The method of asymptotic homogenization is used to find the dynamic effective properties of a metamaterial consisting of two alternating layers of fluid, repeating periodically. As well as the effective wave equation, the method gives the effective equation of motion and constitutive relation in a natural way. When the material properties are such that resonant effects can be present in one of the layers, it is found that the metamaterial changes dynamically from a metafluid with anisotropic density and isotropic stiffness at low frequency to one with anisotropic stiffness when the frequency is near to one of the local resonances. In this region of frequency, the resulting metamaterial is not a pentamode material and thus does not belong to the class of metafluids that can be transformed to an isotropic fluid by a coordinate transformation.
The subject of metamaterials has received considerable attention over recent years (Smith et al. 2004). They are typically described as materials with properties that are difficult to achieve in nature but can be obtained by tailoring the sub-wavelength structure of the material (Pendry et al. 2006). In principle, a metamaterial could be defined for any physical property that is governed by a wave equation and applications have included negative refraction (Veselago 1968; Shelby et al. 2001), sub-wavelength imaging (Smith et al. 2003; Zhu et al. 2010), cloaking (Pendry et al. 2006; Norris 2008) and locally resonant sonic materials (LRSMs) (Liu et al. 2000).
The use of transformation acoustics implies that acoustic cloaking requires metafluids that have either anisotropic inertia, anisotropic stiffness (in the form of a pentamode material) or a mixture of both (Norris 2008, 2009). In fact, this transformation property has sometimes been used to define what is meant by a metafluid, since it includes all metamaterials that can be transformed into an isotropic fluid under a diffeomorphism. It has been demonstrated numerically that inertial cloaks (formed from a metafluid with effective anisotropic inertia and isotropic stiffness) can be achieved using layers of isotropic fluid at the microscale (Torrent & Sanchez-Dehesa 2008).
Complementary to the work on metamaterials is the subject of homogenization, which involves taking a partial differential equation with rapidly oscillating coefficients associated with a microscale and finding effective equations that govern the response on a much larger macroscale. For elasticity and acoustics, the interest has often been in finding the effective properties of composites and a number of different methods have been developed to achieve this. These include static variational methods (Hashin & Shtrikman 1963; Christensen 1979), multiple scattering theory (Waterman & Truell 1961; Varadan et al. 1978) and effective or self-consistent field methods (Sabina & Willis 1988; Kanaun & Levin 2003). Many of these methods rely on guessing the symmetry and form of the effective equations from the start.
For composites with a periodic microstructure, the method of asymptotic homogenization (MAH) can be used, which needs no a priori assumption about the form of the effective equations. It has been used extensively for static problems (Sanchez-Palencia 1980; Bakhvalov & Panasenko 1989) and exploits asymptotic expansions in ε, the ratio of the micro- to the macroscale in the material. This makes it ideally suited to dynamic problems, where there are natural definitions of these scales in terms of the wavelengths, and it has been used for low-frequency homogenization by Parnell & Abrahams (2006) and Andrianov et al. (2008). The case of high-contrast inclusions embedded in an elastic matrix, when higher Bloch modes may become part of the low-frequency response, has been studied in Cherdantsev (2009) and Smyshlyaev (2009). Recently, Craster et al. (2010) showed how to extend the method to high frequencies and derive an effective partial differential equation on the macroscale that accurately reproduces the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone.
In this paper, the MAH is applied to a simple periodic material consisting of two alternating layers of fluid. As well as giving the effective wave equation, it is shown that the MAH naturally gives the effective equation of motion and constitutive relation, and the resulting effective material is a metafluid with anisotropic inertia and isotropic stiffness, in agreement with Torrent & Sánchez-Dehesa (2008). It is then assumed that one of the fluids has a wave speed much slower than the other so that resonant effects can be present, even at low frequencies. Using the extension of the method by Craster et al. (2010), it is shown that near the local resonant frequencies, the effective material is a metafluid (in the sense that it is a metamaterial with an effective pressure and no shear stresses) with an anisotropic stiffness; however, it is not a pentamode material. This means that it does not belong to the class of materials that can be transformed to an isotropic (normal) fluid under a coordinate transformation (Norris 2008, 2009).
