UNIVERSIDADE ESTADUAL DE CAMPINAS
Faculdade de Engenharia Mecânica
Vinícius Fonseca Dal Poggetto
Numeri al Investigation on Periodi Plate Stru tures for the Redu tion of Vibrations
Investigação Numéri a de Estruturas de Pla as Periódi as para a Redução de Vibrações
CAMPINAS 2019
Numeri al Investigation on Periodi Plate Stru tures for the Redu tion of Vibrations
Investigação Numéri a de Estruturas de Pla as Periódi as para a Redução de Vibrações
Thesis presented to the School of Mechanical Engineering of the University of Campinas, in partial fulfillment of the requirements for the degree of Doctor in Mechanical Engineering, in the area of Solid Mechanics and Mechanical Design. Tese apresentada à Faculdade de Engenharia Mecânica da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutor em Engenharia Mecânica, na área de Mecânica dos Sólidos e Projeto Mecânico.
Orientador: Prof. Dr. Alberto Luiz Serpa
Este exemplar corresponde à versão final da tese defendida pelo aluno Vinícius Fonseca Dal Poggetto, e orientada pelo Prof. Dr. Alberto Luiz Serpa.
Campinas 2019
Ficha catalográfica
Universidade Estadual de Campinas Biblioteca da Área de Engenharia e Arquitetura
Luciana Pietrosanto Milla - CRB 8/8129
Dal Poggetto, Vinícius Fonseca,
D15n DalNumerical investigation on periodic plate structures for the reduction of vibrations / Vinícius Fonseca Dal Poggetto. – Campinas, SP : [s.n.], 2019.
DalOrientador: Alberto Luiz Serpa.
DalTese (doutorado) – Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica.
Dal1. Ondas elásticas. 2. Placas. 3. Método dos elementos finitos. 4. Metamateriais. 5. Diagramas de bandas. I. Serpa, Alberto Luiz, 1967-. II. Universidade Estadual de Campinas. Faculdade de Engenharia Mecânica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Investigação numérica de estruturas de placas periódicas para a
redução de vibrações
Palavras-chave em inglês:
Elastic waves Plates
Finite element method Metamaterials
Band diagrams
Área de concentração: Mecânica dos Sólidos e Projeto Mecânico Titulação: Doutor em Engenharia Mecânica
Banca examinadora:
Alberto Luiz Serpa [Orientador] José Roberto de França Arruda José Maria Campos dos Santos Carlos De Marqui Junior
Adriano Todorovic Fabro
Data de defesa: 19-11-2019
Programa de Pós-Graduação: Engenharia Mecânica Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0003-0862-6270 - Currículo Lattes do autor: http://lattes.cnpq.br/1597915564959559
Faculdade de Engenharia Mecânica
Comissão de Pós-Graduação em Engenharia Mecânica
Departamento de Mecânica Computacional
TESE DE DOUTORADO
Numeri al Investigation on Periodi Plate Stru tures for the Redu tion of Vibrations
Investigação Numéri a de Estruturas de Pla as Periódi as para a Redução de Vibrações
Autor: Vinícius Fonseca Dal Poggetto Orientador: Prof. Dr. Alberto Luiz Serpa
A Banca Examinadora composta pelos membros abaixo aprovou esta Tese: Prof. Dr. Alberto Luiz Serpa, Presidente
DMC/FEM, Universidade Estadual de Campinas - UNICAMP Prof. Dr. José Roberto de França Arruda
DMC/FEM, Universidade Estadual de Campinas - UNICAMP Prof. Dr. José Maria Campos dos Santos
DMC/FEM, Universidade Estadual de Campinas - UNICAMP Prof. Dr. Carlos De Marqui Junior
DEA/EESC, Universidade de São Paulo - USP Prof. Dr. Adriano Todorovic Fabro
DEM/FT, Universidade de Brasília - UnB
A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vida acadêmica do aluno.
I would like first to acknowledge all the possibilities God has provided so I could do what I love most through all these years.
Thanks also go out to my supervisor, Alberto Serpa, who always kindly sup-ported me through the last ten years.
I would also like to thank my father, mother, and brother for all the love and support through all tough decisions that I have had to make in the past few years to be able to pursue one more dream.
Finally, I would like to thank all my close friends who have kept my spirits up over the past years and always led me to believe I could achieve anything.
O principal objetivo dessa tese é investigar o comportamento da propagação de ondas em metamateriais periódicos, com foco na comparação entre diferentes opções de modelagem e técnicas para prever a ocorrência de bandas proibidas. Uma introdução a es-truturas periódicas, ondas elásticas e à teoria de placas é apresentada. A principal técnica de cálculo de bandas proibidas usada é a Expansão em Ondas Planas, que inicialmente é aplicada usando equações clássicas de ondas elásticas e um modelo clássico de placa de Kirchhoff. Também é proposta a utilização dessa técnica em conjunto com o modelo de placa de Mindlin com inclusões circulares, sendo as equações correspondentes demon-stradas. Para se usar a Expansão em Ondas Planas, as propriedades dos materiais devem ser expandidas usando uma série de Fourier. A versão mais comum dessa expansão (bidi-mensional, representando inclusões circulares em uma estrutura quadrada) é conhecida. Porém, quando se utilizam equações tridimensionais, a expansão bidimensional pode não ser representativa. Para se abordar esse problema, uma nova expansão tridimensional das propriedades dos materiais (inclusões esféricas em uma estrutura cúbica) foi desenvolvida nessa tese. Bandas proibidas são calculadas usando um código MATLAB próprio. Re-sultados para ambos os modelos bidimensional e tridimensional são avaliados usando o Método dos Elementos Finitos (com MSC Nastran) e o Método de Ondas de Elementos Finitos (implementações MATLAB considerando placas de Mindlin e elementos sólidos). Elementos Espectrais também são apresentados e sua utilização para a obtenção de um diagrama de bandas de estruturas tridimensionais é avaliada utilizando uma técnica de otimização de parâmetros dimensionais para um modelo representativo tomando como base a relação entre deslocamentos e forças de uma célula periódica.
Palavras-chave:
ondas elásticas, placas, método dos elementos finitos, metamateriais, diagramas de ban-das.
The main objective of this thesis is to investigate the behavior of wave propaga-tion in periodic metamaterials, with a focus on the comparison between different modeling options and techniques to predict the occurrence of band gaps. A background on periodic lattices, elastic waves, and plate theory is given. The main technique regarding band gap calculation used is the plane wave expansion, which is initially applied using classical elas-tic wave equations and a classical Kirchhoff plate model. The utilization of this technique in conjunction with the Mindlin plate model with circular inclusions is proposed, and the corresponding equations are derived. For using the plane wave expansion method, material properties should be expanded using the Fourier series. The most common version of this expansion (two-dimensional, representing circular inclusions on a square lattice) is known. However, when three-dimensional equations are used, the two-dimensional expansion may not be representative. To address this issue, a novel three-dimensional expansion of ma-terial properties (spherical inclusions on a cubic lattice) was derived in this thesis. Band gaps are calculated using a proprietary MATLAB code. Results for both two-dimensional and three-dimensional models are assessed using the finite element method (with MSC Nastran) and the wave finite element method (MATLAB implementations considering Mindlin plates and solid elements). Spectral elements are also presented, and their use for obtaining a band diagram for three-dimensional structures is evaluated using a tech-nique that uses dimensional parameters optimization of a representative model taking as reference the displacement and force ratios of a periodic cell.
