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In this paper, we theoretically analyse wave propagation in two canonical problems of interest: fluid-filled thermo-visco-elastic slits and fluid-loaded thermo-visco-elastic plates. We show that these two configurations can be studied via the same pair of dispersion equations with the aid of the framework developed in [https://doi.org/10.1098/rspa.2022.0193], which incoporates thermal effects. These two problems are further interrelated, since in the short wavelength limit (relative to the slit/plate width) the respective modes are governed by the same dispersion equation, commonly known as the Scholte--Stoneley equation. It is the Scholte-type modes that are mainly analyzed in this paper. We illustrate results when the fluid is water, although the theory is valid for any Newtonian fluid. Both `hard' and `soft' solids are compared, with the emphasis being placed on the importance of thermo-viscoelastic effects, particularly when stress relaxation is considered. Two main recent works are discussed extensively, namely [https://doi.org/10.1121/1.5078528] for slits and [https://doi.org/10.1103/PhysRevE.103.063002] for loaded plates, both of which do not incorporate viscoelastic mechanisms. We show how the consideration of viscoelasticity can extend the results discussed therein, and explain the circumstances under which they arise.

We discuss the decomposition of the tensorial relaxation function for isotropic and transversely isotropic Modified Quasi-Linear Viscoelastic models. We show how to formulate the constitutive equation by using a convenient decomposition of the relaxation tensor into scalar components and tensorial bases. We show that the bases must be symmetrically additive, i.e they must sum up to the symmetric fourth-order identity tensor. This is a fundamental property both for isotropic and anisotropic bases that ensures the constitutive equation is consistent with the elastic limit. We provide two robust methods to obtain such bases. Furthermore, we show that, in the transversely isotropic case, the bases are naturally deformation-dependent for deformation modes that induce rotation or stretching of the fibres. Therefore, the Modified Quasi-Linear Viscoelastic framework allows to capture the non-linear phenomenon of strain-dependent relaxation, which has always been a criticised limitation of the original Quasi-Linear Viscoelastic theory. We illustrate this intrinsic non-linear feature, unique to the Modified Quasi-Linear Viscoelastic model, with two examples (uni-axial extension and perpendicular shear).

We present a unified framework for the study of wave propagation in homogeneous linear thermo-visco-elastic (TVE) continua, starting from conservation laws. In free-space such media admit two thermo-compressional modes and a shear mode. We provide asymptotic approximations to the corresponding wavenumbers which facilitate the understanding of dispersion of these modes, and consider common solids and fluids as well as soft materials where creep compliance and stress relaxation are important. We further illustrate how commonly used simpler acoustic/elastic dissipative theories can be derived via particular limits of this framework. Consequently, our framework allows us to: i) simultaneously model interfaces involving both fluids and solids, and ii) easily quantify the influence of thermal or viscous losses in a given configuration of interest. As an example, the general framework is applied to the canonical problem of scattering from an interface between two TVE half spaces in perfect contact. To illustrate, we provide results for fluid-solid interfaces involving air, water, steel and rubber, paying particular attention to the effects of stress relaxation., Comment: Accepted in Proceedings of the Royal Society A

Microstructural models of soft tissue deformation are important in applications including artificial tissue design and surgical planning. The basis of these models, and their advantage over their phenomenological counterparts, is that they incorporate parameters that are directly linked to the tissue's microscale structure and constitutive behaviour and can therefore be used to predict the effects of structural changes to the tissue. Although studies have attempted to determine such parameters using diverse, state-of-the-art, experimental techniques, values ranging over several orders of magnitude have been reported, leading to uncertainty in the true parameter values and creating a need for models that can handle such uncertainty. We derive a microstructural, hyperelastic model for transversely isotropic soft tissues and use it to model the mechanical behaviour of tendons. To account for parameter uncertainty, we employ a Bayesian approach and apply an adaptive Markov chain Monte Carlo algorithm to determine posterior probability distributions for the model parameters. The obtained posterior distributions are consistent with parameter measurements previously reported and enable us to quantify the uncertainty in their values for each tendon sample that was modelled. This approach could serve as a prototype for quantifying parameter uncertainty in other soft tissues.

