Material parameters in calculations.
\r\n\tIn sum, the book presents a reflective analysis of the pedagogical hubs for a changing world, considering the most fundamental areas of the current contingencies in education.
",isbn:"978-1-83968-793-8",printIsbn:"978-1-83968-792-1",pdfIsbn:"978-1-83968-794-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"b01f9136149277b7e4cbc1e52bce78ec",bookSignature:"Dr. María Jose Hernandez-Serrano",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10229.jpg",keywords:"Teacher Digital Competences, Flipped Learning, Online Resources Design, Neuroscientific Literacy (Myths), Emotions and Learning, Multisensory Stimulation, Citizen Skills, Violence Prevention, Moral Development, Universal Design for Learning, Sensitizing on Diversity, Supportive Strategies",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"September 14th 2020",dateEndSecondStepPublish:"October 12th 2020",dateEndThirdStepPublish:"December 11th 2020",dateEndFourthStepPublish:"March 1st 2021",dateEndFifthStepPublish:"April 30th 2021",remainingDaysToSecondStep:"3 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"Dr. Phil. Maria Jose Hernandez Serrano is a tenured lecturer in the Department of Theory and History of Education at the University of Salamanca, where she currently teaches on Teacher Education. She graduated in Social Education (2000) and Psycho-Pedagogy (2003) at the University of Salamanca. Then, she obtained her European Ph.D. in Education and Training in Virtual Environments by research with the University of Manchester, UK (2009).",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"187893",title:"Dr.",name:"María Jose",middleName:null,surname:"Hernandez-Serrano",slug:"maria-jose-hernandez-serrano",fullName:"María Jose Hernandez-Serrano",profilePictureURL:"https://mts.intechopen.com/storage/users/187893/images/system/187893.jpg",biography:"DPhil Maria Jose Hernandez Serrano is a tenured Lecturer in the Department of Theory and History of Education at the University of Salamanca (Spain), where she currently teaches on Teacher Education. She graduated in Social Education (2000) and Psycho-Pedagogy (2003) at the University of Salamanca. Then, she obtained her European Ph.D. on Education and Training in Virtual Environments by research with the University of Manchester, UK (2009). She obtained a Visiting Scholar Postdoctoral Grant (of the British Academy, UK) at the Oxford Internet Institute of the University of Oxford (2011) and was granted with a postdoctoral research (in 2021) at London Birbeck University.\n \nShe is author of more than 20 research papers, and more than 35 book chapters (H Index 10). She is interested in the study of the educational process and the analysis of cognitive and affective processes in the context of neuroeducation and neurotechnologies, along with the study of social contingencies affecting the educational institutions and requiring new skills for educators.\n\nHer publications are mainly of the educational process mediated by technologies and digital competences. Currently, her new research interests are: the transdisciplinary application of the brain-based research to the educational context and virtual environments, and the neuropedagogical implications of the technologies on the development of the brain in younger students. Also, she is interested in the promotion of creative and critical uses of digital technologies, the emerging uses of social media and transmedia, and the informal learning through technologies.\n\nShe is a member of several research Networks and Scientific Committees in international journals on Educational Technologies and Educommunication, and collaborates as a reviewer in several prestigious journals (see public profile in Publons).\n\nUntil March 2010 she was in charge of the Adult University of Salamanca, by coordinating teaching activities of more than a thousand adult students. She currently is, since 2014, the Secretary of the Department of Theory and History of Education. Since 2015 she collaborates with the Council Educational Program by training teachers and families in the translation of advances from educational neuroscience.",institutionString:"University of Salamanca",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"University of Salamanca",institutionURL:null,country:{name:"Spain"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"23",title:"Social Sciences",slug:"social-sciences"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"301331",firstName:"Mia",lastName:"Vulovic",middleName:null,title:"Mrs.",imageUrl:"https://mts.intechopen.com/storage/users/301331/images/8498_n.jpg",email:"mia.v@intechopen.com",biography:"As an Author Service Manager, my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"6942",title:"Global Social Work",subtitle:"Cutting Edge Issues and Critical Reflections",isOpenForSubmission:!1,hash:"222c8a66edfc7a4a6537af7565bcb3de",slug:"global-social-work-cutting-edge-issues-and-critical-reflections",bookSignature:"Bala Raju Nikku",coverURL:"https://cdn.intechopen.com/books/images_new/6942.jpg",editedByType:"Edited by",editors:[{id:"263576",title:"Dr.",name:"Bala",surname:"Nikku",slug:"bala-nikku",fullName:"Bala Nikku"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"878",title:"Phytochemicals",subtitle:"A Global Perspective of Their Role in Nutrition and Health",isOpenForSubmission:!1,hash:"ec77671f63975ef2d16192897deb6835",slug:"phytochemicals-a-global-perspective-of-their-role-in-nutrition-and-health",bookSignature:"Venketeshwer Rao",coverURL:"https://cdn.intechopen.com/books/images_new/878.jpg",editedByType:"Edited by",editors:[{id:"82663",title:"Dr.",name:"Venketeshwer",surname:"Rao",slug:"venketeshwer-rao",fullName:"Venketeshwer Rao"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"4816",title:"Face Recognition",subtitle:null,isOpenForSubmission:!1,hash:"146063b5359146b7718ea86bad47c8eb",slug:"face_recognition",bookSignature:"Kresimir Delac and Mislav Grgic",coverURL:"https://cdn.intechopen.com/books/images_new/4816.jpg",editedByType:"Edited by",editors:[{id:"528",title:"Dr.",name:"Kresimir",surname:"Delac",slug:"kresimir-delac",fullName:"Kresimir Delac"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"63533",title:"Metal-Matrix Embedded Phononic Crystals",doi:"10.5772/intechopen.80790",slug:"metal-matrix-embedded-phononic-crystals",body:'\nIn the last two decades, the propagation of elastic waves in periodic composite materials, known as phononic crystals, has attracted increased attention due to their unique physical properties. These properties include phononic bandgaps which define a frequency range where elastic wave propagation is forbidden [1]. The existence of phononic bandgaps enables a variety of potential applications, such as noise and vibration insulation. Two mechanisms cause the formation of bandgaps: Bragg scattering [1] and local resonance [2]. In Bragg scattering, the associated wavelength with the phononic bandgap is of the same order as the periodicity of the structure. This means a huge lattice constant is needed to obtain phononic bandgaps for the low frequency range, which limits applications. To achieve local resonance, the associated wavelength needs to be two orders of magnitude smaller than the Bragg bandgap. A locally resonant bandgap is related to the resonance frequency associated with scattering units and depends less on the periodicity and symmetry of the structure. Therefore, it overcomes the limitation of Bragg bandgaps and permits bandgaps suitable for low frequencies.
\nThe existence of low-frequency locally resonant bandgaps (LRBGs) gives rise to the application of locally resonant phononic crystals (LRPCs) in the reduction of low-frequency vibration and noise [3]. However, it has been a challenging task due to the long wavelength and weak attenuation of the low-frequency waves. In the past years, many LRPCs were proposed to obtain bandgaps in low frequency range or broaden bandgaps in the low frequency range [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. More recently, the propagation of Lamb waves (elastic waves) in phononic crystal plates, which is based on a local resonance mechanism, has attracted attention due to their potential applications in filters, resonators, waveguides, and for vibration insulation. In general, phononic-crystal plates can be classified into two types according to their structural features: flat plates and stubbed plates. The flat phononic-crystal plate consists of a periodical array of holes or periodic inclusions of foreign material in a homogeneous plate [5]. Many previous studies focused on this type of phononic-crystal plates to investigate the bandgap properties. It is not easy to obtain lower frequency phononic bandgaps using a flat phononic-crystal plate. Stubbed phononic-crystal plates can be classified into two types: the single-sided stubbed phononic-crystal plate and the double-sided stubbed phononic-crystal plate. The single-sided stubbed phononic-crystal plate [3] consists of a square array of stubs on one side of a homogeneous plate [6]. Subsequently, many researchers investigated the effects of material properties and geometric parameters on the BG of the single-sided stubbed phononic-crystal plate and found that the mass and the geometric parameter effect can help tune the bandgap into low frequencies [8]. Later, a novel single-sided stubbed phononic-crystal plate was proposed. It consists of a square array of stubs on one side of a two-dimensional binary locally resonant phononic plate. It can increase of the relative bandwidth over the classical single-sided stubbed plate [12]. The double-sided stubbed phononic-crystal plate consists of a square array of stubs on both sides of a homogeneous plate. It can reduce the bandgap compared to a single-sided stubbed phononic-crystal plate [9]. Recently, a novel double-sided stubbed phononic-crystal plate was proposed. Unlike the classical double-sided stubbed phononic-crystal plate, it can shift the bandgap into the lower frequency range [13].
