Wave propagation and directionality in two-dimensional periodic lattices considering shear deformations

The effects of shear deformation on analysis of the wave propagation in periodic lattices are often assumed negligible. However, this assumption is not always true, especially for the lattices made of beams with smaller aspect ratios. Therefore, in the present paper, the effect of shear deformation on wave propagation in periodic lattices with different topologies is studied and their wave attenuation and directionality performances are compared. Current experimental limitations make the researchers focus more on the wave propagation in the direction perpendicular to the plane of periodicity in micro/nanoscale lattice materials while for their macro/mesoscale counterparts, in-plane modes can also be analyzed as well as the out-of-plane ones. Four well-known topologies of hexagonal, triangular, square, and Kagomé are considered in the current paper and their wave propagation is investigated both in the plane of periodicity and in the out-of-plane direction. The finite element method is used to formulate the governing equations and Bloch’s theorem is used to solve the dispersion relations. To investigate the effect of shear deformation, both the Timoshenko and Euler-Bernoulli beam theories are implemented. The results indicate that including shear deformation in wave propagation has a softening effect on the band diagrams of wave propagation and moves the dispersion branches to lower frequencies. It can also reveal some bandgaps that are not predicted without considering the shear deformation.

A field experiment on surface wave isolation by partially embedded periodic pile barriers

The bandgap characteristics of periodic structures show a great application prospect in seismic isolation and vibration mitigation. A T-shaped partially embedded periodic pile barrier is designed and studied based on the Bloch’s theorem. The proposed structure is fabricated and tested in field experiment to validate the simulation and the isolation performance. Good agreements are observed between the measured and the corresponding simulated results which present effective isolation for surface waves. The influences of the embedded length in the soil and the arrangement of the pile are investigated. The results show that the embedded length is the key parameter affecting the vibration attenuation. The smaller the embedded length, the lower the frequency of attenuation zone, and thanks to the embedded length; the proposed barrier exhibits better performances than the fully embedded and non-embedded barriers in maximum bandgap width, tunability, and feasibility in engineering. Gradient distribution of the embedded length leads to a wider frequency range of attenuation.

Multiscale Modelling and Mechanical Anisotropy of Periodic Cellular Solids with Rigid-Jointed Truss-Like Microscopic Architecture

This paper investigates the macroscopic anisotropic behavior of periodic cellular solids with rigid-jointed microscopic truss-like architecture. A theoretical matrix-based procedure is presented to calculate the homogenized stiffness and strength properties of the material which is validated experimentally. The procedure consists of four main steps, namely, (i) using classical structural analysis to determine the stiffness properties of a lattice unit cell, (ii) employing the Bloch’s theorem to generate the irreducible representation of the infinite lattice, (iii) resorting to the Cauchy–Born Hypothesis to express the microscopic nodal forces and deformations in terms of a homogeneous macroscopic strain field applied to the lattice, and (iv) employing the Hill–Mandel homogenization principle to obtain the macro-stiffness properties of the lattice topologies. The presented model is used to investigate the anisotropic mechanical behavior of 13 2D periodic cellular solids. The results are documented in three set of charts that show (i) the change of the Young and Shear moduli of the material with respect to their relative density; (ii) the contribution of the bending stiffness of microscopic cell elements to the homogenized macroscopic stiffness of the material; and (iii) polar diagrams of the change of the elastic moduli of the cellular solid in response to direction of macroscopic loading. The three set of charts can be used for design purposes in assemblies involving the honeycomb structures as it may help in selecting the best lattice topology for a given functional stiffness and strength requirement. The theoretical model was experimentally validated by means of tensile tests performed in additively manufactured Lattice Material (LM) specimens, achieving good agreement between the results. It was observed that the model of rigid-joined LM (RJLM) predicts the homogenized mechanical properties of the LM with higher accuracy compared to those predicted by pin-jointed models.

Dynamics of Piezo-Embedded Negative Stiffness Mechanical Metamaterials: A Study on Electromechanical Bandgaps

Dynamics of periodic materials and structures have a profound historic background starting from Newton’s first effort to find sound propagation in the air to Rayleigh’s exploration of continuous periodic structures. This field of interest has received another surge from the early 21st century. Elastic mechanical metamaterials are the exemplars of periodic structures that exhibit interesting frequency-dependent properties like negative Young’s modulus, negative mass and negative Poisson’s ratio in a specific frequency band due to additional feature of local resonance. In this research, we present the modeling of piezo-embedded negative stiffness metamaterials by considering a shunted inductor energy harvesting circuit. For a chain of a finite number of metamaterial units, the coupled equation of motion of the system is deduced using generalized Bloch’s theorem. Successively, the backward substitution method is applied to compute harvested power and the transmissibility of the system. Additionally, through the extensive non-dimensional study of this system, the proposed metamaterial band structure is investigated to perceive locally resonant mechanical and electromechanical bandgaps. The results explicate that the insertion of the piezoelectric material in the resonating unit provides better tun-ability for vibration attenuation and harvested energy.

