Topological transport of Lamb waves in pillared phononic crystals 

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Doubly negative property in an asymmetric double-sided pillared metamaterial

In Sec. 1.2, we demonstrate that in the single-sided pillared metamaterials the combination of the bending and the compressional resonances of the pillars can generate the negative effective mass density and the torsional mode of the pillars can contribute to the negative effective shear modulus. As proposed in Ref. [55], one practical scheme to achieve the doubly negative property is to combine two different substructures, each supporting a different resonant mode. Therefore, it is suggested that we can assemble the previously discussed two single-sided pillared metamaterials to form a double-sided system within which the negative effective mass density from one pillar and the negative effective shear modulus from another pillar are turned specially to the common frequency interval. Then, the doubly negative property can be expected in the merged structure.

Doubly negative property in merged structure

To achieve this, we develop two different single-sided pillared metamaterials in analogy to the previously two systems, denoted as SPM-A in figure 1.6(a) constructed by a periodic array of pillar A and as SPM-B in figure 1.6(b) constructed by a periodic array of pillar B. The parameters of pillar A are optimized to possess a low frequency band gap featuring the negative effective mass density and to ensure that the frequency of the torsional resonance of pillar B can fall into this band gap. Regarding SPM-A configuration, the diameter and the height of the pillar A were chosen to be d = 80μm and h = 200μm. The corresponding band structure is displayed in figure 1.6(d). It can be seen that a low frequency band gap occurs in between 5.19MHz and 5.47MHz. As concluded in Sec. 1.2.2, it should be accredited to the negative effective mass density generated by the combination of the bending and the compressional resonances of pillar A. The local resonances can also be evidenced by the flatness of the dispersion curves around the lower limit of the band gap and the eigenmodes at point M of the BZ, labelled as points C and D in figure 1.6(d). The deformation and displacement fields of the eigenmodes displayed in figure 1.6(h) illustrate that they are the second-order bending resonance and the first-order compressional resonance of pillar A. Further, the effective mass density matrix components are evaluated and their evolution against the excitation frequency is shown in figure 1.6(e). Both components turn negative in between 5.32MHz and 5.49MHz which is in good agreement with the forbidden band that goes from 5.19MHz to 5.47MHz with a small discrepancy of about 2.5% at the lower edge. With regard to SPM-B configuration, the parameters of pillar B are the same as the one proposed in Sec. 1.2.3. For comparison, the band structure is displayed in figure 1.6(f). The eigenmodes of the unit cell at points E, F and G are displayed in figure 1.6(h). Afterwards, to build a double-sided pillared metamaterial, we attach pillar A and pillar B to the top and the bottom sides of a thin plate respectively as shown in figure 1.6(c). Clearly, the merged structure is asymmetric about the mid-plane of the plate. To validate this approach to obtain the doubly negative property, we have computed the band structure of this asymmetric double-sided pillared metamaterial (ADPM). The dispersion curves are displayed in figure 1.6(g). As expected, an isolated negative-slope branch (red) appears in between 5.28MHz and 5.35MHz. In addition, we display the eigenmodes of the unit cell at some points (labelled from point C’ to point G’ in order) in figure 1.6(h).
Comparing the band structures of these three kinds of pillared metamaterials allows understanding the formation of this newly isolated branch. At point M of the BZ, the bending and the compressional modes (point C and point D) in figure 1.6(d) slightly shift to point C’ and point D’ in figure 1.6(g) upon attachment of pillar B to the bottom side of the thin plate. For both resonances, the displacement fields of ADPM displayed in figure 1.6(h) show that the deformation of pillar B is very small at the compressional resonance and even null at the bending resonance. This suggests that pillar B acts merely as an inert mass attached to the plate that simply shifts the resonant frequencies of pillar A. Accordingly, the frequency interval of the negative effective mass density generated by the resonances C’ and D’ of pillar A in ADPM also shifts and now appears in between 5.21MHz and 5.48MHz instead of 5.19MHz to 5.47MHz in SPM-A, but the overall mechanism leading to the negative effective mass density is the same for both pillared metamaterials.

Doubly negative property in a symmetric double-sided pillared metamaterial

It has been evidenced in Sec. 1.3 that the doubly negative property can be obtained by assembling two different single-sided pillared metamaterials. One features the negative effective mass density that results from the combination of the bending and the compressional resonances and the other one generates the negative shear modulus that comes from the torsional mode. What’s more, we need to ensure that the eigenfrequency of the torsional mode of one pillar can fall inside the low frequency band gap induced by the other pillar. And to meet such requirement, these two pillars usually have different dimensions. Thus, it might raise the question that what would happen in a symmetric double-sided pillar metamaterial that can be recognized as a more specific case. In this section, we are going to discuss a symmetric double-sided pillared metamaterial that enables the enlargement of the width of the low frequency band gap as reported in Refs. [76,83,84,95]. We show that the doubly negative property can be obtained through an appropriate choice of the dimensions of the pillars and the plate.

