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Key Concepts: Neutral Inclusions

11/11/2020

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PhD student Eleanor Russell discusses an important concept in the development of robust advanced materials

1    Inclusions: The good, the bad and the ugly
In material science, we refer to a material domain as a matrix and an inclusion as an object inside the matrix with different material properties.
Inclusions of all different shapes and sizes can be used to intentionally alter the effective properties of a composite material. For example, reinforcing brittle concrete with steel rods to increase its strength or, on a smaller scale, embedding highly conductive fibres throughout a less conductive matrix to improve heat dissipation. In both cases, the material properties of the matrix and inclusions complement each other and result in a more practical composite material.

​On the other hand, sometimes engineers are forced to incorporate inclusions into their designs; adding windows along the side of commercial aircraft for instance. In this case, the material properties can be affected in undesirable (and occasionally catastrophic) ways.

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Fragment of the first production Comet at the Science Museum in London. Image credit: Krelnik, CC BY-SA 3.0 , via Wikimedia Commons
​A famous example is the world's first jetliner, the Comet, in 1952. The pressurised cabins were designed with large, squared-off windows causing pressure to repeatedly build-up at the corners. At high altitudes the corners of a window could be subjected to pressure up to three times higher than the rest of the cabin. Unfortunately, in several cases, this resulted in fatal structural damage. To avoid this problem, aircraft are now designed with the rounded, porthole-style windows we see today, however, neutral inclusions could offer an alternative approach.
2    Neutral Inclusions
​Neutral inclusions were first introduced by Mansfield in 1953 for holes in plates [1]. The idea is to provide inclusions with an additional coating to remove any undesired perturbations from the physical fields in the matrix. A perturbation is any deviation of a field from its state when the inclusion is not present. 
For example, we could surround the squared-off windows of the Comet with an additional coating to relieve the pressure from its corners. The required properties of the coating will depend on both the matrix (the cabin wall) and the inclusion (the window). If the necessary properties are satisfied the combination of the inclusion and its coating is referred to as a neutral inclusion and the fields in the matrix become unperturbed.

​All we need now is to determine the required properties within the coating. To show how we approach this problem we consider perturbations within the temperature field of a matrix.
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2.1    Neutral Inclusions: thermal conductivity example
​In the following example we consider a steady-state, two-dimensional problem. The steady-state assumption removes any time-dependence from the solution. As a result, the only material property we need to consider is the thermal conductivity​, denoted \(k\). The thermal conductivity of a material measures its ability to conduct heat through diffusion.
​
We are interested in the temperature field, \(T(\mathbf{x})\), which must satisfy the heat equation. For this example, we assume the thermal conductivity is isotropic (independent of direction) and homogeneous (independent of position) and is therefore a scalar. As a result, together with the steady-state assumption, the heat equation reduces to Laplace's equation, given by:
$$\begin{equation}
\nabla ^2 T(\mathbf{x}) = 0,
\label{laplace}
\end{equation}$$
​​where \(\nabla ^2\) is the Laplace operator.​
​Consider the linear temperature field in Figure 1, where the black lines represent lines of constant temperature called isotherms. This temperature field is linear with respect to \(x\) and therefore has the form \(T(x)=\alpha x+\beta\) for some constants \(\alpha\) and \(\beta\).

