as long-wavelength homogenization, and will be detailed formally in the next section. Note that continuum mechanics by itself can be seen as the result of a long-wavelength homogenization, in that no material is actually homogeneous. The basic hypothesis in deriving governing equations indeed is that infinitesimal elements are sufficiently small to be considered to occupy geometrical points, but

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Fig. 2.5 a Dispersion relation for acoustic waves in homogeneous isotropic fluids. b Schematic representation of the dispersion relation of a sonic crystal showing Bragg gaps, computed along one propagation direction. c Dispersion relation of longitudinal and transverse waves in isotropic homogeneous elastic solids. d Schematic representation of the dispersion relation of a phononic crystal (plane strain problem) showing Bragg gaps, computed along one propagation direction

sufficiently big to contain hundreds of inhomogeneities (atoms, molecules, unit cells, natural fibers, grains...) whose actual local behavior is averaged. The linear dispersion of (what we consider to be) homogeneous solids is obtained just because the scale of such inhomogeneities is normally orders of magnitude smaller than the smallest wavelength considered. In Fig. 2.5c it is depicted the dispersion relation for a homogeneous isotropic elastic solid. In this case, owing to the vectorial nature of the displacement field, as opposed to the scalar pressure field of acoustics, three branches can be distinguished emanating from the origin (three rigid body motions are indeed allowed when ω = 0). They correspond to the longitudinal compressional waves having slope c p and the two (superimposed) transverse modes having slope cs . For phononic crystals in the long-wavelength limit, three linear branches emanating from the origin are similarly expected. Note that, owing to the anisotropy of the lattice, there is in principle no reason why two of them should have the same slope, as happens for isotropic homogeneous solids. In Fig. 2.5d, a schematic representation of the real dispersion branches of a prototypical phononic crystal is depicted. To keep the drawing as simple as possible, the plain strain hypothesis is introduced without loss of generality in the considerations that will follow. Such hypothesis reduces the non null components of the displacement vector to two, with the consequence of reducing the number of branches emanating from the origin to two also. These modes can be identified as quasi-longitudinal and quasi-transverse, based on the polarization being

2.6 Long-Wavelength Homogenization

31

predominantly longitudinal or transverse. Owing to the different slopes at the origin, Bragg gaps will open at distinct frequencies when considering each different mode. In Fig. 2.5 a light gray area highlights the opening of the first gap for the slower quasi-transverse mode, while in dark gray it is highlighted the frequency band corresponding to forbidden propagation of the quasi-longitudinal mode. Note that part of this gap overlaps with the second light gray gap, thus creating a frequency range (dashed in the Figure) where no waves can propagate at all. This type of gaps will be referred to as full bandgaps, as opposed to the single mode bandgaps, as the first light gray one, where only one mode (in this case the quasi-longitudinal), can propagate.

2.6 Long-Wavelength Homogenization In the context of the practical implementation of cloaking devices, artificial crystals can be used as building blocks to produce the so-called metamaterials, i.e., as previously mentioned, composites with effective medium properties that go beyond those of their constituent materials. In general, it is thus of interest to develop a method to retrieve these effective medium properties and, in doing so, produce a more rigorous definition of them. The problem setting is the one depicted in Fig. 2.6: a body is composed by a repetition of unit cells whose geometry is assumed, without loss of generality, to be a cube with side . The intuitive idea is that, if the characteristic length of the unit cells (microscale) is much smaller than any other characteristic length of our problem (macroscale), the spatial dependency of the fields solving the governing differential equations with inhomogeneous material properties can be separated into “fast” variations at the microscale level and “slow” variations at the macroscale level. In general, the slow variation can be the result of slow spatial gradients in

Fig. 2.6 Schematic representation of a microstructured body, along with the representation of the two sets of spatial variables x and y = x/ used in the two-scale asymptotic expansion approach for homogenization of locally periodic microstructures

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2 Wave Propagation in Periodic Media

the geometry/composition of the unit cells and/or by the external boundary conditions/initial conditions. From a mathematical point of view this can be expressed by defining a local variable y = x/, such that the material properties can be expressed as (referring to the elastodynamic problem): C = C(x, y) ρ = ρ(x, y)

(2.89)

where in the limit for  → 0 (separation of scales) for each choice of x, that is treated as a parameter, C and ρ are locally periodic in the variable y, and the fields can be expressed as: u = u(x, y, t) ε = ε(x, y, t) (2.90) σ = σ (x, y, t) The idea is that, as  → 0, one is progressively less interested in the local fast variations of the fields, that can be smoothed out by local averaging on the unit cell , i.e.:  1 < u > (x, t) = u(x, y, t)d | |  1 < ε > (x, t) = ε(x, y, t)d (2.91) | |  1 < σ > (x, t) = σ (x, y, t)d | |

where x is again seen as a parameter. The goal of homogenization is to find governing equations for the averaged (in following also called homogenized or macro) variables. In general, the resulting equations will show at each x location the same mathematical structure of the governing equations of a homogeneous material: this allows to compute the constitutive equation of a fictitious equivalent material that can replace locally the considered portion of microstructure. If the original material properties do not have any explicit dependency on x, than it follows that the overall body can be replaced by the equivalent homogeneous material computed at one single point x (this is the case of perfectly periodic phononic crystals extending to infinite), otherwise, interpolating the results at successive locations x will produce a slowly varying homogenized material distribution. Let us now consider in detail the problem of homogenizing the material properties of a perfectly periodic phononic crystal. The governing equations are (harmonic time dependence implied):

2.6 Long-Wavelength Homogenization

33

⎧ 2 ⎪ ˆ y) ⎨∇ · σˆ (x, y) = −ρ( y)ω u(x, σˆ (x, y) = C( y) : εˆ (x, y) ⎪ ⎩ ˆ y) + ∇ uˆ T (x, y)] εˆ (x, y) = 21 [∇ u(x,

(2.92)

i.e. the momentum conservation equation, the constitutive equation and the compatibility equation. Note that, as mentioned before, due to the perfect periodicity of the material (thought infinite in extension) the material properties do not explicitly depend on the macroscale variable. It is known that such a problem admits solutions in the form of Bloch waves: ⎧ ⎪ ˆ ˜ y)e−iκ·x y) = u( ⎨u(x, (2.93) σˆ (x, y) = σ˜ ( y)e−iκ·x ⎪ ⎩ εˆ (x, y) = ε˜ ( y)e−iκ·x ˜ y), σ˜ ( y) and ε˜ ( y) variables are periodic functions with the same periodwhere the u( icity of the lattice. It is thus understood that the characteristic length of the macroscale “slow” spatial variation of the solution is set in this problem by the wavevector κ. Plugging the Bloch wave ansatz into the governing equations, the following is obtained: ⎧ 2 ⎪ ˜ y) ⎨(∇ − iκ) · σ˜ ( y) = −ρ( y)ω u( (2.94) σ˜ ( y) = C( y) : ε˜ ( y) ⎪ ⎩ ˜ y) + [(∇ − iκ⊗)u( ˜ y)]T } ε˜ ( y) = 21 {(∇ − iκ⊗)u( Let us build a family of ansatz solutions for the fields as a perturbation expansion of powers of : ⎧ ⎪ ˜ y) = u˜ 0 ( y) +  u˜ 1 ( y) +  2 u˜ 2 ( y) + ... ⎨u( (2.95) σ˜ ( y) = σ˜ 0 ( y) +  σ˜ 1 ( y) +  2 σ˜ 2 ( y) + ... ⎪ ⎩ 2 ε˜ ( y) = ε˜ 0 ( y) +  ε˜ 1 ( y) +  ε˜ 2 ( y) + ... with every function in the summation being a periodic function with the same periodicity of the lattice. We hope that in the limit for  → 0 (scale separation), we can find equations governing the behavior of the averaged leading terms, thus providing the homogenized equations. Note that in this context the scale separation implies in practical terms that the wavelength λ = 2π/|κ|, which sets the characteristic length of the macroscale, is much bigger than the side of the unit cell (thus the name of "long wavelength" homogenization, or “quasi-static” limit). For every choice of κ, a set of ω j (κ) satisfy the governing equations along with the corresponding fields. Each ω j (κ) is expanded in power series too: j

j

j

ω j (κ) = ω0 (κ) + ω1 (κ) +  2 ω2 (κ) + ...

(2.96)

Plugging the expansion into the compatibility equations, and equating terms corresponding to the −1st and 0th powers of  the following are obtained:

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2 Wave Propagation in Periodic Media

1 [∇ y u˜ 0 ( y) + ∇ y u˜ 0T ( y)] = 0 2 1 ε˜ 0 ( y) = {∇ y u˜ 1 ( y) − iκ ⊗ u˜ 0 ( y) + [∇ y u˜ 1 ( y) − iκ ⊗ u˜ 0 ( y)]T } 2

(2.97)

where it has been recognized that for a function not explicitly depending on x, it follows that ∇ = ∇ y /. The first equation implies that u˜ 0 is a linear function of y. Considering that by definition it should also be a periodic function, it follows that u˜ 0 actually does not depend on y: u˜ 0 = constant =< u˜ 0 >

(2.98)

This implies that in the limit for  → 0, the leading term produces a displacement field in the form of a plane harmonic wave with constant amplitude. Averaging the second equation over the unit cell and considering Eq. 2.98, the following is obtained: i < ε˜ 0 >= − (κ⊗ < u˜ 0 > + < u˜ 0 > ⊗κ) 2

(2.99)

where it has also been exploited the fact that the average over a period of the gradient of a periodic function is null. Taking the −1st order of the momentum conservation equation one obtains: (2.100) ∇ y · σ˜ 0 = 0 which combined with the 0th order constitutive equation σ˜ 0 ( y) = C : ε˜ 0 ( y) and Eq. 2.972 , gives:     1 T ∇ y u˜ 1 ( y) − iκ⊗ < u˜ 0 > +[∇ y u˜ 1 ( y) − iκ⊗ < u˜ 0 >] =0 ∇ y · C( y) : 2 (2.101) that can be rearranged into:   ∇ y · C( y) : ∇ ys u˜ 1 ( y) = −∇ y · (C( y) :< ε˜ 0 >)

(2.102)

where ∇ ys u˜ 1 ( y) indicates the symmetric part of the gradient of u˜ 1 ( y). This equation can be seen as a differential equation for u˜ 1 relating it to < ε˜ 0 >, and has the same structure of an elastostatic problem for u˜ 1 with periodic boundary conditions and in presence of inelastic strains (like in thermal expansion problems). In particular, the imposed inelastic strain is the macroscopic strain < ε˜ 0 > (changed in sign). Performing the average of the 0th order constitutive equation, the following is obtained: ⎛ ⎞     1 1 < σ˜ 0 >= ⎝ C( y) : ∇ ys u˜ 1 ( y) d . (2.103) C( y)d ⎠ :< ε˜ 0 > + | | | |



2.6 Long-Wavelength Homogenization

35

Since the symmetric part of the gradient of u˜ 1 depends linearly on < ε˜ 0 > through Eq. 2.102, both terms on the right hand side of Eq. 2.103 are linear combinations of the elements of < ε˜ 0 >. This means that a linear relationship between < σ˜ 0 > and < ε˜ 0 > is sought: (2.104) < σ˜ 0 >= Chom :< ε˜ 0 > Having obtained such linear relationship, we can interpret it as a linear elastic constitutive equation with Chom being the homogenized elasticity tensor (independent of y) relating the averaged leading terms of the expansion. Finally, considering the 0th order momentum conservation equation, the following is obtained: j,2 (2.105) − ρ( y)ω0 (κ) < u˜ 0 >= ∇ y · σ˜ 1 − iκ · σ˜ 0 , that averaged over the unit cell provides: ⎛ ⎝ 1 | |



⎞

  j,2 ρ( y)d ⎠ ω0 (κ) < u˜ 0 >= iκ · Chom :< ε˜ 0 > .

(2.106)



Upon substitution of the explicit definition of < ε˜ 0 > (Eq. 2.99): ⎛ ⎝ 1 | |



⎞

  j,2 ρ( y)d ⎠ ω0 (κ) < u˜ 0 >= κ · Chom · κ < u˜ 0 >

(2.107)



Expressing the wavevector as κ = κn, this can be interpreted as the Christoffel equation for plane waves in a general anisotropic linearly elastic medium: c2j < u˜ 0 >=

(n · Chom · n) < u˜ 0 > ρ hom

(2.108)

! j 1 with ρ hom = | | ρ( y)d the homogenized density and c j = ω0 /κ the propagation speed of the leading term plane wave. If the dependence of u˜ 1 ( y) on < ε˜ 0 > can be explicitly found from Eq. 2.102 and substituted into Eq. 2.103, than the homogenized elasticity tensor can be computed, and Eq. 2.108 can be used to obtain the dispersion relation of the phononic crystal in the long wavelength limit (the one depicted in orange in Fig. 2.5d), without actually computing the full dispersion relation as seen in the previous section. For this purpose, a set of elementary cell problems can be solved in the form:     ∇ y · C( y) : ∇ ys wi j ( y) = ∇ y · C( y) : ei j

(2.109)

36

2 Wave Propagation in Periodic Media

where wi j is the -periodic displacement resulting from solving an elastostatic problem with an applied thermal expansion in the form of one of the elements of the canonical basis of the symmetric second order tensors: ei j =

1 (ei ⊗ e j + ei ⊗ e j ). 2

(2.110)

Since the homogenized strain can be written as: < ε˜ 0 >=



ε˜ i j ei j

(2.111)

i j,

and by virtue of the Eq. 2.102 (that, as mentioned, represents an elastostatic problem with a thermal strain applied equal to − < ε˜ 0 >), the first order displacement can be written as a linear combination of the wi j ( y): u˜ 1 = −



ε˜ i j wi j ( y)

(2.112)

i, j

which substituted into Eq. 2.103, provides the formula for the components of the homogenized elasticity tensor: Cihom jkl

1 = | |

 "  #  Ci jkl ( y) − C( y) : ∇ ys wkl i j d .

