Session: 01-01-04: Elastic and Acoustic Metamaterial

Paper Number: 99500

99500 - Ray Theory for Wave Propagation in Graded Metamaterials 

Periodic metamaterials exhibit fascinating wave propagation behavior due to their ability to steer and attenuate waves. While extensive research in recent decades has focused on waves in periodic metamaterials, the design space extends far beyond periodic architectures. Spatially graded architectures composed of smoothly varying unit cells can easily be manufactured. Furthermore, graded metamaterials show high potential to manipulate waves in unprecedented ways, such as for broadband attenuation and wave guiding along curved trajectories. However, the design space of spatial grading is largely unexplored due to a lack of efficient modeling tools. Bloch wave analysis requires the assumption of periodicity while finite element modeling requires high spatial and temporal resolutions, which is inefficient for inverse design applications.

To address this gap in modeling capabilities, we develop a ray theory for modeling high frequency wave propagation in spatially graded mechanical metamaterials. Ray theory is a classical approximation for wave propagation originally developed in the context of optics, where it is known as geometric optics. Subsequently, it emerged as a powerful tool in a wide breadth of fields, from quantum mechanics to cosmology to fluid mechanics. Specifically in solid mechanics, ray theory is most well known as a foundation for classical seismology. A key feature of ray theory is its capability to efficiently approximate high frequency wave solutions in heterogeneous media, which motivates its popularity in such a wide range of fields.

Despite the clear demonstration of ray theory as a powerful and mature tool in many fields, it has not yet been practically applied in the context of mechanical metamaterials. In this work, we derive and implement a practical ray theory framework for locally periodic metamaterials, under the assumption that unit cells vary slowly compared to the wavelengths of interest. Under the assumption of local periodicity, a system of ray tracing differential equations is derived, which depend on the local dispersion relations. The amplitude and phase of the wave field are easily obtained along ray trajectories. Thus, ray theory provides an efficient means of obtaining approximate wave solutions in graded metamaterials.

We demonstrate results on a series of examples including a graded mass-spring network with analytical dispersion relations as well as a graded truss metamaterial, which requires numerical evaluation of local dispersion relations. The ray solutions show close agreement with transient finite element simulations. Due to the efficiency and accuracy of ray theory, we anticipate that it will provide a platform to explore the design space of graded metamaterials to achieve fascinating wave propagation behavior.

Presenting Author: Charles Dorn ETH Zürich

Presenting Author Biography: Charles Dorn is currently a postdoctoral fellow at ETH Zürich, where his research focuses on wave propagation in mechanical metamaterials. In 2021, he completed his PhD at Caltech on the mechanics and design of reconfigurable origami structures. He obtained Master's degrees from École Polytechnique in solid mechanics in 2018 and from Caltech in space engineering in 2017.

Authors:

Charles Dorn ETH Zürich
Dennis Kochmann ETH Zürich