​Let the matrix in Figure 1 have conductivity \(k_m\). Notice in Figure 2 that when a circular inclusion with conductivity \(k_i\), where \(k_i \neq k_m\), is embedded in the matrix, the field becomes perturbed and is no longer linear.
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Figure 1: Linear temperature field with respect to x-coordinate
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Figure 2: Inclusion leading to perturbations
By adding a coating with conductivity \(k_c\), as illustrated in Figure 3, we can remove the perturbations from the matrix and recover the linear temperature field
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Figure 3: Configuration for neutral inclusion in polar coordinates, x = [r,θ].
​The temperature fields in the matrix, coating and inclusion are denoted by \(T_m(\mathbf{x}),\ T_c(\mathbf{x})\) and \(T_i(\mathbf{x})\) respectively and must all satisfy Laplace's equation. We solve (1) for all temperature fields to find:
$$ \begin{equation}
\begin{split}
&T_m(r,\theta) = \left(\alpha r + \dfrac{B_m}{r}\right) \sin \theta, \\
&T_c(r,\theta) = \left(A_c r + \dfrac{B_c}{r}\right) \sin \theta, \\
&T_i(r,\theta) = A_i r \sin \theta,
\label{temp fields}
\end{split}
\end{equation} $$​
where \(B_m,\ A_c,\ B_c\) and \(A_i\) are constants. ​​
​We wish to find a solution where \(B_m=0\) as this leads to a linear temperature field throughout the matrix given by \(T_m(r,\theta)=\alpha r \sin \theta = \alpha x\) (since \(x=r \sin \theta\)).
​
Let the radius of the inclusion and coating be given by \(r_i\) and \(r_c\) respectively. Solving the temperature fields in (2) with perfect contact conditions across each interface (continuity of heat flux and temperature) we find that \(B_m=0\) is satisfied when
\begin{equation}
D^2 = \dfrac{(k_m-k_c)(k_i+k_c)}{(k_i-k_c)(k_m+k_c)} \qquad \text{where} \qquad D= \dfrac{r_i}{r_c}
\label{k cond}
\end{equation}​
PictureFigure 4: Neutral Inclusion leading to unperturbed field in the matrix
Therefore, the two radii and the conductivities of all three components are dependent on each other. Figure 4 shows the resulting temperature fields when the condition in (3) is satisfied. We see that the temperature field in the matrix is unperturbed and once again linear with respect to \(x\) as required.

​
2.2    Neutral inclusions: extensions and limitations

​As we are not restricted to a single coating, we can extend the work in the previous section with as many additional coatings as we wish. For example, a bi-layer neutral inclusion where an inner coating insulates its core region, protecting the inclusion from external forces, ​and an outer coating counteracts any perturbations caused by the insulator. Since the inclusion may have any material properties, the bi-layer neutral inclusion acts as a cloak for the specific problem it is designed for.

In terms of limitations, firstly, an isotropic solution may not exist for a given field. For example, to find a neutral inclusion for a pressure field we must relax the isotropic assumption. Secondly, a neutral inclusion is not a perfect cloak as it is tailored for a specific configuration with fixed external forces. Therefore, if we change the nature of the external forces, the properties within each coating will no longer achieve the desired effects.

​A perfect cloak is one which, regardless of the external forces, leads to unperturbed fields in the matrix. Transformation theory is used to design perfect cloaks and other interesting concepts. We will discuss these in a future blog post.
[1] E. H. Mansfield, Neutral holes in plane sheet - reinforced holes which are elastically equivalent to the uncut sheet, The Quarterly Journal of Mechanics and Applied Mathematics, 1953, 6(3), 370–378
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Into the New Academic Year

10/7/2020

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As we move into a new academic year we reflect on the difficulties and achievements of recent months and look forward to the challenges and opportunities ahead.
Like many, we have had an eventful and challenging second semester and summer in 2019.  The lockdown which began in March has meant adapting to online teaching and home working, with some group members also balancing work commitments with the demands of home schooling.

Here is some of the news from our group during this most unusual time.
Moving on
​

Congratulations to Georgia Lynott, who successfully completed her PhD this year! Georgia's viva was conducted online in June, and the group got together virtually to celebrate.  We hope that we can celebrate in person in the near future!  Final year PhD student Marianthi Moschou has also recently submitted her thesis and is awaiting her viva.
Conferences and collaborations
​

In January, Raphael Assier spent a very productive week at the Laboratoire de Mécanique et d'Acoustique in Marseille, working on homogenisation with Bruno Lombard and Cédric Bellis.

In February PhD student Erik Garcia Neefjes travelled to Australia where he spoke on "Wave Propagation in Thermo-Visco-Elastic Continua" at KOZWaves and explored waves in other ways, too!  You can read his blog about the trip here.
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Also in February, PhD student, Marianthi Moschou, was a runner up in the Smith Institute TakeAim competition with her entry, "Noise Pollution: Our Enemy Against Green Aviation".

After lockdown, conferences moved online, and on the 25th of May, Postdoc Matt Nethercote spoke online at Days on Diffraction 2020.  The title of his talk was "High Contrast Approximation for Penetrable Wedge Diffraction".

