Anisotropy and Dispersion of Elastic Waves in Periodically Multilayered Anisotropic Media

Significance 

Periodically multilayered anisotropic media, also known as layered phononic crystals, have attracted significant research because of their unique ability to precisely control and manipulate wave propagation. These media are constructed by the repeated stacking of unit cells, each consisting of layers of anisotropic materials, which exhibit different mechanical properties in different directions. Layered phononic crystals offer the potential to develop advanced materials and devices with tailored mechanical properties, which enables applications including waveguides, vibration isolators, and noise reduction systems. Moreover, the ability to control wave propagation in anisotropic media can lead to significant advancements in non-destructive testing, seismic wave mitigation, and the design of acoustic metamaterials. Despite these promising applications, several challenges arise in studying elastic wave propagation in periodically multilayered anisotropic media. The primary challenge is in the inherent complexity of anisotropic materials. Unlike isotropic materials, where properties are uniform in all directions, anisotropic materials exhibit direction-dependent properties, which makes any analysis significantly more complicated. This complexity is compounded by the periodic nature of the media, which introduces additional boundary conditions and wave interactions that must be accounted for.

One of the key challenges is accurately modeling the wave propagation in such media. Traditional methods often fall short in handling the boundary conditions and the coupling of wave modes that occur in periodically multilayered structures. The interactions between the wave modes within neighboring periods, particularly at high frequencies, further complicate the analysis. Additionally, the dispersion and anisotropy characteristics of elastic waves in these media are not well-understood for materials with arbitrary anisotropy (triclinic materials). Another significant challenge is the experimental validation of theoretical models. The periodicity and anisotropy of the media require sophisticated experimental setups and measurement techniques to capture the wave propagation characteristics accurately. The development of reliable numerical methods that can handle the high computational demands of simulating wave propagation in such complex media is also vital. To address these challenges, new study published in Composite Structures, led by Professor Y.Q. Guo, doctoral candidate Q.Q. Li, and engineer B.R. Peng from the College of Civil Engineering and Mechanics at Lanzhou University, designed a combination of state-space formalism and the Floquet-Bloch theorem, along with the method of reverberation-ray matrix (MRRM). The new approach provides a comprehensive understanding of the anisotropy and dispersion characteristics of elastic waves in periodically multilayered anisotropic media.

The researchers modeled the periodically multilayered anisotropic media as a series of unit cells, each consisting of layers of anisotropic materials. They considered the general case where the materials exhibit triclinic (arbitrary anisotropy) properties. They also derived the dynamic governing equations and conditions for the constituent layers using the state-space formalism. This formalism allowed them to describe the wave propagation within each layer in a systematic manner. The Floquet-Bloch theorem was employed to handle the periodic boundary conditions inherent in the multilayered media. This theorem is crucial for describing the periodic relations between neighboring unit cells.

The authors used MRRM to derive the general dispersion equations for the media. The method is effective in both low and high-frequency ranges and is particularly suited for analyzing the coupled waves in arbitrary directions within the periodically multilayered anisotropic media. The researchers fully characterized the anisotropic and dispersion properties of the elastic waves by solving these dispersion equations for all possible wavenumber components. Moreover, the team implemented a computational program to perform numerical simulations, which validated their theoretical models. They used an exemplified periodically bilayered medium consisting of calcium molybdate (CaMoO4) and sapphire (Al2O3) layers with distinct anisotropic properties. They calculated the slowness and phase velocity curves for elastic waves propagating on three coordinate planes: XOZ, YOZ, and XOY. These calculations provided insights into the anisotropic characteristics of the waves. Additionally, they explored the effects of varying the anisotropic parameters of the constituent materials on the slowness and phase velocity curves. This analysis helped them understand how different degrees of material anisotropy influence wave propagation.

The researchers also computed various dispersion curves, including frequency-wavenumber, frequency-wavelength, and frequency-phase velocity curves. These curves were analyzed for waves propagating in directions perpendicular, parallel, and oblique to the layering. They employed a root-searching technique combining bisection and golden section methods to solve the dispersion equation and obtain the required curves.

The authors showed that the slowness curves on the XOZ and YOZ planes are periodically distributed along the thickness direction due to the periodic stacking of layers. The least positive period decreases with increasing frequency, indicating more complex interactions at higher frequencies. Moreover, the phase velocity curves show that as the absolute phase velocity along the thickness direction decreases, the curves become denser, particularly at higher frequencies. This densification corresponds to the periodic nature of the media and the interaction between wave modes within overlapping periods. They studied the effect of varying the anisotropic parameters of the constituent materials and shed it affects the shape and symmetry of the slowness curves. For example, increasing or decreasing specific parameters like δ36 and δ45 breaks the anti-symmetric property of the two quasi-Shear (qS) waves, reflecting the unique influence of the interfacial and periodic boundary conditions.