\nHowever, because there are three different modes for a plate, and they can only be coupled separately to special resonating modes of the resonators, the gaps for in-plane and out-of-plane plate modes can hardly be overlapped with each other in lower frequencies due to the coupling of the spring-mass system of the resonator, and thus the lower complete bandgaps (below 100 Hz) are difficult to obtain. As a result, the BGs of these disused PC plates are usually located in the frequency range above 300 Hz. However, the frequency of most of the ambient vibration in practical cases is distributed over a wide frequency range from 20 to 250 Hz. Also, the broad bandgaps for in-plane and out-of-plane plate modes can hardly be overlapped with each other in same frequency range due to the weak coupling between the resonator mode and the plate mode, and thus the broad complete bandgaps are difficult to obtain too. At the same time, a common feature of phononic-crystal plates is matrix structures made of nonmetallic materials. However, most mechanical structures typically consist of metals, and a phononic-crystal plate with a nonmetallic material matrix can hardly be used for vibration and noise reduction in mechanical engineering. Recently, a metal-matrix embedded phononic crystal which consists of periodic double-sided stepped resonators deposited on a two-dimensional phononic plate within a steel matrix was proposed. It is found that the bandwidth is increased, but the opening location of the bandgap is higher (1000 Hz) [19]. In other words, the extension of the bandwidth into the lower frequency range remains a challenging task for a metal-matrix embedded phononic crystal.
\nThus, how to adjust the in-plane and out-of-plane gaps overlapping with each other in the lower frequencies (below 100 Hz) or increasing the in-plane and out-of-plane gaps in the same range is an important issue for the control of the vibration in a practical case.
\nIn this chapter, two kinds of metal-matrix embedded phononic crystals are introduced. The one is called the lower frequency complete bandgap metal-matrix embedded phononic crystals [14] because it can produce lower frequency complete bandgap, and the other is called the broad complete bandgap metal-matrix embedded phononic crystals [20] because it can produce broad complete bandgap.
\nIn order to investigate the band properties of the metal-matrix embedded phononic crystals, a series of calculations on dispersion relations and transmission spectra are conducted with FEM based on the Bloch theorem. For the calculation of the dispersion relations, the governing field equations for elastic wave propagation in solids are given by
\nwhere ρ is the mass density, t is the time, ui is the displacement, cijkl are the elastic constants, and xj (j = 1, 2, 3) represents the coordinate variables x, y, and z, respectively. According to the Bloch theorem, in the FEM formulation, the displacement field can be expressed as
\nwhere u is the displacement at the nodes and r is the position vector located at the boundary nodes. The Bloch wave vector k = (kx, ky) is the wave vector limited to the first Brillouin zone of the reciprocal lattice. Since the metal-matrix embedded phononic crystal plate is periodic in the xy-direction and finite in the z-direction, only the unit cell (shown as in Figure 1(b) or Figure 3(b)) needs to be considered in the FEM calculation when the Bloch theorem is adopted on the boundaries between the unit cell and its two adjacent cells, given by
\nwhere the elastic displacement vector is denoted by u; the position vectors are denoted by x, y, and z; and kx and ky are the Bloch wave vectors limited in the irreducible first Brillouin zone (shown as in Figure 1(c) or Figure 3(c)). The Bloch calculation gives the eigenfrequencies and the corresponding eigenvectors, and then the dispersion relationships can be obtained by changing the wave vector in the first irreducible Brillouin zone.
\n(a and b) Schematic of the part and the unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals, respectively. (c) The corresponding first irreducible Brillouin zone (red region) and the high-symmetry M, Γ, and X.
To further demonstrate the existence of the bandgaps of the metal-matrix embedded phononic crystals, the transmission spectra for a structure with finite units along the x- or y-direction is calculated by using FEM. The acceleration excitation source is incident from one side (left side) of the finite structure and propagates along the x- or y-direction. The corresponding transmitted acceleration is recorded on another side (right side) of the structure. The transmission spectrum is defined as
\nwhere αo and αi are the output and input accelerations of the metal-matrix embedded phononic crystals, respectively. Finally, the transmission spectra can be obtained by changing the excitation frequency of the incident acceleration.
\nThe lower frequency bandgap metal-matrix embedded phononic crystal [14] was composed of a square array of composite taper stubs on both sides of a two-dimensional binary locally resonant PC plate which composes an array of rubber fillers embedded in the steel plate. Figure 1(a) and (b) shows part of the proposed structure and its unit cell, respectively.
\nIn the lower frequency bandgap metal-matrix embedded phononic crystals, the taper stub is composed of the A taper cap and the B which is located on the taper A. The geometrical parameters of the structure are defined as follows: the diameter of the rubber filler, the steel plate thickness, and the lattice constant are denoted by D, e, and a, respectively; the height and the diameter of the taper stub are denoted by h (hA for taper A and hB for taper B) and d (the upper diameter of taper stub is denoted by dup, and the lower diameter is denoted by dlow), respectively. The material parameters used in the calculations are listed in Table 1. The taper A and the taper B are rubber and steel, respectively.
\nMaterial | \nMass density (kg/m3) | \nYoung’s modulus (106N/m2) | \nPoisson’s ratio | \n
---|---|---|---|
Steel | \n7800 | \n210,000 | \n0.29 | \n
Rubber | \n1300 | \n0.1175 | \n0.47 | \n
Material parameters in calculations.
Through the use of the finite element method, the systems described in Figure 1 were studied numerically. The band structures and displacement vector fields were computed according to the Bloch theorem. The single-unit cell (as shown in Figure 3(b)) is determined by the periodicity of the structure. The following structure parameters are used: D = 8 mm, e = 1 mm, a = 10 mm, h = 5 mm (hA = hB = 2.5 mm), dup = 9 mm, and dlow = 5 mm, respectively.
\nIt can be observed that there are 13 bands within 0–200 Hz in Figure 2(a). Besides the traditional plate modes, which are the in-plane modes (mainly the symmetric Lamb modes, such as modes S2) and the out-of-plane modes (mainly the antisymmetric Lamb modes, such as mode A2), lots of flat modes (such as modes S1, A1, F2), which are the resonant modes of the composite taper stubs, can be found. The bandgaps (one in-plane bandgap, one out-of-plane bandgap, and one complete bandgap), as a result of the coupling of the two kinds of modes mentioned above, appear. The in-plane bandgap (blue-dashed area: the frequency bands in which no in-plane modes) is due to the coupling between the in-plane modes (modes S2) and the corresponding flat modes (modes S1). It ranges from 53 to 93 Hz (between the fifth and eighth bands). The absolute bandwidth of it is 40 Hz. The out-of-plane bandgap (green-dashed area: the frequency bands in which no out-of-plane modes) is due to the coupling between the out-of-plane modes (mode A2) and the corresponding flat modes (mode A1). It ranges from 59 to 154 Hz (between the sixth and ninth bands), and the absolute bandwidth is 95 Hz; the complete bandgap (red-dashed area: the frequency bands in which neither in-plane modes nor out-of-plane modes) is due to the overlap between the in-plane bandgap and the out-of-plane bandgap. It ranges from 59 to 93 Hz (between the sixth and eighth bands). The absolute bandwidth of it is 34 Hz. It can be observed that the location of the bandgap shifts into lower frequency (below 100 Hz), but the bandwidth is very narrow.