We look forward to a full programme of online conferences and seminars this semester.  Regular fixtures include the ICMS Virtual Seminar Series: Waves in Complex Continua (Wavinar), which is organised by Anastasia Kisil.  You can find a list of regular seminar series here, and upcoming conferences here. ​
Publications
​

There have been several new publications from the group this year.  An ongoing list of publications can be found here.

A few recent highlights include a new review paper in National Science Review, co-authored by Professor I. David Abrahams, Anastasia Kisil and several others [1].   This review was sparked by discussions at the Isaac Newton Institute’s 2019 research programme, "Bringing pure and applied analysis together via the Wiener-Hopf technique, its generalisations and applications".  Another recent paper from Anastasia, with Matthew Colbrook (DAMPT Cambridge) recently appeared in  Proc. R. Soc. A.: A Mathieu function boundary spectral method for scattering by multiple variable poro-elastic plates, with applications to metamaterials and acoustics [2].
Zeshan Yousaf and Will Parnell collaborated with Prasad Potluri from the Department of Materials here at the University of Manchester, and Michael Smith from the Department of Applied Mathematics and Theoretical Physics, University of Cambridge.  Their paper examining the properties of polymer filled syntactic foams was published in April [3].  You can read a 3-minute précis of their work here.

Recent work from Matt Nethercote and Raphael Assier with David Abrahams, 
High-contrast approximation for penetrable wedge diffraction, appeared in the IMA Journal of Applied Mathematics [4].
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Engagement

Before lockdown we took our Making Music With Maths KS1&2 workshop to IntoUniversity Manchester North.  IntoUniversity is an organisation that supports young people from disadvantaged backgrounds to attain either a university place or another chosen aspiration.  We had great fun learning about sound and vibration, and making noisy kazoos!  We also brought our Seeing Sounds KS3 workshop to several local high Schools, where students got hands-on using musical instruments to work out the speed of sound.  Information on schools outreach can be found here.
We had several live events planned for the summer, including Big Bang Fair, Glasgow Science festival and more locally, BlueDot festival.  In celebration of the International Year of Sound 2020, our hands-on exhibit would have asked the question “how big is sound?” and offered visitors the chance to explore the scale of sound waves, and find out why this matters to noise reduction.  Sadly of course, in-person events were cancelled due to the Covid-19 coronavirus.  However, we have enjoyed taking part in online engagement events, including CocoMAD 2020 and Glasgow Science Festival’s Hands-On(line).  The International Year of Sound has been extended into 2021, and you can find the event calendar here.
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During lockdown, we have created many online learning resources about materials and waves, including several experiments with household objects that we recorded at home on a mobile phone!  You can find them here, and on our YouTube channel, newly launched this year.  Also on our YouTube channel you can find short clips of our excellent PhD students talking about their research.
For another recent project, we teamed up with illustrator and comedian, John Cooper, to create short illustrated explainer videos about our research.  You can find them on our YouTube channel, and read about the creative process on our blog, and on John’s.
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Looking Forward
​

Earlier this year we welcomed Matt Nethercote back to the group! Matt completed his PhD with Raphael Assier last year, and returned for a PostDoc with Anastasia Kisil.  This month we welcomed two new PhD students to the group: Matthew Riding, who has started his PhD with Anastasia Kisil, and Mark Mesbur, who is jointly supervised by Will Parnell and Professor Paola Carbone in the Department of Chemical Engineering.  We look forward to a busy academic year 2020-2021, meeting the challenges of online teaching, and remote-working. 
[1] I D Abrahams, X Huang, A Kisil, G Mishuris, M Nieves, S Rogosin, I Spitkovsky, Reinvigorating the Wiener-Hopf technique in the pursuit to understand processes and materials, National Science Review, 2020, nwaa225, https://doi.org/10.1093/nsr/nwaa225

[2] M. J. Colbrook. and A. V. Kisi, A Mathieu function boundary spectral method for scattering by multiple variable poro-elastic plates, with applications to metamaterials and acoustics, Proc. R. Soc. A., 2020, 47620200184, https://doi.org/10.1098/rspa.2020.0184

​
[3 ]Z. Yousaf, M. Smith, P. Potluri, W. Parnell, Compression properties of polymeric syntactic foam composites under cyclic loading, Composites Part B: Engineering, 2020, 186, 107764, 
 https://doi.org/10.1016/j.compositesb.2020.107764

​[4]​ M A Nethercote, R C Assier, I D Abrahams, High-contrast approximation for penetrable wedge diffraction, IMA Journal of Applied Mathematics, 2020, 85, 3, 421-466,   https://doi.org/10.1093/imamat/hxaa011
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Key Concepts: Fourier Transforms and Signal Processing

8/11/2020

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PhD student Tom White explains one of the key mathematical tools for understanding complex sound fields.