Keywords:
1.1 Structure-borne and airborne sound . . . 17
1.2 Thesis sequence and main contributions . . . 21
3.1 Periodic cell in the physical lattice with basis vectors (a1, a2) . . . 26
3.2 First Brillouin zone in the reciprocal lattice with basis vectors (b1, b2) . . 27
3.3 Two-dimensional periodic cell with circular inclusions . . . 29
3.4 Three-dimensional periodic cell with spherical inclusions . . . 30
4.1 Propagating wave . . . 32
4.2 One-dimensional bar element and differential element . . . 35
4.3 Differential element for transverse waves . . . 37
4.4 Beam in pure bending . . . 39
4.5 Differential element for bending waves . . . 40
4.6 Equilibrium for shear force and bending moment . . . 40
4.7 Three-dimensional model with spherical inclusions . . . 44
4.8 Two-dimensional model with circular inclusions . . . 45
4.9 Consecutive periodic cells . . . 47
4.10 Reciprocal lattice with wave vectors k and k . . . 48
4.11 First Brillouin zone (two-dimensional) . . . 49
4.12 Contour using the irreducible Brillouin zone (two-dimensional) . . . 49
4.13 First Brillouin zone (three-dimensional) and IBZ contour . . . 50
5.1 Plate degrees-of-freedom . . . 51
5.2 Rotational degrees-of-freedom . . . 52
5.3 Plate stresses . . . 54
5.4 Plate forces and moments . . . 56
5.5 Plate model with circular inclusions . . . 59
7.1 Plate finite element . . . 72
7.2 Solid finite element . . . 77
8.1 Consecutive periodic cells with forces and displacements . . . 85
8.2 Partitioned nodes for rectangular plate mesh . . . 86
8.3 Equilibrium of node bl . . . 89
8.4 Partitioned nodes for rectangular solid mesh . . . 90
9.1 Frame components with forces/moments in nodes i and j . . . 97
9.2 Components of local x axis with respect to global coordinates . . . 99
9.3 Representative periodic cell composed of spectral elements . . . 101
10.1 PWEM band diagrams (two-dimensional elastic model) . . . 106
10.2 PWEM band diagrams (thin plate, Kirchhoff model) . . . 107
10.3 PWEM band diagrams (thin plate, Mindlin model) . . . 107
10.4 PWEM band diagrams (thick plate, Kirchhoff model) . . . 108
10.5 PWEM band diagrams (thick plate, Mindlin model) . . . 108
10.6 Periodic cell plate mesh in the xy plane . . . 109
10.7 WFEM band diagrams (Mindlin model) . . . 110
10.8 Eigenmodes for position X . . . 111
10.9 Eigenmodes for position M . . . 111
10.10 Numerical experiment performed using FEM . . . 112
10.11 Plate mesh for calculating the z-direction displacement ratio . . . 112
10.12 Displacement ratio and band gaps (thin plate) . . . 113
10.13 Displacement ratio and band gaps (thick plate) . . . 114
10.14 Displacements at 531 Hz obtained using Nastran . . . 115
10.15 Displacements at 785 Hz obtained using Nastran . . . 115
10.16 Displacements at 1039 Hz obtained using Nastran . . . 116
10.17 PWEM band diagrams (three-dimensional elastic model) . . . 117
10.18 Different views of the periodic cell solid mesh . . . 118
10.19 WFEM band diagram (three-dimensional model) . . . 118
10.23 Displacement ratio and band gaps (three-dimensional model) . . . 121
10.24 Equivalent FE model representation . . . 122
10.25 Periodic cell with clamped sides . . . 122
10.26 Force ratio between cell center node and resultant at a single side . . . 123
10.27 Dynamic stiffness at center node for periodic cell . . . 123
10.28 Representative periodic cell composed of 12 spectral frame elements . . . . 124
10.29 Optimized force ratio and dynamic stiffnes for periodic cell . . . 125
10.30 Real part and minimum of imaginary part for kx . . . 125
10.31 Real part and minimum of imaginary part for ky . . . 126
10.32 Real part and minimum of imaginary part for kz . . . 126
10.33 Real part and minimum of imaginary part for kd . . . 126
1.1 Main contributions summary . . . 19
6.1 PWEM eigenproblem size comparison . . . 70
10.1 Material properties used in results computation . . . 105
10.2 Band gap calculations comparison for two-dimensional models . . . 109
10.3 Band gap calculations comparison for Mindlin plates . . . 110
10.4 Attenuation regions comparison for Mindlin plates . . . 116
10.5 Band gap calculations comparison for three-dimensional model . . . 119
10.6 Attenuation regions comparison for three-dimensional model . . . 121
10.7 Equivalent periodic cell optimization results . . . 124
10.8 Band gap calculations comparison for equivalent three-dimensional models 127 C.1 PWEM band gap convergence for the thin Mindlin plate . . . 157
C.2 PWEM band gaps convergence for the thick Mindlin plate . . . 158
C.3 IPWEM band gap convergence for the thin Mindlin plate . . . 159
C.4 IPWEM band gaps convergence for the thick Mindlin plate . . . 160
C.5 Band gaps convergence for the two-dimensional elastic model . . . 161
C.6 Band gaps convergence for the Kirchhoff plate model . . . 161
C.7 PWEM band gap convergence for the three-dimensional elastic model . . . 162
ABS Acrylonitrile-butadiene-styrene DOF Degree-of-freedom
EPWEM Extended plane wave expansion method FEM Finite element method
FPWEM Fast plane wave expansion method IPWEM Improved plane wave expansion method LDPE Low-density polyethylene
LRPC Locally resonant phononic crystal MM Metamaterial
PC Phononic crystal
PWEM Plane wave expansion method SEM Spectral element method
WFEM Wave finite element method WSEM Wave spectral element method
1 INTRODUCTION 17
1.1 Wave propagation . . . 17
1.2 Thesis subjects and research scope . . . 18
1.3 Objectives and original contributions of the thesis . . . 19
1.4 Thesis structure . . . 20
2 LITERATURE REVIEW 22 2.1 Overview . . . 22
2.2 Metamaterials and phononic crystals . . . 22
2.3 Plane wave expansion method . . . 23
2.4 Finite element method . . . 24
2.5 Wave finite element method . . . 24
2.6 Spectral element method . . . 25
3 PERIODIC LATTICES 26 3.1 Basic definitions . . . 26
3.2 Fourier series of periodic material properties . . . 28
3.2.1 Periodic material properties for circular inclusions . . . 29
3.2.2 Periodic material properties for spherical inclusions . . . 30
4 ELASTIC WAVES 32 4.1 Background on waves . . . 32
4.2 Types of waves . . . 34
4.2.1 Longitudinal waves . . . 35
4.2.2 Quasi-longitudinal waves . . . 37
4.2.3 Transverse (shear) waves . . . 37
4.2.4 Bending waves . . . 39
4.3 Propagating and evanescent waves . . . 42
4.4 General wave propagation equations . . . 43
5 PLATE THEORY 51
5.1 Introduction . . . 51
5.2 Kinematics of deformation . . . 52
5.3 Plane stress theory . . . 53
5.4 Equilibrium equations . . . 55
5.4.1 Mindlin plate . . . 57
5.4.2 Kirchhoff plate . . . 57
5.5 Motivation for working with Mindlin or Kirchhoff plates . . . 58
6 PLANE WAVE EXPANSION 60 6.1 Introduction . . . 60
6.2 Plane wave expansion for the two-dimensional elastic model . . . 60
6.3 Plane wave expansion for the three-dimensional elastic model . . . 64
6.4 Plane wave expansion for the Kirchhoff plate model . . . 66
6.5 Plane wave expansion for the Mindlin plate model . . . 68
6.6 Eigenproblem size comparison . . . 70
6.7 Improved plane wave expansion method . . . 70
7 FINITE ELEMENT METHOD 72 7.1 Introduction . . . 72
7.2 Mindlin plate elements . . . 72
7.2.1 Stiffness matrices . . . 74
7.2.2 Jacobian matrix . . . 75
7.2.3 Mass matrix . . . 76