(2.113)



On contrast, in this work, Eq. 2.108 is instead used to compute the elements of the homogenized elasticity tensor (thus sidestepping the solution of the elastostatic problems of Eq. 2.109 and the use of Eq. 2.113), after computation of the homogenized density by spatial average of ρ( y) and after evaluation of the slope of the dispersion branches in the long wavelength limit, computed through the Finite Element method (Eq. 2.88). Before concluding, we note that the method just presented fails when there are parts of the unit cell that are clamped or connected to the ground by spring foundations. In those cases, indeed, the presence of such boundary conditions prevents the occurrence of rigid body motions, thus eliminating the branches emanating from the origin, leading to a long-wavelength behavior characterized by evanescent waves only [16]. In such cases, the present derivation with asymptotic expansion can be slightly modified to produce effective material properties beyond the quasi-static limit, as shown in [17, 18].

2.7 Further Readings

37

2.7 Further Readings The interested reader can find further details on acoustic waves in homogeneous and inhomogeneous fluids in the book by Ginsberg [1], while classical textbooks like Graff’s [3] and Musgrave’s [2] ones provide detailed fundamental knowledge on wave propagation in elastic solids. The part of this chapter devoted to artificial crystals (both sonic and phononic crystals) is largely inspired by the books by Laude [5] and Banerjee [6]. Long-wavelength homogenization with two scale asymptotic expansion, was firstly made popular in the context of periodic media by the seminal monograph by Bensoussan, Lions and Papanicolau [19]. Among the vast literature on the subject, the content of this Chapter has been inspired also by the book by Milton [20] and the review by Hassani and Hinton [21, 22]. The interested reader can find more on homogenization schemes that work also for non periodic media in [23] and [24], where H and G convergence are treated.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Ginsberg JH (2018) Acoustics: a textbook for engineers and physicists, vols 1 and 2. Springer Musgrave MJ (2003) Crystal acoustics. Acoustical Society of America, New York Graff KF (2012) Wave motion in elastic solids. Courier Corporation Christoffel EB (1877) Annali Matematica Pura Applicata (1867-1897) 8(1):193 Laude V (2015) Phononic crystals: artificial crystals for sonic, acoustic, and elastic waves, vol 26. Walter de Gruyter GmbH & Co KG Banerjee B (2011) An introduction to metamaterials and waves in composites. Taylor and Francis Brillouin L (1953) Wave propagation in periodic structures: electric filters and crystal lattices, vol 2. Dover Publications Hussein MI, Leamy MJ, Ruzzene M (2014) Appl Mech Rev 66(4) Mokhtari AA, Lu Y, Srivastava A (2019) J Mech Phys Solids 131:167 Yilmaz C, Hulbert GM, Kikuchi N (2007) Phys Rev B 76(5):054309 Kushwaha MS, Halevi P, Dobrzynski L, Djafari-Rouhani B (1993) Phys Rev Lett 71(13):2022 Liu Z, Zhang X, Mao Y, Zhu Y, Yang Z, Chan CT, Sheng P (2000) Science 289(5485):1734 Torrent D, Pennec Y, Djafari-Rouhani B (2015) Phys Rev B 92(17):174110 Elford DP, Chalmers L, Kusmartsev FV, Swallowe GM (2011) J Acoust Soc Am 130(5):2746 Pendry JB, Holden AJ, Robbins DJ, Stewart WJ (1999) IEEE Microw Theory 47:2075 Movchan A, Poulton CG, Botten LC, Nicorovici N, McPhedran RC (2001) SIAM J Appl Math 61(5):1706 Craster R, Kaplunov J, Pichugin A (2010) Proc Roy Soc A 466(2120):2341. https://doi.org/ 10.1098/rspa.2009.0612 Nemat-Nasser S, Willis JR, Srivastava A, Amirkhizi AV (2011) Phys Rev B 83(10):104103 Papanicolau G, Bensoussan A, Lions JL (1978) Asymptotic analysis for periodic structures. Elsevier Milton GW (2002) The theory of composites. Cambridge University Press Hassani B, Hinton E (1998) Comput Struct 69(6):707 Hassani B, Hinton E (1998) Comput Struct 69(6):719 Allaire G (1992) SIAM J Math Anal 23(6):1482 Jikov VV, Kozlov SM, Oleinik OA (2012) Homogenization of differential operators and integral functionals. Springer Science & Business Media

Chapter 3

Transformation Acoustics

Abstract The purpose of this chapter is to review the background knowledge on transformation methods that is then used in the following of the book. Consider a mapping χ between points X in a so-called undeformed virtual domain  and points x in the so-called deformed domain ω (see Fig. 3.1). The deformation gradient F is a two-point second order tensor whose components are defined by: Fi J =

∂ xi . ∂XJ

(3.1)

Capital letters are employed for quantities in  while lower case letters for variables in ω. At first order, the deformation gradient represents how infinitesimal segments d X emanating from a point in the undeformed configuration are linearly mapped into segments in the deformed ones: d x = Fd X + o(|d X|).

(3.2)

In the context of finite deformations it represents how infinitesimal fibers of the body change from the material configuration to the current spatial configuration at a given point, while in the context of differential geometry it represents the tangent map between the vectors tangent to curves passing through the considered point [1]. The determinant of F is J , the Jacobian of the transformation, that represents the ratio of volume elements between the deformed and undeformed configurations [2]: dv . (3.3) J = det(F) = dV A standard polar decomposition of the tensor F can be applied as: F = VR

(3.4)

with V the symmetric and positive definite left-stretch tensor defined by V 2 = F F T , F T being the transpose of F, and R the orthogonal rotation tensor [2]. According to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. E. Quadrelli and F. Braghin, Acoustic Invisibility for Elliptic Objects, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-22603-8_3

39

40

3 Transformation Acoustics

Fig. 3.1 Schematic representation of the singular transformation between points in the undeformed domain and deformed domain, along with the transformed equation

this decomposition, infinitesimal fibers at a point are first rigidly rotated according to R, and then stretched according to V (Fig. 3.2). Indeed, the set of infinitesimal tangent vectors d X at given point in the undeformed domain identifies a spherical infinitesimal neighborhood around X, that is mapped through χ into an ellipsoidal neighborhood of x in the deformed one. This can be understood considering that, being V in the set of positive definite, symmetric second order tensors, it can be represented according to the orthonormal basis of its eigenvectors: V = λ1 n 1 ⊗ n 1 + λ2 n 2 ⊗ n 2 + λ3 n 3 ⊗ n 3

(3.5)

where λi are the eigenvalues and ni are unit vectors pointing in the principal directions of V . The infinitesimal elements that after the rotation R are oriented along the socalled principal stretch directions ni , do not change their orientation upon application of V and are just stretched of an amount λi , that is called principal stretch owing to the fact that represent the ratio of their actual length over the original length. The principal directions of V thus are the axis of the ellipsoidal neighborhood in ω, as represented in 3.2. The polar decomposition can be thus written in the basis of the principal stretch directions as: F=

 

λi ni ⊗ ni

 

i

 ni ⊗ N i

(3.6)

i

where the N i are unit vectors defined by: ni = R N i

(3.7)

and represent the original orientation of the infinitesimal tangent vectors aligned with ni before the rotation R. Note that R is a two point tensor, as F is.

3.1 Inertial Cloaking

41

Fig. 3.2 Kinematics of finite deformations: an infinitesimal spherical neighborhood of the point X in the original undeformed configuration is mapped into an ellipsoidal neighborhood around the point x = χ(X) via the tangent map F = V R

3.1 Inertial Cloaking The virtual undeformed domain is assumed to be filled with a homogeneous and isotropic fluid (density ρ0 , bulk modulus B0 ), such that solutions of the acoustic wave equation:   1 1 ∂2 P ∇X P = X ∈ (3.8) ∇X · ρ0 B0 ∂t 2 represent free wave propagation. Again, the capital P = P(X) is used to indicate pressure in the undeformed domain, while the symbol ∇ X is used for the nabla operator in the undeformed coordinates. Upon application of the map X = χ −1 (x) as a coordinate transformation, the previous can be written as [3]:   1 1 ∂2 P , J ∇ · J −1 V 2 ∇ P = ρ0 B0 ∂t 2

x ∈ ω,

P = P(χ −1 (x))

(3.9)

with ∇ the differential operator for the gradient in the deformed coordinates. Indeed, by recalling that Nanson’s Formula implies (J −1 Fa B ),a = 0 [2, 3] and using the chain rule of derivation P,B = (F T ) Bi P,i = P,i Fi B :  J

J −1 Fa B Fi B P,i ρ0



F ,a = J (J −1 Fa B ),a i B P,i + J J −1 ρ0



   1 1 P,i Fi B Fa B = P,B ρ0 ρ0 ,a ,B

(3.10) which shows in component form the equivalence between the left hand sides of Eqs. 3.8 and 3.9. At this point, one can notice that equation Eq. 3.9 is formally equivalent to the acoustic wave equation assuming inhomogeneous and anisotropic inertial properties: 1 ∂2 p , x ∈ ω. (3.11) ∇ · (ρ −1 ∇ p) = B ∂t 2

42

3 Transformation Acoustics

By comparison, it follows that the following material properties: B = B0 J

ρ −1 =

V2 Jρ0

(3.12)

allow to obtain on domain ω the same solutions of the acoustic wave equation in the virtual undeformed domain, but deformed onto the real space according to p(x) = P(χ −1 (x)). We note here that the transformed material properties in Eq. 3.12 firstly appeared in [4], following from the analogy between the acoustic wave equation in two dimensions and the Maxwell equations for the transverse electric polarization, for which Transformation Theory was previously introduced in [5]. Later on, this type of acoustic cloak was called inertial by Norris [3], to highlight the use of anisotropic inertia. Perfect cloaking is obtained when the map is the identity outside ∂ω+ , that represents the outer boundary of the cloak, and while the inner boundary ∂ω− is mapped to a point (ref. Figure 3.1). Near cloaking is obtained mapping ∂ω− onto a small object with vanishing scattering cross section, avoiding thus a singular map [6].

3.2 Pentamode Materials and Pentamode Cloaking Consider the constitutive equation of a linear elastic solid: σ =C:ε

(3.13)

the fourth order elasticity tensor C is a linear map between the second order strain tensor ε and the second order stress tensor σ . Given the symmetries of the strain and stress tensors, they are both in the six dimensional set of symmetric second order tensors, for which an orthogonal basis can be found computing the eigentensors ε∗ of C: i = 1, ..., 6 (3.14) λi εi∗ = Cεi∗ with λi being the eigenvalue associated to the eigentensor εi∗ . No summation is implied. A pentamode material is an elastic solid that has five out of the six eigen values equal to zero. According to the spectral decomposition, its elasticity tensor can thus be written as: C = KS⊗ S (3.15) where S is the eigentensor associated to the non-null eigenvalue and normalized such that S : S = 3. It follows that whatever is the state of strain ε, only one state of stress can be generated: σ = K ε˜ S = − pS (3.16) where it has been called ε˜ the component of ε along S, i.e. ε˜ = S : ε. With the symbol p it has been indicated the so-called pseudo-pressure. Due to the symmetries of S it follows that:

3.2 Pentamode Materials and Pentamode Cloaking

43

p = −K S : ∇u → p˙ = −K S : ∇ u˙

(3.17)

where u is the displacement vector, and the dot over variables is used for partial time derivatives. Combining the previous with the conservation of linear momentum ρ · u¨ = ∇ · σ , the following governing equation for the pseudo-pressure is obtained:   K S:∇ ρ −1 S∇ p = p¨

(3.18)

where it has been assumed that ∇ · S = 0, such that ∇ · ( pS) = S∇ p. It can be shown that the transformed acoustic wave equation: J ∇ · (J −1 V 2

1 1 ¨ ∇ P) = P ρ0 B0

(3.19)

is equivalent to [3]: J P : ∇(J −1 P −1 V 2

1 1 ¨ ∇ P) = P ρ0 B0

(3.20)

for every possible symmetric, positive definite and divergence free second order tensor P. It then follows that for a pentamode characterized by: K = J B0 ;

S = P;

ρ = ρ0 J P V −2 P

(3.21)

the pseudo-pressure exactly matches the behavior of the acoustic pressure in the deformed domain: p(x) = P(χ −1 (x)). Note that the choice of S = P is arbitrary, and thus Eq. 3.21 provides infinite possible solution to the acoustic cloaking problem. When S is chosen to be the second order identity tensor, the pentamode is exactly equivalent to a conventional fluid, the pseudo pressure representing the acoustic pressure and K the bulk modulus: the pure inertial cloak is in this way recovered. If S is chosen to be proportional to the left stretch tensor as S = hV , then the density becomes scalar and a pure pentamode cloak is obtained. Because of the hypothesis on the structure of P, this can only happen if hV is divergence free. A circumstance in which this is always true is when the transformation is a pure stretch, i.e. V = F = F T and h is selected to be J −1 . Indeed, as previously mentioned [2] it is always true that ∇ · (J −1 F) = 0. The following is thus obtained: K = J B0

S = J −1 V

C = KS⊗ S

ρ = ρ0 J −1

(3.22)

Note that, since: V = λ1 n 1 ⊗ n 1 + λ 1 n 2 ⊗ n 2 + λ 3 n 3 ⊗ n 3

(3.23)

J = λ1 λ 2 λ 3

(3.24)

and

44

3 Transformation Acoustics

then, for a pure pentamode cloak it follows that: S=

1 1 1 n1 ⊗ n1 + n2 ⊗ n2 + n3 ⊗ n3 λ2 λ3 λ1 λ3 λ1 λ2

(3.25)

and Ci jkl = B0 λ1 λ2 λ3 Si j Skl .

