Viscoelastic Locally Resonant Double Negative Metamaterials with Controllable Effective Density and

Identify the factors that determine its performance and investigate the effect of these factors on its relevant performance by developing a mathematical model and applying appropriate techniques analysis and simulation I need to achieve the aim as describe earlier from either one of these designs which ever is easy to approach and also need guidance to write literature on passive and active metamaterials if possible.

Materials with a subwavelength structure and which have properties allowing the concept of negative refraction to be realised, were first investigated for electromagnetic waves [1�4]. The refractive index (n) of a homogeneous electromagnetic material is a function of its permittivity ?r and permeability ?r (n = �??r?r ). When both parameters are positive, the refractive index is positive real. When a wave transitions between two materials with positive real refractive indices, the change of direction across the boundary is as would be expected for conventional refraction, i.e. the incident and refracted waves lie on opposite sides of the boundary normal. When the parameters have opposite signs, the refractive index is complex and the medium partially blocks the propagation of the incident wave. Veselago [5] concluded that if both of these parameters could be made negative simultaneously, a negative refractive index material could result, i.e. n is negative real.

When a wave transitions between mediums with opposite signed real refractive indices, Snell�s law leads to a refracted wave which lies on the same side of the boundary normal as the incident wave. Pioneering work has shown that double negative materials can be constructed by combining two materials, one with the ability to provide negative permittivity and the other negative permeability [3]. The negative values for both parameters result from low frequency resonances built into a periodic structure. More recently metamaterials have been extended to pressure waves [6�18]. The mechanical analogues of permittivity and permeability are respectively, the modulus of elasticity and density. It follows that the refractive index for a pressure wave in a homogeneous medium is a function of density ? and modulus of elasticity ? (n = �??/? ). Thus negative refraction for a pressure wave in a homogeneous elastic medium occurs when the modulus of elasticity and density are simultaneously negative. For acoustics, a material with the properties of negative density and modulus of elasticity can be constructed using either a single resonant structure [16] or a combination of two different structural units [17,18]. An array of Helmholtz resonators connected to a one-dimensional transmission medium is described by Fig. 1a. An array of Helmholtz resonators connected to a transmission medium has previously been shown to provide negative effective modulus of elasticity in a designed frequency band [8,12].

Design from the Literature

These relationships have been derived through analogy with earlier work on electromagnetic metamaterials. Other arrangements have been shown to posses either a negative effective modulus of elasticity [6,13] or density [14,15]. In the single negative band these systems thus have a complex refractive index which acts to partially block the propagation of a pressure wave through the medium. In these studies the Helmholtz resonator has been modelled as a mass connected to a stationary reference point through a stiffness element and to the transmission medium through a damping element.

An array of Helmholtz resonators connected to a transmission medium can be approximated by the lumped parameter spring�mass� damper system illustrated in Fig. 1b. By applying D�Alembert�s principle to the individual masses and assuming a harmonic signal, this system can be represented by a homogeneous effective system consisting of a simple cascade of masses connected by parallel stiffness and damping elements, such as is described by:

The mass and stiffness of the effective homogeneous system are derived as Eqs. (2) and (3) respectively.

The mass term is a complex function, which can be divided into its real and imaginary terms. The real part (4) acts as the actual effective mass of the system and the imaginary term (5) acts as an additional loss term, which is effectively equivalent to a damper attached to a stationary point, i.e. this loss term is only a function of the motion of the respective point in the transmission system. The other dispersive loss term is given by Eq. (6) and is equal to the loss factor of the original transmission system.

�An active metamaterial with a controllable negative effective mass The stationary reference point intrinsic to acoustic Helmholtz resonators, results from the motion of the fluid mass in the connecting neck, against the compressible fluid contained within the attached volume. If the volume is encased in a rigid shell, the motion of the shell can be ignored, resulting in the stationary connection. In solid materials the equivalent modulus of elasticity of the transmission material is generally much higher than the equivalent modulus for most fluids and properties such as the Poisson effect become significant. Under these conditions the creation of this rigid connection is difficult, particularly in three-dimensional structures. However, a mass with a single elastic connection to the transmission medium can be relatively easily introduced into a solid material.

An active metamaterial with a controllable negative effective mass and stiffness The Skyhook controlled system described by Fig. 1c is adaptable but still only capable of providing a single negative effective parameter. To create an adaptable system with double negative properties the control system is extended to include the motion of the (n ? 1)th and (n + 1)th transmission masses. This modified control force fcn = kc (xn?1 + xn+1 ? 2xrn ) + cc (x?n?1 + x?n+1 ? 2x?rn ) leads to Eqs. (9) and (10) for the mass and combined damping�stiffness of the effective system.

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Fig. 2. The displacement transmission across the first and fourth masses (m) which form part of a series of four units connected by stiffness and damping elements (a), i.e. an ambient system with no LR structures (red line); and for the ambient system with the addition of actively controlled masses (mr) attached in parallel and designed to emulate Helmholtz resonators (black line). The sign of the real part of the dispersion curve for the effective mass of the actively controlled system is plotted in (b). The grey regions mark the single negative region, i.e. the region where the real part of the effective mass of the controlled system is negative. The model parameters are: m = 0.01 kg, mr = 0.1 kg, k = 3000 N/m, kr = 1000 N/m, kc = 2100 N/m, c = 0.1 Ns/m, cr = 0.002 N s/m, cc = 0.02 N s/m.

(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.) Both the effective mass (11) and stiffness (14) are dispersive and can become negative in a prescribed frequency band for certain mass, stiffness and damping combinations. For a homogeneous system of the form described by Eq. (1), the equation for the refractive index of an elastic medium can be generalised to a function of the mass and stiffness of the discrete elements. For such a system a negative refractive index occurs when the effective mass and stiffness are simultaneously negative. An array of actively controlled masses connected to a transmission medium described in a lumped parameter form thus constitutes a potential material with a negative refractive index in a prescribed frequency band, i.e. a double negative metamaterial.

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