7.3 Hexahedral solid elements . . . 77
7.3.1 Stiffness matrix . . . 79
7.3.2 Jacobian matrix . . . 80
7.3.3 Mass matrix . . . 80
8 WAVE FINITE ELEMENT METHOD 85
8.1 Basic outline . . . 85
8.2 Two-dimensional wave finite element method . . . 86
8.3 Three-dimensional wave finite element method . . . 90
8.4 Eigenproblem formulation . . . 93
9 SPECTRAL ELEMENT METHOD 96 9.1 Frame elements . . . 96
9.2 Dynamic stiffness matrix . . . 97
9.2.1 Spectral element for rods . . . 97
9.2.2 Spectral element for shafts . . . 97
9.2.3 Spectral element for beams . . . 98
9.3 Assembly . . . 99
9.4 Wave spectral element method . . . 100
10 RESULTS AND DISCUSSION 105 10.1 Initial considerations . . . 105
10.2 Band diagrams using two-dimensional elastic and plate models . . . 105
10.2.1 Plane wave expansion method . . . 106
10.2.2 Wave finite element method . . . 109
10.2.3 Displacement ratio comparison . . . 112
10.3 Band diagrams using three-dimensional models . . . 117
10.3.1 Plane wave expansion method . . . 117
10.3.2 Wave finite element method . . . 118
10.3.3 Displacement ratio comparison . . . 120
10.4 Band diagrams using spectral elements . . . 122
10.4.1 Optimization problem . . . 122
10.4.2 Equivalent spectral element model dispersion curves . . . 125
11.3 Concluding remarks . . . 130
11.4 Future work . . . 131
BIBLIOGRAPHY 132 APPENDICES 138 Appendix A Derivations of equations of Chapter 3 138 A.1 Equations of Section 3.2.1 . . . 138
A.2 Equations of Section 3.2.2 . . . 140
Appendix B Derivations of equations of Chapter 6 145 B.1 Equations of Section 6.2 . . . 145
B.2 Equations of Section 6.3 . . . 148
B.3 Equations of Section 6.4 . . . 150
B.4 Equations of Section 6.5 . . . 152
B.5 Equations of Section 6.7 . . . 154
Appendix C Plane wave expansion convergence 157 C.1 Two-dimensional models . . . 157
C.2 Three-dimensional models . . . 161
Appendix D Improved plane wave expansion application 163 D.1 Improved plane wave expansion in the three-dimensional case . . . 163
1
INTRODUCTION
1.1 Wave propagation
The area of wave propagation has been studied for a long time. Mathematical description of waves and physical experiments for demonstrating associated phenomena have drawn considerable attention, since applications are found in several fields of interest such as vibroacoustics, noise, vibration in industrial and automotive areas, and seismol-ogy, to name a few. The comprehension of wave properties is paramount to the design, testing, and application of solutions in many fields of expertise. One of the most impor-tant areas concerning vibration attenuation is the development of materials that present interesting properties specifically designed towards objectives such as wave filtering and soundproofing.
A significant field of study of wave propagation is building acoustics. The structural acoustic process is usually divided into four main stages [13]: generation (origin of an oscillation or disturbance), transmission (transfer of energy between source and a structure), propagation (distribution of energy throughout this structure), and radiation (power is transmitted to adjacent fluid from the structure). The field of physics which deals with these four phenomena in solid structures is named structureborne sound -considering audible frequencies (usually up to 20 kHz). Meanwhile, propagation through the air is called airborne sound. The source of disturbance can generate vibrations via impact or transmit sound through the air (structure-borne and airborne, respectively).
In both cases, energy propagates from the source to the receiver (a person, for instance) through structures such as the floor, ceiling, and walls. A fraction of energy is transmitted directly, while another fraction is propagated via flanking transmission between adjacent structures. A schematic example is given in Figure 1.1.
Regardless of the path, vibrations radiate energy into the air, which then carries energy as sound to the receiver. Modifying the path of vibration propagation is often used when modification of the noise source is impractical or not economically feasible [4].
From the many types of waves propagating in various media, flexural waves are of the greatest significance [17] in the structure-fluid interaction, since it involves significant displacements in the normal direction of the propagation media. This vibration can effectively disturb adjacent fluids, thus facilitating power radiation. Therefore, two-dimensional structures having mainly transverse displacements - such as plates - are of particular interest in the sense of vibration reduction.
Low and mid-frequency range vibration has been regarded as a severe problem concerning both residential comfort and environmental noise pollution. Masonry walls and concrete slabs can generally provide good vibration attenuation at mid to high fre-quencies, but not at low to mid frequencies. At such frefre-quencies, the mass density law (which indicates that the acoustic transmission in a material is inversely proportional to the product of its thickness, the mass density, and the sound frequency) indicates inferior noise reduction. Thus, the development of structures that can provide enough attenuation of mechanical vibrations without a substantial increase in mass is of great interest. Com-posite materials have demonstrated interesting properties in isolating vibrations when these materials present a characteristic of periodicity in its physical construction in one or more directions, thus being an active subject of research.
More details regarding the main research subjects and objectives are given in the next sections.
1.2 Thesis subjects and research scope
The target of this thesis is the numerical investigation of the behavior of wave propagation in plate structures. When considering composite materials, one has a specific interest in periodic media, since this type of material seems promising in the field of passive vibration attenuation.
Material properties in periodic media are described, and its mathematical de-scription is presented and investigated. In order to model the media in which the prop-agation occurs, classical elastic waves are presented, as well as plate theory considering Kirchhoff and Mindlin plates.
For the characterization of wave propagation, several techniques may be em-ployed to investigate its nature. First, the plane wave expansion method is presented, considering two-dimensional and three-dimensional media. Later, to verify the results, the wave finite element method is used. Also, considering three-dimensional periodic me-dia, spectral elements are proposed as an alternative for obtaining wave characterization.
Results using different models and techniques are assessed and compared be-tween themselves and those obtained using commercial software.
1.3 Objectives and original contributions of the thesis
The aim of this work is the numerical investigation of periodic plate structures using two- and three-dimensional models, analyzing the physical phenomena involved in the generation of wave interaction that results in band gaps and proposing numerical approaches for dealing with the problem of wave characterization in such media.