For waves propagating perpendicular to the layering, the frequency domain is divided into alternating passbands and stopbands. The wavenumber exhibited periodicity, while the wavelength and phase velocity show multivalued behavior, becoming asymptotic to frequency lines corresponding to zero wavenumber. Additionally, waves propagating parallel to the layering exhibited dispersion characteristics similar to Lamb waves in laminated composite plates. The lowest three modes propagate from zero frequency, while higher-order modes have cutoff frequencies above which they propagate. Moreover, for waves propagating obliquely, the dispersion characteristics combine those of perpendicular and parallel propagation. Mode repulsion and transition occur due to the zone folding effect of periodicity, and some higher-order modes can propagate within narrow frequency ranges below their cutoff frequencies. The researchers validated their theoretical findings and computational program by comparing the phase constant spectra of elastic waves propagating perpendicularly to the layering with results obtained using the finite element method. The excellent agreement between the two sets of results confirmed the accuracy of their approach.

Professor Y.Q. Guo and team findings are significant because they provided a detailed and comprehensive characterization of the anisotropic and dispersion characteristics of elastic waves in periodically multilayered anisotropic media. Combining the state-space formalism, Floquet-Bloch theorem, and the MRRM, the researchers successfully developed a robust framework to analyze wave propagation in these complex materials. Phononic crystals are materials engineered to control and manipulate wave propagation. The authors’ findings contribute to the design and optimization of layered phononic crystals by providing a detailed understanding of how elastic waves propagate in anisotropic, periodically structured media. This can lead to the development of advanced materials with tailored properties for specific applications. Moreover, the ability to predict and control wave propagation in anisotropic multilayered media can significantly impact the design of materials for vibration isolation, noise reduction, and waveguiding. According to the authors, with better understanding the influence of anisotropic parameters, engineers can design materials with specific wave propagation characteristics, leading to improved performance in various applications such as acoustic filters, waveguides, and metamaterials. Furthermore, the study has important implications for non-destructive testing and structural health monitoring. With the understanding how elastic waves propagate in anisotropic media, more accurate and reliable methods can be developed for detecting defects and monitoring the health of structures. This is important in aerospace, civil engineering, and manufacturing, where the integrity of materials and structures is critical. With the design of better materials for vibration isolation and noise reduction, the findings can contribute to improved infrastructure and public safety. For example, in earthquake-prone regions, materials designed to mitigate seismic waves can protect buildings and save lives. In conclusion, Professor Y.Q. Guo and colleagues provided comprehensive analysis of elastic wave propagation in periodically multilayered anisotropic media, the study will have significant implications in the fundamental understanding and practical applications of wave control, and it’s potential to drive advancements in material design, infrastructure safety, and economic development.

Anisotropy and Dispersion of Elastic Waves in Periodically Multilayered Anisotropic Media - Advances in Engineering

About the author

Y.Q. Guo is currently a professor at the College of Civil Engineering and Mechanics, Lanzhou University. He was born in 1979 in Inner Mongolia, China. He received his Master degree in 2005 from Lanzhou Jiaotong University and his Ph.D. degree in 2008 from Zhejiang University, China. He joined Lanzhou University in 2008 and since then was the principal investigator of two research projects from the National Natural Science Foundation of China, one project from the China Post-doctoral Science Foundation, and one project from Lanzhou University. He published more than 20 journal papers and 2 book chapters. His research activities relate to the dynamics of structures and elastic wave propagation in solids, such as periodic layered and framed structures, composite and smart structures.

About the author

Q.Q. Li is currently a doctoral candidate at the College of Civil Engineering and Mechanics, Lanzhou University and also a teacher at the College of Urban and Rural Construction, Shanxi Agricultural University. He was born in 1989 in Gansu, China. He received his Bachelor degree in 2013 and Master degree in 2016, respectively, both in Civil engineering from Lanzhou University. The main research interest is the propagation and control of elastic waves in periodic layered media.

About the author

B.R. Peng is currently an engineer at the College of Civil Engineering and Mechanics, Lanzhou University. He was born in 1974 in Gansu, China. He received his Bachelor degree in 1999 in structural engineering from Southwest Jiaotong University and his Master degree in 2008 in bridge engineering from Lanzhou Jiaotong University, respectively. His research interest includes the experimental measurement and the numerical analysis of static and dynamic behaviors of periodic structures.

Reference

Y.Q. Guo, Q.Q. Li, B.R. Peng, Anisotropy and dispersion of elastic waves in periodically multilayered anisotropic media, Composite Structures, Volume 324, 2023, 117558,

Go to Composite Structures

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