\nBand structures of (a) the lower frequency complete bandgap metal-matrix embedded PC plate, (b) the transition PC plate, and (c) the classical PC plate. The insets are the schematic view of the unit cell of the corresponding structure. The red, blue, and yellow shadow regions denote the complete, in-plane, and out-of-plane bandgaps, respectively.
As a comparison, we also calculated the band structures of the transition PC plate composed of double-sided composite taper stubs deposited on a homogeneous steel plate and the classical PC plate which was proposed by Assouar [9]. They are shown in Figure 2(b) and (c), respectively. Their complete bandgaps (red-dashed areas) are both due to the overlap between the second in-plane bandgap and the first out-of-plane bandgap. It can be found clearly that the introduction of the proposed structure gives rise to a significant lowering of the opening location of the first complete bandgap by a factor of 5.5 compared with the classical PC plate. Compared with the classical PC plate, introducing the double-sided composite taper stubs, the locations of both the in-plane and out-of-plane bandgaps are lowered, but the out-of-plane bandgap is always overlapped with the second in-plane bandgap; when introducing the rubber filler, the location of the in-plane bandgap is kept stationary, and the out-of-plane bandgap is shifted to lower frequency (59 Hz) overlapped with the first in-plane bandgap. Finally, a complete bandgap is generated in lower frequency (below 100 Hz). Therefore, the double-sided taper stub has a direct effect on the lowering of the location of the in-plane bandgaps (53 Hz), and the rubber filler has a direct effect on the lowering of the location of the out-of-plane bandgaps (59 Hz). It makes the out-of-plane bandgaps overlap with the first in-plane bandgap and leads to a complete bandgap in lower frequencies [14].
\nThe broad complete bandgap metal-matrix embedded phononic crystal [18] was composed of a square array of single “hard” cylinder stubs on both sides of a two-dimensional binary locally resonant PC plate which composes of an array of rubber fillers embedded in the steel plate. Figure 3(a) and (b) shows part of the structure and its unit cell, respectively.
\n(a and b) Schematic of the part and the unit cell of the broad complete bandgap metal-matrix embedded phononic crystals, respectively. (c) The corresponding first irreducible Brillouin zone (red region) and the high-symmetry M, Γ, and X.
The metal-matrix embedded phononic crystals with single “hard” cylinder stubs which consist of “hard” stub such as steel stubs contact with rubber filler. The geometrical parameters of the structure are defined as follows: the diameter of the rubber filler, the steel plate thickness, and the lattice constant are denoted by D, e, and a, respectively; the height and the diameter of the stub are denoted by h and d, respectively. The material parameters used in the calculations are listed in Table 1.
\nThrough the use of the finite element method, the systems described in Figure 3 were studied numerically. The band structures and displacement vector fields were computed according to the Bloch theorem. The single-unit cell (as shown in Figure 3(b)) is determined by the periodicity of the structure. The following structure parameters are used: D = 8 mm, e = 1 mm, a = 10 mm, h = 2.5 mm, and d = 7.5 mm.
\nThere are 12 bands within 0–600 Hz (see Figure 4(a)). Besides the traditional plate modes, which are the in-plane modes (such as mode S2) and the out-of-plane modes (such as mode A2), many flat modes (such as modes S1, A1, F1), which are the resonant modes of the single “hard” cylinder stubs, can be found. The bandgaps (one in-plane bandgap, one out-of-plane bandgap, and one complete bandgap) appear due to coupling between the two modes. The in-plane bandgap (Figure 4(a): blue-dashed area) is due to coupling between the in-plane modes S2 and the corresponding flat mode S1. It ranges from 260 to 573 Hz (between the sixth and twelfth bands). The absolute bandwidth is 313 Hz. The out-of-plane bandgap (Figure 4(a): green-dashed area) is due to coupling between the out-of-plane mode A2 and the corresponding flat modes A1. It ranges from 187 to 396 Hz (between the third and eighth bands), and the absolute bandwidth is 209 Hz; the complete bandgap (Figure 4(a): red-dashed area) is due to the overlap between the in-plane bandgap and the out-of-plane bandgap. It ranges from 260 to 396 Hz (between the sixth and eighth bands). The absolute bandwidth is 136 Hz.
\nBand structures of (a) the broad complete bandgap metal-matrix embedded phononic crystals, (b) the transition PC plate, and (c) the classical PC plate. The insets are the schematic view of the unit cell of the corresponding structure. The red, blue, and yellow shadow regions denote the complete, in-plane, and out-of-plane bandgaps, respectively.
As a comparison, we also calculated the band structures of the transition PC plate composed of double-sided composite cylinder stubs deposited on a two-dimensional locally resonant phononic crystal plate that consists of an array of rubber fillers embedded in a steel plate and the classical PC plate which was proposed by Assouar [9]. They are shown in Figure 4(b) and (c), respectively. For the classical phononic-crystal plate, its first complete bandgap (red-dashed area) is caused by the overlap between the second in-plane bandgap and the first out-of-plane bandgap. The associated absolute bandwidth is 29 Hz. For the transition phononic-crystal plate, the bandgaps are lowered after introducing the rubber filler, such that both the out-of-plane bandgap and in-plane bandgap were lowered. However, the out-of-plane bandgap is lowered more and overlaps with two in-plane bandgaps (the first and the second in-plane bandgaps). This causes the complete bandgaps to be increased, but the absolute bandwidth is also narrow (78 Hz). These phenomena confirm that the single “hard” cylinder stub has a special effect on the bandwidth and the bandwidth can be increased by introducing it. This occurs mainly because the in-plane bandgap is increased by introducing the single “hard” cylinder stub in the broad complete bandgap metal-matrix embedded phononic crystals. The absolute bandwidth of the in-plane bandgap is increased by a factor of 3.37 compared with a classical phononic-crystal plate [18].
\nIn order to study the physical mechanism for the occurrence of the lower frequency complete bandgap in the metal-matrix embedded phononic crystals [14, 18], several specific resonance modes (mode A1, mode S1), which correspond to the lower edge of the first bandgap (out-of-plane and in-plane bandgap), and several specific traditional plate modes (mode A2, mode S2), which correspond to the upper edge of the first bandgap (in-plane and out-of-plane bandgaps), are extracted.
\nFigure 5 displays the magnitude of the total displacement vector of a unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals (Figure 2(a)), the transition structure (Figure 2(b)), and the classical structure (Figure 2(c)), respectively. They correspond to the upper and lower edge of the out-of-plane bandgap of each structure. The mode A2 is an antisymmetric Lamb mode of the plate. The steel plate vibrates along the z-axis, while the stub remains stationary. In the frequencies, the antisymmetric Lamb mode will be activated, and the out-of-plane waves propagate through the PC plate in the antisymmetric Lamb mode. When the frequency of the out-of-plane waves is near the first nature frequency of the stub resonator, the resonant mode A1 will be activated. The stub vibrates along the z-direction, and it gives a reacting force to the plate against the plate that vibrates along the z-direction. In that case, the out-of-plane waves are not capable of propagating through the PC plate. As a result, an out-of-plane bandgap is opened. Within the out-of-plane bandgap, the reacting force is still applied on the plate and prevents the propagation of the out-of-plane waves. The frequency of the out-of-plane waves deviates from the nature frequency of the mode A1, according to the modal superposition principle which can be described as
\nwhere z denotes the response of the plate and ηn denotes the modal participation factor of the mode An. The modal participation factor η1 of the mode A1 becomes small, leads the reacting force to becoming weak, and then disappears, and the antisymmetric Lamb mode A2 is released again. As a result, the out-of-plane bandgap is closed. The formation mechanism of the bandgaps is shown in Figure 6. As the frequency of the out-of-plane waves deviates from the nature frequency of the resonator mode, the modal participation factors of it become small, lead the reacting force to becoming weak, and then disappear; the out-of-plane bandgap is closed. It can be concluded that the out-of-plane bandgap of the system is formed due to the coupling between the flat mode A1 and the antisymmetric Lamb mode A2 which is established based on the modal superposition principle.