1. The shape of sound
​Sound travels as small vibrations of particles, and can be modelled by a wave of oscillating pressure. A single sound pressure wave is governed by an amplitude and frequency. The amplitude dictates the strength of the pressure wave and the frequency gives the number of full oscillations completed in one second. For a given point in space, the oscillating pressure looks like the following when measured against time:
Picture
Figure 1. A sound wave with frequency \(f = 1\) and amplitude \(A = 1\).
We can see here that the wave peaks at a pressure of 1, which gives us the amplitude, \(A = 1\), and completes a full oscillation in 1 second meaning the wave has a frequency of \(f = 1\text{Hz}\), and the wave is defined by the cosine function of time, \(t\):
$$ \begin{equation} g(t)=A \cos(2\pi f t)  \label{eq:eq1} \end{equation}$$
A wave of this shape would only produce a simple sound of a single note. In reality the majority of audible sound is the result of many individual waves interacting to create a more complex sound. In Figure 2 we can see how three different sound waves with frequencies \(f = 1, 2\) and \(3 \text{Hz}\) interact to create a sound pressure field.


When the peaks of the waves meet, the pressure field is amplified, and where the the peaks meet the troughs the pressure cancels out. We call this process of overlapping waves to create a non wave-like pressure field superposition. Sound recorded by a microphone is much more likely resemble the pressure field on the right hand side of Figure 2, than the wave in Figure 1. However, for us to look at the complex pressure field it it difficult to identify its component frequencies. In fact, not only can certain sound fields be created by the superposition of waves, but any sound field can be produced by a unique combination of waves.
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Figure 2. The sum of three waves with frequencies, \(f = 1; 2; 3\text{Hz}\)
2. Fourier Transform
In signal processing and mathematical analysis, it is difficult to analyse a time-pressure field as the individual contributing waves behave independently. For example, different frequencies of sound can be absorbed at different rates by a damping material. Therefore, a tool called the Fourier Transform is used to convert the time-pressure field into a frequency dependent function. The transform was first proposed by a French mathematician, Joseph Fourier, in the 19th century. It converts a time-domain pressure field as discussed above, into a frequency-domain function, which is dependent of the frequency \(f\). The transform is given by the equation
$$\begin{equation} G(f)=\int_\infty^\infty g(t) \exp\{2\pi i f t\}\, dt, \label{eq:eq2}\end{equation}$$
where \(\exp\{-\}\) is the exponential operator, and \( i = \sqrt{-1}\).
Initially, the pressure field is given as a time-domain function, where the pressure changes over time. However, after the Fourier transform is applied, this dependence on time is removed, and the solution becomes a function of frequency, peaking where there are contributing frequencies. An important property of the Fourier transformed function is that the sum of two Fourier transforms is equal the Fourier transform of the sum of two waves. Therefore, we can analyse the behaviour of our Fourier transformed function as sound behaves differently at each frequency, and then transform the final product back into the time domain to get the resulting sound field.

