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Wave propagation in fractionally damped nonlinear phononic crystals

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Abstract

Studying wave propagation in phononic crystals (PCs) in the presence of energy dissipation is a crucial step toward the precise dynamic modeling of periodic structures. Fractional calculus is an appropriate tool to reach a more perceptive idea of energy dissipation compared to other damping models. Therefore, in this work, we aim to provide a semi-analytical model for wave propagation in fractionally damped nonlinear PCs. For this purpose, the method of multiple scales is used to solve the governing equations of PCs, and the nonlinear dispersion relations of fractionally damped monoatomic chains and lattices are obtained. The Caputo definition of fractional derivatives is used to model damping. Besides providing new insight into the energy dissipation in PCs, the results of this research emphasize the importance of considering nonlinearities in modeling periodic materials, especially because the propagation frequency in nonlinear crystals is amplitude-dependent. The obtained results are validated with numerical modeling of fractionally damped PCs.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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  1. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

    Soroush Sepehri, Mahmoud Mosavi Mashhadi & Mir Masoud Seyyed Fakhrabadi

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  1. Soroush Sepehri
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  3. Mir Masoud Seyyed Fakhrabadi
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Appendices

Appendix 1: First Brillouin zone (FBZ) and irreducible Brillouin zone (IBZ) of a square lattice

The high symmetry points of \(G, X,\) and \(M\) along the edges of the IBZ are defined in Table

Table 1 The coordinates of high symmetry points for a 2D lattice with square topology in Cartesian and reciprocal coordinates
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1.

The wave vectors are then defined on the closed-loop of \(G - X - M - G\) along the edges of the IBZ. Also, FBZ and IBZ of a square lattice are depicted in Fig. 

Fig. 18
figure 18

FBZ (black dots) and IBZ (gray triangle) for a square lattice

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18.

Appendix 2: Numerical validation of dispersion curves when \(t \to \infty\)

To validate the analytic dispersion curve at very large times (\(t \to \infty\)) numerically, several simulations similar to Sect. 4 are performed. The analytic dispersion curves are obtained using a very large value of \(T_{1}\) in Eq. (22), reducing the dispersion frequency to \(\Omega \cong \omega_{0} + \varepsilon \mathcal{H}_{2}\). The numerical validation is presented in Fig. 

Fig. 19
figure 19

The numerical validation of the dispersion curve of a weakly nonlinear chain with fractional damping when \(t \to \infty\) for (\(m = 1,\;k = 1,\;\alpha = 0.1,\;\mu = 1,\;\gamma = 0.5,\;\varepsilon = 0.05,\;A = 2,\;h = 0.005,\;t_{l} = 100\)) along with the magnified frequency range of \(\Omega = \left[ {1.85, 2} \right]\)

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19. For a better comparative view, the dispersion curve of a linear undamped chain is also illustrated in a red dashed line. It is numerically verified that unlike the PCs with linear damping, the dispersion curve of a fractionally damped nonlinear chain does not converge to the dispersion curve of the linear chain even at very large times. However, due to the weak nature of the nonlinearities of the system, the curves are formed very close to each other. Therefore, the magnified dispersion curves at larger wavenumbers are also plotted separately.

Appendix 3: The \(\Omega - T^{*}\) planes of dispersion curves of 1D and 2D structures

More detailed information about the time-dependent behavior of the dispersion curves in 1D and 2D crystals can be obtained by investigating the corresponding planes of \(\Omega - T^{*}\) of the schematic 3D dispersion curves. They are presented in Fig. 

Fig. 20
figure 20

The \(\Omega - {\mathbf{T}}^{*}\) planes of 3D dispersion curves of a 1D and b 2D nonlinear fractionally damped PCs

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20a, b, respectively. The gray lines in Fig. 20a, b denote the maximum frequency of the linear PC without considering the effect of fractional damping. It is observed that the peak point exhibits a time-independent behavior, attaining constant values at different times. On the other hand, the time-dependent evolution of the dispersion curves of the fractionally damped nonlinear chain can be clearly noticed by comparing the frequencies at points \(A\) and \(B\). It is observed that as the time passes, the dispersion curves of the fractionally damped nonlinear PCs are converged to the ones corresponding to the linear undamped chains/lattices.

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Sepehri, S., Mashhadi, M.M. & Fakhrabadi, M.M.S. Wave propagation in fractionally damped nonlinear phononic crystals. Nonlinear Dyn 110, 1683–1708 (2022). https://doi.org/10.1007/s11071-022-07704-z

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