We design a class of spatially inhomogeneous heat spreaders in the context of steady-state thermal conduction leading to spatially uniform thermal fields across a large convective surface. Each spreader has a funnel-shaped design, either in the form of a trapezoidal prism or truncated cone, and is forced by a thermal source at its base. We employ transformation-based techniques, commonly used to study metamaterials, to determine the require thermal conductivity for the spreaders. The obtained materials, although strongly anisotropic and inhomogeneous, can be accurately approximated by assembling isotropic, homogeneous layers, rendering them realisable. An alternative approach is then considered for the conical and trapezoidal spreaders by dividing them into two or three isotropic, homogeneous components respectively. We refer to these simple configurations as neutral layers. All designs are validated numerically both with and without the effects of thermal contact resistance between interfaces. Such novel designs pave the way for future materials that can manipulate and control the flow of heat, helping to solve traditional heat transfer problems such as controlling the temperature of an object and energy harvesting.

We describe an effective active cloaking strategy for the scalar Helmholtz equation in three dimensions where multipole active sources are located at the vertices of the Platonic solids. A "silent zone" is created interior to the imaginary Platonic solid and only the incident field remains in a defined region exterior to the silent zone and active source configuration. This distribution of the sources ensures that the implementation of the cloaking strategy is extremely efficient. In particular, once the multipole source amplitudes required at a single source location are determined, the other source amplitudes can be calculated by simple post-processing involving multiplication of the multipole source vector by a rotation matrix. The general nature of the problem means that the technique is relevant to any scalar wave field, including both acoustics and electromagnetism., Comment: 22 pages, 11 figures

Originating in the field of biomechanics, Fung's model of quasi-linear viscoelasticity (QLV) is one of the most popular constitutive theories employed to compute the time-dependent relationship between stress and deformation in soft solids. It is one of the simplest models of nonlinear viscoelasticity, based on a time-domain integral formulation. In the present study, we consider the QLV model incorporating a single scalar relaxation function. We provide natural internal variables of state, as well as a consistent expression of the free energy to illustrate the thermodynamic consistency of this version of the QLV model. The thermodynamic formulation highlights striking similarities between QLV and the internal-variable models introduced by Holzapfel and Simo. Finally, the dissipative features of compressible QLV materials are illustrated in simple tension.

Recently, hollow thermoplastic microspheres, such as Expancel made by Nouryon, have emerged as an innovative filler material for use in polymer-matrix composites. The resulting all-polymer syntactic foam takes on excellent damage tolerance properties, strong recoverability under large strains, and favourable energy dissipation characteristics. Despite finding increasing usage in various industries and applications, including in coatings, films, sealants, packaging, composites for microfluidics, medical ultrasonics and cementious composites, there is a near-complete absence of statistical geometrical information for Expancel microspheres. Further, their mechanical properties have not yet been reported. In this work we characterise the geometrical quantities of two classes of Expancel thermoplastic microspheres using X-ray computed tomography, focused ion beam and electron microscopy. We also observe the spatial distribution of microspheres within a polyurethane-matrix syntactic foam. We show that the volume-weighted polydisperse shell diameter in both classes of microsphere follows a normal distribution. Interestingly, polydispersity of the shell wall thickness is not observed and in particular the shell thickness is not correlated to the shell diameter. We employ the measured geometrical information in analytical micromechanical techniques in the small strain regime to determine, for the first time, estimates of the Young's modulus and Poisson's ratio of the microsphere shell material. Our results contribute to potential future improvements in the design and fabrication of syntactic foams that employ thermoplastic microspheres. Given the breadth of fields which utilise thermoplastic microspheres, we anticipate that our results, together with the methods used, will be of use in a much broader context in future materials research., Comment: 19 page, 7 figure, 3 tables