2. The two-layer system
In this section, the MAH is applied to a material composed of two layers of different fluid, repeating periodically. Figure 1 shows a schematic of the geometry: the fluid interfaces are taken to be normal to the x-axis, fluid 1 has thickness R and the whole system is periodic with period l. The system is taken to be two-dimensional (for the moment) though the extension to three dimensions is trivial.
Taking fluid 1 to have density ρ1 and bulk modulus κ1 (and similarly for fluid 2), the linearized constitutive relations and equations of motion relating acoustic pressure, p, and velocity, v, in each fluid are given by (Kinsler et al. 2000) 2.1for harmonic excitations proportional to . These lead to the wave equations, 2.2where ki=ω/ci denotes the wavenumber in fluid i, and is the corresponding acoustic wave speed.
It is now assumed that the wavelength in each fluid is much longer than the period of the microscale, l. New coordinates, (ξ1,ξ2)=(x/l,y/l), are introduced for the microscale and the small parameter, ε=k2l, defined. The essence of the method is to assume that p and v are functions of both the microscale coordinates, ξi, and a new set of coordinates on the macroscale, X=(X,Y), where 2.3
Following Parnell & Abrahams (2006), L(ε) will be taken (for the moment) to have the asymptotic form: 2.4Expanding the derivatives using the chain rule, viz., 2.5leads to the scaled constitutive relations and equations of motion, 2.6and associated wave equations 2.7All the equations have been expressed in terms of the material properties of fluid 2 using the non-dimensional parameters 2.8and 2.9
In addition to the differential equations, p and vx must also be periodic at the microscale and continuous across the interface ξ1=r=R/l. For convenience, the domain of ξ corresponding to fluid 1 will be donated and that for fluid 2 by .
Expanding 2.10aand 2.10bleads to a hierarchy of equations that need to be satisfied at each order in ε. Focusing on the wave equation (for the moment), the leading order equation is 2.11This has a general solution 2.12for arbitrary functions f and g. Since ξ1 is restricted to the range (0,1) owing to the periodicity, the functions f and g can be expanded in a Fourier series 2.13There are no restrictions on ξ2 thus p0 will be unbounded unless it is independent of ξ2. It is easy to see that this will happen at all orders thus p (and v) can be assumed to independent of ξ2 from now on.
Since p is continuous, 2.14The x component of the velocity, vx, must also be continuous across the interfaces. From the second equation of equation (2.6), this gives the boundary conditions 2.15where denotes the boundary of . The leading order solution is thus 2.16where A is an arbitrary function of X only. It will be found from the order ε2 equations that A(X) must satisfy an effective wave equation.
At order ε, the wave equation gives 2.17Following Parnell & Abrahams (2006), it is convenient to define 2.18then 2.19and continuity of p1 across the interfaces together with the periodicity leads to 2.20
Continuity of vx (through the asymptotic expansion of equation (2.6)) then gives the boundary conditions 2.21These are satisfied if 2.22aand 2.22bwhere 2.22c 2.22d and 2.22e Thus the solution at this order is expressed in terms of the (arbitrary) constants, a, d0, d1 and the arbitrary function, B(X).
The order ε2 problem then leads to the effective wave equation at the macroscale. From the expansion of the wave equation, 2.23with the boundary conditions that p2 is continuous across the interfaces and 2.24
Integrating equation (2.23) over ξ1 from 0 to r, and adding to it m times, the integral from r to 1 gives 2.25Substituting in equation (2.25) for the form of the solution for p1 from equation (2.18) then leads to the effective equation for A(X), 2.26
Since p≃A(X)+O(ε), equation (2.26) is also the effective wave equation for the pressure at leading order, which solves the mathematical homogenization problem. To use the system as a homogenized medium, however, it is also necessary to know the effective velocity and there is physical insight to be gained from a knowledge of the homogenized constitutive equation. For these reasons, attention is now focused on the asymptotic expansions of equation (2.6). The second of these has already been used to set the boundary conditions for the wave problem and leads to the hierarchy of equations: 2.27
The first of these equations gives V0=0 since p0 is independent of ξ. It can then be seen from the second equation that, although vx is independent of ξ to leading order, vy changes discontinuously when ξ1=r. Thus, even in this simple case, taking the low-frequency limit of the equations of motion on its own does not lead to physics that is independent of the microscale at leading order and the concept of homogenized variables must be introduced. These are taken to be the average over the microscale of the physical variables, i.e. 2.28aand 2.28b
The expansion of the constitutive relation (2.6) in terms of ε leads to the set of equations 2.30Integrating the equation for p0 from 0 to r, and using V0=0, gives 2.31A similar equation is obtained by integrating p0 from r to 1. Thus 2.32Since the x component of V2, V 2x, must be continuous across the interfaces, this leads to the homogenized constitutive relation 2.33
It can be seen that the homogenized equations of motion and constitutive relation are those for a metafluid with anisotropic density and isotropic stiffness. This system has been used before as an inertial cloak (Torrent & Sánchez-Dehesa 2008) and these expressions for the effective modulus and inertia tensor agree with those given previously. That composites of this type typically lead to an anisotropic inertia is well known (Milton & Willis 2007). It is straightforward to show that equation (2.29) with equation (2.33) correctly reproduce the effective wave equation (2.26).