(3.26)

Thus it follows also that if the elasticity tensor of the pure pentamode material is expressed in the basis of the principal stretch directions, only the components Ci jkl with i = j and k = l are non-null. This corresponds to the symmetries of the elasticity tensor of an orthotropic elastic solid with zero shear resistance (see Example 2.1). In Voigt notation: ⎡

λ1 1 1 ⎡ ⎤ ⎢ λ2 λ3 λ3 λ 2 σ1 ⎢ 1 λ2 1 ⎢ ⎢σ2 ⎥ ⎢ ⎢ ⎥ ⎢ λ3 λ1 λ3 λ1 ⎢σ3 ⎥ 1 λ3 ⎢ ⎥ = B0 ⎢ ⎢ 1 ⎢σ4 ⎥ ⎢ λ ⎢ ⎥ λ λ ⎢ 2 1 1 λ2 ⎣σ5 ⎦ ⎢ 0 0 0 ⎢ σ6 ⎣ 0 0 0 0 0 0

⎤ 0 0 0⎥ ⎡ ⎤ ⎥ ε1 ⎥ ε2 ⎥ 0 0 0⎥ ⎢ ⎥ ⎥⎢ ⎢ ⎥ ⎢ ε3 ⎥ ⎥ 0 0 0⎥ 2ε4 ⎥ ⎥⎢ ⎢ ⎥⎣ ⎥ 2ε5 ⎦ 0 0 0⎥ ⎥ 0 0 0⎦ 2ε6

(3.27)

000

Computing the Christoffel tensor along the principal stretch directions, as was detailed in Example 2.1, one finds:

 n1

⎡ λ ⎤ 1 00 ⎢ ⎥ = B0 ⎣ λ2 λ3 ; 0 0 0⎦ 0 00

⎡

0

 n2

0 ⎢ λ2 = B0 ⎣0 λ1 λ3 0 0

0

⎤

⎥ 0⎦ ;

⎡

 n3

0

⎤ 00 0 ⎢0 0 0 ⎥ = B0 ⎣ λ3 ⎦ 00 λ1 λ2 (3.28)

Note that, due to the fact that the elasticity tensor is five times singular, the Christoffel tensor results two times singular, for each choice of the propagation direction. More than that, the only non null eigenvalue for the three principal directions is that associated with the longitudinal polarization, and considering: ρ=

ρ0 λ1 λ2 λ3

(3.29)

the longitudinal wave speed along the principal directions is: cn1 = c0 λ1 ;

cn2 = c0 λ2 ;

cn3 = c0 λ3

(3.30)

3.2 Pentamode Materials and Pentamode Cloaking

45

that relates the spatial stretch along one direction with the wave speed in that direction. In particular, if the space is “elongated” in a particular direction, than the wave speed should increase in that direction with respect to the background fluid speed c0 , while in the same way, if the space is “compressed” in one direction, the wave should slow down (incidentally, this also provides a nice parallel between the behavior of an anisotropic medium and that of a stretched space with a non flat metric in that V −2 represents the push forward of the metric tensor of the undeformed space on the deformed one). The pentamode material has thus only one slowness surface, that is an ellipsoid with axis aligned with the principal directions of V (that are also the axis of the ellipsoid neighborhood of material fibers around d x). Once again, this represents a generalization of the behavior of a fluid: in this case the medium carries only one longitudinal mode, but with anisotropic wave propagation properties.

Example 3.1 Acoustic Mirage via Transformation Acoustics with Pure Pentamode Material As a simple example of application of Transformation Acoustics, let us consider the situation depicted in Fig. 3.3: the undeformed virtual domain is infinite in directions X and Y , and extends infinitely in the direction of the negative Z coordinate, while it is truncated with a perfectly reflecting boundary in the direction of the positive Z coordinate. Our aim is to substitute the fluid domain filling the portion of space corresponding to the positive Z values, with a material that occupies a smaller region of space, as shown in Fig. 3.3, while maintaining the same acoustical behavior in the rest of space. In this way, a listener that assumes the whole space filled by the original fluid and is waiting for echoes produced from a reflecting surface on top of this smaller material region, is fooled into thinking that such boundary is located further than it actually is. The map χ in this situation can be set to be the identity for the portion of  that corresponds to the negative Z values, while for the positive halfspace: ⎧ ⎪ ⎨x = X y=Y ⎪ ⎩ z = βZ

(3.31)

with β < 1 a contraction coefficient. The deformation gradient is: ⎡

⎤ 10 0 F = ⎣0 1 0 ⎦ 0 0 β,

(3.32)

which is obviously symmetric, representing a pure stretch: V = F. In this case the principal directions of stretch are the Cartesian directions, and the only non unitary stretch is the one in the z direction: λz = β, thus:

46

3 Transformation Acoustics

Fig. 3.3 Underformed domain  and deformed domain ω for the computation of the material properties realizing an acoustic mirage. The wavefronts of a plane wave reflected by the mirror placed on top of both domains are represented too. Note that in ω the wavefronts are not perpendicular to the propagation direction, witnessing dispersion due to the anisotropy in the material properties

Fig. 3.4 a Velocity surface for the pentamode implementing the acoustic mirage with β = 0.3. b Slowness surface. c Wavefronts

⎡

ρ0 ρ= ; β

K = B0 β;

⎤ 1/β 0 0 S = ⎣ 0 1/β 0⎦ 0 0 1

(3.33)

In Fig. 3.4a the velocity surface is plotted assuming the wave propagation speed in the original fluid to be unitary and β = 0.3. Figure 3.4b represent the slowness surface. Being the stretches in the x y plane unitary, the velocity and slowness surface have unitary circle contours onto this plane, while a minimum for the velocity is shown in the z direction where the wave has lower speed according to the shirking of space. Figure 3.4c shows wavefronts emanating from a point source inside the pentamode.

Pentamode materials where firstly introduced in 1995 by Milton and Cherkaev [7], in a work in which they showed that any positive definite fourth order elasticity tensor characterized by the usual major and minor symmetries can be obtained as the effective static material property of a mixture of two high contrast homogeneous materials: one extremely stiff, and one extremely compliant. In particular, a diamond lattice of point-joints between double-cone trusses in a vacuum matrix (see Fig. 3.5) was proposed as a possible realization of a isotropic pentamode. The fact that trusses

References

47

Fig. 3.5 Schematic representation of a cubic unit cell of the diamond lattice of hinge-like joints that realizes a pentamode microstructure in three dimensions

meet at a single point implied a hinge-like connection that reduces the shear resistance to zero. The interest in pentamode materials was renewed in 2008 by Norris [3, 8], where he showed that anisotropic pentamodes could be employed to obtain acoustic cloaking, generalizing the findings of Cummer and Schuring [4] related to inertial cloaking. The actual practicability of three dimensional pentamode materials was later on demonstrated by Kadic et al. [9, 10], that adopting dip-in three-dimensional DLW optical lithography showed that the ratio between static bulk and shear moduli could be pushed to several orders of magnitude by manufacturing tiny junctions. Later on [11], they also demonstrated that the great separation between the static moduli, leads to the appearance of a single mode bandgap, where the pentamode microstructure behave dynamically as a fluid, in the sense that only longitudinal waves are free to propagate in the bulk of the crystal. Anisotropic [12] and two dimensional [13, 14] versions of the pentamode material have also been introduced.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Marsden JE, Hughes TJ (1994) Mathematical foundations of elasticity. Courier Corporation Holzapfel G (2000) Nonlinear solid mechanics. A continuum approach for engineering. Wiley Norris AN (2008) Proc Roy Soc A 464:2411 Cummer SA, Schurig D (2007) New J Phys 9(3):45 Pendry JB, Schurig D, Smith DR (2006) Science 312:1780 Kohn RV, Shen H, Vogelius MS, Weinstein MI (2008) Inverse Prob 24(1):015016 Milton GW, Cherkaev A (1995) J Eng Mater T 117:483 Norris AN (2009) J Acoust Soc Am 125(2):839 Kadic M, Bückmann T, Stenger N, Thiel M, Wegener M (2012) Appl Phys Lett 100:191901 Bückmann T, Thiel M, Kadic M, Schittny R, Wegener M (2014) Nat Commun 5(1):1 Martin A, Kadic M, Schittny R, Bückmann T, Wegener M (2012) Phys Rev B 86:155116 Kadic M, Schittny R, Bückmann T, Wegener M (2013) New J Phys 15:023029 Layman CN, Naify CJ, Martin TP, Calvo DC, Orris GJ (2013) Phys Rev Lett 111(2):024302 Chen Y, Liu X, Hu G (2015) Sci Rep 5:15745

Chapter 4

Transformation Acoustics in Elliptic Coordinates

Abstract In this chapter, a method based on transformation acoustics is introduced to tackle systematically the design of pentamode cloaks aiming at reducing the acoustic scattering of elliptical obstacles.

4.1 Defining Transformations in Elliptic Coordinates Let us consider the problem of finding the pentamode material properties that can be used to reduce the acoustic signature of an elliptic obstacle. The geometry of the problem is depicted in Fig. 4.1: the horizontal and vertical semiaxes of the obstacle will be called H and V , with V > H . These univocally define the geometry of the inner surface of the cloak, while the outer one is set by the two semiaxes V O and H O, with V O > H O. The idea is to exploit transformation acoustics by mapping the inner surface of the cloak onto a smaller surface with an ideally vanishing scattering cross section. To do so, it seems reasonable to describe such transformation adopting a curvilinear coordinate system in which coordinate lines correspond to ellipses, such that the inner surface of the cloak can be easily described as one of such coordinate lines. Note that this is exactly the same approach usually adopted for axisymmetric obstacles exploiting polar coordinates. The elliptic coordinate system will be thus employed. In general, the elliptic coordinates are indicated with μ and ν and are linked with Cartesian coordinates by:  x = a sinh μ sin ν (4.1) y = a cosh μ cos ν. The covariant base vectors can be thus evaluated to be: ⎧ ∂x ∂y ⎪ ⎨ gμ = ex + e y = a sin ν cosh μex + a cos ν sinh μe y ∂μ ∂μ ⎪ ⎩ g ν = ∂ x ex + ∂ y e y = a sinh μ cos νex − a sin ν cosh μe y , ∂ν ∂ν

(4.2)

from which follows the metric tensor: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. E. Quadrelli and F. Braghin, Acoustic Invisibility for Elliptic Objects, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-22603-8_4

49

50

4 Transformation Acoustics in Elliptic Coordinates

Fig. 4.1 Schematic representation of an elliptic obstacle surrounded by a cloak. Reprinted with permission from [1]

[gi j ] =

  2 0 a (sinh2 μ + sin2 ν) , 0 a 2 (sinh2 μ + sin2 ν)

(4.3)

with the contravariant metric tensor being: ⎤

⎡

1 2 2 2 ⎢ [g i j ] = ⎣ a (sinh μ + sin ν) 0

0

⎥ ⎦ 1 . 2 2 a 2 (sinh μ + sin ν)

(4.4)

The associated contravariant base vectors, finally, are: ⎧ 1 ⎪ ⎪ ⎨ gμ = gμ |g μ |2 1 ⎪ ⎪ . ⎩ gν = gν |g ν |2

(4.5)

Note that, ∀ν : cos2 ν + sin2 ν = 1 =

x2 y2 + . a 2 cosh2 μ a 2 sinh2 μ

(4.6)

Thus, a fixed choice of μ produces coordinate lines that correspond to ellipses with vertical semi-axis given by a cosh μ and horizontal semi-axis equal to a sinh μ (see Fig. 4.2). The distance of the two foci with respect to the origin of the Cartesian frame is given by:  (4.7) (a cosh μ)2 − (a sinh μ)2 = a which is independent on the choice of μ. This implies that such ellipses all share the same foci. The value of μ is related to the ratio of horizontal semi-axis over vertical

4.1 Defining Transformations in Elliptic Coordinates

51

Fig. 4.2 The set of coordinate lines of the elliptic coordinate system is shown along with the schematic representation of the undeformed domain  and the deformed domain ω. The relationship between the related geometrical quantities and the elliptic coordinates is also shown. Reprinted with permission from [2]

semi-axis, which is equal to tanh μ. This implies that when μ → ∞ the difference between the two semiaxes tends to zero, while for μ → 0 the ellipses become more and more elongated, and tend to the straight line of length 2a that joins the two foci. On contrast, ∀μ: cosh2 μ − sinh2 μ = 1 =

y2 x2 − a 2 cos2 ν a 2 sin2 ν

(4.8)

that, for a fixed choice of ν, corresponds to the equation of the hyperbola. Independently on the choice of ν, the focal position is:  a 2 cos2 ν + a 2 sin2 ν = a

(4.9)

which means that coordinate lines corresponding to fixed values of ν are confocal hyperbolae. As mentioned, this coordinate system is used here exploiting the fact that the inner surface of the cloak can be described as the set of points corresponding to a fixed value of μ. In particular, with reference to Fig. 4.2, we will use a set of elliptic coordinates (μ = R, ν = ) to define points in the undeformed configuration, and a set of elliptic coordinates (μ = r, ν = θ ) to indicate points in the deformed configuration. The use of the letters r, R and θ,  is made to force an intuitive parallel with polar coordinates: indeed the variables θ,  represent an angle (that is equal to zero for points along the positive y axis, and grows up to 2π in the counterclockwise direction) while the variables r, R take only positive values and behave similarly to the polar distance. Capital and plain letters are used, as was done in Chap. 3, to distinguish between the two configurations. With these definitions, and referring again to Fig. 4.2, the two domains can be described as:

52

4 Transformation Acoustics in Elliptic Coordinates

 = {X : (X, Y ) = (a sinh(R) sin , a cosh(R) cos ), R ∈ [R1 , R3 ],  ∈ [0, 2π ]}

(4.10) and ω = {x : (x, y) = (a sinh(r ) sin θ, a cosh(r ) cos θ ), r ∈ [R2 , R3 ], θ ∈ [0, 2π ]} , (4.11) with R1 , R2 , R3 real numbers satisfying 0 < R1 < R2 < R3 . It can be seen that the inner surface of the cloak ∂ω− is defined by r = R2 , while its outer surface ∂ω+ = ∂+ is set by r = R = R3 . The direct mapping x = χ (X) can be thus written as:  r = g(R) (4.12) θ = and inverse transformation X = χ −1 (x): 

R = f (r ) =θ

(4.13)

with the inner surface of the cloak ∂ω− mapped by R1 = f (R2 ) to the inner surface of the undeformed domain ∂+ . The parameter R1 is chosen to be a small non vanishing value, to obtain a reduction of the scattering cross section (the ellipse corresponding to R1 is contained into the ellipse representing the obstacle) while avoiding the singularity in the mapping resulting from perfect cloaking. This perfectly mirrors what is usually done for axisymmetric cloaks, except by the fact that in this case R1 → 0 produces a boundary ∂− that shrinks to a vertical line joining the two foci, instead of shrinking to a point. This implies that perfect cloaking cannot be achieved (the goal is thus a reduction of scattering), and that the performance of the cloak will be highly dependent on the direction of incidence. Technically speaking, it could be more appropriate then to talk about “near-cloaking”, since perfect invisibility is ruled out in principle. Nonetheless, we will still loosely use the word cloak to refer to each device that surrounds an object and aims at reducing its detectability. Indeed, in the limit for R1 = 0 the scattering cross section of ∂− vanishes only for waves having wavevector aligned with the vertical axis, while for waves traveling in the horizontal direction the scattering cross section corresponds to that of a line having length 2a. With this respect, the horizontal and vertical incidence correspond to the worst and best case scenarios in terms of acoustic performance of the cloak, respectively. The ratio a/V is thus strictly related to the maximum reduction of scattering that can be achieved in the worst case scenario, and is fully determined by the shape of the obstacle to be cloaked. Indeed, if we imagine that the obstacle surface is assigned in terms of semiaxes H and V , these two are related to the focal position a by:

4.1 Defining Transformations in Elliptic Coordinates

53

Fig. 4.3 a Relationship between the shape of the obstacle, expressed in terms of H/V , and the equivalent geometrical size reduction a/V . b Rotation of the principal stretch directions for points on the inner boundary of the cloak. Reprinted with permission from [2]

⎧ ⎪ ⎨a =

H V = sinh(R2) cosh(R2 ) H ⎪ ⎩ R2 = atanh V

,

(4.14)

i.e. for each obstacle shape defined by H/V , a single a/V parameter is associated. In the geometrical acoustics limit [3], this latter ratio is proportional to the ratio between the scattered powers of the cloaked and uncloaked scenarios, in case of horizontal incidence and R1 → 0. Ultimately, this means that the ratio a/V is linked to the best achievable performance in the worst case incidence scenario, for a given assigned geometry of the obstacle. The relationship between a/V and H/V is depicted in Fig. 4.3a. From the definition Eq. 4.11 also follows a restriction on the choice of the outer shape of the cloak. Indeed, when choosing R3 only one of the two parameters defining the outer surface V O and H O can be freely chosen, by inverting one of the followings: V O = a cosh(R3 ) (4.15) H O = a sinh(R3 ); and the other will be fully determined by: HO = tanh(R3 ). VO

(4.16)

This restriction follows from the fact that the outer surface must be confocal to the inner surface. At this point, the deformation gradient can be computed to be: F=

1 g ⊗ G R + gθ ⊗ G f  (r ) r

(4.17)

54

4 Transformation Acoustics in Elliptic Coordinates

where, again, capital and plain letters have been used to distinguish between base vectors in the undeformed and deformed configurations. The transpose of the deformation gradient can be obtained computing its components as [4]: (F T ) A a = gab F b B G AB ,

(4.18)

from which it can be written as: sinh2 (r ) + sin2 θ 1 sinh2 (r ) + sin2 θ G R ⊗ gr + G ⊗ gθ . 2 2 f (r ) sinh ( f (r )) + sin θ sinh2 ( f (r )) + sin2 θ (4.19) The computation of the transpose of the deformation gradient allows ultimately the computation of the left stretch tensor by its definition V 2 = F F T : FT =



sinh2 (r ) + sin2 θ 1 sinh2 (r ) + sin2 θ r g gθ ⊗ gθ . ⊗ g + f  (r )2 sinh2 ( f (r )) + sin2 θ r sinh2 ( f (r )) + sin2 θ (4.20) Since the elliptic coordinate system is an orthogonal system, as witnessed by the fact that the metric tensor is diagonal, V is diagonal and expressed in terms of an orthogonal base, which implies that the principal directions of stretch are easily found to be: 1 g n1 = |gr | r , (4.21) 1 gθ n2 = |g θ | V2 =

i.e. the principal stretch directions are aligned everywhere with the local tangent to the coordinate lines. This is of great importance because, as seen in Chap. 3, the elastic tensor of a pure pentamode cloak results to be that of a singular orthotropic material, when expressed according to the basis of the principal stretch directions, and thus the “principal directions of anisotropy” will be everywhere tangent to confocal ellipses and hyperbolae. The left stretch tensor is thus written as:

where

V = λ1 n 1 ⊗ n 1 + λ2 n 2 ⊗ n 2 ,

(4.22)

 1 |g | sinh2 (r ) + sin2 θ r  =  λ1 = F r R |G R | f (r ) sinh2 ( f (r )) + sin2 θ  |g θ | sinh2 (r ) + sin2 θ θ = λ2 = F  |G  | sinh2 ( f (r )) + sin2 θ

(4.23)

are the principal stretches, according to the definition given in Chap. 3. The deformation gradient is accordingly rewritten as:

4.1 Defining Transformations in Elliptic Coordinates

55

F = V R = (λ1 n1 ⊗ n1 + λ2 n ⊗ n2 )(n1 ⊗ N 1 + n2 ⊗ N 2 ) = λ1 n1 ⊗ N 1 + λ2 n2 ⊗ N 2 .

(4.24) This allows to retrieve the rotation tensor R. Indeed, by comparing Eq. (4.24) with Eq. (4.17), the principal stretch directions in the undeformed configuration N i can be found as: GR GR = N1 = R |G R | |G | (4.25)  G G N2 = = |G  | |G  | and the rotation tensor is: R = n1 ⊗ N 1 + n2 ⊗ N 2 =

|G R | |G  | g ⊗ GR + g ⊗ G. |gr | r |g θ | θ

(4.26)

This highlights the fact that, in general, the transformation is not a pure stretch: indeed ni = N i , with the scalar product n1 · N 1 = cos α allowing the computation of the rotation angle α: cos α =

sin2 (θ ) cosh(r ) cosh( f (r )) + cos2 (θ ) sinh(r ) sinh( f (r ))  .  sinh2 (r ) + sin2 (θ ) sinh2 ( f (r )) + sin2 (θ )

(4.27)

Figure 4.3b is used to provide a visual insight in the nature of the rotation. On the set of coordinate lines, the boundaries of  and ω domains are highlighted in black. A given point on ∂− is considered, which is mapped to the point of ∂ω− that lies on the same hyperbola. The unit vectors corresponding to the principal directions of stretch in the undeformed configuration N i and in the deformed configuration ni are also depicted. Note that, since the vectors N 1 and n1 are, by definition, the unit vectors tangent to such hyperbola, the rotation is a consequence of the curvature of the hyperbolae near the focus. Nonetheless, far from the foci the hyperbolae tend asymptotically to straight lines, and this fact can be exploited to show that under certain hypotheses the rotation remains bounded everywhere in the cloak to low values. Indeed, if g(R) > R for each point in , the points X ∈ ∂− are those expected to be characterized by the maximum rotation, as can be seen on Fig. 4.3b. The maximum rotation is thus primarily related to the choice of R1 , that sets how far the points on ∂− are with respect to the foci, and to the original geometry of the obstacle to be cloaked, on which depends the location of the foci with respect to ∂ω− , as expressed by Eq. 4.14. Thus, the bigger H/V , the smaller would be the region of space internal to ∂ω− where hyperbolae show high curvature, and the bigger R1 the farther will be the boundary ∂− from such region. It follows that, once the shape of the target is assigned, appropriate choices of R1 lead to low values of maximum rotation and mean rotation over the domain of the cloak. Under this hypothesis, the transformation can be considered to be quasi-symmetric (or near pure stretch) and the material properties can be computed to be that of a pure pentamode material:

56

4 Transformation Acoustics in Elliptic Coordinates

sinh2 ( f (r )) + sin2 θ sinh2 (r ) + sin2 θ 2 sinh (r ) + sin2 θ 1 K = B0 J = B0 λ1 λ2 = B0  f (r ) sinh2 ( f (r )) + sin2 θ −1 S= J V   sinh2 ( f (r )) + sin2 θ sinh2 ( f (r )) + sin2 θ =  n1 ⊗ n1 + f  (r )  n2 ⊗ n2 sinh2 (r ) + sin2 θ sinh2 (r ) + sin2 θ C = KS⊗ S (4.28) We remark that this implies an approximation, in that pure pentamode cloaks can be in principle obtained only for transformations involving pure stretch and no rotation, as explained in Chap. 3. Note also that being the problem bidimensional, it should be more appropriate to speak about bimode material, i.e. a material with two easy way to deform. We will instead loosely continue to use the word pentamode for both the 2D and 3D settings. ρ = ρ0 J −1 = ρ0 (λ1 λ2 )−1 = ρ0 f  (r )

4.2 Selected Examples of Transformations in Elliptic Coordinates The freedom in choosing the function R = f (r ) for any given geometry of the problem set by the choice of the parameters R1 , R2 and R3 , allows for infinite possible combinations of material properties’ distribution. In the following, some examples of special transformations will be introduced and discussed. Before doing so, it is worth highlighting that, since the scale factors of the elliptic coordinate system are equal (|G R | = |G  |, |gr | = |g θ |), the elasticity tensor is simplified in the following way: ⎡λ

1

⎤

⎤

⎡ 1 |g | |G |  r 1 ⎥ ⎢ f  (r ) |G R | |g θ | ⎥ ⎢ |G R | |g θ | = B0 ⎢ 0⎥ 1 f  (r ) ⎦ ⎣ |gr | |G  | 0 0 0 ⎤ ⎡ ⎤ 1 0⎥ Kr 1 0 ⎥ = B0 ⎣ 1 K θ 0⎦  f (r ) 0⎦ 0 0 0 0 0

1 0

⎢ λ2   ⎢ λ2 C = B0 ⎢ ⎣1 λ1 0 0 ⎡ 1 ⎢ f  (r ) = B0 ⎢ ⎣ 1 0

0

⎥ ⎥ 0⎥ ⎦ 0

(4.29)

where it has also been introduced the definition of K r and K θ as the elements on the first diagonal, normalized with respect to the bulk modulus of the background

4.2 Selected Examples of Transformations in Elliptic Coordinates

57

Fig. 4.4 a Maximum rotation angle αmax computed for H/V ∈ [0.2, 1] and R1 /R2 ∈ [0.2, 1], when a linear mapping is chosen in order to have homogeneous elastic moduli. b Averaged rotation angle μα . c Equivalent geometrical reduction v/v for the vertical semi-axis. d Equivalent gometrical reduction h/H . Reprinted with permission from [2]

fluid. Equation 4.29 shows that the elements of the elasticity tensor only depend on the first derivative of f (r ), this resulting in ease of definition of such function upon requirements on the elements of C.

4.2.1 Spatially Independent Elasticity Tensor From Eq. 4.29, it follows that the dependency of the elasticity tensor on the spatial coordinates can be eliminated choosing f (r ) = A1r + A2 , with A1 and A2 arbitrary constants that have to be found to satisfy: 

The result is: f (r ) =

f (R2 ) = R1 f (R3 ) = R3 .