New connections between static elastic cloaking, low frequency elastic wave scattering and neutral inclusions are established in the context of two dimensional elasticity. A cylindrical core surrounded by a cylindrical shell is embedded in a uniform elastic matrix. Given the core and matrix properties, we answer the questions of how to select the shell material such that (i) it acts as a static elastic cloak, and (ii) it eliminates low frequency scattering of incident elastic waves. It is shown that static cloaking (i) requires an anisotropic shell, whereas scattering reduction (ii) can be satisfied more simply with isotropic materials. Implicit solutions for the shell material are obtained by considering the core-shell composite cylinder as a neutral elastic inclusion. Two types of neutral inclusion are distinguished, \textit{weak} and \textit{strong} with the former equivalent to low frequency transparency {and the classical Christensen and Lo generalised self-consistent result for in-plane shear from 1979. Our introduction of the \textit{strong neutral inclusion} is an important extension of this result in that we show that standard anisotropic shells can act as perfect static cloaks, contrasting previous work that has employed "unphysical" materials.} The relationships between low frequency transparency, static cloaking and neutral inclusions provide the material designer with options for achieving elastic cloaking in the quasi-static limit., Comment: 30 pages, 2 figures

Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterise wave propagation through an inhomogeneous material, the most crucial parameter is the effective wavenumber. For this reason, there are many published studies on how to calculate a single effective wavenumber. Here we present a proof that there does not exist a unique effective wavenumber; instead, there are an infinite number of such (complex) wavenumbers. We show that in most parameter regimes only a small number of these effective wavenumbers make a significant contribution to the wave field. However, to accurately calculate the reflection and transmission coefficients, a large number of the (highly attenuating) effective waves is required. For clarity, we present results for scalar (acoustic) waves for a two-dimensional material filled (over a half space) with randomly distributed circular cylindrical inclusions. We calculate the effective medium by ensemble averaging over all possible inhomogeneities. The proof is based on the application of the Wiener-Hopf technique and makes no assumption on the wavelength, particle boundary conditions/size, or volume fraction. This technique provides a simple formula for the reflection coefficient, which can be explicitly evaluated for monopole scatterers. We compare results with an alternative numerical matching method.

The mechanical response of syntactic foams comprising hollow thermoplastic microspheres (HTMs) embedded in a polyurethane matrix were experimentally examined under uniaxial compressive strain. Phenomenological strain energy models were subsequently developed to capture both the axial stress-strain and transverse strain response of the foams. HTM syntactic foams were found to exhibit increased small-strain stiffness with reduced density, revealing a highly tuneable and extremely lightweight syntactic foam blend for applications. The foams were also found to become strongly compressible at large strains and possess a high threshold for plastic deformation, making them a robust alternative to hollow glass microsphere syntactic foams. The non-standard transverse strain relationship exhibited by HTM syntactic foams at high filling fractions was captured by Ogden-type strain energy models. The thermal characteristics of these syntactic foams were also explored with Differential Scanning Calorimetry testing which showed that HTMs have a negligible impact on the thermal characteristics of the matrix., Comment: 31 pages, 14 figures, 3 tables

For over 70 years it has been assumed that scalar wave propagation in (ensemble-averaged) random particulate materials can be characterised by a single effective wavenumber. Here, however, we show that there exist many effective wavenumbers, each contributing to the effective transmitted wave field. Most of these contributions rapidly attenuate away from boundaries, but they make a significant contribution to the reflected and total transmitted field beyond the low-frequency regime. In some cases at least two effective wavenumbers have the same order of attenuation. In these cases a single effective wavenumber does not accurately describe wave propagation even far away from boundaries. We develop an efficient method to calculate all of the contributions to the wave field for the scalar wave equation in two spatial dimensions, and then compare results with numerical finite-difference calculations. This new method is, to the authors' knowledge, the first of its kind to give such accurate predictions across a broad frequency range and for general particle volume fractions., Comment: Supplementary material is available as an ancillary file (can be downloaded from a link on the right of the screen). Code to reproduce results in this paper is available in https://github.com/arturgower/EffectiveWaves.jl