\nThe total displacement vector fields of the modes (resonant mode A1 and antisymmetric Lamb mode A2) (a) correspond to Figure 2(a), (b) correspond to Figure 2(b), and (c) correspond to Figure 2(c).
The formation mechanism of the out-of-plane bandgaps of the lower frequency complete bandgap metal-matrix embedded phononic crystals.
The opening location of the out-of-plane bandgap is determined by the nature frequency of the resonant mode A1. The vibration process of the resonant mode A1 can be understood as a mass-spring system whose frequency is determined by the formula
\nwhere K1 is the spring stiffness and M1 is the lump mass. For the classical structure, the rubber stub (denoted by A) acts as a spring, and the cap steel stub (denoted by B) acts as a mass (M1 = MB, where MB denotes the mass of stub B). For the lower frequency complete bandgap metal-matrix embedded phononic crystals, it can be found that the displacement fields are distributed in the whole stub and manifest an “analogous-rigid mode” of the whole stub, since the whole stub bodily moves along the z-axis with weak constrain, and the natural frequency is not zero. In this case, the rubber filler acts as a spring, and the whole stub acts as a mass; thus, the frequency is shifted to a lowest frequency range. It can be concluded that the out-of-plane bandgap is adjusted into lower frequency range by the rubber filler.
\nFigure 7 displays the magnitudes of the total displacement vectors of a unit cell of the lower frequency complete bandgap metal-matrix embedded phononic crystals (Figure 7(a)), the transition structure (Figure 7(b)), and the classical structure (Figure 7(c)), respectively. They correspond to the upper and lower edges of the first in-plane bandgap of each structure. The mode S2 is a symmetric Lamb mode of the plate. The steel plate vibrates along the xy-plane, while the stubs “swing” in the opposite direction. In the frequencies, the symmetric Lamb mode S2 will be activated, and the in-plane waves propagate through the PC plate in the symmetric Lamb mode. When the frequency of the in-plane waves is near the first nature frequency of stub resonator, the resonant mode S1 (flat mode) will be activated. The stub “swings” along a plane which is vertical to the xy-plane, and it gives a reacting force to the plate to prevent the plate to vibrate along the xy-plane. In that case, the in-plane waves are not capable of propagating through the PC plate. As a result, an in-plane bandgap is opened. Within the in-plane bandgap, the reacting force is still applied on the plate and prevents in-plane waves that propagate. The frequency of the in-plane waves deviates from the nature frequency of the resonant mode S1, according to the modal superposition principle. Which can be described as
\nwhere xy denotes the response of the plate and ηn denotes the modal participation factor of the mode Sn. The modal participation factor η1 of the mode S1 becomes small, leads the reacting force to becoming weak, and then disappears, and then the symmetric Lamb mode S2 is released again. As a result, an in-plane bandgap is closed. It can be concluded that the first in-plane bandgap of the system is formed due to the coupling between the flat mode S1 and the symmetric Lamb mode S2 which is established according to the modal superposition principle. The opening location is determined by the nature frequency of the resonant mode S1. The resonant mode S1 can also be understood as a “mass-spring” system whose frequency is determined by the formula
\nwhere K2 is the tensional stiffness of the spring and M2 is the lump mass. For the three PC plates, the rubber stub A mainly acts as a spring, and the cap steel stub B acts as a mass (M2 = MB). As the stiffness of the taper rubber stub is weaker than that of the cylinder stub, the location of the in-plane bandgap is adjusted into lower frequency range.
\nThe total displacement vector fields of the modes (resonant mode S1, symmetric Lamb mode S2) (a) correspond to Figure 2(a), (b) correspond to Figure 2(b), and (c) correspond to Figure 2(c).
We can conclude from the above investigations that the resonator in the metal-matrix embedded phononic crystals can be considered as a spring-mass system, as shown in Figures 6 and 7. The spring-mass system includes two subsystems. The first one is the K1-M1 subsystem (also refer to Figure 6). Its local resonance mode is coupled with the antisymmetric Lamb mode of the plate according to the modal superposition principle, and then the out-of-plane bandgap is generated. The other one is the K2-M2 subsystem (also Figure 7). Its local resonance mode is coupled with the symmetric Lamb mode of the plate according to the modal superposition principle and is responsible for the formation of the in-plane bandgaps. The opening locations of the in-plane and out-of-plane BGs depend on the natural frequencies of the two subsystems.
\nAs for the classical and the transition metal-matrix embedded phononic crystals, the steel stub (stub B) acts as a mass, and the rubber stub (stub A) acts as a spring, which leads to the coupling between the K1-M1 subsystem and K2-M2 subsystem. Therefore, it is difficult to adjust the in-plane and out-of-plane bandgaps separately. In the lower frequency bandgap metal-matrix embedded phononic crystals, the rubber filler acts as the stiffness K1, the stub A acts as the stiffness K2, the whole stub acts as the mass M1, and the stub B acts as the mass M2. As a result, the coupling between the K1-M1 subsystem and the K2-M2 subsystem can be decoupled. Moreover, the mass M1 is magnified due to the “analogous-rigid mode” of the whole stub. Therefore, the out-of-plane bandgap can be adjusted into the lowest frequency range. Additionally, the stiffness K2 can be reduced by introducing the taper stub, and thus the in-plane bandgaps can be adjusted into the lowest frequency range. As a result, the two bandgaps can be overlapped with each other in the lowest frequency, and finally a lowest complete bandgap is formed [14].
\nIn order to study the physical mechanism for the occurrence of a broad complete bandgap in the metal-matrix embedded phononic crystals, several specific resonance modes (modes A1 and S1), which correspond to the lower edge of the first bandgap (out-of-plane and in-plane bandgap), and several specific traditional plate modes (mode A2 and S2), which correspond to the upper edge of the first bandgap (in-plane and out-of-plane bandgaps), are extracted.
\nFigure 8 displays the magnitude of the total displacement vector of a unit cell of the broad complete bandgap metal-matrix embedded phononic crystals (Figure 8(a)), the transition PC plate (Figure 8(b)), and the classical PC plate (Figure 8(c)), respectively. They correspond to the upper and lower edge of the out-of-plane bandgap of each structure. The mode A2 is an antisymmetric Lamb mode of the plate. The steel plate vibrates along the z-axis, while the stub remains stationary. With regard to the frequencies, the antisymmetric Lamb mode will be activated, and the out-of-plane waves propagate through the phononic-crystal plate in the antisymmetric Lamb mode. When the frequency of the out-of-plane waves is near the first nature frequency of the stub resonator, the resonant mode A1 will be activated. The stub vibrates along the z-direction and produces a reacting force for the plate, and the plate vibrates along the z-direction. In this case, the out-of-plane waves are not capable of propagating through the phononic-crystal plate. As a result, an out-of-plane bandgap is created. The bandwidth is determined by the coupling between the resonance mode A1 and the traditional plate A2 mode.
\nThe total displacement vector fields of the modes (resonant mode A1 and antisymmetric Lamb mode A2) (a) correspond to Figure 4(b), (b) correspond to Figure 4(b), and (c) correspond to Figure 4(c).
The coupling strength between the resonance mode A1 and the traditional plate A2 mode is determined by the formula
\nwhere sT is the vibration amplitude of the stub and ko is the spring stiffness.
\nFor the classical structure, the formation mechanism of the out-of-plane bandgap is shown in Figure 9(b). The hard stub (rigid body) vibrates along the z-direction and generates a reacting force through the soft stub (flexible body) to the plate against the plate vibrating along the z-direction. The soft stub acts as a spring, while the hard stub acts as a mass (ko = ksc, mo = mh, where mh denotes the mass of the hard stub and ksc denotes the compression stiffness of the soft stub) (see Figure 10(b)).