First, well-known techniques (plane wave expansion method, finite and spectral element methods, and wave finite element method) and models (classical elastic wave equations, Kirchhoff and Mindlin plates) are presented. After detailing relevant techniques and models and having their limitations identified, three main improvements are proposed: • the plane wave expansion method is employed considering Mindlin plates with con-centric circular inclusions made of two different materials in a third material matrix, taking advantage of their local resonance effect; the resulting band diagrams and effects of inclusions are investigated;
• in the case of three-dimensional periodic media with spherical inclusions, material properties must be modeled accordingly; thus, a new Fourier series calculation for material properties considering spherical inclusions is derived and assessed consid-ering three-dimensional wave propagation;
• finally, a new model for approximating three-dimensional media using spectral ele-ments for modeling a representative periodic cell is obtained via optimization con-sidering the model displacement and force ratio curves; dispersion relations are then calculated and compared to the attenuation observed in a three-dimensional model. Table 1.1 summarizes contributions in different chapters, which are then vali-dated/compared using the wave finite element method and MSC Nastran.
Table 1.1: Main contributions summary
Contribution Chapters
Plane wave expansion applied to three-dimensional
3, 6 elastic media with spherical inclusions
Plane wave expansion applied to
6 Mindlin plates with circular inclusions
Equivalent spectral element
9, 10 frame structure
1.4 Thesis structure
In Chapter 2, a discussion of relevant literature about the research topics is presented. In Chapter 3, a brief background regarding periodic lattices and periodic properties is presented. After explaining basic concepts, material properties are expressed in terms of its Fourier series for circular inclusions. Also, a novel Fourier series of material properties for spherical inclusions is given.
Chapter 4 presents basic concepts about elastic waves and its generic solution form for periodic media. Waves in solid media are also formulated. Plate theory is given in Chapter 5, deriving equations using kinematics of deformation, constitutive equations, and equilibrium equations. Thus, Kirchhoff and Mindlin plates formulations are presented. Chapter 6 presents the plane wave expansion method and its application to out-of-plane and in-plane motion considering a simplification for approximating three-dimensional media as two-three-dimensional media. The plane wave expansion method is also applied to the three-dimensional general case and Kirchhoff plates. Also, a new model for the band diagram calculation in plates considering circular inclusions is presented, being suitable for the use in thick plates. This new result is obtained from the application of the plane wave expansion method to the Mindlin plate model.
The finite element method can also be used to obtain dispersion relations in various types of structures. Thus, its basic usage is presented in Chapter 7, defining degrees-of-freedom, shape functions, and the derivation of stiffness and mass matrices for plate and solid elements. Using the results from the previous chapter, Chapter 8 outlines the wave finite element method, deriving equations for two-dimensional and three-dimensional wave propagation. Corresponding eigenproblems for determining dispersion curves in periodic plates and three-dimensional media are presented. A brief explanation regarding spectral elements is given in Chapter 9 considering frame elements, as well as a corresponding wave spectral element method.
The results are presented and discussed in Chapter 10. Conclusions and future work suggestions are given in Chapter 11. All relevant equations derivations are presented in Appendices A and B. Results indicating the convergence of the methods are presented in Appendix C, and an explanation regarding improvements on the plane wave expansion method are given in Appendix D.
Figure 1.2 represents the organization of the thesis and corresponding contri-butions highlighted in each chapter.
2
LITERATURE REVIEW
2.1 Overview
A general overview of the literature on the topics of wave mechanics, periodic materials, and the most common analysis techniques is presented in this chapter. Although the given list is not intended to be exhaustive, these references are sufficient to cover relevant concepts to this work and to highlight the current status of research in the field. Periodic structures are composite materials that present periodicity in its phys-ical construction and have been thoroughly discussed in the literature since they can present interesting properties in isolating vibrations. Brillouin [6] presents a classical ap-proach to wave propagation in periodic structures and uses a mathematical background to analyze their behavior in a mixture of systems, such as electric lines and crystal lattices. Also, a wide variety of methods for analyzing free and forced wave motion in periodic structures have been developed over the years, as shown by Mead [39].
Using local resonators and the property of periodicity, it may be possible to obtain frequency ranges where the transmissibility of elastic waves is reduced - thus at-tenuating forced motions - named band gaps. Band gap regions are defined as frequency ranges where vibrations (elastic or acoustic waves) are completely prohibited for all val-ues of the Bloch wave vector which defines the direction of propagation of a wave and its wavelength in a single periodic cell [27].
2.2 Metamaterials and phononic crystals
Wave propagation in periodic structures has been an active subject of research for a long time [6]. The literature on acoustic metamaterials and phononic crystals is extensive, including several one-dimensional, two-dimensional, and three-dimensional ex-amples [14]. A review of applications, including lensing, imaging, and cloaking is also available [12].
Phononic crystals (PCs) are periodic structures which present band gaps due to its periodicity effects, based on the Bragg mechanism (destructive interferences). Thus, the band gap can be obtained at a frequency which relates to the wavelength of the structure.
Liu et al. [36] have demonstrated, based on the idea of local resonance, the existence of band gaps at frequencies with two orders of magnitude lower than related wavelengths using sonic crystals. The resulting resonance profile can be explained as the resulting interference between the local resonance and the continuum background in which it is embedded [23]. This effect was first theoretically described by Fano [18] when studying autoionizing resonances in atoms.
A locally resonant phononic crystal (LRPC) or periodic metamaterial (MM) is an artificially engineered periodic composite structure, where resonators (small inclusions compared to the involved wavelengths) interact with the hosting media, creating a band gap as a coupled effect of local resonance and waves propagating in the structure. This effect typically creates band gaps at lower frequencies than those that would be observed in the PCs.
Several works have demonstrated useful applications for MMs, especially in the low-frequency range. Acoustic metamaterial panels consisting of thin elastic membranes fixed by a rigid plastic grid, with a small weight attached to the center of each grid have been shown to significantly reduce sound transmission in the 50 Hz to 1000 Hz range [64]. Periodic structures using a honeycomb configuration with added local resonant structures have been demonstrated in [9] to be a lightweight alternative for acoustic insulation.
2.3 Plane wave expansion method
The plane wave expansion method (PWEM) has been applied to the case of one-dimensional periodic media (such as periodic beams with attached resonators) in many works. For instance, Xiao et al. [61] have demonstrated the effect of flexural wave propagation in locally resonant beams with multiple periodic arrays of spring-mass resonators. In another work, Zhang et al. [67] apply the same idea to phononic crystals using Timoshenko beams to calculate the bending vibration characterization using the PWEM. Essential explanations regarding the PWEM applied to mechanical systems can be found in [3].
An alternative form named improved plane wave expansion method (IPWEM) has shown a better convergence for the method [31] - which may be computationally slow due to the use of Fourier series for expanding properties with a considerable mismatch between different materials. Studies on the convergence of the method for one-dimensional structures have also been performed comparing the PWEM, IPWEM, and the transfer matrix method [45].
In the case of two-dimensional propagation studies, local resonators have been shown to improve attenuation significantly at lower frequencies when compared to band gaps obtained using simple interference [10]. Thick plates with periodically attached spring-mass resonators have also been demonstrated as potential sound insulators [19]. Wilm et al. [59] have applied the PWEM to study out-of-plane propagation of elastic waves in phononic crystals using quartz rods embedded in an epoxy matrix. Also, Romero-García et al. [51] showed that the extended plane wave expansion method (EPWEM) offers ad-vantages over the PWEM, informing about the evanescent (spatially damped) behavior of localized modes in point defects inside periodic structures. Point defects in thin plates and their associated bending waves also had their correlation analyzed using the IPWEM
in [66]. Also, a version of the improved fast plane wave expansion method (IFPWEM) has been presented by [63], comparing the computational efficiency of this method to the traditional fast plane wave expansion method (FPWEM).
