# The mathematics of seismic metasurfaces

*D. J. Colquitt (University of Liverpool), A. Colombi (Imperial College), R. V. Craster (Imperial College), and P. Roux (ISTerre Université de Grenoble 1 Joseph-Fourier),**S. R. L. Guenneau (Institut Fresnel)*

Although initially developed for optics, the field of metamaterial research has expanded rapidly and now includes the development of novel materials for applications in acoustics and elasticity. Metamaterials, as synthetic composite materials with a structure such that they exhibit properties not usually found in natural materials, now form a highly activing research area across all the physical sciences.

Recently, the applicability of metamaterials to seismology has sparked the interest of geophysicists in the development of novel methods to control surface waves. Given the interest in this emerging area, there is a need to study the properties of the solutions to fundamental canonical problems. We will present some recent results for two relevant problems and develop the theoretical framework for these seismic metamaterials. We will initially consider a periodic array of resonators on an elastic plate providing explicit expressions for the dispersion equation and solutions. This then motivates the more complicated problem of periodic arrays of resonators on elastic half-spaces.

We will analyse the dispersive properties of these structures and present explicit solutions in the metamaterial regime. We will also examine the transmission problem for surface waves and present a novel method for the control of surface waves on elastic solids.

*Given at the 3rd Workshop on Seismic Metamaterials, University of Bologna, in May 2017.*

# The foundations of seismic metamaterials: Plates and Half-spaces

*D. J. Colquitt (University of Liverpool), A. Colombi (Imperial College), R. V. Craster (Imperial College), and P. Roux (ISTerre Université de Grenoble 1 Joseph-Fourier)*

Metamaterials barely existed fifteen years ago, but now forms a major research area. Although initially developed for optics, the field of metamaterial research has expanded rapidly and now includes the development of novel materials for applications in acoustics and elasticity. More recently, the applicability of metamaterials to seismology has sparked the interest of geophysicists in the development of novel methods to control surface waves. Given the interest in this emerging area, there is a need to study the properties of the solutions to fundamental canonical problems. We will present some recent results for two relevant problems and develop the theoretical framework for these seismic metamaterials. We will initially consider a periodic array of resonators on an elastic plate providing explicit expressions for the dispersion equation and solutions. This then motivates the more complicated problem of periodic arrays of resonators on elastic half-spaces. We will analyse the dispersive properties of these structures and present explicit solutions in the metamaterial regime. We will also examine the transmission problem for surface waves and present a novel method for the control of surface waves on elastic solids.

*Given at the "Metamaterials beyond photonics" workshop at the ICMS, Edinburgh in June 2016.*

# The mathematical foundations of seismic metamaterials

*D. J. Colquitt (University of Liverpool), A. Colombi (Imperial College), R. V. Craster (Imperial College), and P. Roux (ISTerre Université de Grenoble 1 Joseph-Fourier)*

Metamaterials, as synthetic composite materials with a structure such that they exhibit properties not usually found in natural materials, now form a major emerging research area that barely existed before 2000. The first metamaterials were developed in optics and the field has now expanded to include acoustics and solid mechanics. Much more recently, the interest in metamaterials for seismological applications has begun to increase rapidly. Many approaches focus on the use of arrays of resonators to control the propagation of surface waves. Given the interest in this emerging area, there is a need to study the properties of the solutions to fundamental canonical problems. In the present talk, we will examine two such problems and provide the required theoretical background for these seismic metamaterials. We begin by discussing a periodic array of resonators attached to an elastic plate. Explicit expressions for the dispersion equation and solution are derived and we explore the interaction of the resonators with waves propagating in the plate. This naturally leads to the more complicated problem of a resonant array atop an elastic half-space. Again, explicit results are obtained and we discuss the interaction of elastic waves with the resonators.

*Given at the British Applied Mathematics Colloquium, University of Oxford on Wednesday 6th April 2016. *

# The control of elastic waves by multi-scale elastic metamaterials

*D. J. ColquittDepartment of Mathematical Sciences, University of Liverpool*

Over the past fifteen years, electromagnetic metamaterials have been extensively studied with tens of thousands of scholarly works devoted to their investigation. In contrast, multi-scale structures capable of manipulating the propagation of elastic waves have received relatively little attention. Nevertheless, elastic metamaterials have a broad range of potential applications from cloaking, seismic protection, and surface acoustic waves in radio transmitters and other electronic devices.

In this talk we will discuss several recent developments in the field of discrete elastic metamaterials. We will introduce the method of high frequency homogenisation which allows the effective material parameters of our elastic metamaterials to be extracted in dynamic regimes were the wavelength is comparable to the micro-structural scale. Several different classes of elastic metamaterials will be identified based on the classification of the partial differential equations governing their effective behaviour in a range of dynamic regimes. The novel notion of parabolic metamaterials will be introduced and analysed in detail.

We will also consider some applications of elastic metamaterials including cloaking for SH- and flexural waves, and the control of elastic surface waves by multi-resonant arrays.

*Given at the **Computational and Applied Mathematics Seminar, Department of Mathematics, Brunel University on Thursday 21st January 2016.*