\nFormation mechanism of the bandgap for the out-of-plane bandgap of (a) the transition PC plate, (b) the classical PC plate, and (c) the broad complete bandgap metal-matrix embedded phononic crystals.
The equivalent theoretical model of the resonator for formation mechanism of the out-of-plane bandgap of (a) the transition PC plate, (b) the classical PC plate, and (c) the broad complete bandgap metal-matrix embedded phononic crystals.
For both the broad complete bandgap metal-matrix embedded phononic crystals and the transition PC plate, it can be found that the displacement fields are distributed in the whole stub (see Figure 8(a) and (b)), respectively. This results in an “out-of-plane analogous-rigid mode” because the whole stub vibrates along the z-axis (out-of-plane) with a weak constraint, while the frequency is non-zero. The displacement fields of its eigenmodes are distributed throughout the whole stub. This means that the whole stub body moves along the z-direction like a rigid body moves in rigid mode. However, the natural frequency is not 0, and the whole stub is constrained by the rubber filler. Therefore, we refer to the concept “rigid mode” and call these types of vibration modes for the whole stub the “out-of-plane analogous-rigid mode.” The formation mechanisms for the out-of-plane bandgap of the two structures are shown in Figure 9(a) and (c), respectively.
\nThe whole stub vibrates in the z-direction (out-of-plane) and generates a reacting force. F1 and F2 respond to the broad complete bandgap metal-matrix embedded phononic crystals and the transition PC plate, respectively. F1 = k0.s1 = kRT.s1, where kRT is the transverse stiffness of the rubber filler and s1 is the vibration amplitude of the whole single “hard” stub; F2 = k0.s2 = kRT.s2, where s2 is the vibration amplitude of the whole composite stub through rubber filler to the plate against the plate vibrates along the z-direction (see Figure 9(a) and (c)).
\nIn this case, the rubber filler acts as a spring, and the whole stub acts as a mass. The broad complete bandgap metal-matrix embedded phononic crystals includes mo = 2mh and mo = 2ms + 2mh, where ms denotes the mass of soft stub and mh denotes the mass of hard stub. It is shown in Figures 9(c) and 10(c) that the frequency is shifted to the lowest frequency range. Compared to the transition PC plate, where the soft stub contacts the rubber filler such that it represents a flexible constraint to the rubber filler (see Figure 9(a)), the hard stub contacts the rubber filler in the broad complete bandgap metal-matrix embedded phononic crystals to produce a rigid constraint for the rubber filler (see Figure 9(c)). This causes the spring stiffness k0 (k0 = kRT—the longitudinal stiffness of the rubber filler) to increase, while the lump mass becomes smaller (2ms + 2mh > 2mh). This, in turn, not only causes the opening location of the out-of-plane bandgap to shift to higher frequencies but also makes the force F1 larger than the force F2. As a result, the out-of-plane bandwidth becomes wider. We conclude that, after introducing the rubber filler, an out-of-plane analogous-rigid mode of the stub was produced, which can reduce the location of the out-of-plane bandgap. Hence, the introduction of a single “hard” stub increases the bandwidth of the out-of-plane bandgap by enhancing the stiffness of the out-of-plane analogous-rigid mode of the stub.
\nFigure 11 displays the magnitudes of the total displacement vectors of a unit cell of the broad complete bandgap metal-matrix embedded phononic crystals, the transition PC plate, and the classical PC plate. The figures correspond to the upper and lower edges of the first in-plane bandgap of each structure. The mode S2 is a symmetric Lamb mode of the plate. The steel plate vibrates along the xy-plane (in-plane), while the stubs swing in the opposite direction. With respect to the frequencies, the symmetric Lamb mode will be activated, and the in-plane waves can propagate through the phononic-crystal plate in the symmetric Lamb mode. When the frequency of the in-plane waves approaches the first natural frequency of the stub resonator, the resonant mode S1 will be activated. The stub vibrates in the yz-plane and produces a reacting force to the plate while vibrating in the xy-direction. In that case, the in-plane waves are not able to propagate through the PCs. As a result, an in-plane bandgap opens up. The bandwidth is determined by the coupling between the resonance mode S1 and the traditional plate mode S2.
\nThe total displacement vector fields for the modes (resonant mode S1, symmetric Lamb mode S2) (a) correspond to Figure 4(a), (b) correspond to Figure 4(b), and (c) correspond to Figure 4(c).
The coupling strength between the resonance mode S1 and the traditional plate mode S2 is determined by the formula
\nwhere sL is the vibration amplitude of the stub and ki is the tensional stiffness of the spring.
\nFor both the transition and the classical phononic-crystal plates, the formation mechanism of the in-plane bandgap is shown in Figure 12(a) and (b), respectively.
\nFormation mechanism process of the bandgap for the in-plane bandgap of (a) the “transition” PC plate, (b) the classical PC plate, and (c) the broad complete bandgap metal-matrix embedded phononic crystals.
It can be found that the stubs “swing” in the xy-plane and produce a reacting force (F2 = ki.sL = kST.sL, where kST is the transverse stiffness of the soft stub and sL is the vibration amplitude of the whole stub) through the soft stub (flexible body) to the plate against the plate vibrating along the xy-plane. The soft stub (flexible) acts as a spring (ki = ks), while the hard stub (rigid body) acts as a mass (mi = mh), as shown in Figure 13(a) and (b), respectively. Hence, both location and bandwidth of the first in-plane bandgap in these structures are the same.
\nThe equivalent theoretical model of the resonator for formation mechanism of the in-plane bandgap of (a) the “transition” PC plate, (b) the classical PC plate, and (c) the broad complete bandgap metal-matrix embedded phononic crystals.
For the broad complete bandgap metal-matrix embedded phononic crystals, it can be found that the displacement fields are distributed across the entire stub (Figure 11(a)) to cause an “in-plane analogous-rigid mode” because the whole stub vibrates along the xy-plane (in-plane) with a weak constraint, while the frequency is non-zero. It can be observed that the displacement fields of its eigenmodes are distributed throughout the whole stub. This means that the whole stub body moves in the xy-plane like a rigid body moves in rigid mode. However, the natural frequency is not 0, and the whole stub is constrained by the rubber filler. Therefore, we refer to the concept of rigid mode and call this type of vibration mode of a whole stub the “in-plane analogous-rigid mode.” The formation mechanism of the bandgap is shown in Figure 12(c). The whole stub vibrates in the xy-plane and produces a reacting force (F1 = ki.sL = kRL.sL, where kRL is the longitudinal stiffness of the rubber filler and sL is the vibration amplitude of the whole stub) through the rubber filler to the plate against the plate vibrating in the xy-plane.
\nIn this case, the “rubber filler” acts as a spring, where the longitudinal stiffness of the rubber filler acts as the spring, while the whole stub acts as a mass, where mi = mh, and mh denotes the mass of hard stub (see Figure 13(c)). Compared to the transition PC plate, in which the soft stub contacts the rubber filler such that it gives a flexible constraint to the rubber filler, this leads to the soft stub acting as a spring (see Figure 13(c)). The hard stub contacts the rubber filler in the broad complete bandgap metal-matrix embedded phononic crystals, which creates a rigid constraint to the rubber filler that causes the rubber filler to act as a spring (see Figure 13(c)). This causes the spring stiffness ki to increase in the broad complete bandgap metal-matrix embedded phononic crystals such that the opening location of the in-plane bandgap is shifted toward higher frequencies. However, it also makes the force F1 larger than force F2 and causes the in-plane bandwidth to become broader. We conclude that, after introducing the rubber filler and the single “hard” stub simultaneously, an “in-plane analogous-rigid mode” of the stub is produced, which can increase the bandwidth of the in-plane bandgap.
