Tunable elastodynamic band gaps

The effect of nonlinear elastic pre-stress on coupled compressional and vertically polarised shear elastic wave propagation in a two-dimensional periodic structure is investigated. The medium consists of cylindrical annuli embedded on a periodic lattice in a uniform host material. An identical inhomogeneous deformation is imposed in each annulus and the theory of small-on-large is used to find the incremental wave equations governing subsequent small-amplitude elastic waves. The plane-wave-expansion method is employed in order to determine the permissable eigenfrequencies. It is found that the application of pre-stress has a significant effect on the band structure, allowing stop bands to be controlled. The sensitivity of the choice of constitutive behaviour is studied and it is shown that the fundamental shear wave mode is largely unchanged for the class of strain energy functions considered here, whereas the compressional mode is considerably more sensitive to this choice. © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Improvements in engineering and technology rely greatly on advanced, complex materials, which often possess intricate microstructure, permitting macroscopic behaviour that is not present in naturally occurring materials. Elastodynamic and acoustic phononic media present a broad range of opportunities for directing waves [1][2][3] and being able to design these materials carefully in order to enable wave focusing, filtering and directivity is greatly advantageous. Of specific importance is the notion of tunable, configurable or re-configurable phononic media, which have static and dynamic material properties that can be tuned in real time. Such materials have many obvious advantages over media with properties that are fixed upon manufacture. A standard approach is to modify the microstructure and then understand how this modification affects the macroscopic response on a static and dynamic level. A number of opportunities to enable this have been presented. Early work in tuning the acoustic response of such media focused on modifying the band gap properties of phononic media by rotating cylinders [4][5][6].
Several groups have considered using the influence of magnetic or electric fields as a means to perform external control over a crystal [7,8]. In [9], an electric field deforms annular cylinders that are embedded in air, piezoelectric materials are studied in [10][11][12] and electro-rheological materials in [13]. Thermal effects have also been proposed but the effect is usually fairly weak and so a phase transition effect is usually required [14,15]. These mechanisms for control provide the advantage of a phononic switch in such materials. Mechanical mechanisms to create photonic switches have also recently been considered, see e.g. [16,17].
It is well-known that nonlinear elastic pre-stress affects the propagation of subsequent linear elastic waves in the medium [18] and this effect has been exploited in order to modify the band-structure of elastodynamic phononic media. In [19,20] an inhomogeneous beam was considered and it was shown that the band structure can be manipulated effectively using pre-stress. It has recently been noted that the response of a nonlinearly elastic material can be extremely sensitive to the choice of constitutive behaviour [21].
An electrical bias has been employed in order to control deformation and hence band gaps in layered and fibre reinforced media [22,23]. Experiments in one dimensional structures were performed by [24]. By making use of numerical simulations, via the finite-element method, the mechanical tunability of threedimensional structures has also recently been studied [25].  focus of investigation in [26][27][28][29]. In [30] elastic microstructural helices were employed to control sound.
In this article a new mechanism for the control of band gaps in the full elastodynamic context is introduced, based on the work in [31], which was restricted to the scenario of antiplane elastic waves. In that work, incompressible hyperelastic annular cylinders were embedded periodically in an elastic medium and their deformation ensured that band gaps could be switched on and off. Further however, it was shown, based on the theory of hyperelastic cloaking [32][33][34][35] that some materials permit deformation in a manner such that in a normal setting from a geometric viewpoint they would possess band gaps but the prestress prohibits this; these materials were termed phononic cloaks.
The theory of small-on-large was employed to derive the equations governing the propagation of superposed antiplane elastic waves and the plane-wave-expansion technique was extended in order to accommodate the inhomogeneous deformation present in the cylinders.
Here the same geometry as considered in [31] is employed, as depicted in Fig. 1 but here the annular cylinders are permitted to be compressible in order to accommodate the propagation of compressional waves.

Deformation and governing incremental wave equations
As illustrated in Fig. 1 the medium consists of a twodimensional square array (with period ℓ) of nonlinear isotropic elastic annular cylinders each with initial inner and outer radii R 0 and R 1 respectively, density ρ 1 and linear elastic bulk and shear moduli κ 1 and µ 1 respectively, embedded inside a homogeneous elastic host medium of density ρ 0 and linear elastic bulk and shear moduli κ 0 and µ 0 . The unit cell is depicted in its stress-free state in Fig. 1(a). All annuli are assumed to be deformed identically. The deformation consists of an inflation (leading to an inhomogeneous radial deformation) and extension along its axis such that, crucially here, the outer radius of the cylinder remains unchanged, r 1 = R 1 .
This results in a deformed configuration as depicted in Fig. 1(b). In principal the host medium can be linear or nonlinearly elastic but here we shall consider it as linear. Subsequent elastodynamic wave propagation depends on the initial deformation and the governing equations are determined by appealing to the theory of small-onlarge [18].
It is envisaged that the deformation results from inserting a stiff cylindrical inclusion with elastic properties µ 2 and κ 2 and density ρ 2 into the inner region of the annulus.