R3 − R1 (r − R2 ) + R1 . R3 − R2

(4.30)

(4.31)

58

4 Transformation Acoustics in Elliptic Coordinates

Fig. 4.5 a Rotation angle α computed for the linear mapping for a cloak characterized by H/V = 0.8, R1 /R2 = 0.6 and V O = 1.5V . b Direct transformation r = g(R). c Curves obtained inside the cloak applying the direct transformation to straight lines. d Normalized density distribution ρ/ρ0 inside the cloak. d–e Normalized bulk moduli K r and K θ in principal directions. Reprinted with permission from [2]

Fig. 4.6 a Maximum rotation angle αmax computed for H/V ∈ [0.2, 1] and R1 /R2 ∈ [0.2, 1], when a map is chosen in order to have the bulk moduli following a power law, β = −10. b Averaged rotation angle μα . Reprinted with permission from [2]

As will be shown in Chap. 5, a spatially independent elasticity tensor is very interesting for the microstructure implementation of the cloak, since the spatial gradient of wave speed is completely related to one parameter only, i.e. the scalar density. To quantify the non-ideality of the solution following from considering the material properties to be those of a pure stretch transformation when instead the rotation is non null, the rotation angle averaged on the surface of the cloak μα , and the maximum rotation angle αmax , will be taken as figures of merit. As mentioned in the previous, the design variables that univocally define such indicators are the R1 /R2 ratio, that sets the relative size of ∂− with respect to ∂ω− , and the H/V ratio,

4.2 Selected Examples of Transformations in Elliptic Coordinates

59

Fig. 4.7 a Rotation angle α computed with the power law for a cloak characterized by H/V = 0.8, R1 /R2 = 0.6 and V O = 1.5V . b Direct transformation r = g(R). c Curves obtained inside the cloak applying the direct transformation to straight lines. d Normalized density distribution ρ/ρ0 inside the cloak. d–e Normalized bulk moduli K r and K θ in principal directions. Reprinted with permission from [2]

which specifies the ellipticity of the obstacle to be cloaked. The variability of the figures of merit upon variation of the design variables is represented in Fig. 4.4a, b. To each combination of the two design variable, it also corresponds an associated “geometric reduction” v/V of the vertical semi-axis and a h/H for the horizontal semi-axis, that are depicted in Fig. 4.4c, d. Since in the geometrical acoustic limit the ratio of the scattered power in the cloaked scenario over the scattered power in the uncloaked scenario tends to the ratio of the cross section of ∂− over ∂ω− [3], the two parameters shown in Fig. 4.4c, d can be taken as a measure of the acoustical performance of the cloak in the worst and best case incidence scenario, respectively. In conclusion, Fig. 4.4 allows to understand, for each choice of the design parameters, the associated performance in terms of ideal scattering reduction, and the associated non-ideality introduced by the non-null rotation. As a matter of example, let us consider an obstacle characterized by V = 1, H/V = 0.8, and a cloak whose geometry is set by V O = 1.5V and whose performance are set by R1 /R2 = 0.6: Fig. 4.5a depicts the local distribution of rotation angle α. A dashed line is used to indicate ∂− . As expected, since r = g(R) > R everywhere (see Fig. 4.6b) the maximum of α is obtained on the inner boundary of the cloak and decreases rapidly as one moves towards the exterior. Figure 4.5c shows how straight lines in  are deformed according to χ : note that these would also represent ray paths inside the cloak, would α be zero everywhere. Figure 4.5d–f Allow to inspect the spatial distribution of normalized density ρ/ρ0 and bulk moduli K r , K θ inside the cloak.

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4 Transformation Acoustics in Elliptic Coordinates

4.2.2 Bulk Moduli Following a Power Law The goal is now set to determine the arbitrary constant A1 such that the definition of the elements of the elasticity tensor follows: ⎧  β ⎪ r ⎪ ⎨ K θ = A1 R2 1 ⎪ ⎪ ⎩ Kr = Kθ

(4.32)

with assigned β. This implies: 



f (r ) = A1

r R2

β (4.33)

which is seen to be a differential equation for f (r ), which can be solved to find: A1 f (r ) = β +1



r R2

β+1

R2 + A 2

(4.34)

that guarantees the desired power law for the bulk moduli. As seen in the previous example, we are left with the two arbitrary constants A1 and A2 to be determined imposing the boundary conditions f (R2 ) = R1 , f (R3 ) = R3 , which give: (R3 − R1 )(β + 1) A1 = R2



R3 R2

β+1

 −1 (4.35)

R2 K1. A 2 = R1 − β +1 As an example, Fig. 4.6 shows αmax and μα upon different choices of H/V and R1 /R2 , for β = −10. In Fig. 4.7, a problem characterized by the same design parameters chosen for the numerical example of Fig. 4.5 is now solved using the power law with chosen β = −10. As can be seen in Fig. 4.7b, c, since the slope of g(R) is very low near ∂− , the whole region interested by high curvature in the hyperbolae is mapped to points that are not far from ∂ω− . This reduces the average rotation inside the cloak, with respect to that obtained in the previous example. Overall, this mapping is thus characterized by less non ideality, with respect to the one characterized with spatially homogeneous elasticity tensor. On the flip of the coin, this is paid with a more complicated material distribution, as can be seen looking at Fig. 4.7c–e. Note that, in this case, r = g(R) is not bigger than R for all R, as can be seen from the fact that points near ∂+ are displaced towards the inner surface of the cloak. From this follows that αmax is not the same in Figs. 4.4a and 4.6a, despite the fact that all the design parameters are the same.

4.3 Numerical Test Cases

61

4.2.3 Spatially Independent Density As mentioned, the fact that the elements of the elasticity tensor depend only upon f  (r ) makes the choice of the transformation very easy to be done upon requirements on the elasticity tensor. Conversely, the following equation: ρ = ρ0 f  (r )

sinh2 ( f (r )) + sin2 θ sinh2 (r ) + sin2 θ

(4.36)

is much harder to be solved for f (r ) when assigning a desired density distribution. Note that the solution may even not exist. For example, consider the problem of finding the transformation that guarantees a homogeneous density distribution. Considering that there is no axial symmetry in the problem, the dependence on θ in the anisotropic wave propagation speeds:  vr =

 Kr ; ρ

vθ =

Kθ ρ

(4.37)

cannot reasonably be eliminated. Moreover, remembering that the components of C cannot depend on θ by definition, it follows that the dependency of θ for the density is required, and the density cannot be independent with respect to the spatial coordinates.

4.3 Numerical Test Cases As mentioned, the fact that the pure pentamode material distribution is used even though the rotation is not strictly zero everywhere, implies that the actual performance of the cloak are not the ideal ones. In order to assess the difference between the actual behavior and the ideal one, a set of Finite Element simulations are conducted, sending a plane wave against the cloak. In particular, the design parameters are chosen to be H/V = 0.85, R1 /R2 = 0.1 and V O = 1.5V , that, along with the choice of the linear mapping, lead to a maximum rotation angle αmax = 50◦ and an averaged rotation μα = 7◦ (see Fig. 4.8a). With such high maximum and mean rotation angles, the aim is set to verify whether the scattering reduction is still obtained, even when the small rotation condition is not met. Different wavelengths λ in the range V /λ ∈ [0.5, 3] and different angles of incidence are considered for the plane wave impinging on the cloak, and for each case the scattered intensity is integrated to compute the Total Scattering Cross Section T SC S [3] as: Psc (4.38) TSCS = Iinc

62

4 Transformation Acoustics in Elliptic Coordinates

Fig. 4.8 Numerical assessment of the cloak with design parameters H/V = 0.85, R1 /R2 = 0.1 and V O = 1.5V . a Distribution of rotation angle α computed with the linear transformation. b Ratio of computed scattered power over the scattered power in the uncloaked scenario. The solid green line refer to the obstacle with the cloak, while the dashed black line is the reference behavior, obtained for a rigid obstacle shaped as ∂− . Vertical incidence. c 45◦ angle incidence. d Incidence from the horizontal direction. Reprinted with permission from [2]

Fig. 4.9 Numerical assessment of the cloaking performance (following from Fig. 4.8): computed fields in terms of acustic pressure and pseudo-pressure normalized with respect to the amplitude of the incident wave, V /λ = 2. a–c Vertical incidence. d–f 45◦ incidence. g–i Horizontal incidence. Reprinted with permission from [2]

4.3 Numerical Test Cases

63

Fig. 4.10 Numerical assessment of the carpet cloak (V /λ = 2, H/V = 0.85, R1 /R2 = 0.01.). a Distribution of rotation angle inside the cloak. b Uncloaked case. c Cloaked scenario. d Reference. Reprinted with permission from [2]

where Psc is the scattered power and is Iinc the intensity of the incident plane wave. For each combination of incident direction and wavelength, a simulation is also performed for the rigid obstacle without the cloak and a rigid obstacle shaped as ∂− , that constitute thus the uncloaked scenario and the reference behavior, respectively. Figure 4.8 shows the comparison between the T SC S computed for the cloaked scenario and for the reference scenario. Both of them are normalized with respect to the T SC S calculated at the same frequency and for the same direction of incidence in the uncloaked scenario, providing thus a ratio of scattered powers. The case of vertical incidence (best case scenario in terms of direction of incidence) is addresses in Fig. 4.8b, while Fig. 4.8c, d refer to incidence at 45◦ and horizontal incidence (worst case scenario), respectively. Despite the fact that the rotation has been completely neglected in choosing the material properties’ distribution, still a ≈ 80% broadband reduction in scattered power is observed for vertical incidence, and a ≈ 50% reduction for the horizontal case. In Fig. 4.9 the calculated acoustic fields normalized with respect to the amplitude of the incident sound are shown, for V /λ = 2. It can be tempting to reduce the parameter R1 /R2 to get an improvement in the acoustic performance. However, due to the nature of the elliptic coordinate system that was previously discussed, as R1 further decreases the improvement in the acoustic performance is progressively decreased because of the small associated variation in v/V and h/H , while at the same time the rotation angle increases, increasing the non-ideality introduced by designing the cloak as a pure pentamode. This implies that for each problem an optimal value for R1 can be found, where the reduction of acoustic scattered power (especially in case of horizontal incidence) has ha maximum. On contrast, the fact that for R1 → 0 the equivalent acoustic behavior of the cloaked obstacle tends to that of a line joining the foci suggests that extremely low values of R1 can be used for carpet cloaking [5–9], where the presence of a rigid boundary is exploited to make the obstacle undetectable. Indeed, if it behaves acoustically as a flat line, placing it on a flat rigid surface will make its reflections indistinguishable from those of the perfect mirror on which is placed. Let us then consider to choose R1 = 0.01 and place half of  in contact with a flat surface. In Fig. 4.10d, the reference scenario is shown, where it can be seen that ∂− is almost overlapping to the rigid boundary. Figure 4.10a shows the distribution of the

64

4 Transformation Acoustics in Elliptic Coordinates

rotation angle α, reaching a maximum over 70◦ . Despite such high rotation angle, that make the transformation far from being quasi-symmetric, a comparison between Fig. 4.10b, c, where the bare obstacle and the cloak case are depicted, shows how the cloak is capable of reducing the spread in the directions of the scattered intensity.

References 1. Quadrelli DE, Casieri MA, Cazzulani G, La Riviera S, Braghin F (2021) Extreme Mech Lett 49:101526 2. Quadrelli DE, Cazzulani G, La Riviera S, Braghin F (2021) J Sound Vib 512:116396 3. Ginsberg JH (2018) Acoustics: a textbook for engineers and physicists, vol 1 and 2. Springer 4. Marsden JE, Hughes TJ (1994) Mathematical foundations of elasticity. Courier Corporation 5. Li J, Pendry JB (2008) Phys Rev Lett 101(20):203901 6. Popa BI, Zigoneanu L, Cummer SA (2011) Phys Rev Lett 106:253901 7. Zigoneanu L, Popa BI, Cummer SA (2014) Nat Mater 13(4):352. https://doi.org/10.1038/ nmat3901 8. Bi Y, Jia H, Lu W, Ji P, Yang J (2017) Sci Rep 7(1):1. https://doi.org/10.1038/s41598-01700779-4 9. Bi Y, Jia H, Sun Z, Yang Y, Zhao H, Yang J (2018) Appl Phys Lett 112(22):1. https://doi.org/ 10.1063/1.5026199

Chapter 5

Design and Experimental Validation of an Elliptic Cloak

Abstract In this chapter we use the method introduced in Chap. 4 to design a nonaxisymmetric cloak and produce an underwater experimental validation of its functioning.

5.1 Problem Setting As it has been shown in Chap. 4, the application of transformation acoustics in elliptic coordinates allows for the design of pure pentamode cloaks for scattering reduction of elliptical targets. With reference to the nomenclature introduced in Chap. 4, the linear map: R3 − R1 (r − R2 ) + R1 (5.1) f (r ) = R3 − R2 was shown to lead to spatially homogeneous elastic moduli, according to: ⎡R − R ⎤ ⎤ 3 2 ⎡ ⎤ 1 1 0 Kr 1 0 ⎢ R3 − R1 ⎥ ⎢ f  (r ) 1 0⎥   ⎢ ⎥ ⎥ R3 − R1 ⎥ = B0 ⎣ 1 K θ 0⎦ . C = B0 ⎢ ⎣ 1 f  (r ) 0⎦ = B0 ⎢ 1 0⎦ ⎣ 0 0 0 R3 − R2 0 0 0 0 0 0 (5.2) This peculiar map is here chosen because it makes things easier when trying to design the microstructure that practically implements the cloak. Indeed, the practical implementation is usually done by discretizing the continuous spatial variation of material properties in small pieces, that are then filled with a locally periodic microstructure which is properly designed to exhibit an equivalent behavior as the one prescribed by the discretized distribution. Having a spatially homogeneous elasticity tensor, one only has to care about a gradient in the inertial properties (instead of both elasticity and density). Since there are well known unit cell geometries [1–3] that can be used to implement pentamode materials with independent tailoring of ⎡

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. E. Quadrelli and F. Braghin, Acoustic Invisibility for Elliptic Objects, PoliMI SpringerBriefs, https://doi.org/10.1007/978-3-031-22603-8_5