The theory of quasi-linear viscoelasticity (QLV) is modified and developed for transversely isotropic (TI) materials under finite deformation. For the first time, distinct relaxation responses are incorporated into an integral formulation of nonlinear viscoelasticity, according to the physical mode of deformation. The theory is consistent with linear viscoelasticity in the small strain limit and makes use of relaxation functions that can be determined from small-strain experiments, given the time/deformation separability assumption. After considering the general constitutive form applicable to compressible materials, attention is restricted to incompressible media. This enables a compact form for the constitutive relation to be derived, which is used to illustrate the behaviour of the model under three key deformations: uniaxial extension, transverse shear and longitudinal shear. Finally, it is demonstrated that the Poynting effect is present in transversely isotropic, neo-Hookean, modified QLV materials under transverse shear, in contrast to neo-Hookean elastic materials subjected to the same deformation. Its presence is explained by the anisotropic relaxation response of the medium., Comment: 16 pages, 16 figures

To what extent can particulate random media be characterised using direct wave backscattering from a single receiver/source? Here, in a two dimensional setting, we show using a machine learning approach that both the particle radius and concentration can be accurately measured when the boundary condition on the particles is of Dirichlet type. Although the methods we introduce could be applied to any particle type. In general backscattering is challenging to interpret for a wide range of particle concentrations, because multiple scattering cannot be ignored, except in the very dilute range. Across the concentration range from 1% to 20% we find that the mean backscattered wave field is sufficient to accurately determine the concentration of particles. However, to accurately determine the particle radius, the second moment, or average intensity, of the backscattering is necessary. We are also able to determine what is the ideal frequency range to measure a broad range of particles sizes. To get rigorous results with supervised machine learning requires a large, highly precise, dataset of backscattered waves from an infinite half-space filled with particles. We are able to create this dataset by introducing a numerical approach which accurately approximates the backscattering from an infinite half-space., Comment: Supplementary material is given at the end of the paper. Code to create the data and results are available from: https://github.com/jondea/MultipleScattering.jl and https://github.com/gowerrobert/MultipleScatteringLearnMoments.jl

We formally deduce closed-form expressions for the transmitted effective wavenumber of a material comprising multiple types of inclusions or particles (multi-species), dispersed in a uniform background medium. The expressions, derived here for the first time, are valid for moderate volume fractions and without restriction on the frequency. We show that the multi-species effective wavenumber is not a straightforward extension of expressions for a single species. Comparisons are drawn with state-of-the-art models in acoustics by presenting numerical results for a concrete and a water-oil emulsion in two dimensions. The limit of when one species is much smaller than the other is also discussed and we determine the background medium felt by the larger species in this limit. Surprisingly, we show that the answer is not the intuitive result predicted by self-consistent multiple scattering theories. The derivation presented here applies to the scalar wave equation with cylindrical or spherical inclusions, with any distribution of sizes, densities, and wave speeds. The reflection coefficient associated with a half-space of multi-species cylindrical inclusions is also formally derived., Comment: The supplementary material has simpler self-contained formulas, it is an ancillary file and can be downloaded from a link on the right. The code to reproduce all graphs is provided in https://github.com/arturgower/EffectiveWaves.jl

Hyperelastic materials possess the appealing property that they may be employed as elastic wave manipulation devices and cloaks by imposing pre-deformation. They provide an alternative to microstructured metamaterials and can be used in a reconfigurable manner. Previous studies indicate that exact elastodynamic invariance to pre-deformation holds only for neo-Hookean solids in the antiplane wave scenario and the semi-linear material in the in-plane compressional/shear wave context. Furthermore, although ground cloaks have been considered in the acoustic context they have not yet been discussed for elastodynamics, either by employing microstructured cloaks or hyperelastic cloaks. This work therefore aims at exploring the possibility of employing a range of hyperelastic materials for use as antiplane ground cloaks (AGCs). The use of the popular incompressible Arruda-Boyce and Mooney-Rivlin nonlinear materials is explored. The scattering problem associated with the AGC is simulated via finite element analysis where the cloaked region is formed by an indentation of the surface. Results demonstrate that the neo-Hookean medium can be used to generate a perfect hyperelastic AGC as should be expected. Furthermore, although the AGC performance of the Mooney-Rivlin material is not particularly satisfactory, it is shown that the Arruda-Boyce medium is an excellent candidate material for this purpose.