Annular cylinder inflation
The deformation of the annular cylinder is conveniently described in the form where the usual convention is followed, using upper (lower) case variables for the reference (deformed) configuration. Deformation is imposed such that Principal stretches are A second order nonlinear ordinary differential equation (ODE) is determined from the first equation of static equilibrium divT = 0, the second and third being trivially satisfied by the assumed form of deformation. The ODE takes the form where W is the strain energy function governing the constitutive behaviour of the annulus and g is some function that is known only when W is prescribed. This ODE for R(r) is subject to the boundary conditions (2). In general the ODE (4) cannot be integrated analytically except for six classes of strain energy functions [36]. The issue with these materials types is that they are not considered physically realistic. Therefore two alternative strain energy functions shall be considered here that have been proposed as realistic models of compressible nonlinear elastic materials. First the form introduced by Levinson and Burgess [37] is employed: which was been proposed as a compressible extension of the incompressible neo-Hookean strain energy function and this shall be referred to here as LB. Note that λ 1 = κ 1 − 2µ 1 /3. The second form considered is a compressible extension of the Mooney-Rivlin strain energy function [38]  which will be referred to as CMR. Note that W CMR incorporates the The deformation (R(r)) is determined numerically within MATLAB by writing the ODE (4) as a system of two first order ODEs, before employing the function bvp4c. The deformation is plotted in Fig. 2. As can be seen, the form of the deformation is largely independent of the choice of strain energy function, and the compressible deformation is only slightly different to the incompressible case. The function R(r) is then used within the incremental equations that govern the propagation of elastodynamic waves. The deformation is characterised not only by the axial stretch, but also by the volume change given by where V i and V f are the initial and final volumes of the annular section. The deformed inner radius is therefore given by For each case, in order to ensure that the annulus is deformed appropriately the necessary radial pressure is applied on the outer surface and the inner expansion is driven by the insertion of the stiff cylindrical inclusion. This is also accompanied by an axial stretch ζ and chosen volume change V . In principal these two parameters can be chosen so that the pressure exerted on the outer boundary of the annulus is zero, which is a more physically satisfying condition in terms of fixing the annuli inside a phononic medium. The annular (nonlinear) region is chosen to be significantly more compliant than the host region and it is assumed that the host material remains stress-free.

Incremental waves
Assume now that in-plane small-amplitude linear elastic waves propagate through the deformed structure. For geometrical reasons it is convenient to consider the displacements with respect to the cylindrical polar coordinate system and then transform to Cartesian coordinates when quasi-periodicity of the Bloch waves is imposed. In this case, the incremental displacement from the deformed configuration is given by u = (u r (r, θ )e r + u θ (r, θ )e θ )e −iωt , (9) where e r and e θ are unit vectors in the radial and azimuthal directions and where time-harmonic dependence has been assumed; ω is the angular frequency. It is straightforward to show that within the annular cylinder, in the periodic cell, the incremental wave equation is [18] where ζ is the push forward of the incremental nominal stress, given by ζ = M : γ in which γ = grad u and It can be shown that (see [18]) with no sum over repeated indices. Furthermore, provided that All other combinations give M ijkℓ = 0. In the host and inclusion media the governing wave equations are those of linear isotropic elastodynamics with isotropic elastic moduli κ 0 , µ 0 and κ 2 , µ 2 and densities ρ 0 and ρ 2 respectively. It transpires that for the study of in-plane waves there are four non-zero elements of ζ that are of interest, which are ζ rr = M rrrr γ rr + M rrθ θ γ θ θ , ζ rθ = M rθ rθ γ θr + M rθ θ r γ rθ , ζ θr = M θrθr γ rθ + M θrrθ γ θr , ζ θ θ = M θ θ rr γ rr + M θ θ θ θ γ θ θ .
The stress ζ zz is non-zero but does not influence the propagation of in-plane waves. The following two incremental wave equations are generated: In order to study Bloch-Floquet wave propagation through the deformed periodic structure, it is now necessary to transform (10) E.G. Barnwell ζ xy = M xyxx γ xx + M xyxy γ yx + M xyyx γ xy + M xyyy γ yy , (18) ζ yx = M yzxx γ xx + M yxxy γ yx + M yxyx γ xy + M yxyy γ yy , (19) ζ yy = M yyxx γ xx + M yyxy γ yx + M yyyx γ xy + M yyyy γ yy (20) where M ijkℓ given here with respect to the Cartesian basis are specified in terms of their cylindrical coordinate system counterparts in the Appendix.

Implementation of the modified plane-wave-expansion technique
The implementation of the plane-wave-expansion method is carried out in much the same manner as described in [31] expect that here instead of a scalar problem, the pair of coupled Eqs. (15)-(16) governing the (coupled) shear and pressure waves must be dealt with. First, the reciprocal lattice is introduced: where m, n span the integers. Introducing the Bloch wave vector K, displacements are written in the quasi-periodic form noting that the sum here is a double sum over m and n (spanning all integers). The (periodic) incremental moduli are written in (periodic) plane-wave form, e.g. (24) and analogously for all other M ijkℓ and the density ρ(x) with planewave (Fourier) coefficients m ijkℓ [G] and R[G] respectively.
Using (22)-(23) as well as the material property expansions such as (24) in the governing equations with the forms (17)- (20) generates an infinite homogeneous linear system for the compo- where, having truncated the system with m and n ranging from −N max to N max , the vector U[G] is written in the short-hand form . . .
It should also be noted that B depends only on the Fourier coefficients of the density, i.e. R[G]. The matrix eigenvalue problem (25) is solved using the MATLAB eig function and this enables the dispersion relation K = K(ω) to be determined.