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5 Design and Experimental Validation of an Elliptic Cloak

elastic and inertial properties, the choice of the linear map thus tremendously reduces the burden implied in the design of the microstructure, in that each periodic crystal that has to fill a single block of the discretized material property distribution does not have to be designed from scratch, but only the parameters controlling the density have to be changed with respect to the unit cell design implementing neighboring blocks. This benefit is felt even more in case of non axisymmetric cloaks: the fact that the θ dependency cannot be eliminated indeed implies that the number of distinct discretizing blocks to be filled with a distinct artificial crystal is much higher than in the axisymmetric case [3–5], thus any means that can reduce the computational burden related to the design of the microstructure should be adopted. Note that, at least three different scales can be identified in this problem: the macroscale, whose characteristic length is represented by the wavelength and/or the size of the cloak, a mesoscale, whose characteristic length is associated to the size of the discretization of material properties, and a microscale, whose characteristic size is represented by the characteristic length of the unit cell of the artificial crystal that implements the material properties inside each block (note that the term block will be used to identify the unitary element of the material property discretization while unit cell is used to refer to the unitary element of the artificial crystal). In principle, following what seen in Chap. 2, if these three scales are separated, i.e. the size of the blocks is negligible with respect to the size of the macroscale, and the size of the unit cell is negligible with respect to the size of the blocks, the microstructure could be designed in such a way that it exactly matches the material properties prescribed by transformation acoustics. Nonetheless, this would imply, at least, some fabrication issues (and related costs) coming from the infinitesimal size of unit cells with respect to the overall size of the specimen. In practical implementations [3, 4] is thus usually considered that the mesoscale and microscale coincide, i.e. each block is typically filled by one unit cell only. This will imply a second level of approximation (other than the local rotation being non-null) in our design, whose impact will be further detailed in the following. The choice of the linear map ensures that g(R) > R, ∀R, from which follows that the low rotation requirement will be fulfilled choosing a geometry with a moderate ellipticity. This ensures that hyperbolic coordinate lines tend as fast as possible to straight lines inside . Note that asking for low ellipticity does not constitute a serious limit on the applicability of the method in underwater applications for submarine stealth, in that the cross section of such vehicles, despite not being exactly axisymmetric, never shows high ellipticity (primarily for reasons related to structural resistance of the hull [6]). The shape of the obstacle for the experimental validation is thus selected to be characterized by H/V = 0.93, which matches typical ratios for the considered underwater applications. As mentioned in Chap. 4, assigning such ratio completely defines the parameter R2 , which is one of the three parameters that are required to compute the material properties. The other two are R3 , that completely defines the outer boundary of the cloak, and R1 , that sets the shape of ∂− and thus the acoustic performance. Indeed:

5.1 Problem Setting

67

Fig. 5.1 a B0 K r as a function of the percentage vertical thickness (V O − V )/V % and the percentage geometrical reduction (v − V )/V %. b B0 K θ . c K θ /K r . d ρmin , minimum of the mass density distribution. e ρmax , maximum of the mass density distribution. f ρmax /ρmin . Reprinted with permission from [7]

cosh(R1) h v = ; = tanh (R1 ) V cosh (atanh(H/V )) v cosh(R3) HO VO = ; = tanh (R3 ) V cosh (atanh(H/V )) VO

(5.3)

The choice of the two remaining design parameters can be made upon inspection of the maps in Fig. 4.4, that provide information on the rotation angle and on the equivalent geometric reduction of semiaxes, and on the maps in Fig. 5.1. Indeed, Fig. 5.1a, b represent the values of the diagonal entries of the elasticity tensor as a function of the variation of the design parameters, when the background fluid is considered to be water (B0 = 2.25 [GPa], ρ0 = 1000 [kg/m 3 ]). Since providing just the mere value of R1 and R3 hides the geometrical significance of the selected configuration, each choice of the design parameters is represented in the graph with the equivalent pair of parameters (V O − V )/V , that is related to the choice of R3 and represents the percentage thickness of the cloak, and (v − V )/V , that is related to the choice of R1 and represents the percentage equivalent reduction in the vertical semi-axis which is linked, as seen in Chap. 4, to the ideal acoustic performance in the worst-direction-of-incidence case scenario. Figure 5.1c represents K θ /K r , that summarizes how extreme is the anisotropy of the cloak (recall that anisotropy is only found in the elastic properties). In Fig. 5.1d, e, the minimum and maximum density that is computed inside the cloak is shown for each possible configuration, while in

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5 Design and Experimental Validation of an Elliptic Cloak

Fig. 5.1f the ratio between the two represents the level of inhomogeneity in the cloak (again, recall that the spatial inhomogeneity only depends on the inertial properties). To avoid the problem of joining more pieces together, it is decided to fabricate the specimen in a single piece. This implies that the whole microstructure must be obtained with the same base material, which is selected to be Aluminum (ρ = 2700 (kg/m3 ), E = 70 (GPa), ν = 0.3). This, in turns, sets bounds on the obtainable equivalent material properties. In particular, since the equivalent density is obtained with a spatial average on the unit cell (Chap. 2), and the only other material phase that will be considered in the unit cell design is void, the maximum and minimum densities obtained by varying the filling fraction of Aluminum inside the cell set a constraint on the values of the design variable R1 and R3 . Note indeed that a decrease of R1 , i.e. a decrease of (v − V )/V produces an increase in inhomogeneity factor ρmax /ρmin (Fig. 5.1f) while decreasing R3 , i.e. (V O − V )/V , shifts up both the maximum and minimum values of ρ (Fig. 5.1d, e). Keeping thus in mind such constraints and looking at Fig. 5.1, a ratio R1 /R2 = 7.8/12 and a ratio R3 /R2 = 1.244 are selected, corresponding to: v−V ≈ −40%; V

VO −V ≈ 47%. V

(5.4)

In general, a cloak with a smaller thickness V O−V can be obtained fabricating the V cloak with a denser metal, like Titanium, while lower values of v can be achieved fabricating the cloak in more pieces, each one fabricated with a base material with different density, allowing to extend the constraints on the inhomogeneity factor.

5.2 Microstructure Design and Validation Once the design parameters are set, the resulting material properties can be computed. Figure 5.2a shows the distribution of mass density ρ, while in Fig. 5.2b it is shown the distribution of the local rotation angle α, where it can be verified that is everywhere bounded to values lower than 5◦, suggesting that the transformation can be considered to be quasi-symmetric. Following from this fact, we can consider the acoustic scattering from the cloak to be close to the ideal one, i.e. the scattering from an elliptical object shaped as ∂− . To get an insight in the ideal expected behavior of the cloak, Finite Elements simulations are conducted to compute the ratio of the scattered power from the reference obstacle ∂− over that from the uncloaked osbtacle ∂ω− , for both the best and worst case scenarios in terms of direction of incidence. They are plotted in Fig. 5.2c. Note that the frequency dependence of the incident wave is now expressed in terms of κ L c , being L c a characteristic length of the microstructure that will be introduced below. This is done for ease of comparison with results that will be shown later. At this point, it is sufficient to note that, as expected, when the wavelength gets small (geometrical acoustic limit) the ratio of scattered powers tends to the ratio of the semiaxes: v/V for horizontal incidence and h/H for vertical incidence.

5.2 Microstructure Design and Validation

69

Fig. 5.2 a Mass density distribution. b Rotation angle α. c Scattered power computed for ∂− over scattered power computed for ∂ω− , representing the ideal cloaked over uncloaked acoustic scattering reduction. Vertical and horizontal incidence are considered. Reprinted with permission from [7]

Fig. 5.3 a Topology of the unit cell of the centered rectangular lattice adopted to implement the pentamode material properties required in the cloak. b Discretization of the surface of the cloak to obtain the discretized density distribution. Reprinted with permission from [7]

Along with the computed density distribution shown in Fig. 5.2a goes the computation of the target pentamode elasticity tensor: ⎡ ⎤ ⎡ ⎤ Kr 1 0 0.91 2.25 0   C = B0 ⎣ 1 K θ 0⎦ = ⎣2.25 5.55 0⎦ [G Pa] 0 0 0 0 0 0

(5.5)

The subsequent step is the previously mentioned discretization of the spatial density distribution and the choice of a suitable unit cell that, once homogenized, is characterized by equivalent material properties equal to those needed in each block. Note that this implies, in principle, to solve the problem of finding a microstructure for assigned ρ hom , Chom , which is the inverse of the direct computation of the homogenized properties. While for the direct problem a solution exists in closed form, as was shown in Chap. 2, in general there is no analytical way to proceed to solve the inverse one, which is thus be tackled adopting numerical parametric structural optimization schemes. In the most general sense these consist in choosing a priori a given topology for the unit cell, that is fully described by a set of geometrical parameters on which optimization routines iterate, in order to find their best combination, which ultimately leads the the minimization of a selected objective.

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5 Design and Experimental Validation of an Elliptic Cloak

The chosen unit cell geometry is depicted in Fig. 5.3a, along with the specification of its geometrical parameters. Note that it is a rectangular unit cell of a centered rectangular unit lattice (see Example 2.2): this implies that, due to its symmetries, whatever is the choice of the parameters, the homogenized elasticity tensor will always show the required orthotropic symmetry (see Chap. 2, and Example 2.1). The unit cell geometry is inspired by well-known pentamode designs [1–3] that aim at obtaining the required low equivalent shear moduli and an independent control on inertial and elastic properties. Indeed, in its simplest form, it is made by an Aluminum frame of struts shaped as an irregular honeycomb, where thin joints between the struts are used to achieve the low shear resistance, as was originally suggested in [8]. To the oblique sides of this frame, big appendages are attached that are intended to control the inertial properties. Indeed, an increase of the size of these masses above a certain value, does not cause a further increase in the stiffness of the microstructure anymore, but only changes the Aluminum filling fraction. With respect to the works that directly inspired this design [3, 5], additional degrees of freedom are here used to describe the shape of the appendages: this is done to allow for a more precise control on density, which in our application is the only parameter that varies throughout the cloak surface. The optimization is conducted using standard Genetic Algorithm routines, already implemented in the commercial software Matlab. At each iteration, a fitness value based on the distance between the computed homogenized properties of the considered lattice and the target ones is computed for each single element of the generation, and a new generation is built with the usual evolutionary-inspired methods from the parameters of the best performing elements of the previous generation. Note that, as explained in Chap. 2, the elements of the homogenized elasticity tensor are computed as hom = c2py ρ hom C yy 2 hom C xhom x = c px ρ

2 )2 − (C hom − C hom )2 /4 − G hom C xhom ρ hom2 (cq2 p − cqs y = xx yy

(5.6)

G hom = cs2 ρ hom where c py and c px are the slopes of the longitudinal dispersion branches in the long wavelength limit, computed for wavevectors aligned to the the y and x Cartesian directions respectively, as defined in Fig. 5.3(a). In the same way, cs is the phase speed computed for the shear mode in either the x or y direction, and cq p , cqs are the quasi-longitudinal and quasi-shear phase speeds in the long wavelength limit for wavevectors aligned with the 45o direction (see Example 2.1). As mentioned before, the homogenized density ρ hom , is just the density of Aluminum multiplied by the filling fraction of the material on the surface of the unit cell. Note that, while the orthotropic elasticity tensor of the artificial crystal is expressed with respect to its local x and y frame, the orthotropic target elasticity tensor is expressed in the orthogonal frame defined by the principal directions of stretch, i.e. tangent to confocal ellipses and hyperbolae. Let us consider that, subjected to the hypothesis that the size of the

5.2 Microstructure Design and Validation

71

unit cell is sufficiently small with respect to the local curvature of the ellipses, and that we can find a method to slightly deform it around the obstacle while preserving angles and ratios of lengths, each cell can be accommodated in the cloak aligning its own x and y directions respectively to the tangential and normal directions of an elliptical coordinate line. The goal of the optimization routine is thus to obtain: ⎧ hom ⎪ → ρD ⎪ ⎪ρ ⎪ hom ⎪ ⎪ ⎨C yy → B0 K r C xhom x → B0 K θ ⎪ ⎪ ⎪C xhom ⎪ y → B0 ⎪ ⎪ ⎩G hom → 0

(5.7)

where ρ D is the value of the density of the considered block after discretization of the continuous distribution. The fact that the homogenized elastic moduli have to be the same for each block is exploited to reduce the computational burden associated to the optimization. Indeed, if the unit cell geometry represented in Fig. 5.3a really allows for independent control on elastic and inertial properties, a common irregular honeycomb frame implementing the required Chom can be optimized preliminarily once and for all the blocks, while a second optimization can be employed just to adjust the size of the masses in each block for fine tuning of the density. A first optimization is thus conducted by considering all the geometrical parameters fixed except by φ and L/H . These parameters univocally determine the aspect ratio L x /L y of the unit cell and set the common hexagonal frame that will be kept fixed in the following steps. The initial fixed value of the other parameters is set manually to approximately match the filling fraction that corresponds to the mean density in the cloak. A fitness function is then set to be: hom − B0 K r | |C hom − B0 K θ | |C yy + . (5.8) J = xx B0 K θ B0 K r The extra diagonal term of the elastic tensor and the shear modules are not included in J , since it is verified that keeping small the size of the junctions, the value of C xhom y √ naturally tends to K r K θ and G tends to values that are of two orders of magnitude smaller than K θ . To ensure that this condition is met, the parameters t1, f i x and t2, f i x are kept fixed and small for all the steps of the design of the microstructure. This is also done to allow a smooth connectivity between adjacent cells, that otherwise cannot be ensured a priori. Once φ and L/H are found by solving this first Genetic Algorithm routine, they are then considered as fixed parameters for the following stages. As mentioned, they set the aspect ratio L x /L y , that is a key factor in the choice of the discretization of the density distribution. Indeed, as was previously discussed, the discretization has to be performed in such a way that it allows the “housing” of the microstructure in place, guaranteeing that the x and y directions of each cell are tangential and normal to the local elliptic coordinate line and that no distortions are introduced in the shape of the cell in such process, in order to not to lose the

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5 Design and Experimental Validation of an Elliptic Cloak

optimality of the design in terms of homogenized properties. To accomplish for the first requirement, it is sufficient to define the discretization of the cloak adopting increments dr and dθ in the elliptic coordinate system. The resulting blocks will be naturally delimited by coordinate lines. To accomplish for the second requirement, it is at least necessary that the aspect ratio of the blocks, i.e. the ratio L θ /L r , should be exactly the same as the aspect ratio of the unit cell, set after the first optimization stage: Lx Lθ = . (5.9) Lr Ly To accomplish this task, one can take advantage of the fact that the scaling factors of the elliptic coordinate system are equal (Chap. 4). Indeed, the lengths L θ and L r are linked to the increments dθ and dr by: L θ = dθ |g θ | L r = dr |gr |.

