Vertical deformation of member OA (x10-4).
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\n'}],latestNews:[{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"},{slug:"intechopen-s-chapter-awarded-the-guenther-von-pannewitz-preis-2020-20200715",title:"IntechOpen's Chapter Awarded the Günther-von-Pannewitz-Preis 2020"}]},book:{item:{type:"book",id:"2211",leadTitle:null,fullTitle:"Wireless Sensor Networks - Technology and Protocols",title:"Wireless Sensor Networks",subtitle:"Technology and Protocols",reviewType:"peer-reviewed",abstract:"This book explores both the state-of-the-art and the latest developments in wireless sensor networks technology. 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He was first appointed as Assistant Professor and then promoted to Associate Professor at North South University in 2011 and later on Professor. While in that post he was also the coordinator of EEE program. During 2012-2017, he was an Associate Professor at Universiti Teknologi Brunei (UTB), Brunei Darussalam (QS World University ranking 379). He received his B.Sc. degree in Electrical and Electronic Engineering from BUET (Bangladesh), his M.Sc. degree in Digital Communication from Loughborough University, UK and PhD in Wireless Communication from Newcastle University, UK. He has taught several courses in communications, electronics and signal processing at KUET, Khulna University, BRAC University, and UKM (Malaysia) during his career. He has published over 90 peer-reviewed journals and conference papers, and is the author/editor of 16 (sixteen) academic books such as Towards Cognitive IoT Networks (Springer, 2020), Communication Systems for Electrical Engineers (Springer, 2018), Spectrum Access and Management for Cognitive Radio Networks (Springer, 2016), Coding for MIMO-OFDM in Future Wireless Systems (Springer, 2015), Advances in Sensor Networks Research (Nova publisher, USA, 2014) and 10 (ten) book chapters. He has presented invited talks in Bangladesh and Malaysia and has served as a member of the program committee for more than 50 international conferences. He is on the editorial board of several international journals such as IEEE Communications Magazine, IEEE, USA, IET Wireless Sensor Systems (IET-WSS), and so on. Dr. Matin is a member of the IEEE, IEEE Communications Society (IEEE ComSoc), and several other international organizations. 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This can be addressed by the adoption of advanced construction technology which may lead to appropriate design, project delivery, suitable procurement monitoring, high quality control and careful commissioning. Advanced technologies include 3D printing, composite materials, computer vision, modular construction, substructure works and temporary structures. Structural design will also closely affect construction costs, quality and safety. The book aims to publish current practice or recent advances in both building construction and structures including infrastructures such as bridges, tunnels and offshore platforms. The focus of chapters can be construction systems and operations; structural use of emerging materials including composite; behavioral analysis of structural systems; structures under extreme conditions such as strong winds, earthquakes, fires and blasts; performance-based structural engineering; Energy modeling; life-cycle structural engineering; structural health monitoring; and sustainability in construction and structure; green building.
",isbn:"978-1-83881-141-9",printIsbn:"978-1-83881-140-2",pdfIsbn:"978-1-83881-142-6",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"df2ad14bc5588577e8bf0b7ebcdafd9d",bookSignature:"Mr. Alireza Kaboli and Dr. Sara Shirowzhan",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10110.jpg",keywords:"Information Modeling, Strand7, Abaqus, Sensing technology, Remote sensing, Virtual reality, Augmented reality, Advanced GIS applications, Computational intelligence, Big data, Analytics and prediction, Smart infrastructure, Off-site construction, Modular building, Safety in construction, Non-combustible cladding, Fa?ade types, Estimating and contracts, Earth moving activities, Emission in construction, Construction haulage management",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 5th 2019",dateEndSecondStepPublish:"November 28th 2019",dateEndThirdStepPublish:"February 1st 2020",dateEndFourthStepPublish:"May 13th 2020",dateEndFifthStepPublish:"July 11th 2020",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!0,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"313050",title:"Mr.",name:"Alireza",middleName:null,surname:"Kaboli",slug:"alireza-kaboli",fullName:"Alireza Kaboli",profilePictureURL:"https://mts.intechopen.com/storage/users/309192/images/system/309192.png",biography:null,institutionString:null,position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"UNSW Sydney",institutionURL:null,country:{name:"Australia"}}}],coeditorOne:{id:"273838",title:"Dr.",name:"Sara",middleName:null,surname:"Shirowzhan",slug:"sara-shirowzhan",fullName:"Sara Shirowzhan",profilePictureURL:"https://mts.intechopen.com/storage/users/273838/images/system/273838.jpeg",biography:"Dr. Sara Shirowzhan completed her Ph.D. in Geomatic Engineering at UNSW and her strong capability in designing and conducting high-quality multi-disciplinary research has been demonstrated by solving complex problems facing digital representation and analysis of the built environment and civil engineering infrastructures. 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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"}},{type:"book",id:"3621",title:"Silver Nanoparticles",subtitle:null,isOpenForSubmission:!1,hash:null,slug:"silver-nanoparticles",bookSignature:"David Pozo Perez",coverURL:"https://cdn.intechopen.com/books/images_new/3621.jpg",editedByType:"Edited by",editors:[{id:"6667",title:"Dr.",name:"David",surname:"Pozo",slug:"david-pozo",fullName:"David Pozo"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"39156",title:"Carbon Nanotubes Under Simple Tension and Torsion – Molecular/Structural Mechanics and the Finite Element Method",doi:"10.5772/51070",slug:"carbon-nanotubes-under-simple-tension-and-torsion-molecular-structural-mechanics-and-the-finite-elem",body:'\n\t\tThe intended applications of carbon nanotubes have steadily increased since their discovery by Ijima [4]. They range from the nanoscale, as in the tip of an atomic electron microscope, to the macroscale, as in the preliminary design of the space elevator cable.
