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A Lumped Model for Rotational Modes in Phononic Crystals

We present a lumped model for the rotational modes induced by the rotational motion of individual scatterers in two-dimensional phononic crystals comprised of square arrays of solid cylindrical scatterers in solid hosts. The model provides a physical interpretation of the origin of the rotational modes, reveals the important role played by the rotational motion in the band structure, and reproduces the dispersion relations. The model increases the possibilities of wave manipulation in phononic crystals. In particular, expressions, derived from the model, for eigen-frequencies at high symmetry points unambiguously predict the presence of a new type of Dirac-like cone at the Brillouin center, which is found to be the result of accidental degeneracy of the rotational and dipolar modes.

Although rotational motion has been extensively observed in EPCs [12,14,17], to the best of our knowledge, it has not yet been modeled. Without a model, it is difficult to accomplish the task of establishing the relationship between rotational modes and the microstructure of an EPC, which is crucial in utilizing the rotational modes to achieve intriguing wave propagation properties with potential applications. In the past, it seems that the focus of work on EPCs was on controlling the usual longitudinal and transverse modes [10,11,[13][14][15] in the crystal rather than on the physics of the rotational modes. Without a clear understanding of the rotational motion, however, it is difficult to fully understand the wave propagation properties in an EPC. 3 In this paper, we establish a model that describes the rotational modes in two-dimensional EPCs. We show that a simple lumped model captures the essence of the physics of the rotational modes, as well as offers a reasonably good prediction of the band structures of EPCs. Our model reveals the inherent link between the properties of the rotational modes and the microstructure of the EPC, and allows us to engineer an EPC that exhibits an unusual Dirac-like cone [6,20,21] at the Brillouin center. The newly observed Dirac-like cone is obtained by exploiting the rotational motion to achieve accidental degeneracy of a rotational mode and dipolar modes.

Lumped Model
The EPC considered in this work is a square array of cylindrical solid inclusions with radii R embedded in another solid matrix. The lattice constant is a , which is also the length unit.
The elastic wave propagation in such a system is characterized by the displacement of each point in the unit cell. We focus on the wave propagation along the X  direction and propose the lumped model shown in Fig. 1. 1 m and 2 m are the masses of the matrix and the scatterer, respectively, and the interaction forces between these masses are modeled by massless "springs" connecting them [22,23]. In fact, these springs are "continuous" springs, but for simplicity and without loss of generality, we demonstrate in Fig. 1 by using two individual springs connected to the scatterer at two arbitrary points, "A" and "B", on the boundary of the scatterer. To study the low frequency rotational modes, where the rotational vibration is more likely to occur in the scatterer about its axis, the mass 2 m is considered to be a finite-sized cylinder with radius R and a moment of inertia, 2 I , rather than a point mass like 1 m . The 4 finite-size assumption was also used in the study of granular systems [24,25], where the moment of inertia comes into play. A rotational mode naturally satisfies 0   u , which means that the transverse wave is required to excite the rotational motion as 0 l   u for longitudinal waves. In the low frequency limit, it can be shown that along the X  direction, only transverse waves are coupled with rotational motion. Therefore, we consider only transverse motion here as the translational movement. As a result, the displacements of the centers of masses are confined along the vertical direction and denoted by 21 i u  and 2i u for 1 m and 2 m , respectively, where i is the index of the unit cell.
The inter-mass force arises because of the relative distortion between the adjacent masses.
There are two sources for the restoring force exerted on 1 m in the i-th unit cell. One is the translational movement, i.e., the relative displacement between the centers of the mass and the scatterers nearby, which can be expressed as: where K is the spring constant and is related to both the bulk and shear moduli of the matrix [23] . The other source is introduced by the rotational movements of the two neighboring scatterers and takes the form: the rotated angle of the scatterer from its equilibrium position and 2 ,1 is the polar angle before deformation. Similarly, the total restoring force exerted on can be written as for the rotational part.
Newton's second law gives us the equations of motion for 1 m and 2 m : If we suppose that the vibration is time harmonic and invoke the Bloch theorem, we can transform Eq. (0) into the following secular equation:

1) Band structure and eigenstates
We test the validity of the lumped model by using a real EPC: a square array of steel cylinders with radius 0.2 Ra  embedded in epoxy. The mass density and longitudinal and transverse velocities inside the epoxy are, respectively, Thus, the value of K can be evaluated from 22 4 Km branch [26], in which the scatterer and the matrix move as a whole in the zero-frequency limit and the movement is concentrated on the scatterer at the Brillouin zone boundary.
Different from Figs. 2(b) and 2(c), pure rotation of the scatterer is found in Fig. 2(d), which is in accordance with the lumped model as frequency 2   is the natural angular frequency of an oscillating rotating cylinder and is a function only of the moment of inertia. The mode on the same band but at the Brillouin zone boundary is exhibited in Fig. 2(e), where the rotational motion of the scatterer is preserved but is weak compared with the strong translational displacements of the matrix. The matrices on the left and right sides are moving in the opposite directions as if they drive the rotational motion of the scatterer. We call this type of mode "in-phase", which is a contrast to the mode plotted in Fig. 2(g), where the rotation of the scatterer is not in line with the translational displacement of the matrix and we call it the 8 "out-of-phase" mode. Figure 2(f) demonstrates a mode that has a similar pattern of the "optical-type" mode [26], where the scatterer and the matrix are moving both vertically but in opposite directions. This, again, agrees with the lumped model prediction, which gives the typical frequency, i.e., 3   , of an optical mode of a diatomic chain [26]. In fact, this mode is the transverse state of the two degenerated dipolar modes. The other one depicted in Fig. 2   the other is a rotational mode. This is opposite to the case shown in Fig. 2(a).

2) Zero gap: a Dirac-like cone
Since the magnitudes of 2   and 3   depend on the scatterer's size, it is possible to make them equal by carefully tuning the radius of the cylinder. Thus, an interesting result, the zero gap width, is achievable as shown in Fig. 4(a), where the radii of the steel rods are 0.177 .
As expected, we find the rotational mode and the dipolar modes occur simultaneously at a dimensionless frequency 01 2 0.826 t ac   at the point. A bit surprising result is that in the vicinity of this frequency, the dispersion relations become linear as plotted in an enlarged view in Fig. 4(b). An equi-frequency surface at a frequency slightly below 0  , i.e., , is plotted in Fig. 4(c), with the blue circles indicating the results calculated by COMSOL. These circles form a perfect and large circle drawn as a red solid curve, which implies that the dispersion relation is isotropic. Similar results are also obtained for the upper branch but are not shown here. The isotropic behavior of the equi-frequency surface somehow contradicts our common understanding that the dispersion relation of an EPC in a square lattice is in general anisotropic [15,16,27]. In fact, this special characteristic is a result of accidental degeneracy and ensures that the dispersion relation near the degenerate frequency can be described in terms of a linear cone that intersects a flat sheet in the vicinity of the zone center, which is also called a "Dirac-like" cone and the vertex of the cone is called a Dirac-like point [21]. This behavior looks very similar to the Dirac-like cone of electromagnetic [20] and acoustic waves [6], where the monopolar and dipolar bands meet at the  point. Dirac-like cones were also found in an EPC when the accidental degeneracy of a  the dipolar and quadrupolar modes occurs and was understood from an effective medium perspective [14]. In that case, the effective mass density eff  , and the effective stiffness 44 1 C , are simultaneously zero at the Dirac-like point. Here, for the first time, we discover Dirac-like cones for elastic waves, which is a result of the accidental degeneracy of rotational and dipolar modes. The occurrence of the Dirac-like point cannot be interpreted by conventional effective medium theory [11,14], however, because the wavelength in the steel rods is longer than that in the epoxy matrix, deeming the conventional effective medium  Fig. 4(b)). Interestingly, a longitudinal dipolar mode with its vibration direction parallel to k is found. The imaginary part of mode "B" is two orders of magnitude smaller than its real part and is not presented. The eigenmode along the M  direction is plotted in Figs. 4(g) and 4(h) for the real and imaginary parts of the mode that belongs to the lower branch of the linear cone (marked as "C" in Fig. 4(b), whose 0.05

Conclusions and Outlook
In this work, we propose a simple lumped model that can account for the local rotation of the scatterer in certain two-dimensional EPCs. The model is useful for a physical understanding of the lowest transverse and rotational bands. It also provides a good prediction of the dispersion relations. With the model, it is convenient to go beyond the translational degree of freedom and utilize the rotational mode to achieve interesting wave propagation properties.
As an example, we demonstrate the occurrence of a new type of Dirac-like cone, which is a result of the accidental degeneracy of the rotational and dipolar mode at the Brillouin zone center. This Dirac-like cone possesses isotropic dispersion relations that can be excited only by transverse waves in the vicinity of the Dirac-like point even for a square lattice, which might lead to interesting elastic wave transport properties.