(5.10)

Lθ dθ = Lr dr

(5.11)

dθ Lx = . dr Ly

(5.12)

Since |gr | = |g θ |, it follows that:

that also ultimately leads to:

This means that a constant fixed ratio of the increments dθ and dr can be adopted to discretize the density distribution (note that this is different to what would happen in case of polar coordinates in axisymmetric cloaking, since a constant dr and dθ in that case leads to blocks that change their aspect ratio as one moves away from the origin). Exploiting the symmetries of the problem, only one quarter of the cloak should be designed, which means that the increments can be related to the number of cells in radial and tangential directions over one quarter of cloak (Fig. 5.3b) as: π/2 Nr Lx dθ = = , dr N θ R3 − R2 Ly

(5.13)

that finally provides a constraint on the number of unit cells to be optimized. Note that that Nr and Nθ can only take integer values, while L x /L y is arbitrary and comes from the results of the first stage of the optimization, aimed at obtaining the desired elasticity tensor. The exact solution may thus not exist, thus a percentage error is defined as   L θ /L r − L x /L y · 100. (5.14) e% = L x /L y

5.2 Microstructure Design and Validation

73

Fig. 5.4 a Error on the equal-aspect-ratio condition computed for different combinations of Nr and Nθ . b Acoustic performance in terms of reduction of scattered power for different discretizations. c Resulting discrete density values inside each block for the choice Nr = 9 and Nθ = 17. Reprinted with permission from [7]

For each considered value of Nr an optimal choice for Nθ can be found that minimizes such error, as shown in Fig. 5.4a. To select among the optimal pairs, one has to consider a trade off between two opposing objectives. Indeed, the higher the total number of unit cells Nr × Nθ , the smaller will be the discretization size with respect to the characteristic length of the gradient of density, leading to a better approximation of the continuous distribution and, ultimately, leading to acoustic performance that are closer to the expected ones. On the flip of coin, the lower the total number of cells, the lower the computational burden implied in the optimization. The actual values of Nr and Nθ are thus chosen performing a numerical simulation of the scattering produced by a set of distinct cloaks, each one composed by a piece-wise constant distribution of density, according to the different possible optimal pairs. The idea is to choose the minimum number of total cells that guarantees an acoustical behavior sufficiently close to the ideal one. In Fig. 5.4b, the normalized scattered power in the geometrical acoustic limit is shown for different configurations. A value of Nr = 9 and Nθ = 17 is thus chosen, providing a total of 153 cells over one quarter of the ellipse to be optimized (as mentioned, this is much higher than the number of cells needed for validation of an axisymmetric cloak [3, 4]). In Fig. 5.4c it is represented the value of density in each block of the discretization: each line in the graph corresponds to one of the nine “radial layer” of the cloak, and is composed by steps representing the 17 blocks each layer is composed of. Note that the gradient of density over one single layer is much smaller than that obtained across each layer. This fact is also exploited to further minimize the computational effort required in the optimization of the microstructure. Indeed, before running the optimization for each block, a further preliminary Genetic Algorithm routine is run to obtain the geometry of the cell corresponding to the minimum density in each layer. For this purpose, the cost function is modified as follows: hom − B0 K r | |ρ hom − ρ D | |C hom − B0 K θ | |C yy + + (5.15) J = xx B0 K θ B0 K r ρD

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5 Design and Experimental Validation of an Elliptic Cloak

Table 5.1 Mean value, deviation and percentage error on the homogenized material properties over the entire cloak target μ σ e% C xhom x C yhom C xhom y G ρ hom

5.552 · 109 Pa 0.9119 · 109 Pa 2.25 · 109 Pa – –

5.555 · 109 Pa 0.9091 · 109 Pa 2.209 · 109 Pa 7.052 · 107 Pa –

1.712 · 107 3.421 · 106 9.593 · 106 4.754 · 106 –

Pa Pa Pa Pa

0.235 0.416 1.82 – 0.094

A part from the parameters that are considered fixed, all other parameters are allowed to vary in a wide range, and the iterations are stopped only after a maximum of 150 generations or 40 consecutive generations without appreciable decrease in the cost. This allows to get as close as possible to the goal. After this step, the remaining cells of each layer are optimized, allowing considerable variations only in the parameter tr , which is the one that mostly affects density without affecting elasticity, and constraining all other parameters to take only small changes around the values of the neighboring cell in the same layer. It is observed that in this way good results are obtained in less than 5 iterations. In Table 5.1, some figures of merit regarding the accuracy of the results of the optimization are reported. For each element of the elasticity tensor, the mean value μ and the associated standard deviation σ are computed over the values obtained for all the 153 cells. A mean percentage error is computed, being lower than 0.5% for the diagonal entries, and lower than 2% for the extradiagonal one, that was not included in the cost function. The mean value of the obtained shear modulus shows to be two orders of magnitude less than C xhom x (note that being the objective a minimization of G, no percentage mean error is computed in this case). The mean error on density is the lowest and is computed to be less than 0.1%. At this stage, what is left to be done to assemble the cloak is to join all the unit cells in the prescribed order and “house” them in the corresponding block around the elliptical target. As mentioned, this should be done in a way that preserves the shape of the cell, in order to not to lose the optimality of the topology that was previously computed. For this task, it is not sufficient to guarantee that the aspect ratio of the blocks is the same as the aspect ratio of the cell, but a way that ensures to preserve angles and relative distances between elements is needed to compute the coordinates of each point of the unit cell in the elliptic coordinate system. This means that a map:  r = r (x, y) θ = θ (x, y)

(5.16)

is needed that relates in a conformal way points (x, y) of the cell in its own Cartesian frame (Fig. 5.3a), to points (r, θ ) in the cloak. Note that, since we want to locally accommodate the unit cells in such a way that their vertical direction coincides with the radial one and the horizontal with the tangential one, the mapping can be

5.2 Microstructure Design and Validation

simplified as:

75

 r = r (y) θ = θ (x).

(5.17)

This will map horizontal straight lines onto ellipses and vertical straight lines onto hypernbolae from the cell to the cloak. Considering one single cell, whose four corners are identified by the letters from A to D, its coordinates are specified in the Cartesian reference frame as: ⎧ (x A , y A ) = (0, 0) ⎪ ⎪ ⎪ ⎨(x , y ) = (L , 0) B B x (5.18) ⎪ , y ) = (L (x C C x, L y) ⎪ ⎪ ⎩ (x D , y D ) = (0, L y ). Consider also the desired location of such cell in the cloak in terms of the ri , θi coordinates of the four corners in the elliptic coordinate system. Let us assume to use a simple linear map between the coordinates as: ⎧ rD − rA ⎪ (y − y A ) ⎨r = r A + yD − y A θ − θA ⎪ ⎩θ = θ A + D (x − x A ) xD − xA

(5.19)

that guarantees that vertices and boundaries of the cell are correctly mapped onto vertices and boundaries of the block. The deformation gradient of the mapping can be computed to be: F=

θD − θ A rD − rA g ⊗ Ey + g ⊗ Ex yD − y A r xD − xA θ

(5.20)

which turns out to be: F=

dr |gr | dθ |gθ | Lr Lθ er ⊗ E y + eθ ⊗ E x = er ⊗ E y + eθ ⊗ E x . Ly Lx Ly Lx

(5.21)

Because of the condition on the aspect ratio expressed by Eq. 5.9 and imposed by the optimal choice of Nr , Nθ , it follows that: F = λiso R

(5.22)



dr dθ a sinh2 (r ) + sin2 (θ ) = a sinh2 (r ) + sin2 (θ ) Ly Lx

(5.23)

with λiso =

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Fig. 5.5 Schematic drawing of the conformal map that allows to accommodate the microstructure around the obstacle without introducing distortions of shape. Reprinted with permission from [9]

and R = er ⊗ E y + eθ ⊗ E x

(5.24)

which means that the transformation only implies a rigid rotation and an istropic expansion/contraction, i.e. it is a conformal map that preserves angles and relative length between elements, without distorting the geometry of the cell. Note that the same is valid for any assembly of unit cells, in that considering Nr and Nθ cells at ones, both the numerator and denominator of λiso is multiplied by the same quantity: Nr dr |gr | Nθ dθ |gθ | Nr L r Nθ L θ er ⊗ E y + eθ ⊗ E x = er ⊗ E y + eθ ⊗ E x Nr L y Nθ L x Nr L y Nθ L x (5.25) This means that the stage of accommodation of the unit cells around the obstacle can be made at once, after assembly of one quarter of cloak, as illustrated in Fig. 5.5. Adopting a global Cartesian coordinate system (X, Y ) to identify the position of points in the assembly of cells, the position of each point in the cloak can be computed to be: ⎧ rD − rA θD − θ A ⎪ (y − y A )) sin(θ A + (x − x A )) ⎨ξ = a sinh(r A + yD − y A xD − xA (5.26) − r − θ r θ A D A ⎪ ⎩η = a sinh(r A + D (y − y A )) sin(θ A + (x − x A )), yD − y A xD − xA F=

according to a Cartesian system (ξ, η) defined over the cloak. Note that a linear map between Cartesian and elliptic coordinates leads to a conformal map only because of the nice property of the elliptic coordinates system of having equal scaling factors. The assembled cloak is shown in Fig. 5.6a. Although the homogenized shear resistance is not strictly zero, by consequence of the great contrast of the shear modulus with respect to the other elements of the elasticity tensor, a Bragg gap is opened for the shear mode at frequencies that still correspond to a linear behavior for the longitudinal branches. This behavior corre-

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77

Fig. 5.6 a Rendering of the assembled cloak. b Photo of the manufactured cloak. Reprinted with permission from [7]

Fig. 5.7 Static deformation of the cloak under external hydrostatic pressure, when the inner boundary is considered to be fixed. On the left the modulus of the displacement field computed for the microstructure is shown, while on the right the computation is conducted on the homogenized material distribution. Reprinted with permission from [7]

sponds to what was called in Chap. 2 a single mode gap, in that those frequency values correspond to free propagation of the longitudinal mode only. With this respect, far from boundaries or excitation points the artificial crystal characterized by a single longitudinal mode behaves like a fluid, and it thus is in the single mode gap that the microstruture exhibits its pentamode-like behavior [10]. In the case at hand, all the designed cells are characterized by a common single mode bandgap between 1.79 < κ L c < 2.46, being L c the biggest L θ value computed for the assembly (i.e. the tangential length of the blocks in the outer layer), and κ the wavelength in the background fluid. Such normalized wavenumber is thus introduced as a measure of separation between the macroscale and the microscale. As a first numerical assessment of the designed specimen, one may want to verify that the overall microstructure behaves as close as possible to its homogenized counterpart. Indeed, we know that the homogenized properties that are retrieved inspecting the dispersion diagram are actually representative of an infinite periodic repetition

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5 Design and Experimental Validation of an Elliptic Cloak

Fig. 5.8 Normal modes of the designed cloak. For each pair, the left mode is computed with the microstructured cloak, while the right one is that obtained with the respective homogenized model. Reprinted with permission from [7]

of unit cells, while because of the assumed equivalence between the mesoscale and microscale, the cloak is not even locally periodic, being made by unit cells that are all distinct from the others. More than that, the optimization of the dispersion branches was carried in the unit cell Cartesian frame, and despite having taken care of deforming the microstructure in a conformal way while accommodating it around the cloak, one can ask whether actually the obtained cloak really matches the homogenized behavior after such deformation. To check separately the elastic behavior from the inertial one, we firstly perform a static analysis. This is conducted applying a hydrostatic load on the outer surface of the cloak, while the inner side is taken as a fixed boundary. The results obtained on the microstructure are then compared to the results obtained performing the same analysis on a cloak made of blocks filled with the piece-wise continuous homogenized properties. The results are show in Fig. 5.7 in terms of the modulus of the displacement vectors normalized over the maximum observed on the microstructure. The two fields show good agreement, witnessing that the homogeneous elasticity tensor defined in the elliptic coordinate system correctly reproduces the behavior of the designed microstructure.