2.4 Finite element method
The finite element method (FEM) is a traditional method which is commonly used for calculating displacements, stresses, and strains in mechanical structures, and the literature on the subject is vast [11]. Many formulations for static and dynamic analysis are available [5], including implementations using MATLAB [20, 28]. Also, due to the size of the problems that the FEM may deal with, many different methods may be used to reduce the size of the problem and to obtain approximate solutions [50].
The thin-plate theory has been used for studying effective properties and acous-tic radiation in metamaterials [32] and for low-frequency (100 Hz to 1000 Hz) sound ab-sorption [40]. Experimental realizations of acoustic metamaterials have been shown to be able to overcome the mass density law of sound attenuation for low frequencies (100 Hz to 1000 Hz) [65]. Lamb wave propagation in a single elastic metamaterial layer has also been studied, and its dispersion curves (relations between the wavelength of a wave propagating in a medium and its frequency) show a strong dependence on the ratio between the mate-rial’s effective Young’s and shear modulus [52]. Also, the FEM has been used to determine dispersion relations in both two-dimensional and three-dimensional cases; a detailed ex-planation of its application in periodic media in an anisotropic material is presented in [24]. A dynamic finite element method is formulated and compared with other techniques in [35], such as the finite-difference time-domain method and the PWEM, considering a two-dimensional structure.
Topology optimization techniques have also been used to optimize the fre-quency response of systems considering the case of fluid-structure interaction [57] and to determine optimal discretization of periodic cells which lead to maximum photonic [43] and phononic [34] band gaps, considering both isotropic and anisotropic media.
As for practical applications, Khelif et al. [25] have computed the band dia-grams of a structure consisting of cylindrical pillars and correlated them to the transmis-sion of surface waves in a finite length array of pillars using the FEM, showing band gaps in frequencies lower than those expected from the Bragg condition.
2.5 Wave finite element method
The wave finite element method (WFEM) uses numerical wave modes to com-pute dispersion curves of waves traveling along periodic structures [42]. It may also be used to calculate the low and mid-frequency forced response of complex periodic elastic structures with lower computational cost using wave modes as representation basis [41].
Some applications of the WFEM in one-dimensional cases include the calcula-tion of forced responses for waveguides using low order models [16], an applicacalcula-tion to an aircraft fuselage-like lightweight periodic array of resonant devices [53], and comparisons between the WFEM and more precise frequency-dependent models using rod elements [46]. This method has also shown good correlation to physical experiments [44].
A two-dimensional approach to the WFEM for obtaining dispersion relations in homogeneous structures is given in [37]. On the other hand, several numerical ill-conditioning issues have been pointed out, and some solutions for them have also been developed considering coupled structures [42] and a basis reduction technique applied with wave decomposition [16].
2.6 Spectral element method
Another possible option for element-based modeling in dynamical analyses is the spectral element method (SEM), which is extensively described by Lee et al. in [29]. It has the advantage of obtaining an exact representation of the dynamic response of a structure in the whole frequency domain using a single element (no need for mesh refinement), even though these solutions are readily available only for simple elements, such as rods and beams [15]. Band gap opening mechanisms are thoroughly explained using the SEM beam elements with attached resonators in [62], where the influence of position, width, and wave attenuation performance of band gaps is investigated.
Applications in the civil engineering area may also greatly benefit from the use of the SEM. The dynamic behavior of tall structures has been thoroughly pointed out as potentially hazardous [33]. Several proposals have been made regarding this issue, including periodic foundations [2] and tuned mass dampers [55]. The occurrence of band gaps in slender elastic structures has been observed [7] and also when considering locally resonant foundations [26].
The use of lumped parameters as an approximation to more complex systems has also been suggested as an effective way to determine the dispersion relation in simple systems [22]. A complete three-dimensional frame model has been proposed to represent skyscraper structures considering frequency-dependent models with the capability of rep-resenting the dynamic response of structures, but with a computational effort limitation [48]. In this work, an optimization problem in beams using a resonator composed of two masses (one fixed to the end of the beam and another one suspended by a spring) to determine the best distribution between these masses in order to maximize the width of the formed band gap for a given frequency range of interest was proposed, and its results indicate a favorable use of the SEM for simplifying the modeling of complex structures.
3
PERIODIC LATTICES
3.1 Basic definitions
In order to represent a periodic material and be able to use the plane wave expansion method, it is necessary to describe its structure using periodic lattices [3, 24]. A general two-dimensional structure can be obtained using its physical lattice basis vectors a1 and a2, as depicted in Figure 3.1, with a periodic cell indicated by dashed lines.
Figure 3.1: Periodic cell in the physical lattice with basis vectors (a1, a2)
For the general three-dimensional case, the lattice basis vectors, which describe the general form of periodicity for a given structure, are given by a1, a2, and a3. Thus, a general physical lattice point may be written as a linear combination of such basis vectors, i.e.,
R = n1a1+ n2a2 + n3a3, (3.1.1)
where n1, n2, and n3 are integers.
The reciprocal lattice is defined using basis vectors b1, b2, and b3, and the scalar product (·), such that
ai· bj = 2πδij, (3.1.2)
where δij represents the Kronecker delta, given by
δij = 1, if i = j, 0, otherwise. (3.1.3)
Equation (3.1.2) states that vectors ai and bj are orthogonal for i 6= j and the product between their lengths yields the constant 2π when i = j.
A general two-dimensional reciprocal lattice is shown in Figure 3.2, and the dashed lines indicate a region named the first Brillouin zone in the reciprocal lattice. The reciprocal lattice basis vectors b1and b2 are orthogonal to vectors a2 and a1, respectively.
Figure 3.2: First Brillouin zone in the reciprocal lattice with basis vectors (b1, b2) A lattice point in the reciprocal lattice is given by the reciprocal lattice vector G, written as
G = m1b1+ m2b2+ m3b3, (3.1.4)
for any integers m1, m2, and m3.
An important fact that will be used later is that the product G · R yields an integer number times 2π. This result can be shown by using different integers over G and R, and by applying the scalar product between G and R, which yields
G · R = (m1b1+ m2b2 + m3b3) · (n1a1+ n2a2+ n3a3) . (3.1.5) Using Equation (3.1.2), the orthogonality property can be applied, and the product of Equation (3.1.5) simplifies to
G · R = 2π(m1n1+ m2n2+ m3n3) . (3.1.6)
Since m1, m2, m3, n1, n2, and n3 are all integers, their product and sum is also an integer. Thus, the scalar product G · R equals an integer times 2π, which implies that
eiG·R = ei2π(m1n1+m2n2+m3n3) = 1 , (3.1.7)
where i =√−1 is the imaginary unit number.
The reciprocal lattice vector may also be written in terms of its cartesian coordinates (Gx, Gy, and Gz) by using the unitary vectors ˆi, ˆj, and ˆk, as in
If a rectangular physical lattice of sides Lx, Ly, and Lz is used, the lattice
vector is given by
R = nxLxˆi + nyLyˆj + nzLzk ,ˆ (3.1.9)
where n1, n2, and n3are replaced by nx, ny, and nz, respectively. In this case, the reciprocal
lattice vector simplifies to
G = 2π Lx nxˆi + 2π Ly nyˆj + 2π Lz nzk ,ˆ (3.1.10)
for any integers nx, ny, and nz.
