Former subsections discussed how different kinds of special local resonators can modify a rigid mass and a deformable spring to achieve frequency-dependent Meff and Keff, respectively. In this section, detailed analysis on their behaviors and utilization to metamaterials will be discussed. As shown in Fig. 2.7, a series of lumped elements containing the aforementioned local resonators can be interpreted as a simple mass-spring lattice having effective properties. Although similar to the periodic mass-spring system discussed in Section 2.2, the unbounded effective parameters from the local resonators are the key fundamentals for anomalous wave phenomena in metamaterials. In fact, for such lattice with effective parameters, the governing Bloch relation in Eq. (2.5) can be modified to
2 2
4 sin
eff eff 2
M K qd
. (2.24)
To examine how the parameters affect the wave propagation, further studies with dispersion relations formed by each effective parameter are provided.
First, a dispersion curve and its corresponding Meff are shown in Fig. 2.8(a). Here, Keff is remained constant to investigate the effect of only Meff . As observed from Eq. (2.17), Meff starts from Mmmm, which is the static total mass value at 0. Then, Meff experiences unbounded range of values near the resonance frequency (r). Also, it should be noted that Meff rapidly increases below r and then starts from negative infinity just above r. After recovering to positive values from the rapid increase, Meff converges to Mm, which is the outer mass value. Owing to the trait of the Meff curve, the dispersion curve experiences
“resonance gap” near r. Specifically, q first becomes complex just below r due to the increased Meff term. Above r, q becomes purely imaginary due to
Meff’s negative value. The former gap can otherwise be named “Bragg-like bandgap” owing to the complex value of q. All these observation can be confirmed from Eq. (2.24) and Fig. 2.8(a).
Similarly, another dispersion curve and its corresponding Keff is shown in Fig.
2.8(b). In contrast to Meff, Keff first experiences negative value just below r,
and then starts from positive infinite value just above r. Because of this fact, the induced dispersion curve exhibits similar shape but different location of the resonance gap. Although the Bloch wavenumber q has both the complex and purely imaginary value near r, passband starts from r on contrary to the Meff case.
This is because in Eq. (2.24), q can have real values as long as Keff term is positive. On the other hand, for the Meff case, q became complex when Meff exceeded a certain value.
Now, by combining only the negative regions of these two, double negativity phenomenon is investigated in Fig. 2.9. Double negativity is considered to be the most fascinating trait of the metamaterials for their unusual features that do not exist in nature. As can be observed in the figure, the negative branch has opposite signs between the phase velocity
sin( / 2) / 2
eff p
eff
K qd
V d
M qd
, (2.25)
and the group velocity as in Eq. (2.12). This means that the phase momentum direction is opposite to the direction of energy propagation. To visualize such propagation effects of metamaterials, configurations of dynamic shapes of elements are provided in Fig. 2.10.
Fig. 2.10(a) shows the wave propagating shapes for double-positivity case, which is similar to that of normal materials. The wave phase moves (defined by phase velocity) in the same direction as the wave propagating direction (defined by group
velocity). In Fig. 2.10(b), not only decaying but also fluctuation of the lattice occur simultaneously due to the complex q, composed of both real and imaginary value.
This Bragg-like bandgap has attenuation factor which is determined by the imaginary component. On the other hand, the resonance gap phenomenon is shown in Fig. 2.10(c). Here, in contrast to the former bandgap, the wave field has only the decaying motion without fluctuation because q has only the imaginary value. Thus, this gap may be more efficient in some applications. Finally, the double-negativity inducing negative phase velocity is described in Fig. 2.10(d). The peculiarity of double negativity comes from manipulating the wave to propagate in the desired direction while guiding the phase to propagate in the opposite direction. This characteristic can otherwise be described to have negative refractive index.
Overall, with delicately designed local resonators to exclusively tune each elastic parameter, the design domain for wave manipulation can be extensively broadened.
Fig. 2.1 1-D periodic mass-spring lattice.
Fig. 2.2 (a) Dispersion relation of the mass-spring lattice. (b) Dispersion relation of the irreducible Brillouin zone with real (black solid line) and imaginary (purple dotted line) components.
Fig. 2.3 2-dimensional simple mass-spring model with different periodicity and spring constants in the x and y directions.
Fig. 2.4 (a) Dispersion surface for the 2-dimensional lattice. The irreducible Brillouin zone is denoted with Greek letters Г, X, and Ϻ. (b) Dispersion surface of the first Brillouin zone. The black contours are EFCs that show the wavenumber components at the target frequencies denoted in the figure. The arrows denote the group velocity vectors.
Fig. 2.5 Schmatic drawing for a dipolar resonator and its equivalent rigid mass with effective mass property.
Fig. 2.6 Schematic drawing for a monopolar mechanical resonator and its equivalent spring model having effective stiffness.
Fig. 2.7 Combined lumped element model with local resonators and its equivalent simple mass-spring model with effective parameters.
Fig. 2.8 (a) Dispersion curve for a simple mass-spring lattice with frequency dependent effective mass and constant stiffness. (b) Dispersion curve for a simple mass-spring lattice with constant mass and frequency dependent effective stiffness.
Fig. 2.9 Dispersion curve for a simple mass-spring lattice with frequency dependent effective mass and stiffness specially tuned to have overlapped negative region to yield negative curve.
Fig. 2.10 Displacement plots of the particles for (a) double-positivity, (b) single- negativity where q is complex, (c) single-negativity where q is purely imaginary, and (d) double-negativity cases with respect to time variance where t1<t2<t3. The group and phase velocities are denoted by Vg and Vp, respectively. All plots are calculated by a finite lattice composed of 30 mass elements.