Continuum Modelling and Analysis of a Class of One and Two Dimensional Elastic Metamaterials with Local Rotation
Creators: Schiavone, Antonio J.
Publisher: name: University of Alberta Library
Year: 2022

<p>The group of materials classified as &quot;metamaterials&quot; have accrued great interest in the scientific community as of late for their potential to revolutionize several multidisciplinary applications. Metamaterials are defined as synthetic/man-made materials which have been engineered to possess a number of desired unusual, and often counterintuitive properties which do not occur naturally. The inception of metamaterials into engineering science was in the field of optics when a material exhibiting an apparent negative index of refraction was designed. Following this, so-called &quot;optical metamaterials&quot; were researched and implemented in the field of electromagnetic cloaking, as well as utilized to design superlenses with sub-wavelength resolution. Recently a subclass of metamaterials known as elastic metamaterials has become of great interest to engineering scientists. This is a large class of materials which exhibits one or more unusual elastic properties such as negative Poisson&#39;s ratio, negative effective stiffness, negative shear modulus, and many more. Elastic metamaterials have potential for application in the fields of impact mitigation, shock absorption, wave attenuation, energy shielding, and wave guiding, to name a few.\\<br>
In order to facilitate the use of this new class of materials, it is of paramount importance to possess the ability to predict the behaviour of these materials under specific, as well as sufficiently general loading conditions. There are two main ways to do this; the first of which is experimentally, through trial and error, and the second is analytically by creating a mathematical model capable of predicting both material behaviour and effective properties under specific loading conditions. This thesis will focus on the latter method.\\<br>
There exists a myriad of mathematical techniques for material characterization, some of these techniques include homogenization methods, unit cell design, discrete modelling, and continuum modelling. This thesis will focus on the continuum modelling of a class of elastic metamaterials with local rotational effects. Typically, when local effects need to be considered in the framework of a continuum, the micropolar continuum model is the first avenue people explore. In this thesis it will be shown that this model is incapable of capturing all of the salient features present in both one, and two dimensional elastic metamaterials belonging to this class.\\<br>
In this thesis a series of continuum models are developed with increasing generality. First, in the third chapter, a micropolar-type continuum model is derived for a specific one-dimensional double negative metamaterial capable of exhibiting negative mass and/or negative modulus under certain loading frequencies when subject to harmonic loads. This is done by analyzing a discrete structure, obtaining the equations of motion, and then making a continuous approximation to bring the discrete model to the continuum framework. This model is used to evaluate the transient response of a specific one-dimensional semi-infinite elastic metamaterial when subject to an axial impact. In the fourth chapter, a higher order continuum model is developed in a manner very similar to the methodology presented in the third chapter, but with a higher order derivative of the rotational variable $\theta$. This model is then generalized to an entire class of materials, even though it is developed using a representative discrete structure. Harmonic wave propagation is then studied in the same one-dimensional elastic metamaterial that was modelled in the third chapter using this new model, leading to the determination of the stop and passing bands, as well as the determination of the dispersion relation governing the wave propagation. This new model is then compared to both the model in the third chapter, as well as the discrete model to determine the range of suitability. In the fifth chapter a model for a two-dimensional class of elastic metamaterials with local rotation is developed in a slightly different way than in the previous two chapters. In this chapter a set of constitutive laws for the relevant class of materials is proposed, and then a representative discrete metamaterial is modelled, and approximated as a continuum to prove suitability of the model. This model is then used to study harmonic longitudinal (P) and transverse (S) wave propagation in the material, which covers all cases of general two-dimensional wave propagation. The stop and passing bands, as well as the dispersion relations were determined for both wave propagation schemes and the effect of local rotation was analyzed. The sixth chapter uses the model developed in the fifth chapter to study surface wave propagation in a new continuum with local rotation. The dispersion relation of the surface wave is obtained, as well as expressions for the decay parameters, $b<em>1$ and $b</em>2$. The behaviour of the general dispersion relation, as well as some simplified cases are investigated. It is found that surface waves propagating through a continuum with local rotation are dispersive even when the local rotational effects are small compared to the translational ones. Two parameters governing general wave propagation, $f$ and $g$ are identified. The parameter $f$ controls the height of frequency peaks in the dispersion relation, and the parameter $g$ controls the location of the second peak. Furthermore, for values of $f \approx 1$ or greater, surface waves are found to propagate with wavespeeds significantly lower than $c_R$, a phenomenon unique to this continuum. Finally, the motion of the particles residing on the surface of this continuum is determined to be elliptical when subject to surface wave propagation, similar to classical Rayleigh wave propagation.</p>