\nWhen the broad out-of-plane bandgaps and broad in-plane bandgaps, which were simultaneously increased by the single “hard” stub, overlap, a wider complete bandgap, in which the out-of-plane and in-plane Lamb waves are prohibited, is created.
\nIt can be concluded that the bandwidth of the bandgap is determined by the resonator mode. After introducing the rubber filler and the single “hard” stub simultaneously, two new resonator modes are produced. One resonator mode is the in-plane analogous-rigid mode, in which the whole stub vibrates along the in-plane plate. The other resonator mode is the out-of-plane analogous-rigid mode, in which the stub vibrates along the out-of-plane plate. They both increase the in-plane and out-of-plane bandgaps, respectively. Two increased bandgaps overlap to produce a broad complete bandgap, in which both the out-of-plane and in-plane Lamb waves are prohibited. The rubber filler shifts the two kinds of bandgaps simultaneously toward low frequency such that a broad complete bandgap for low frequencies is obtained.
\nIn order to investigate the effect of the stubs on the complete bandgaps of the metal-matrix embedded phononic crystals [14, 18], we studied the influence of the stub height on the first complete bandgap.
\nFigure 14 displays the evolution of the first complete bandgap as a function of the steel-stub height, hS. We can find that, with the increase of the steel-stub height, both the lower and upper edges of the first complete shift to a lower frequency range firstly and then move to higher frequency range. For example, when the steel-stub height is less than or equal to 3 mm, with the increase of the steel-stub height, the lower edge shifts to lower frequencies, but when the rubber stub height is larger than 3 mm, the lower edge frequency shifts to higher frequencies as the steel-stub height increases.
\nThe evolution of the first complete BG in the lower frequency complete bandgap metal-matrix embedded phononic crystals as a function of the steel-stub height with D = 8 mm, dup = 9 mm, dup = 5 mm, h = 5 mm, e = 1 mm, and a = 10 mm, respectively.
In order to investigate the effect of single “hard” stubs on the complete phononic bandgaps of the novel metal-matrix PCs, we studied the effect of the steel-stub height on the first complete bandgap. Figure 15 displays the evolution of the first complete bandgap as a function of the steel-stub height h.
\nThe evolution of the first bandgap in the proposed phononic-crystal plate with “hard” stubs as a function of steel-stub height with D = 8 mm, d = 7.5 mm, e = 1 mm, and a = 10 mm, respectively.
We find that, with the increase of the steel-stub height, the location of the first complete bandgap shifts to a lower frequency range before it shifts to a higher frequency range. Furthermore, the bandwidth of the bandgaps becomes broad. For example, when the steel-stub height is below or equal to 3.5 mm, after increasing the steel-stub height, the location of the bandgaps shifts to lower frequencies. However, when the steel-stub height exceeds 3.5 mm, the location of the bandgaps shifts to higher frequencies as the steel-stub height increases.
\nIn this chapter, metal-matrix embedded phononic crystals consisting of double-sided stubs (single stubs and composite stubs), which are deposited on a two-dimensional locally resonant phononic crystal plate that consists of an array of rubber fillers embedded in a steel plate, are introduced. The following summaries were drawn [14, 18]:
\nThe spring-mass system of the resonator can be decoupled by introducing the rubber filler, and then the out-of-plane bandgap and the in-plane bandgap can be adjusted into the same lowest frequency range. The out-of-plane bandgap and the in-plane bandgap can be overlapped with each other. As a result, a lower frequency complete bandgap which ranges from 59 to 93 Hz is obtained in the metal-matrix embedded phononic crystals.
\nBoth the out-of-plane and the in-plane phononic bandgap increase after introducing single “hard” cylinder stubs. When introducing the rubber filler and the single “hard” stub simultaneously, two new kinds of resonance modes are produced: an in-plane analogous-rigid modes, where the whole stub vibrates in-plane with the plate, and the other new resonance mode is the out-of-plane analogous-rigid mode, where the whole stub vibrates out-of-plane with respect to the plate. The out-of-plane bandgap increases for the out-of-plane analogous-rigid mode, and the in-plane bandgap increases for the in-plane analogous-rigid mode. Both the out-of-plane and the in-plane analogous-rigid modes are mainly formed due to introducing the single “hard” cylinder stub. The in-plane and out-of-plane bandgaps overlap and produce a broad complete bandgap in the metal-matrix embedded phononic crystals. Within this broad complete bandgap, both the in-plane and out-of-plane Lamb waves are prohibited. The absolute bandwidth of the bandgap for the proposed structure is five times higher than for the classic double-sided stubbed metal-matrix phononic-crystal plate.
\nThe effect of the stub on the bandgaps is investigated. Results show that the location of the bandgaps can be modulated in a significant lower frequency range, and the bandwidth can be expanded in a considerable large frequency range by introducing different composite taper stubs, and the bandwidth can be expanded to a larger frequency range by introducing different “hard” stubs.
\nThe proposed structure provides an effective way for phononic crystals to obtain wide complete bandgaps and lower frequency complete bandgaps (below 100 Hz), which have a potential application in the low-frequency vibration reduction in a practical case.
\nThis chapter was supported by the Project of the National Science Foundation of China (No. 51674199).
\nBiological particles on the nanometer and submicrometer scale, such as proteins, lipids, nucleic acids, exosomes, and metabolic content, have attracted much attention as biomarkers for diagnosing diseases from biologically generated fluids such as blood, urine, and lymph. These biomarkers are now understood to be fundamental to healthy intercellular communication and can be produced in diseased cells. Label free Raman spectroscopy is useful for verification of biological samples ranging from nanoscale to millimeter size, such as tissue [1, 2], cells [3, 4, 5], bacteria [6, 7], exosomes [8, 9], and proteins [10, 11]. After incident laser emission with a single wavelength, Raman spectroscopy can identify biomarkers with the spectral peak position as a fingerprint because the molecular vibrations of the sample are represented by spectra due to inelastic scattering. Electromagnetic enhancement can be achieved on rough surface of metal such as a gold or silver nanoparticle that causes amplification of the light by local surface plasmon resonance (LSPR) effects [12, 13]. A “hot spot” is formed on the surface of the SERS particles, and the Raman signal is dramatically increased at the nano-sized gap. Surface-enhanced Raman spectroscopy (SERS) is an approach for cell analysis and identification that applies a wide range of chemical spectroscopy to nanometer-sized biomarkers. Recent studies on monomolecular scales have been made possible through surface-enhanced Raman techniques [14, 15, 16, 17, 18]. According to finite element method (FEM) analysis, when the colloid is separated by 2 nm between a diameter of 30 nm colloids, a “hot spot” is formed which gives a surface-enhanced effect of about 108 degrees [13]. In biomedical applications, biomarkers suitable for these nanogaps are very rare, and due to the size and shape of biomolecules, research on nanogap and signal enhancement of the SERS structure is needed to optimize the LSPR effect.
\nIn this chapter, we fabricated SERS substrate based on ZnO nanorods and improved the SERS effect by forming selective growth clustering of gold nanoparticles, which could be formed in specific condition of ZnO nanorod-based SERS substrate. To control the porosity and gold nanostructure, the length and density of the ZnO nanostructures and the thickness of the deposited gold were modified morphologically. The SERS enhancement mechanism was described based on finite element analysis. Cell viability was also evaluated to determine the presence or absence of toxicity for cancer cell applications. In other bio-applications, we demonstrate early diagnostic possibilities with Raman signals and statistical analyses from nano-sized biomarkers of intractable inflammatory diseases that cause patient pain.