3.2 Fourier series of periodic material properties
In the following derivations, the general position vector r is described using its cartesian coordinates (x, y, and z), denoted by
r = xˆi + yˆj + zˆk . (3.2.1)
If a material possesses spatial periodicity regarding its physical properties, it is possible to expand such property, denoted by p = p(r), using a Fourier series, i.e.,
p(r) =X G
p(G)eiG·r, (3.2.2) where the sum is taken over all G, i.e., every possible combination of integers which yield all G (see Equation (3.1.4)). It is interesting to notice that if the periodic property has its position vector incremented by a spatial period (r 7→ r + R), one has
p(r + R) =X G
p(G)eiG·(r+R)=X G
p(G)eiG·reiG·R. (3.2.3)
Using Equation (3.1.7), it is verified that
p(r + R) = X G
p(G)eiG·r1 = p(r) , (3.2.4) which ensures the periodicity of properties.
Each property p(G) from Equation (3.2.2) can be expressed as [27]
p(G) = 1 V
Z
V
p(r)e−iG·rdV , (3.2.5)
3.2.1 Periodic material properties for circular inclusions For the two-dimensional case, Equation (3.2.5) reduces to
p(G) = 1 A
Z
A
p(r)e−iG·rdA , (3.2.6)
where A represents the area of the domain of integration.
Consider now a periodic cell with side L, constituting a square lattice, depicted in Figure 3.3. The outer portion of the cell is the matrix (index m), the outer circular
inclusion (index o) has outer radius roand inner radius ri, and the inner circular inclusion
(index i) has radius ri.
Figure 3.3: Two-dimensional periodic cell with circular inclusions
It is possible to calculate each component p(G), as in Equation (3.2.5), for any given material properties. A particular case G = 0 (nx = ny = 0 in Equation (3.1.10),
since no nz is present in the two-dimensional case) must be considered, which represents
the Fourier component at the origin of the reciprocal lattice. Since inside each region (i, o or m), the material property is constant (pi, po or pm, respectively), p(G = 0) represents
an average of properties of each material weighted by area, i.e.,
p(G = 0) = Ai Api+ Ao A po+ Am A pm, (3.2.7) where Ai = πr2i, Ao = πro2− πri2, Am = L2− πr2o, and A = L2.
The general case (G 6= 0) is given by
p(G) = 2π AG (po− pm)roJ1(Gro) + (pi− po)riJ1(Gri) , (3.2.8)
where G = |G|, and J1 is the Bessel function of the first kind of order one. All derivations related to Equation (3.2.8) are presented in Appendix A.
Equation (3.2.8) must be used to determine the coefficients of the Fourier series that represents the spatial periodicity of materials according to Equation (3.2.2). In practical computer calculations, the series must be truncated at a sufficiently large number of terms to ensure convergence.
The simple case of a circular inclusion consisting of a single material may be represented with ri = 0 (inner circular inclusion vanishes), and ro representing the radius
of the inclusion.
3.2.2 Periodic material properties for spherical inclusions
Consider now a three-dimensional periodic cubic cell with side L, constituting a lattice of three possible materials, as previously described in Section 3.2.1, but now considering spherical inclusions, as indicated in Figure 3.4.
Figure 3.4: Three-dimensional periodic cell with spherical inclusions
It is now possible to calculate each component p(G), as in Equation (3.2.5), for any given material properties. A particular case G = 0 (nx = ny = nz = 0 in Equation
(3.1.10)) must be considered. Since inside each region, the material property is constant,
p(G = 0) represents the average of properties of each material weighted by volume, i.e., p(G = 0) = Vi V pi+ Vo V po+ Vm V pm, (3.2.9) where Vi = 43πri3, Vo = 43πr3o− 43πr 3 i, Vm = L3− 43πro3, and V = L3.
The general case (G 6= 0) is given by p(G) = 2π 2 V G Γ(3 2) Γ(−1 2) ∞ X n=0 (2n+1)Γ(− 1 2 + n) Γ(5 2 + n) (po−pm)r2oJn+2 12 r oG 2 +(pi−po)ri2Jn+2 12 r iG 2 , (3.2.10) where Γ is the gamma function, G = |G|, and Jn+12 is the Bessel function of order n +12, being n the summation index. All derivations are presented in Appendix A. To the best of the author’s knowledge, this formulation has not yet been presented in the literature and represents a contribution of this thesis.
Equation (3.2.10) is used in a similar way of Equation (3.2.8). The simple case of a spherical inclusion consisting of a single material may be represented with ri = 0
(inner spherical inclusion vanishes), and ro representing the radius of the inclusion.
This chapter presented the concepts regarding the periodicity of materials, which are fundamental for understanding the interaction between waves for creating de-structive interferences, which leads to band gap formation. Also, for the use of the plane wave expansion method, it is necessary to have material properties already expressed us-ing a convenient Fourier series expansion. Later in this thesis, the two-dimensional version (circular inclusions in a square matrix) will be employed, as also the three-dimensional version (spherical inclusions in a cubic matrix).
4
ELASTIC WAVES
4.1 Background on waves
Consider the case of a wave propagating along an infinite medium, with a portion of its domain depicted in Figure 4.1. This figure indicates several time instants which show a single frequency wave traveling from left to right. After an instant t0 = 0, a corresponding wavelength is marked, with a white circle representing the wavefront.
x
x = 0 t0 = 0
t1 > t0
t2 > t1
t > t2
Figure 4.1: Propagating wave
For the current section, u(x, t) will be used to describe a generic displacement in a coordinate x at a time t. The left end of the wave has its displacement u(0, t) imposed using a sinusoidal displacement, described as
u(0, t) = U sin ωt , (4.1.1) where ω is the angular frequency of the excitation, and U is an amplitude. After an initial time t = 0, the wavefront starts traveling forward - in the positive x direction. After some time ∆t, the wavefront will have traveled a horizontal distance x.
Since this fixed x point will have its displacement described by the same sinu-soidal input, but with a lagging ϕ(x) phase, its displacement can be described by
u(x, t) = U sin(ωt − ϕ(x)) , (4.1.2) where ϕ(x) = ω∆t is the phase by which a given point at a distance x lags from the left end (x = 0).
Suppose now this wavefront propagates to the right with a constant speed c, named phase speed. The total distance x traveled in the time interval ∆t is given by
x = c∆t. Thus, in the same time interval, both distance x and lag phase ϕ(x) have been
produced, which are related by
∆t = ϕ(x)
ω = x
c. (4.1.3)
Therefore, the general displacement of a point in the position x for a time t is given by u(x, t) = U sin ωt −ωcx . (4.1.4)
The term ω/c is called wavenumber, represented by k, i.e.,
k = ω
c . (4.1.5)
The relation between k and ω is named dispersion relation. If the phase speed
c is independent of the wave frequency ω, then k and ω are related linearly. In this case,
all waves propagate with the same speed, independent of their frequencies, and the wave is called non-dispersive.