3. The locally resonant case
It is now assumed that c1≪c2 so that local resonant effects can be observed even when the wavelength in fluid 2 is much longer than the period of the microscale. To this end 3.1is set. The equations in fluid 2 are the same as in the previous case and the boundary conditions are unchanged. To simplify matters, L(ε)=ε is taken (L2 had no effect on the homogenized equations to leading order) and, since the perturbation equations in fluid 2 are still forced Laplace equations, the solution is assumed to be independent of ξ2.
If the ansatz (2.10) is taken and the wave equations expanded as before, the leading order system only gives a non-zero solution when α=αn=2nπ/r. This is analogous to the high frequency case (Craster et al. 2010) and it is to be expected that the MAH will only give the effective equations close to these frequencies. Following Craster et al. (2010), it is assumed that 3.2
At leading order, the wave equation in fluid 1 is then 3.3The leading order equation for fluid 2 is still Laplace's equation, (2.11), and the boundary conditions are the same as before thus 3.4
At order ε, 3.5and p1 satisfies equation (2.17) in . Before solving these equations, a consistency condition can be found which allows a1=0 to be deduced (Craster et al. 2010): since P0 satisfies the same differential equations as p0, 3.6Thus (using the orthogonality of P0 and ∂P0/∂ξ1 in ) 3.7Similarly, using the form of the equations in , 3.8Adding equation (3.7) to m times equation (3.8) and using continuity of P0, p1 together with the boundary conditions (2.15) and (2.21) shows that a1≡0.
The wave equation in fluid 1 at order ε2 is 3.10together with equation (2.23) in fluid 2. As in the high frequency case (Craster et al. 2010), finding the consistency condition at this order leads to the effective equation for A(X). Multiplying equation (3.10) by P0 and subtracting p2 times the equation for P0, then integrating over , gives 3.11Similarly, 3.12
Adding equation (3.11) to m times equation (3.12) and using the continuity of pressure and normal velocity then gives the required differential equation, 3.13It is also straightforward to find the form of the solution for p2 and it is given by 3.14with 3.15aand 3.15band D(X), E(X) and G(X) are arbitrary functions of X.
As before, the homogenized pressure and velocity will be taken as the average values over the period of the microstructure. Thus 3.16
Expanding the equation of motion in fluid 1 leads to the same hierarchy of equations as before. Since the integral of V0 over is zero, using equation (2.28b) and the form of the solutions for p0 and p1 gives 3.17
Expanding the constitutive relation for fluid 1 in ε leads to the system of equations 3.18Integrating the last of these over gives 3.19Also, 3.20Using the continuity of V 2x together with the form of p2 then gives 3.21leading to the effective constitutive relation 3.22where ωn=2πnc1/R. Thus, near the resonant frequencies ωn (n≠0), the system behaves effectively as a metafluid with both anisotropic density and anisotropic stiffness. In fact, if the densities in the two fluids are the same (m=1), it is dynamically a metafluid with isotropic density but anisotropic stiffness. As the shear stress is everywhere zero, it is still essentially a fluid; however, the pressure is no longer related to the gradient of velocity in the usual way. The results are easily extended to three dimensions noting the symmetry of Y and Z. Again it is straightforward to show that these effective equations of motion and constitutive relation reproduce the effective wave equation (3.13).