\nMost of the research to fabricate SERS-based chips focuses on optimizing the surface of substrates through nanomaterials and nanostructures synthesized using sophisticated techniques such as lithographic patterning or high-temperature processes [16, 19, 20, 21, 22]. Other research groups deposit Au/Ag nanoparticles on papers [23, 24, 25] or coat metal on a Si nanowire structure [26] to make a porous SERS substrate suitable for biological or liquid samples. Such Si nanowires are too dependent on the substrate and are difficult to combine with common cell culture substrates such as glass and petri dishes, due to their amorphous and manufacturing nature. In the case of paper-based SERS substrate, porosity and nanogaps could not be adjusted. On the other hand, if a ZnO nanorod-based platform is introduced, the substrate can be manufactured at a temperature below 100°C. Furthermore, homogeneous nanostructures can be formed without any lithography process on amorphous substrates such as glass and plastic, which are common in bioscience applications [27, 28].
\nTo make the SERS substrates, the Si wafer were scribed with a size of 1 × 1 cm2 for substrate of ZnO nanorods initially. It was cleaned in ethanol and deionized (DI) water for 5 min, respectively. The 30 nm ZnO seed layer was deposited on the surface of as-prepared samples by using RF magnetron sputtering for 5 min under 100 W power to grow the vertically aligned ZnO nanorods utilizing by the hydrothermal synthesis. The ZnO growth solution was prepared by dissolving 10 mM zinc nitrate hexahydrate (Sigma-Aldrich Co.) and 0.9 mL of ammonium hydroxide (Sigma-Aldrich Co.) in 30 mL DI water. A homogeneous aqueous solution was achieved using mildly stirred vortexer for 5 min at room temperature. Then, the as-prepared samples were immersed into the aqueous solution in an oven at 90°C for 50 min.
\nAfter ZnO growth, the substrates were cleaned with DI water and dried with nitrogen gas. Finally, the ZnO nanorods (NRs) were coated with Au using a thermal evaporator (Alpha Plus Co., Ltd., Korea). The thickness monitor for 100 and 200 nm deposition was standardized. The morphological and structural properties of the Raman measured samples were observed by using a field-emission scanning electron microscope (FE-SEM) (S-4700, HITACHI, Japan) with 15 kV beam voltage. The procedure of the experiment including the measurement analysis is schematically shown as a diagram in Figure 1.
\nSchematic of the experiment involving zinc oxide nanostructure-based SERS substrate fabrication.
To obtain adequate porosity for the solution sample, the ZnO seed layer was modified and deposited such that the preferential growth direction of the zinc oxide nanorods was within about 10° from vertical. A volume of gold having a height of 100 and 200 nm per unit area was deposited on nanorods having length distribution of 300–450 nm or 500–650 nm, respectively. These four specimens were displayed with FE-SEM images of the 45° tilt view as shown in Figure 2. The top and bottom of the Figure 2e show the substrates with gold deposited (top) and not deposited (bottom) for Figure 2a, and ZnO is fully covered even when only 100 nm of gold is deposited. When the gold deposition is increased to 200 nm, the rod thickness is distributed about 10–30 nm thicker than when the gold deposition is not performed. Also, since the nanorod length distribution has a standard deviation of 50 nm and the deposited gold is clustered at the head of the nanorods, the height distribution of the gold clusters undergoes a similar variation. Therefore, when confocal Raman spectroscopy measurements are focused on the gold clusters, the head size can be a key factor in the Raman enhancement effect.
\nSecondary electron images of the substrate with nanorod length and deposited gold thickness modified for (a) ZnO of length 400 nm with 100 nm deposited Au, (b) 600 nm length ZnO with 100 nm Au, (c) 400 nm length ZnO with 200 nm Au, and (d) 400 nm ZnO with 200 nm Au. (e) Shows the initial difference in covering due to gold coating. All of scales are the same.
The Raman enhancement effect of SERS substrate based on ZnO nanorods was confirmed using 1 mM Rhodamine B drop, and the signals were measured after natural drying. Rhodamine B (RhB, >95%) purchased from Sigma-Aldrich was used as a standard for Raman measurements due to its refined condition. Raman measurements (LabRam Aramis, Horiba) were carried out using a 785 nm diode laser in a confocal geometry with a 0.5 NA, x50 objective lens and beam spot diameter ~1.9 μm. The spectrum of each point was measured in the range of 400–2500 cm−1 with a spectral resolution of 5 cm−1 and an integration time of 30 s at room temperature. The spectra were postprocessed by Savitzky-Golay smoothing, and a third-order polynomial fit to the autofluorescence background was subtracted.
\nFigure 3a shows the enhancement of the Raman signal ranging from 1000 to 1500 cm−1 according to each specimen. The greatest enhancement was observed in 600 nm ZnO nanorods deposited with 200 nm gold, which is a correction to the area of the circle by drying droplet. However, without area correction, a random point of 400 nm in length with the same average gold thickness appeared to give a greater improvement. This difference is explained by the correlation between signal enhancement and sample concentration. The enhancement factor EF follows the equation:
\nwhere CSERS is the concentration of RhB on the ZnO nanorod-coated Au SERS substrate, ISERS is the measured Raman intensity from the nanorod-coated substrate, and Cbare and Ibare are the same quantities on the bare substrate, respectively. The effective concentration of RhB on the dried specimen varies across the sample as the diffusion of the droplet depends upon the porosity of the SERS substrate. In addition, even though samples of the same volume are dropped for all experiments, the value of Cbare/CSERS is dependent upon a correction for area since the initial droplet varies in size. Therefore, EF due to the substrate correction can be seen to be largest in the 200 nm gold-coated nanorods of 600 nm length. This enhancement is independent for each specific peak, as shown in Figure 3b. This measurement shows that the enhancement effect is compared without denaturation of the sample, since the relative ratio between the peaks does not change significantly.
\n(a) Raman signal enhancement of RhB on each substrate. (b) Differences in the enhancement intensity of each specific peak according to the substrate. (c) Differences in the coffee ring effect of the Raman signal depending on the nanostructure. (d) Optical microscope image including Raman acquired point.
In addition, the SERS substrates based on zinc oxide nanorods show no coffee ring effect, as shown in Figure 3c. Due to the rise in concentration at the ring region, the edges of RhB on bare substrates and of RhB on gold thin films show stronger Raman signals. On the other hand, nanorod substrates have larger values in the interior of the deposition ring as shown in Figure 3d.
\nCalculation analysis using FEM was performed to understand the effect of Au cluster size on SERS enhancement. The finite element method (FEM) was used in COMSOL Multiphysics software (COMSOL Inc., USA) to simulate the SERS activities of the electromagnetic fields. A two-dimensional model for metal-coated nanostructured substrates with various metal spherical diameters has been established. The near-field distribution of the electromagnetic field was calculated for given boundary condition to solve the equation of time harmonic Maxwell at the excitation wavelength of 785 nm.
\nFrom secondary electron microscopy images, structures were modeled with 80 and 125 nm heads on nanorods of width 50 nm including gold coating and length 600 nm. The near-field distribution of the electric fields was calculated for 785 nm incident light and parallel plate boundary conditions with symmetry of the electric field. Even if the head width is increased, there is almost no change in the full scale, and the electric field is distributed in the vertical direction of the incident light as shown in Figure 4. The cross-sectional width of incident light is 2 μm, and the density of the nanorods in the region is constant. This suggests that the difference in SERS enhancement by Au thickness is due to concentration of sample at LSPR area.
\nFinite element analysis showing (a) the different factor in LSPR on ZnO nanorod-based SERS substrates with gold head diameters of (b) 80 nm and (c) 125 nm, respectively.
Raman measurements of nanometer-sized biomarkers secreted from living cells require confirmation of the suitability of cells for SERS substrates. The toxic endogenous properties of gold nanoparticles have previously been reported [29], and ZnO nanorods are reported to be toxic to NIH 3 T3 fibroblasts [30]. Therefore, it is necessary to confirm whether the sensing chip is suitable for cell application through the evaluation of cytotoxicity, and cell culture and cell viability tests were carried out as follows.