Whenever the propagating speed c differs between frequencies, there is a non-linear relation between k and ω, and the wave is called dispersive. In both cases, the total energy transported by a wave can be described as a carrier wave and a modulated wave (group wave), and its group speed cg is obtained from the dispersion curve by the relation
[17, 15]
cg =
dω
dk . (4.1.6)
Another way to describe k is using the fundamental relations ω = 2πf , where
f is the frequency in Hz, and c = λf , where λ is the wavelength. Considering these
relations, it is possible to write
k = 2π
λ . (4.1.7)
Thus, substituting Equation (4.1.5) in Equation (4.1.4), a traveling wave can be described by the equation
u(x, t) = U sin(ωt − kx) . (4.1.8) It is more common to use a phasor notation since complex number manipu-lation is more straightforward in terms of derivatives. By this reason, the wave equation will be written as
For the three-dimensional case, the displacement vector u can be written using its cartesian coordinates (ux, uy, and uz), yielding
u = uxˆi + uyˆj + uzk .ˆ (4.1.10)
Using this general version, the displacement of a point may be writen as
u(r, t) = Ue−i(ωt−k·r), (4.1.11)
where k is the Bloch wave vector, which represents a wavenumber in a specific propagating direction, and U = Uxˆi + Uyˆj + Uzk is a three-dimensional amplitude vector.ˆ
4.2 Types of waves
Considering the classical theory of elasticity, the assumptions on small defor-mation theory should be presented. Linear strains ǫ and angular strains γ are related in terms of displacements derivatives [49] as
ǫx = ǫxx = ∂ux ∂x , (4.2.1a) ǫy = ǫyy = ∂uy ∂y , (4.2.1b) ǫz = ǫzz = ∂uz ∂z , (4.2.1c) γxy = 2ǫxy = ∂ux ∂y + ∂uy ∂x , (4.2.1d) γyz = 2ǫyz = ∂uz ∂y + ∂uy ∂z , (4.2.1e) γxz = 2ǫxz = ∂uz ∂x + ∂ux ∂z . (4.2.1f)
Constitutive equations relate stress components σ and strain components ǫ, described by σ = σx σy σz τxy τyz τxz , ǫ = ǫx ǫy ǫz γxy γyz γxz . (4.2.2)
In the general case of linear strains, considering the Poisson effect for a material with Young’s modulus E, shear modulus µ, and Poisson coefficient ν, it is possible to write
ǫx = 1 E σx− νσy− νσz , (4.2.3a) ǫy = 1 E σy − νσx− νσz , (4.2.3b) ǫz = 1 E σz− νσx− νσy , (4.2.3c) γxy = τxy µ , (4.2.3d) γyz = τyz µ (4.2.3e) γxz = τxz µ . (4.2.3f)
Waves will be divided according to the most relevant direction of vibration of the particles in a medium:
• longitudinal and quasi-longitudinal waves: longitudinal vibrations; • transverse and bending waves: transverse vibrations.
The following sections describe wave propagation considering each specific type and the corresponding dispersion relations for every case. For simplicity, a one-dimensional element is taken, and several dispersion relations are obtained.
4.2.1 Longitudinal waves
Suppose a one-dimensional bar element with its axis in the x-direction, as in Figure 4.2. Purely longitudinal waves suppose no transverse displacement is present in the bar. However, for a strictly longitudinal force applied in the longitudinal axis of an element, transverse strains appear due to the Poisson effect.
Figure 4.2: One-dimensional bar element and differential element
In order to restrain transverse strains, constraints should be applied in orthog-onal directions regarding the direction of the force application. If strains in both y and
constraints yield a relation between the longitudinal stress and the longitudinal strain given by
σx = Bǫx, (4.2.4)
where B = E(1+ν)(1−2ν)(1−ν) .
The equation of motion for an infinitesimal element may be written using Newton’s second law. Considering an element with specific mass ρ and cross-section area
S(x), one may write
−σxS(x) + σx+ ∂σx ∂x dx S(x) = ρS(x)dx∂ 2u x ∂t2 . (4.2.5)
This derivation leads to the equation of motion for the infinitesimal element
∂σx
∂x = ρ ∂2u
x
∂t2 . (4.2.6)
Using its corresponding constitutive and strain relations (Equations (4.2.4) and (4.2.1a), respectively), one obtains
∂2u x ∂x2 = ρ B ∂2u x ∂t2 , (4.2.7)
which, after using the displacement definition from Equation (4.1.9), yields
∂2 ∂x2 Uxe−i(ωt−kx) = ρ B ∂2 ∂t2 Uxe−i(ωt−kx) , (4.2.8)
By applying the partial derivatives in the last equation, one obtains −k2U xe−i(ωt−kx) = ρ B(−ω 2)U xe−i(ωt−kx), (4.2.9)
which can be rewritten as
−k2ux =
ρ B(−ω
2)u
x. (4.2.10)
The derivation between Equations (4.2.7) and (4.2.10) is similar to those that will be verified in other types of waves. Using Equation (4.2.10), one obtains the dispersion relation kl(ω) for longitudinal waves as
kl(ω) = ± rρ
B ω . (4.2.11)
propagating waves, in both positive and negative directions, and yields phase speed cl as cl = s B ρ . (4.2.12) 4.2.2 Quasi-longitudinal waves
Quasi-longitudinal waves are obtained if no additional constraints are enforced to prevent transverse strains that arise due to the Poisson effect. In this case, σy = 0 and
σz = 0, which yields the most common constitutive equation for bars
σx = Eǫx. (4.2.13)
Since this is the only difference with respect to longitudinal waves, one obtains the dispersion relation kql(ω) for quasi-longitudinal waves as
kql(ω) = ± rρ
E ω , (4.2.14)
yielding also a non-dipersive relation for purely propagating waves, with speed phase cql
given by
cql = s
E
ρ . (4.2.15)
Longitudinal and quasi-longitudinal waves may be observed in solids, and al-though transverse motion in longitudinal waves may be small, it can radiate sound into adjacent medium due to its perpendicular motion. These same types of waves may be observed in plates due to their membrane effect, yielding symmetric modes.
4.2.3 Transverse (shear) waves
Consider now a beam in the x-direction and transverse displacements in the
z-direction. The equation of motion for a given infinitesimal element may be written by
using the differential element as in Figure 4.3.
Using Newton’s second law, and considering an element with specific mass ρ and cross-section area S(x), one may write
−τxzS(x) + τxz+ ∂τxz ∂x dx S(x) = ρS(x)dx∂ 2u z ∂t2 . (4.2.16)
Thus, it is possible to write
∂τxz
∂x = ρ ∂2u
z
∂t2 . (4.2.17)
Using the corresponding constitutive and strain relations (Equations (4.2.3f) and (4.2.1f), respectively), one obtains
∂2u z ∂x2 = ρ µ ∂2u z ∂t2 , (4.2.18)
which, after using the displacement definition from Equation (4.1.9) - similar to the deriva-tion between Equaderiva-tions (4.2.7) and (4.2.10) - becomes
−k2uz =
ρ µ(−ω
2)u
z. (4.2.19)
Thus, one obtains the dispersion relation ks(ω) for shear waves as
ks(ω) = ± s
ρ
µω . (4.2.20)
This indicates also a non-dispersive relation, with purely propagating waves, in both positive and negative directions, with phase speed cs as
cs = s
µ
ρ . (4.2.21)
Also, considering the relation between Young’s modulus E and shear modulus
µ, given by
µ = E
2(1 + ν), (4.2.22)
it is possible to obtain the relation between speeds cs(Equation (4.2.21)) and cql(Equation
(4.2.15)), i.e., cs cql = s 1 2(1 + ν) < 1 , (4.2.23)
4.2.4 Bending waves
Bending waves are the most relevant type of waves in the process of structure-fluid interaction. These waves involve significant displacements in transverse directions to the propagation [13].