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79

Fig. 5.9 Curl of the displacement field and associated deformation for a pair of selected shear dominated eigenmodes. The curl is normalized with respect to the maximum of the distribution computed on the homogenized cloak. Reprinted with permission from [7]

Once the elastic properties have been checked, the inertial ones can be inspected by comparing the normal modes of the microstructure to those of the homogenized model. To do this, a fixed boundary condition is considered again on the inner surface of the cloak, while the outer one is considered free, and the associated eigenvalue problem is solved. The modes are shown in Fig. 5.8, and are grouped based on their symmetries. For each mode, it is also specified the percentage difference in the computed eigenvalue between the microstructure and the homogenized model. Among the shown eigevectors, some "whispering gallery"-like modes can be observed, appearing every time the periphery of the cloak happens to be a multiple of half the wavelength. These are mainly dominated by compression in the “tangential” direction, plus the radial displacement resulting from Poisson’s effect. Among these, they are also obtained some shear-dominated modes. This is due to the non zero shear modulus that characterizes the actual practical implementation of the cloak with respect to the ideal one prescribed by transformation acoustics. Figure 5.9 represents two examples of such shear dominated modes: the deformation and the curl of displacements computed on the microstructure are compared with those computed on the homogenized counterpart. In the top part of Fig. 5.9, it is represented the same mode that appears in terms of displacements on the right of the second row of Fig. 5.8. This still reminds some kind of whispering gallery mode, but with respect to the others the displacements (indicated with black arrows in the Figure) is such

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5 Design and Experimental Validation of an Elliptic Cloak

Fig. 5.10 a Acoustic performance of the designed cloak in terms of ratio of scattered power over the scattered power in the uncloaked scenario. The yellow line refers to the simulation without damping, while the blue one considers viscous damping inside Aluminum (η = 0.01). The black solid curve is the expected behavior (rigid obstacle shaped as ∂− ), while the black dashed is the behavior of a rigid obstacle shaped as ∂ω+ . b The cloak is now assembled with a 2 × 2 pattern of unit cells in each block of the discretization of the density distribution, reducing L c by a half. Reprinted with permission from [7]

that portions of the cloak just locally rotate with respect to the neighboring ones. Due the the separation in propagation speed between shear and compression waves, these modes appear at higher frequency that the first compression dominated whispering gallery modes, as a consequence of their smaller characteristic wavelength. At even higher frequency, modes like the one appearing on the bottom of Fig. 5.9 can be observed: in this case the wavelength is such small that the mode can develop radially, i.e. the thickness of the cloak is a multiple of half the characteristic wavelength associated to the modes. Inspecting the displacement field and the associated curl, it can be seen that the motion is transversal with respect to the radial direction, suggesting the shear dominated nature of the eigenmode. As a final numerical assessment, the acoustic performance of the cloak has to be checked and compared with the expected one. This is done adopting fully coupled acoustic structural simulations performed with Comsol Multiphysics, that allow to compute the scattered field from the microstructure against plane wave incidence. Figure 5.10a shows the obtained results, in terms of computed scattered power normalized with respect to the scattered power in the uncloaked scenario for vertical incidence. The black line refers to the behavior of an acoustically rigid body shaped as ∂− , i.e. the expected behavior. A yellow line represents the results obtained with the microstructured cloak. For low κ L c values, the obtained behavior appears to be close the expected one, except by the presence of sharp peaks. Those are related to shear resonances of the cloak: indeed, as long as the homogenization is valid, the longitudinal modes are matched with the exterior acoustic waves, and thus cannot create reflections nor trap energy inside the cloak. Conversely, the fact that the shear modulus is different from zero, implies the existence of shear modes, that can be excited by the impinging acoustic wave, and that can partially remain trapped inside

5.3 Experimental Setup

81

the cloak because of the impedance mismatch, when they try to leave the cloak. The effect of these resonances can be attenuated by adding in the model the viscosity of Aluminum, modifying the Young’s modulus as: E = E 0 (1 − iη)

(5.27)

with η = 0.01. The resulting computed scattering is represented as a blue line in Fig. 5.10a: it can be seen that the viscous damping is beneficial in reducing the impact of resonances. For increased values of κ L c , the single mode bandgap range 1.79 < κ L c < 2.46 is encountered. In this range, no resonances are expected, as long as the longitudinal modes are still matched with the exterior fluid, and no propagating shear waves can be observed. This would thus in principle correspond to the best working frequency range, corresponding to the pentamode-like behavior. However, looking at the graph in Fig. 5.10a it can be seen that, although no sharp peaks are obtained, this range correspond to a maximum of the scattered power, witnessing the worst of the performance. This can be explained by the fact that, although no propagating shear waves are expected, evanescent modes are present in the bandgap, and their presence is relevant near the boundaries, i.e. at the separation surface between the fluid and the cloak. The results of Fig. 5.10a show that completely forgetting their presence and identifying the behavior of the microstructure with that of an ideal pentamode, is justified only for infinitely extended domains. When the interaction between a finite sized metamaterial and the surroundings is considered, the full set of Bloch modes has to be considered [11]. Finally, when κ L c is increased even more, the long wavelength homogenization does not hold anymore, and the scattering tends to an asymptotic value, that corresponds to that prescribed by the geometrical acoustics limit, i.e. H O/ h. To show this, a dashed black line is added to the graph, that corresponds to the behavior of an acoustically rigid ellipse shaped as ∂ω+ , i.e. the outer surface of the cloak. It is seen that when the wavelength becomes much smaller than any other characteristic length, the graphs all tend to the same value. The cut-off limit for having good performance is thus solely set by the size of the unit cell with respect to the impinging wavelength, that defines the range where the long wavelength approximation holds. This can be directly checked by building a new numerical model, where the coincidence between the meso and micro scales is relaxed, i.e. each block is now filled by a 2 × 2 pattern of unit cells. This reduces by half the characteristic size L c of a unit cell. The results are shown in Fig. 5.10b where it can be seen that the κ L c value at which the performance start to worsen is equal as before.

5.3 Experimental Setup The designed cloak is produced by wire EDM starting from an Aluminum plate (Fig. 5.5b). The internal boundary of the cloak is left to be joined with the internal Aluminum disc shaped as the obstacle ∂− , in order to implement a rigid bound-

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5 Design and Experimental Validation of an Elliptic Cloak

ary condition and to ease the attachment of the specimen in place in the experimental setup. Indeed, to recreate an acoustically bi-dimensional environment, an acoustically rigid waveguide is needed [12] where to put the cloak and perform the experiments. In designing the waveguide, we follow the pressure compensation technique explained in [13]. Two parallel Aluminum plates are used, having size 10.5λs × 10.5λs × 0.05λs , being λs the wavelength at the central frequency of the signal, which has a spectrum corresponding to 0.45 < κ L c < 1.34 (the working range as deduced by the numerical analysis). The thickness of these plates is chosen to comply with weight requirements, but it is far too low to consider the plates as acoustically rigid boundaries in the underwater environment. Two disks sized as ∂ω− , that represent the expected behavior inside the waveguide, are thus attached at both sides outside the waveguide in correspondence with the specimen. This ideally makes the local pressure distribution at each side of each plate equal, reducing to zero the resulting excitation for plate vibrations [13]. To avoid flooding of the cavities inside each cell, one sheet of EPDM rubber is placed between the specimen and each plate. One sheet of rubber is also added between each compensation disk and the waveguide, in order to maintain the conditions as equal as possible between the interior and the exterior. The assembly composed by the waveguide with the specimen and the compensation disks is hanged vertically in a pool, with equal distance from the free surface and the bottom. The measurement system is illustrated in Fig. 5.11: an omnidirectional hydrophone (co.l.mar. TD0190) hangs from the free surface into the waveguide, and is moved by a gantry robot in such a way that it is possible to scan an area around the specimen. The scanning points are represented onto the area of the waveguide in Fig. 5.12a: the hanging hydrophone is carried around the specimen in such a way to measure both the forward and backward scattering, covering the top half of the waveguide, while the incident signal comes from the left in the Figure. Because of symmetry considerations, the bottom part can be reconstructed mirroring the measured data. The hydrophone is also carried around three circumferences increasing radii (3.84λs , 3.89λs and 3.94λs ). These measurements will be used to compute the scattered intensity and thus, ultimately, the scattered power. Note that rotating by 90◦ C the specimen, two different placements are obtained, that correspond to the best and worst case scenario in terms of direction of incidence, i.e. wavevector aligned with the minor and major semiaxis, respectively. The hydrophone is connected to a B&K type 2650 amp such that the signal is pre-amplified before being acquired by a with a NI-9222 board for post processing. An omnidirectional projector (co.l.mar. TD0720) is used and is fixed at a distance 32λs from the center of the waveguide, such that because of spherical spreading the pressure field at the aperture of the waveguide is as homogeneous as possible in the direction of the thickness of the waveguide, between the two plates. This would ensure that the modes of the waveguide that are characterized by homogeneous pressure along the thickness of the waveguide are the ones that are mostly excited [12]. A KEYSIGHT 33500B signal generator is used to drive the projector, with the driving signal amplified by a REVAC Pro 1200 model AR 446. A similar setup is used for measuring the scattering of elliptical disks shaped as ∂ω− and ∂− , providing a reference for the uncloaked obstacle and expected ideal performance of the cloak, respectively. Even though not

5.4 Results and Discussion

83

Fig. 5.11 Schematic of the measurement system

Fig. 5.12 Schematic of the measurement points

strictly required, the sheets of EPDM are used also in these two cases, in order to allow for comparison between results obtained with as equal as possible boundary conditions.

5.4 Results and Discussion Figures 5.13 and 5.14 provide snapshots of the time histories measured for the best and worst case scenario respectively. Each row constitutes of four subsequent instants of the wave propagation from left to right. Figure 5.13a represents the free wave propagation, i.e. when nothing is placed inside (and outside) the waveguide. As expected, no backscattering or shadows are observed. The same snapshots are repeated Fig. 5.14a for convenience. In Figs. 5.13b and 5.14b, the scattering from the bare obstacle is represented. The two snapshot in the middle of the row clearly

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5 Design and Experimental Validation of an Elliptic Cloak

Fig. 5.13 Measured acoustic fields for incident wave traveling parallel to the major axis. a Free wave propagation from left to right, four snapshots. b Uncloaked obstacle, same four snapshots. c Cloaked obstacle. d Obstacle shaped as ∂− . e Scattered field from the uncloaked obstacle. f Scattered field from the cloak. g Scattered field from the obstacle shaped as ∂− . Reprinted with permission from [7]

5.4 Results and Discussion

85

Fig. 5.14 Measured acoustic fields for incident wave traveling parallel to the minor axis. a Free wave propagation from left to right, four snapshots. b Uncloaked obstacle, same four snapshots. c Cloaked obstacle. d Obstacle shaped as ∂− . e Scattered field from the uncloaked obstacle. f Scattered field from the cloak. g Scattered field from the obstacle shaped as ∂−

show a backscattered wave that diverges from the obstacle, while the last snapshot on the right shows the distortion of the wavefronts in the froward scattering area. Figures 5.13c and 5.14b show the scattering from the cloak. Comparing with the uncloaked scenario, it can be seen that the backscattering is reduced, and so is also the distortions of wavefronts besides the cloaked obstacle. These snapshots can also be compared with those in Figs. 5.13d and 5.14d where the scattering of the “equivalent” obstacle ∂− provides visual information on the ideally expected behavior.

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5 Design and Experimental Validation of an Elliptic Cloak

Note that the full animations can be seen in the Supplementary Information associated to [7]. By looking at the two central snapshot, one can see that the scattering from the equivalent obstacle has a delay with respect to that of the cloak. This suggests that part of the scattering that is observed from the cloak does not come from the internal obstacle (this scattering would be observed at the same time the scattering from ∂− is obtained), but instead is produced at the outer surface of the cloak. This is probably a consequence from the fact that the external surface is matched with “ideal” water (ρ0 = 1000 [kg/m3 ] and c0 = 1500 [m/s]), which can actually differ from the actual experimental conditions. By comparing the shadows, it can be seen how the pressure behind the cloaked obstacle is lower than expected. This can only be associated with absorption in the cloak. Absorption is also probably responsible of the fact that the reflections from the internal obstacle are lower than those seen in the expected behavior, since they are partially absorbed in the cloak while diverging from the inner boundary. Subtracting the measured free field behavior to the one shown in Fig. 5.13b–d, the scattered fields depicted in Fig. 5.13e–g can be obtained, where the previously illustrated phenomenology can be clearly seen. Same is done in Fig. 5.14e–g. The computation of the scattered field is also carried out for the measurements conducted around the three circumferences shown in Fig. 5.12b. As mentioned, this allows the computation of the scattered radial intensity as: Ir = pvr with:

1 vr = − ρ0



∂p dt, ∂r

(5.28)

(5.29)

according to Euler equation. The pressure p in Eq. 5.28 is taken to be the one measured in the middle circumference. The radial gradient of pressure is instead computed with central finite differences adopting the measure on the inner and outer circumference. The error implied in such method can be quantified as [14]: e% = 1 −

sin(κr ) κr

(5.30)

r being the discrete spacing between the circumferences, and κ the wavenumber. Equation 5.30 provides ≈ 2% at the central frequency of the used burst. At each polar location, the instantaneous intensity is then time averaged to obtain the polar distribution of the Decibel (reference 1[μPa]) mean intensity of the scarred signal. Similarly, the radial scattered intensity obtained from the Finite Element simulations shown in Fig. 5.10 are averaged in the same frequency range for the same polar directions. Note that a time domain simulation of the cloak would require a really high computational power, thus the frequency domain numerical simulations are still used. For this reason (among many others regarding the differences between the numerical and experimental set up) it would make non sense to compare directly

References

87

Fig. 5.15 Difference between the experimental measurements and the numerical analysis in terms of Decibel gain computed comparing the cloaked and the uncloaked scattered intensity. Reprinted with permission from [7]

the results obtained for the cloak in the experiments and the numerical analysis. On contrast, what we can do is to compute a Decibel gain of the cloaked vs the uncloaked scenario for the experimental campaign. This represents how much the cloak outperforms the bare obstacle for each scattering direction. A similar indicator can be computed for the cloaked vs uncloaked numerical analysis. The experimental and the numerical gains are then subtracted and shown in Fig. 5.15 for both the incidence directions: for each polar angle these graphs represent whether the benefit of the cloak obtained experimentally over the experimental bare obstacle is higher or lower than that seen in the numerical analysis. A value of 0 [dB] means equal gain in the experiments and the numerical analysis, a value below 0 [dB] indicates that actually the cloak reduces scattering more than expected, while a value above means that the scattering reduction in that direction is lower than expected. Looking at Fig. 5.15 one can see that the fabricated cloak performs better than expected in backward scattering, but worse in forward scattering. This again can be interpreted as the effect of absorption in the cloak, as previously mentioned.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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