Slow sound is a frequently exploited phenomenon that metamaterials can induce in order to permit wave energy compression, redirection, imaging, sound absorption and other special functionalities. Generally however such slow sound structures have a poor impedance match to air, particularly at low frequencies, and consequently exhibit strong transmission only in narrow frequency ranges. This therefore strongly restricts their application in wave manipulation devices. In this work we design a slow sound medium that halves the effective speed of sound in air over a wide range of low frequencies, whilst simultaneously maintaining a near impedance match to air. This is achieved with a rectangular array of cylinders of elliptical cross section, a microstructure that is motivated by combining transformation acoustics with homogenization. Microstructural parameters are optimised in order to provide the required anisotropic material properties as well as near impedance matching. We then employ this microstructure in order to halve the size of a quarter-wavelength resonator (QWR), or equivalently to halve the resonant frequency of a QWR of a given size. This provides significant space savings in the context of low-frequency tonal noise attenuation in confined environments where the absorbing material is adjacent to the region in which sound propagates, such as in a duct. We term the elliptical microstructure `universal' since it may be employed in a number of diverse applications.

Materials with properties that are modulated in time are known to display wave phenomena showing energy increasing with time, with the rate mediated by the modulation. Until now there has been no accounting for material dissipation, which clearly counteracts energy growth. This paper provides an exact expression for the amplitude of elastic or acoustic waves propagating in lossy materials with properties that are periodically modulated in time. It is found that these materials can support a special propagation regime in which waves travel at constant amplitude, with temporal modulation compensating for the normal energy dissipation. We derive a general condition under which amplification due to time-dependent properties offsets the material dissipation. This identity relates band-gap properties associated with the temporal modulation and the average of the viscosity coefficient, thereby providing a simple recipe for the design of loss-compensated mechanical metamaterials.

Serpentine interconnects are highly stretchable and frequently used in flexible electronic systems. In this work, we show that the undulating geometry of the serpentine interconnects will generate phononic band gaps to manipulate elastic wave propagation. The interesting effect of `bands-sticking-together' is observed. We further illustrate that the band structures of the serpentine interconnects can be tuned by applying pre-stretch deformation. The discovery offers a way to design stretchable and tunable phononic crystals by using metallic interconnects instead of the conventional design with soft rubbers and unfavorable damping., Comment: 12 pages, 8 figures

In 1957 Eshelby showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in a homogeneous isotropic host would feel uniform strains and stresses when uniform strains or stresses are applied in the far-field. Of specific importance is the uniformity of Eshelby's tensor S. Following this paper a vast literature has been generated using and developing Eshelby's result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor P=S D^0 arises, where D^0 is the host medium compliance tensor. The tensor P is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled.

The Wiener-Hopf and Cagniard-de Hoop techniques are employed in order to solve a range of transient thermal mixed boundary value problems on the half-space. The thermal field is determined via a rapidly convergent integral, which can be evaluated straightforwardly and quickly on a desktop PC.

This article offers a reappraisal of Fung's method for quasilinear viscoelasticity. It is shown that a number of negative features exhibited in other works, commonly attributed to the Fung approach, are merely a consequence of the way it has been applied. The approach outlined herein is shown to yield improved behaviour, and offers a straightforward scheme for solving a wide range of models. Results from the new model are contrasted with those in the literature for the case of uniaxial elongation of a bar: for an imposed stretch of an incompressible bar, and for an imposed load. In the last case, a numerical solution to a Volterra integral equation is required to obtain the results. This is achieved by a high order discretisation scheme. Finally, the stretch of a compressible viscoelastic bar is determined for two distinct materials: Horgan-Murphy and Gent.