Even though transverse displacements are more significant than longitudinal displacements in the most important cases for beams and plates, stresses are associated with longitudinal strains, because bending induces tensile or compressive stresses. Since the entire behavior of bending waves and their differential equations differ significantly from transverse waves and quasi-longitudinal waves, flexural waves cannot be classified as either of those [17].
Consider a one-dimensional portion of a beam in pure bending, as the one in Figure 4.4.
Figure 4.4: Beam in pure bending The curvature radius rc is given by [49]
1
rc
= M
EI , (4.2.24)
where M is the bending moment, and I is the second moment of area of the transverse section with respect to the neutral axis (σx = 0). If transverse displacements in the
z-direction are given by uz, the curvature radius can be approximated as
1
rc ≈
∂2u
z
∂x2 . (4.2.25)
Thus, using Equations (4.2.24) and (4.2.25), the relation between the bending moment and the second derivative of uz is given by
M = EI∂
2u
z
∂x2 . (4.2.26)
The equation of motion for an infinitesimal element may be written considering an element with specific mass ρ, and cross-section area S(x). By taking V (x) as the shear resultant forces in a section, it is possible to use the differential element as presented in Figure 4.5.
Figure 4.5: Differential element for bending waves
The equation of motion for a given infinitesimal element may be written as +V − V + ∂V ∂xdx = ρS(x)dx∂ 2u z ∂t2 . (4.2.27)
Thus, it is possible to write −∂V
∂x = ρS(x) ∂2u
z
∂t2 . (4.2.28)
Consider now bending moments and shear forces in a differential element, as in Figure 4.6.
Figure 4.6: Equilibrium for shear force and bending moment
It is possible to consider negligible rotational inertia and write the equilibrium equation as X My = − M + ∂M ∂x dx + V + ∂V ∂xdx dx + M = 0 , (4.2.29) which leads to the relation
∂M
∂x = V , (4.2.30)
and consequently, it is possible to write
∂2M
∂x2 =
∂V
∂x . (4.2.31)
Using Equation (4.2.31) in Equation (4.2.28) yields
∂2M
∂x2 = −ρS(x)
∂2u
z
that can be combined with Equation (4.2.26), i.e., ∂2 ∂x2 EI∂ 2u z ∂x2 = −ρS(x)∂ 2u z ∂t2 . (4.2.33)
Considering the product EI as a constant in the last equation, one obtains,
∂4u z ∂x4 = − ρS EI ∂2u z ∂t2 , (4.2.34)
which, by using now the displacement definition from Equation (4.1.9) - similar to the derivation between Equations (4.2.7) and (4.2.10) - yields
k4uz = −
ρS EI(−ω
2)u
z. (4.2.35)
Thus, the dispersion relation may be obtained from
k4 = ρS
EIω
2. (4.2.36)
By defining an auxiliary quantity kb(ω) as
kb(ω) = 4 s ρS EI √ ω , (4.2.37)
it is now possible to determine all four possible wavenumbers for bending waves as
kbp(ω) = ±kb(ω) , (4.2.38a)
kbe(ω) = ±ikb(ω) , (4.2.38b)
where kbp(ω) are named propagating waves, and kbe(ω) are named evanescent waves.
For propagating waves, it is possible now to see that the relation between kb
and ω is not linear, and therefore, characterizes a dispersive relation. Bending waves will have a phase speed cb given by
cb = 4 s EI ρS √ ω . (4.2.39)
The corresponding group speed cbg (Equation (4.1.6)) is directly related to the
4.3 Propagating and evanescent waves
From the results of Equations (4.2.38), it is possible to see that the relation
k(ω) may yield complex-valued wavenumbers. The wave characterization may be done
through the analysis of possible components of k [15]. Thus, it is useful to write k as a complex number, i.e.,
k = Re{k} + i Im{k} , (4.3.1) where Re{k} is the real part of k, and Im{k} is the imaginary part of k. Thus, one might rewrite Equation (4.1.9) to describe the solution as
u(x, t) = Ue−i(ωt−Re{k}x)e−Im{k}x. (4.3.2) Using Equation (4.3.2) it is possible to see that Re{k} is associated with
e−i(ωt−Re{k}x) and is related to the oscillatory nature of the wave, yielding propagating
waves. Also, Im{k} is associated with e−Im{k}x and contributes to the decay of waves, yielding damped solutions, usually referred to as evanescent waves.
Thus, k may be real, imaginary or a complex number - the nature of the wave is associated with this characteristic of k. A band gap region is defined as a frequency band in which only evanescent waves are present, i.e., k is purely imaginary [17]. In this case, a band gap region can be identified when the real part of k divided by π/L is equal to an integer n, being L the length of a periodic cell. This situation yields eiRe{k}L = einπ
LL =
cos(nπ) + i sin(nπ) = ±1. With this result, waves have a purely exponential behavior, thus decaying without an oscillatory nature.
Since this might not be the case, one might also analyze the imaginary part of every wavenumber k [58, 60]. If for a given frequency, all imaginary values of k have absolute values greater than zero, waves are attenuated in such a frequency, even though they may also possess a spatially oscillatory characteristic. The band of frequency in which all waves are attenuated is named width of attenuation zone (WAZ). In either case, the attenuation observed in a finite structure will depend on the number of periodic cells.
4.4 General wave propagation equations
From the elastic dynamics theory, the wave propagation in a general three-dimensional elastic medium [3, 63] is given by
ρ∂ 2u x ∂t2 = ∂ ∂x(λ∇ · u) + ∇ · µ ∇ux+ ∂u ∂x , (4.4.1a) ρ∂ 2u y ∂t2 = ∂ ∂y(λ∇ · u) + ∇ · µ ∇uy + ∂u ∂y , (4.4.1b) ρ∂ 2u z ∂t2 = ∂ ∂z(λ∇ · u) + ∇ · µ ∇uz+ ∂u ∂z , (4.4.1c)
where the Lamé coefficients are denoted by λ and µ, and the nabla operator represents a compact form given by
∇ = ∂ ∂xˆi + ∂ ∂yˆj + ∂ ∂zk .ˆ (4.4.2)
Using Equations (4.1.10) and (4.4.2), it is possible to write the product that represents the divergence of the displacement vector, i.e.,
∇ · u = ∂ux ∂x + ∂uy ∂y + ∂uz ∂z , (4.4.3)
which, after substituting in Equation (4.4.1a), yields
ρ∂ 2u x ∂t2 = ∂ ∂x λ ∂u x ∂x + ∂uy ∂y + ∂uz ∂z + ∇ · µ ∂ux ∂x + ∂ux ∂x ∂ux ∂y + ∂uy ∂x ∂ux ∂z + ∂uz ∂x ! . (4.4.4)
After applying again the divergence using the ∇ operator, one obtains
ρ∂ 2u x ∂t2 = ∂ ∂x λ ∂u x ∂x + ∂uy ∂y + ∂uz ∂z + ∂ ∂x µ ∂u x ∂x + ∂ux ∂x + ∂ ∂y µ ∂u x ∂y + ∂uy ∂x + ∂ ∂z µ ∂u x ∂z + ∂uz ∂x . (4.4.5)
It is now possible to group similar terms according to the first partial deriva-tive. Also, similar equations may be derived for the equilibrium in y and z directions by