Occurrence of isolated negative-slope branch

The elementary unit cell of the symmetric double-sided pillared metamaterial is shown in figure 1.12(a). Two identical pillars are symmetrically arranged on both sides of the plate. The geometric parameters of the unit cell are the same as those of the single-sided pillared metamaterial in Sec. 1.2.2. The corresponding band structure is shown in figure 1.12(b). In contrast to the band structure of the single-sided pillared metamaterial displayed in figure 1.1(b), an isolated propagative negative-slope branch arises inside the complete band gap that opens in between 3.25MHz and 3.76MHz. This is almost twice the width of the band gap obtained in the single-sided pillared metamaterial. This enlargement is due to the strong coupling between the resonances of the double-sided pillars and Lamb waves propagating in the plate [76].
The isolated branch in between 3.53MHz and 3.57MHz divides the band gap into two narrower ones ranging from 3.25MHz to 3.53MHz and from 3.57MHz to 3.76MHz respectively. A zoomed view of the isolated branch is displayed in the inset. The negative slope of this branch throughout the first irreducible BZ cannot be afforded to a band folding effect and rather suggests simultaneous negative effective mass density and modulus resulting from the local resonances of the pillars. In support of this argument, we show in figure 1.12(c) the displacement field at two characteristic points in this branch. At point  of the BZ [point D in the inset of figure 1.12(b)] the displacement field clearly corresponds to a symmetric compressional mode of the double-sided pillars [left panel in figure 1.12(c)] whereas the motion of the double-sided pillars at middle point between  and X is a symmetric bending mode [right panel in figure 1.12(c)]. We show in the following that the former is responsible for the negative effective Young’s modulus whereas the latter, similar to the single-pillared metamaterial, leads to the negative effective mass density.

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Evolution of the double-negative branch against the geometric parameters

Being the consequence of a resonant phenomenon, the double-negative branch of the proposed structure is relatively narrow which may be a drawback for some applications. However, the width of the frequency band where the doubly negative property occurs can be increased through a proper choice of the geometrical parameters of the unit cell [82]. Here, we investigate the influence of the height and the diameter of the pillars as well as the thickness of the plate, on both the width of the forbidden band gap and the negativeslope branch.
Both figure 1.15(a) and (b) display the effect of the dimensions of the pillars on both the low frequency band gap and the negative-slope branch. Increasing the height of the pillar leads to the decrease of the central frequency of the band gap, as well as to the decrease of the range where the doubly negative property occurs. This should be related to the decrease of the compressional resonance frequency for increasing the height of the pillars [96]. Moreover, the lower part of the band gap broadens as the height of the pillars increases.  In contrast, the propagative branch moves closer to the upper limit of the band gap as the height of the pillars increases leading to the closure of the upper part when the height of each pillar is more than about 350μm. Remembering that the effective mass density tends towards zero while keeping negative values when the frequency approaches the upper limit of the band gap [see figure 1.13(b)], it is expected that this structure may behave as a zero-index elastic metamaterial. Actually, in that case the phase velocity in the metamaterial tends to infinity and therefore the refractive index (i.e. ratio of the velocity in the background to the velocity in the metamaterial) goes to zero. This point is further developed below. On the other hand, the bending resonance is very sensitive to the diameter of the pillars [82] and consequently this parameter has a large impact on the width of the band gap that broadens as the diameter increases [figure 1.15(b)]. However, the width of the double-negative branch is very little affected by this parameter and remains equal to about 8.5% of the width of the forbidden band whatever the diameter of the pillars is.

Topological transport in an asymmetric double-sided PPnC

As discussed in chapter 1, in the low frequency regime, A0 and S0 Lamb waves propagating in the plate can be modulated by the bending and the compressional vibration of the pillars. With regard to SH0 mode, its in-plane polarization is perpendicular to the propagation direction that can be well coupled into the torsional motion of the pillars. We have demonstrated that a double-negative branch can occur in a square lattice asymmetric double-sided PPnC by assembling the bending, the compressional and the torsional modes into a common frequency interval [19]. Besides, the wave propagation along ΓX direction at a frequency where the doubly negative property occurs is polarization-dependent. It can be considered as a propagative band for an incident SH0 wave, while it turns to be a forbidden one for an incident A0 (S0) Lamb wave. What’s more, unlike the respective intersection of the other dispersion curves, it is totally isolated inside a complete band gap that can simplify the discussion significantly. In this part, we develop a honeycomb lattice asymmetric double-sided PPnC and investigate the occurrence of the topological edge state in  analogy to QVHE in this system.