In Figure 3 the first graph shows the resulting field of three waves in superposition, with frequencies \(f = 1/2\pi\), \(f = 1\) and \(f = 2\), given by
$$\begin{equation} g(t)=2\cos(t)+\cos(2\pi t)+\cos(4 \pi t). \label{eq:eq3}\end{equation}$$
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Figure 3. Fourier Transform of \(f = \cos(2\pi t) + \cos(4\pi t) + 2 \cos(t)\).
The second graph in Figure 3 shows the Fourier transform of \((\ref{eq:eq3})\). The Fourier transform clearly has spikes at the three contributing frequencies in f, and the relative height of the spikes tells us about the relative strength of each wave to each other. In the initial function \((\ref{eq:eq3})\) we see that the \(f=1/2\pi\) wave has twice the amplitude in as the other two contributing waves, which is reflected in the Fourier transform, with the spike at this frequency being twice as large. The inverse Fourier transform can also be performed, to create a sound profile from a given frequency domain function.
3. Applications of the Fourier transform
In sound analysis the Fourier transform is very useful in helping us understand the composition of sound and how it changes as it passes through certain materials. This is because each sound frequency behaves differently in different materials, so to understand how a given sound changes we must first know which frequencies are contributing to the sound profile. The transform is also very useful in sound editing. If, for example you wanted to remove a high pitched noise from a recording, by taking the Fourier transform the high pitched frequency can be identified. Then by 'squashing' the peak at this high frequency and performing the inverse Fourier transform to return to a time-domain signal, the high pitched note will have been removed.

The Fourier transform is not limited to use in sound processing and analysis: it can be applied to any system where there is known to be an oscillating energy source which can operate at a continuous spectrum of frequencies. For example, the Fourier transform is used in a wide range of spectroscopy techniques such as Nuclear Magnetic Resonance (NMR) spectroscopy which uses the Fourier transform to analyse the magnetic fields of atomic nuclei.
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Illustrating the Research

6/3/2020

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We worked with Illustrator John Cooper to create a short video about metamaterials for noise reduction.
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Still image from the video. Credit: John Cooper.
One of the things researchers often wrestle with is creating clear and succinct explanations of their research for the non-specialist.  It can be extremely helpful to work with others; such as illustrators, teachers and performers, to eliminate jargon and create meaningful outputs that showcase the research.

When we wanted to make a short video about the design of metamaterials for noise reduction devices, we turned to illustrator and comedian John Cooper.

John has worked on projects for the University of Manchester before, creating work for the University’s School Governor Initiative and for the Children’s University of Manchester.  His humorous style lends clarity and informality to a topic.

We started from a blog post about the piece of research in question.  John used it to sketch out an initial storyboard proposal featuring noisy geese!  We then put together an initial script, from which John created a slideshow storyboard.
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Sketches from the initial storyboard proposal. Credit: John Cooper, reproduced with kind permission from the artist
This project was completed during lockdown, so all our discussions were carried out over email or video conferencing.  Keeping the length of the script to a minimum was challenging, but after several iterations we arrived at the final version, which John narrates.

Here’s what John had to say, ‘I really enjoyed this project. The work the department does is fascinating, and it was an exciting challenge in generating visuals to complement their work on noise reduction.  It's good to learn new things while being creative.’
​
Here’s the finished product.  Watch out for those geese!
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3-Minute Papers: Frequency Control with Phononic Crystals

5/22/2020

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We define the mathematical requirements for a special class of material with unusual wave filtering properties that are robust to deformation.
 
In 1995 a group of Spanish physicists demonstrated that a kinetic artwork by artist Eusebio Sempere, consisting of a periodic arrangement of steel rods, selectively attenuated sound at a frequency of 1670 Hz.  Sempere's sculpture, similar in concept to the one pictured below, is an example of a phononic crystal [1].
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A kinetic sculpture by Eusebio Sempere, similar in concept to the one exhibited at the Juan March Foundation in Madrid, which was found to have an acoustic band gap. Image credit: 19Tarrestnom65, distributed under a CC BY-SA licence (https://creativecommons.org/licenses/by-sa/4.0)
What are phononic crystals?

Phononic crystals (PCs) interact with sound and elastic waves in ways not found in conventional materials.  Like Sempere's sculptures, they have periodic structures and typically contain regular arrangements of one material held in a host or matrix material.

Both phononic crystals and acoustic metamaterials allow the manipulation of sound waves, but while metamaterials rely on micro-structures that are very much smaller than the wavelengths in question, phononic crystals typically have repeating units on a larger scale.
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Schematic figure showing the periodic structure of a phononic crystal. From [2].
Frequency band gap

One of the remarkable and useful properties of phononic crystals is the frequency band gap.  This is a range of frequencies of acoustic or elastic waves that cannot propagate through the material.  The PC acts as a selective filter, eliminating specific frequencies of unwanted vibration or noise.