\nBreast cancer cell line of MDA-MB-231 was purchased from the Korean Cell Line Bank (Seoul, Korea). The breast cancer cells MDA-MB-231 were cultured in Dulbecco’s modified Eagle’s minimal essential medium (DMEM; Life Technologies, Inc., Grand Island, NY, USA) supplemented with 10% fetal bovine serum (FBS; Hyclone Laboratories, Logan, UT, USA) and a 1% penicillin–streptomycin solution (Life Technologies, Inc.) in a humidified 5% CO2 incubator at 37°C. Cell viability analyses were based on MTT (Sigma, USA) assays. Cells were plated at a density of 3 × 105 cells/well and incubated for 24 hours. After preparing the Au-ZnO substrate, the cell was treated with 5 mg/ml MTT for 30 min and then dissolved using DMSO. The absorbance was measured at 540 nm with an ELISA microplate reader (Multiskan EX, Thermo Scientific, USA). Figure 5 shows the results of MTT assay of the death of the breast cancer cell line MDA-MB-231 according to SERS substrate condition. The MDA-MB-231 is commonly used in biotechnology applications such as analyzing cancer cell malignancy. From the results, the substrate at this ZnO nanorod and gold deposition conditions is suitable for experiments in live breast cancer cell lines.
\nCell viability of MDA-MB-231 on each substrate by MTT assay.
Interstitial cystitis/bladder pain syndrome (IC/BPS) is a refractory disease that afflicts the vague pelvic pain when the urine enters the bladder and makes frequent urination [31]. There are various treatments based on oral agents [32, 33, 34], but they are still unsatisfactory with frequent recurrence of symptoms and Hunner’s lesions [35]. Parallel with the development of therapeutic technology, early diagnosis of IC/BPS related to quality of life and detection of disease before development to chronic type can minimize patient’s pain and increase treatment effect. Therefore, it is necessary to confirm the possibility of early detection of nanometer biomarkers from urine obtained from interstitial cystitis animal model using SERS substrate. IC/BPS animal models and comparative groups for these experiments were derived using 10-week-old female Sprague Dawley rats. The rats were instilled with 0.2 M HCl for 10 min using a 26-gauge angiocatheter in the bladders of four rats, followed by neutralization and saline wash. Other rats in the comparison group were used as vehicle instead of HCl injection. The voiding pattern was analyzed to confirm the reproducibility of animal modeling as in the previous paper [36]. Rat urine was collected in a 50 mL tube using a metabolic cage. The voiding pattern measured at a week after HCl injection was examined, and the collected urine was used as a sample for Raman measurement. Twenty-four hours of natural voiding patterns in the metabolic cage were recorded and analyzed using AcqKnowledge 3.8.1 software and an MP150 data acquisition system (Biopac Systems, Goleta, CA, USA) at a sampling rate of 50 Hz. The volume change of the obtained raw data of urine was estimated by 1 mL unit as shown in Figure 6a. Irregular frequency of urinary dysfunction caused by bladder inflammation is observed in HCl-treated rats and is consistent with previous animal model studies [35, 36]. Steps and terraces in the graph are the excretion urine and the duration between voiding, respectively. The total amount of control and IC/BPS animal models for approximately 10 hours is 11 and 13 mL, respectively. However, compared with the amount, the frequency is 3 and 6 times. When the unit of step is 0.5 mL, the frequency is 4 and 11 times, and the difference in the voiding frequency is clearly revealed.
\n(a) Measurement of voiding function in control group (blue line) and IC/BPS animal group (orange line) at 7 days after HCl treatment. (b) Optical microscope images of a Raman measurement region diffused from a sample droplet into a nanoporous area and (c) magnified SEM image.
From the identified sample, a drop of 5 μL was applied to the SERS specimen, and the sample was allowed to spread for 60 min. After confirming that the droplets were dry and diffused, they were loaded onto a Raman spectroscope system attached to a microscope (IX-73, Olympus, Japan) and measured. As shown in Figure 6b, the diffused region of the sample can be confirmed by an optical microscope, and the region where the sample is diffused as in (c) can be confirmed to have a nanometer-scale porosity. In this area, Raman spectra were collected using a customized spectrometer (FEX-INV, NOST, Korea) with a 785 nm diode laser as the excitation source. The 1 mW of excitation light was focused on the sample through a 40 ×/0.6 NA objective with spot size ~2.4 μm. The spectrum of each point was measured 8 times in the range of 550 to 1500 cm−1 with a spectral resolution of 1 cm−1 and an integration time of 40 s at room temperature. The Raman spectrum was calibrated by measuring a silicon sample before the Raman measurements. To evaluate the spectral differences between control and IC/BPS of rat urine, principal component analysis (PCA) was introduced. A statistical analysis method of PCA reduces the number of variables in multivariate systems, and all of spectral range was used as variables. All analyses were conducted using XLSTAT 2018 software.
\nAs shown in Figure 7a, brown bars are indicated on the peak, which is the main factor above the graph drawn as the average of total data. The main peaks for the control and IP/BPS samples were observed at 641, 683, 723, 873, 1002, 1030, and 1355 cm−1, which corresponded to C-C twisting mode of tyrosine [37, 38], ring breathing of nucleic acids for G [38, 39] and A [39, 40], C-C stretch of hydroxyproline [37, 41], symmetric ring breathing mode [37, 38, 39, 40, 41] and C-H in-plane bending mode of phenylalanine [37, 41], and CH3CH2 wagging mode of collagen [37, 41], respectively.
\n(a) Averaged Raman spectra for IC/BPS (blue line) and control (green line) of rat’s urine. Standard deviations are painted around the spectra. (b) Principal component analysis results for urine of IC/BPS and control sample.
The peak at 1002 cm−1 has a considerably large value compared to the rest of the data, which is notable in literature referring to other peaks, as they relate to other biologies. To analyze Raman peaks, PCA is utilized as shown in Figure 7b. Clear discrimination and reliable separation between control and IC/BPS groups were observed. By plotting PC2 and PC3, the groups show clear distinctions between IC/BPS urine and the control samples (dotted line). Raman spectrum measurements and PCA analysis showed that it is possible to distinguish between normal and diseased groups using gold-coated ZnO nanorod substrates that can be applied to early disease diagnostic sensing chips.
\nIn summary, we compared the differences in surface-enhanced Raman effect using RhB by adjusting the length and diameter of ZnO nanorods and the volume of deposited gold in ZnO-based SERS substrates. Electron microscopy images showed clustering on top of nanorods during gold deposition and showed nanometer level porosity. As the volume of deposited gold increases, the Raman signal also improves, but as the growth conditions of the nanorods change, the signal intensity also changes. This is because the Raman enhancement factor is determined by the enhancement by the SERS properties of metal and the concentration of the sample. Through the finite element analysis in the two-dimensional plane, signal enhancement was similar for 80 and 125 nm of Au-grain head. It was found that the enhancement of the Raman signal was determined by the wider surface area of gold. It was confirmed that this signal enhancement is made in the vertical direction of the rod, so that only nanometer targets trapped in the porous space can obtain enhanced signals. In addition, the SERS chips based on ZnO nanorods were found to have no coffee ring effect in measuring liquid samples and were also suitable for cell application experiments. An IC/BPS animal model was constructed for the bio-application, the voiding pattern was observed for urinary disease status, and urine was collected from the IC/BPS. The obtained urine was diffused into a ZnO-based SERS chip having nanopores, and Raman was measured in the corresponding region. Statistical analysis of Raman signals obtained from nanometric level area showed that IC/BPS and normal animals were distinguished. Therefore, we can confirm that ZnO nanorod-based SERS has sufficient potential for early disease diagnosis by efficiently detecting nano-sized biomarkers.
\nThis work was supported by the Basic Science Research Program (2018R1D1A1B07048562) and MRC grant (2018R1A5A2020732) through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (MSIT), by the Ministry of Trade, Industry and Energy (MOTIE) under the Industrial Technology Innovation Program (10080726, 20000843), and by a grant of the Korea Health Technology R&D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health and Welfare, Republic of Korea (HI18C2391).
\nThe authors declare no conflict of interest.
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