Artificially folding and polarization-dependent propagation

Firstly, we consider an asymmetric double-sided PPnC arranged in a triangular lattice. The corresponding elementary unit cell together with its first irreducible BZ are displayed in figure 2.4(a). Two distinct pillars denoted as pillar A and pillar B are concentrically connected to a thin plate. The lattice constant and the thickness of the plate are chosen to be a = 231μm and e = 100μm. Pillar A is designed to form a low frequency band gap that features the negative effective mass density by combining the bending and the compressional resonances. The diameter and the height are dA = 120μm and hA = 268μm. Further, the frequency of the torsional resonance of pillar B is optimized to occur inside the above band gap whose diameter and height are dB = 140μm and hB = 160μm. Then, the doubly negative property can be achieved. Afterwards, the band structure is computed and displayed in figure 2.4(b). As expected, a negative-slope branch exists in between 4.408MHz and 4.486MHz. And it would disappear when considering the singlesided PPnC constructed merely by pillar A or pillar B. Therefore, it is exactly a double-negative branch that possesses simultaneously negative effective mass density (in between 4.176MHz and 4.643MHz) and shear modulus (in between 4.408MHz and 4.486MHz). To give more evidences, the eigenmodes at points C, D and E labelled in figure 2.4(b) are depicted in figure 2.4(c). Clearly, the eigenmode at point C is the firstorder torsional resonance of pillar B that contributes to the negative effect shear modulus. The eigenmodes at points D and E are the second-order bending resonance and the first-order compressional resonance of pillar A respectively. Their combination accounts for the negative effective mass density. The color bar shown in figure 2.4(b) depicts the weighting of the torsional deformation of pillar B along the z-axis computed by       2 Pillar B Pillar B curl curl curl z dV dV.

Table of contents :

1 Doubly negative property in double-sided pillared metamaterials 
1.1 Introduction
1.2 Single negative property in single-sided pillared metamaterials
1.2.1 Lamb waves in a periodic structure
1.2.2 Negative effective mass density
1.2.3 Negative effective shear modulus
1.3 Doubly negative property in an asymmetric double-sided pillared metamaterial
1.3.1 Doubly negative property in merged structure
1.3.2 Enlargement of the double-negative branch
1.3.3 Polarization-filter behavior
1.3.4 Mode conversion phenomenon
1.3.5 Pillared metamaterial with chirality
1.4 Doubly negative property in a symmetric double-sided pillared metamaterial
1.4.1 Occurrence of isolated negative-slope branch
1.4.2 Formation of the double-negative branch
1.4.3 Evolution of the double-negative branch against the geometric parameters
1.5 Applications of doubly negative property
1.5.1 Refraction at the outlet of a prism-shaped supercell
1.5.1.1 Asymmetric single-sided pillared metamaterial supercell
1.5.1.2 Symmetric double-sided pillared metamaterial supercell
1.5.2 Cloaking effect in a rectangular supercell with void
1.5.2.1 Chiral asymmetric double-sided pillared metamaterial supercell
1.5.2.2 Symmetric double-sided pillared metamaterial supercell
1.6 Conclusion
2 Topological transport of Lamb waves in pillared phononic crystals 
2.1 Introduction
2.2 Constructing a single Dirac cone and a double Dirac cone
2.3 Topological transport in an asymmetric double-sided PPnC
2.3.1 Artificially folding and polarization-dependent propagation
2.3.2 Emulating QVHE
2.3.2.1 Topological phase transition
2.3.2.2 Valley-protected edge states
2.4 Topological transport in a symmetric double-sided PPnC
2.4.1 Occurrence of the Dirac cones and its evolution against the height of the pillars
2.4.2 Emulating QVHE
2.4.2.1 Topological phase transition
2.4.2.2 Valley-protected edge states of the antisymmetric dispersion curves
2.4.2.3 Valley-protected edge states of the symmetric dispersion curves
2.4.3 Emulating QSHE
2.4.3.1 Topological phase transition
2.4.3.2 Pseudospin-protected edge states
2.4.4 Pseudospin-valley combined edge states
2.5 Conclusion
3 Active control of transmission through a line of pillars 
3.1 Introduction
3.2 Eigenmodes of a line of pillars
3.3 Lamb waves reemitted by a line of pillars
3.4 Control of the transmission through a line of pillars by introducing external sources
3.4.1 Line of pillars with separated modes
3.4.2 Line of pillars with superimposed modes
3.5 Conclusion
General conclusion and perspectives 
Bibliography

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