An active elastodynamic cloak destructively interferes with an incident time harmonic in-plane (coupled compressional/shear) elastic wave to produce zero total elastic field over a finite spatial region. A method is described which explicitly predicts the source amplitudes of the active field. For a given number of sources and their positions in two dimensions it is shown that the multipole amplitudes can be expressed as infinite sums of the coefficients of the incident wave decomposed into regular Bessel functions. Importantly, the active field generated by the sources vanishes in the far-field. In practice the infinite summations are clearly required to be truncated and the accuracy of cloaking is studied when the truncation parameter is modified.

The effective macroscopic response of nonlinear elastomeric inhomogeneous materials is of great interest in many applications including nonlinear composite materials and soft biological tissues. The interest of the present work is associated with a microsphere composite material, which is modelled as a matrix-inclusion composite. The matrix phase is a homogeneous isotropic nonlinear rubber-like material and the inclusion phase is more complex, consisting of a distribution of sizes of stiff thin spherical shells filled with gas. Experimentally, such materials have been shown to undergo complex deformation under cyclic loading. Here, we consider microspheres embedded in an unbounded host material and assume that a hydrostatic pressure is applied in the "far-field". Taking into account a variety of effects including buckling of the spherical shells, large deformation of the host phase and evolving microstructure, we derive a model predicting the pressure-relative volume change load curves. Nonlinear constitutive behaviour of the matrix medium is accounted for by employing neo-Hookean and Mooney-Rivlin incompressible models. Moreover a nearly-incompressible solution is derived via asymptotic analysis for a spherical cavity embedded in un unbounded isotropic homogeneous hyperelastic medium loaded hydrostatically. The load-curve predictions reveal a strong dependence on the microstructure of the composite, including distribution of microspheres, the stiffness of the shells, and on the initial volume fraction of the inclusions, whereas there is only a modest dependence on the characteristic properties of the nonlinear elastic model used for the rubber host., Comment: Keywords: microsphere, composite, pressure-volume curve, buckling, nonlinear elasticity, rubber, Mooney-Rivlin

The active cloak comprises a discrete set of multipole sources that destructively interfere with an incident time harmonic scalar wave to produce zero total field over a finite spatial region. For a given number of sources and their positions in two dimensions it is shown that the multipole amplitudes can be expressed as infinite sums of the coefficients of the incident wave decomposed into regular Bessel functions. The field generated by the active sources vanishes in the infinite region exterior to a set of circles defined by the relative positions of the sources. The results provide a direct solution to the inverse problem of determining the source amplitudes. They also define a broad class of non-radiating discrete sources., Comment: 21 pages, 17 figures

It is shown that nonlinear elastic pre-stress of neo-Hookean hyperelastic materials can be used as a mechanism to generate finite cloaks and thus render objects near-invisible to incoming antiplane elastic waves. This approach appears to negate the requirement for special cloaking metamaterials with inhomogeneous and anisotropic material properties in this case. These properties are induced naturally by virtue of the pre-stress. This appears to provide a mechanism for broadband cloaking since dispersive effects due to metamaterial microstructure will not arise., Comment: 4 pages, 2 figures

A theory is presented showing that cloaking of objects from antiplane elastic waves can be achieved by elastic pre-stress of a neo-Hookean nonlinear elastic material. This approach would appear to eliminate the requirement of metamaterials with inhomogeneous anisotropic shear moduli and density. Waves in the pre-stressed medium are bent around the cloaked region by inducing inhomogeneous stress fields via pre-stress. The equation governing antiplane waves in the pre-stressed medium is equivalent to the antiplane equation in an unstressed medium with inhomogeneous and anisotropic shear modulus and isotropic scalar mass density. Note however that these properties are induced naturally by the pre-stress. Since the magnitude of pre-stress can be altered at will, this enables objects of varying size and shape to be cloaked by placing them inside the fluid-filled deformed cavity region., Comment: 21 pages, 4 figures