It is tempting to identify the effective material in this region as a pentamode material. From Norris (2008), the elasticity tensor of a pentamode material has the form 3.23for some symmetric tensor Sij, which gives an elasticity tensor that automatically has the symmetry 3.24In addition, this form for the elasticity tensor leads to the relationship 3.25between the stress tensor, σij, and the ‘pseudo pressure’, p*.
Noting that the homogenized stress tensor, , is simply given by 3.26if the homogenized stress–strain relationship in Cartesian coordinates, xk, had the form 3.27(where for steady state and summation over repeated indices is assumed) then the homogenized elasticity tensor Cijkl would have components 3.28a 3.28b and 3.28c Here 3.29These components are incompatible with the form (3.23); thus, although the metamaterial has anisotropic stiffness and no shear stresses, it is not a pentamode material. In fact, an elasticity tensor defined in this way would violate the symmetry (3.24) and could not be physical; thus the assumption that the dynamic relations arise from an effective constitutive relation of the form (3.27) is invalid in this frequency range. Other forms for the effective constitutive equations have been considered in Milton & Willis (2007).
4. Discussion and conclusions
The MAH forms a powerful tool for obtaining the effective properties of metamaterials with periodic microstructure. It involves no a priori assumptions about the form of the effective constitutive relations and thus can be used to investigate the nature of the effective materials obtained. In the case of the metamaterial considered, it is clear that its form changes dynamically from a metafluid with anisotropic density and isotropic stiffness at low frequency to one with anisotropic stiffness when the frequency is near to one of the locally resonant frequencies.
The technique used, which is an application of the method of Craster et al. (2010), gives the effective properties near the locally resonant frequencies, ωn. In fact, consideration of the exact solution (Schoenberg & Sen 1983) indicates that, in general, the solution is comprised of a sum of eigenmodes; however, near the ωn, it will be dominated by a single mode and the material should behave as an effective material obeying the effective equations for this mode.
In deriving the effective properties, the periodic boundary conditions have been applied to the microscale variables, ξ, only. The assumption here is that on the large scale, X, everything is approximately constant over the period of the microscale so that the system behaves as an effective material with the effective pressure, , and the effective velocity, . For high-contrast media (e.g. air in water), it is possible that the effective velocities arising in the effective wave equation (either equation (2.26) or equation (3.13)) can be slower than the velocities in the constitutive media and the solution for A(X), the amplitude of the lowest order contribution to the pressure will therefore vary at the microscale. Since X was defined as the scale associated with the wavelength in fluid 2, it is still long so the assumptions leading to equations (2.26) and (3.13) should still hold. The pressure, p, still has to be periodic over period l so presumably the periodic boundary conditions would now have to be imposed on A, leading to a band structure associated with the effective wavenumber. The homogenized variables and would no longer be meaningful however, and it is arguable whether the system should be considered an effective material. This perhaps is the distinction between an effective medium (or composite) and a metamaterial.
Finally, throughout this paper, the resulting metamaterial has been referred to as a metafluid; however, it has been shown that near the local resonant frequencies it is not a pentamode material. This means that it does not belong to the class of materials that can be transformed to an isotropic (normal) fluid under a coordinate transformation (Norris 2008, 2009). If this is used as the strict definition of a metafluid, then it would not be one. However, it has been shown that the resulting metamaterial obeys equations of motion and constitutive equations which are modifications of the usual fluid equations, does not support shear stresses, and the effective pressure is the actual averaged pressure seen on the macroscale (unlike the pseudopressure of some pentamode materials); thus it seems natural to refer to it as a metafluid.
This property of the constitutive relation has implications for cloaking: since the system does not transform to an isotropic fluid near the resonant frequencies, there is no choice of the material properties of the individual layers that can give perfect cloaking in the sense of the transformation technique. Depending on how the material parameters are chosen, either the wave equation or the boundary conditions in the transformed coordinate system will be different from those of an isotropic fluid at these frequencies. Substantial reduction in scattering might still be observed if the deviation in one of these conditions is not too great, but this would have to be checked by simulation. In any event, it seems that the material properties of the individual layers would have to change from the values at low frequency to maintain a significant reduction in scattering.
The author would like to thank Victor Humphrey for interesting discussions on the behaviour of LSRMs. ©Crown Copyright 2011. This was published with the permission of DSTL on behalf of the controller of HMSO.
- Received April 12, 2011.
- Accepted June 14, 2011.
- This journal is © 2011 The Royal Society