Tuning band gaps: results
When presenting results associated with dispersion relations, the band diagrams will be plotted in the usual manner. That is, the frequency of each mode is determined whilst scanning the wavenumber around the edge of the irreducible Brillouin zone [39,2]. Following convention, the three points defining this boundary are denoted Γ (K = 0i + 0j), X (K = 0.5i + 0j) and M (K = 0.5i + 0.5j).
Throughout the results section, the Lamé parameters and density are non-dimensionalised on those of the host material and the wavenumbers on the lattice period ℓ, i.e. for j = 0, 1, 2 The nondimensional frequency therefore becomeŝ where c 0 = √ µ 0 /ρ 0 is the shear wave speed in the host material (and unstressed annulus due to the choice of properties above). The spatial variable is non-dimensionalised on the period, ℓ, i.e.x = x ℓ , y = y ℓ , and so the unit cell is mapped to the region −1/2 <x < 1/2 and −1/2 <ŷ < 1/2. The outer radius of the pre-stressed region remains fixed at R 1 = r 1 = 0.45. This value was chosen since it is large enough to give a sizeable pre-stressed region, but not too large so the pre-stressed region is close to the outer edge of the unit cell. Although it is possible to study the latter case as r 1 gets closer to 0.5, a significant increase in the number of terms in the PWE is required. Since this is not the main focus of the study it is avoided here.
For all calculations, the maximum plane-wave number is taken at N max = 10 (corresponding to 441 plane waves) which has been established to give sufficiently accurate results. The percentage error in the method is estimated by taking a large number of plane waves and is typically below 0.05%. Through the remainder of this paper, the non-dimensional properties in the host region and annulus are chosen to beμ 0 = µ 1 = 1,λ 0 =λ 1 = 2 andρ 0 =ρ 1 = 1. For the inclusion, the material properties are chosen to be stiffer and denser:μ = 5, λ = 30 andρ = 2 in this region.

P-SV mode
In Fig. 3 the effect on the band diagram associated with the coupled compressional/vertically polarised shear waves is illustrated after applying a pre-stress of the form described in Section 2.1 with ζ = 1.4 and volume change V = 0.8 for the LB and CMR materials. A significant change in the band diagram is observed including the appearance of an absolute band gap for the coupled mode. However, when compared to the band diagram of a stress free material with larger inclusion (with radius equal to the inner annulus radius in the deformed state), the stress free material exhibits a significantly larger band gap. This means that materials can be generated in a pre-stressed that permit the propagation of elastodynamic waves in frequency ranges that are not possible in unstressed media with the same geometry.
It is appears that altering the Mooney-Rivlin parameter has no significant effect on the shape of the band structure. However, the frequencies of the bands are altered slightly, illustrating some sensitivity to choice of material.
In Fig. 4 the effective (fundamental) shear and compressional wavespeeds are plotted and band gap widths for the various materials. In these plots, typical values of the pre-stress are chosen as  All materials show generally the same trends with regards to the wavespeed. Although the shear wavespeed for the different materials is largely unchanged, the choice of material properties has a significant effect on the compressional wavespeed. This shows a sensitivity to the strain energy function that is not initially obvious from the band diagrams.
It is clear that there is some dependence of the band gap width on the strain energy function. However, rather interestingly 6 E.G. Barnwell et al. / Extreme Mechanics Letters ( ) all strain energy functions cause the band gap to be switched on at a smaller initial radius than would occur in an initially unstressed medium. It is also interesting to note that applying differing amounts of compression, and keeping ζ = 1, has a large effect on the width of the stop band but does not change the point at which the stop band switches on. The ζ = 1 case is of specific interest since this means that the annulus could be held in situ within the material when imposing initial pre-stress.

Concluding remarks
In this paper the theory of small-on-large has been employed to incorporate pre-stress into a phononic crystal made up of a compressible elastic material. The effect of this pre-stress on the band structure was investigated for two different strain energy functions.
For the coupled pressure and shear mode we have shown that applying pre-stress can have the effect of switching on and off stop bands, giving a mechanism for real time tuning. The sensitivity to different strain energy functions was investigating and it was shown that in some situations the choice of constitutive behaviour has a significant effect on the material response. This effect is more pronounced for the compressional wave mode than the shear mode.
There is scope for future work influenced by the work given here. In particular, it would be interesting to include the SH mode to see if it is possible to create an absolute band gap for all modes. Similarly, it would be interesting to determine if it is possible to obtain a stop band in the pressure mode but allow the same frequencies to propagate in the shear mode. In doing this, by building a finite specimen it may be possible to convert pressure waves from a fluid into shear waves in a (phononic) solid, thus making efficient soundproofing at specific frequencies.