Waves and particles in the crystal

This chapter provides the groundwork necessary to mathematically describe the crystalline solid that is to be the semiconductor used to explore the variety of interactions and cause and chance behaviors that physics builds insights into. The crystalline environment can be portrayed as a space-filling periodic arrangement consisting of a lattice with an atomic basis. The periodic arrangement leads to real space and reciprocal space descriptions with unit cells—the Wigner-Seitz cell and the Brillouin and Jones zones—where a variety of characteristics can be represented. Bloch’s theorem with its modulation function of the plane wave of a quantized wavevector, momentum versus crystal momentum, together with the consequences of symmetries and periodic perturbation in the appearance of bandgaps, is discussed for electron states. Phonons as particles for periodic oscillations of atoms, their modes and various branches, and consequences of ionicity leading to frequency-dependent permittivity are discussed.

ENERGY BAND STRUCTURE OF AN ELECTRON IN A ONE-DIMENSIONAL PERIODIC POTENTIAL

It has been observed that electron in a perfect crystal moves in a spatially periodic field of force due to the ions and the averaged effect of all the electrons. This work shows the investigative work done to determine the energy band structure of an electron in a one-dimensional periodic potential. The application of the Kronig-Penney model was applied to an electron state in a delta-like potential. To fully understand the Kronig-Penney model, the concept of Bloch’s theorem was first introduced to describe the conduction of electrons in solids. It has been found that the periodic potential introduces gaps in the reduced representation with an increasing number of potential well/barrier strengths. It has been observed that the regions of non-propagating states, which give rise to energy band gaps, become larger with decreasing values.

Simultaneous energy harvesting and vibration attenuation in piezo-embedded negative stiffness metamaterial

Mechanical metamaterials are uniquely engineered form of periodically arranged unit cells that exhibit interesting frequency-dependent physical properties like negative effective mass, Young’s modulus and Poisson’s ratio. These extreme engineering properties are beyond the natural properties of a material, which can modulate the propagation of wave. In this article, a mechanical realization of one of these uncommon properties called negative stiffness is emulated through analytical simulation. Wave propagation in metamaterials is contingent on frequency, which in turn results in transmission and attenuation bands. Simultaneous vibration control and energy harvesting can be executed by embedding energy harvesting smart material within the resonating units of the metamaterial. However, this needs careful design studies to outline the range of parameters. In this work, first, the band structure of a piezo-embedded negative stiffness metamaterial is studied using generalized Bloch’s theorem. Subsequently, harvested power along with the transmissibility is computed for a chain of finite number of metamaterial units by using backward substitution method. The results of the parametric studies elucidate that piezo-embedded negative stiffness metamaterial can enhance the performance in terms of vibration attenuation and harvested energy.

Insulators and Semiconductors

The origin of the strange band gaps that arise in the electron density of states in real materials is that the regular arrangement of the nuclei in a crystal lattice creates a periodic potential. This affects the energy levels of the electrons and gives rise to gaps in the energy spectrum. The properties of insulators and semiconductors are very important in modern technology. They are largely due to the Fermi–Dirac statistics obeyed by electrons. Following the discussion of the behavior of free electrons in the previous chapter, this one turns to the properties that emerge when electrons are subject to a periodic potential, as they are in a crystal. The periodic potential leads to Bloch's theorem, band theory, and the prediction of band gaps, which are responsible for the differences in the properties of metals, insulators, and semiconductors. Dopants and their basic effects are discussed.

On the Mechanism of Bandgap Formation in Beams With Periodic Arrangement of Beam-Like Resonators

Metastructures made of spring-mass resonators present a bandgap at the natural frequency of the resonator. This rule cannot be generalized for more complex resonators. This work analyzes the case of a metastructure composed of a periodic arrangement of vertical beams rigidly joined to a horizontal beam. The vertical beams work as resonators, and their natural frequencies play a strong role on the band structure of the whole system, however, different than the case with spring-mass resonators. Since this metastructure can be considered a lattice, Bloch’s theorem is applied to the unit cell and a numerical procedure based on the finite element method permits to obtain the dispersion curves. Illustrative results show the influence of the natural frequencies of the horizontal and vertical beams on the band structure.