The image below illustrates this phenomenon.  Along the x-axis we plot the wave vector​, which describes the physical property of the wave and specifically the direction in which it propagates.  The y-axis corresponds to the frequency.  The area shaded in green highlights a gap in the frequencies through which no wave vector passes, meaning that no waves in this frequency range can propagate in the material.
Picture
Graph of wave vector vs frequency for a phononic crystal. A frequency band gap is highlighted in green. Adapted from [2]
Phononic crystals typically have band gaps that alter when the material is deformed [3].  In the work described here we define the theoretical requirements for soft PCs whose band gaps remain stable under deformation.  Although quite a broad range of work has been published associated with tuning band gaps by employing deformable materials, no work had been carried out to understand if any materials possessed band gaps that were invariant to deformation. This is therefore an unusual phenomenon which could be extremely useful where acoustic or elastic properties that are robust to material distortion are required.
Designing Phononic Crystals: Transformation Elasticity

Phononic crystals and acoustic metamaterials can be designed through a process called transformation acoustics (where sound waves are concerned), or transformation elasticity (the equivalent for elastic waves).

This stepwise process is summarised in the image below.  The desired wave manipulation is first defined in terms of a spatial, or coordinate transformation.  Then the physical properties of a material capable of achieving this wave distortion are calculated.
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Schematic for the stepwise process of transformation acoustics and transformation elasticity.
A necessary condition for phononic crystals with band gaps independent of deformation is that the Lagrangian Elasticity Tensor (ET) of the material remains constant under deformation.
​
The Lagrangian elasticity tensor quantifies the elasticity of a material in the Lagrangian specification: a frame of reference in which a medium (e.g. a fluid) is observed from the point of view of its material particles. The Lagrangian point of view is sometimes described as analogous to observing the flow of a river from a boat floating on the water.  The alternative, Eulerian specification would be like observing from a fixed position on the riverbank [4].
Our Models
​
We derived the properties of soft PCs that fulfil this requirement.  Two examples are described here.  Both require a hyperelastic, or rubber-like matrix phase.  The first model contains an arrangement of stiff aluminium cylinders, and the second contains a regular distribution of cylindrical voids.

The figure below shows the band structure for in-plane and out-of plane waves in the first model.  A is the undeformed case, B shows the case where the material is stretched in the direction of the red arrows, and C depicts a shear deformation.  The shear deformation has negligible effect on band gaps for waves in either direction, and the effect of the stretch deformation is only slight.
Picture
Frequency band structure for elastic waves in SPC with aluminium cylinders embedded in a semilinear hyperelastic matrix. A: undeformed material, B: stretch deformation, C: shear deformation.
​In the case of the material with voids (below), deformation by stretching in the direction of the voids alters the band gap structure for waves in the antiplane direction, but does not affect the band gaps for in-plane waves.  
Picture
Frequency band structure for SPC with cylindrical voids. A: undeformed material, B: stretched in the direction of the cylindrical axis.
​We have shown with this study that it is theoretically possible to design soft phononic crystals that have frequency band gaps which remain unaffected when the material is deformed.  This unusual phenomenon is of particular interest in engineering applications where flexible materials are required.  The next step in the realisation of such materials is the design and synthesis of polymers whose elastic behaviour fits the model.
Read the full paper here: doi.org/10.1098/rspa.2016.0865

  1. R. Martinez-Sala, J. Sancho J. V. Sanchez, V. Gomez J. Llinares and F. Meseguer, Sound attenuation by sculpture, Nature, 1995, 378(16), 241
  2. E.G. Barnwell, W. J. Parnell, and I. D. Abrahams, Anti-plane elastic wave propagation in pre-stressed periodic structures; tuning, band gap switching and invariance, Wave Motion, 2016, 63, 98-110. doi.org/10.1016/j.wavemoti.2016.02.001
  3. K. Bertoldi and M. C. Boyce, Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures, Phys. Rev. B, 2008, 77, 052105. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.77.052105
  4. Lagrangian vs Eulerian frame of reference:​https://www.youtube.com/watch?v=iDIzLkic1pY
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    The Mathematics of Waves and Materials group are are a research group in the Department of Mathematics at the University of Manchester.  We work on the theoretical, numerical and experimental aspects of both materials and waves. See our research page for more information and details.

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