\n\t\t\tIn modeling CNTs, molecular as well as quantum mechanics have been the primary tools for analysis. Also, closed form expressions were developed to study the response of CNTs in different environments [3,8,12].
\n\t\t\tOn the other hand, some attempted to use structural mechanics, and built corresponding finite element models, to study the behavior of CNTs, as evident in several publications [5-8]. Some of these publications simplified the property relations between molecular mechanics and structural mechanics. They assumed the structural bending stiffness EI/a to be a constant and set it equal to the molecular bond bending stiffness C.
\n\t\t\tHowever, in [1], with a simple proof, we showed that the main assumption used by various authors to equate the element bending stiffness to the bond bending stiffness (C= EI/a) does not hold. In addition, in our previous publications (Kasti [1,2]), we related some of the mechanical properties used in molecular and structural mechanics.
\n\t\t\tIn [1], we derived an expression for the axial deformation of zigzag CNTs that accounts for the axial and bending structural stiffnesses under simple tension. While molecular mechanics uses the bond angle between two bonds to describe bond bending deformations, structural mechanics uses the bending within one 3D frame element for this definition. Comparing the deformation equation in structural mechanics to the equivalent equation derived for molecular mechanics, leads us to a “consistent” frame bending stiffness for an infinitely long zigzag CNT. For large diameter tubes, the frame bending stiffness tends to half the bond bending stiffness. This later case is representative of a graphene sheet. For small diameters, EI/a changes with the bond bending stiffness C, the torsional angle φ and the lattice translational index n. The expression for the axial deformation was then expanded to include the torsional stiffness of 3D frame elements. This provided us with an equation for the “elastic modulus” of an infinitely long zigzag CNT. It included the axial, bending and torsional deformations. Then, in [2], we extended the work to the case of simple torsion of zigzag carbon nanotubes. The expression relating the molecular bond bending stiffness C and the structural bending stiffness EI/a was derived. It was found to be different from the case of simple tension. The structural bond bending stiffness was both load and chirality dependent. However, for the particular configuration of a graphene sheet, the relation of simple tension was recovered, namely EI/a=C/2. We concluded the work by presenting the expression for the deformation of the tube when axial, bending and torsional structural stiffnesses are accounted for. We noticed in this case of simple torsion that the axial stiffness couples with the bending and torsion stiffnesses, unlike simple tension.
\n\t\t\tIn a recent paper (Chen et al., [9]), the radial elastic modulus of the original Molecular Structure Mechanics model (MSM) was compared to the one from the Molecular Dynamics (MD) simulation. In that paper, it was pointed to the fact that a modification to the original MSM model was suggested in our previous paper (Kasti, [1]).
\n\t\t\tIn this chapter, we extend our previous work to armchair carbon nanotubes under simple tension. In addition, we summarize the equivalent results for zigzag CNTs under simple tension and torsion.
\n\t\t\tWe start with a brief review of molecular and structural mechanics and we refer to the work of Chang and Gao [3]. Then, the relation between the structural bending stiffness EI/a and the molecular bond bending stiffness C is derived for the case of simple tension. This shows that EI/a depends on the bond bending stiffness C and the torsional angle φ. In the limit of an infinite tube radius, which represents a graphene sheet, we recover the previous relation, i.e., EI/a tends to C/2. Finally, an expression for the Young’s modulus is presented that accounts for the axial, bending and torsional deformations. We conclude the chapter with numerical simulations that validate the results.
\n\t\tThis section deals with the molecular and structural mechanics formulations of bond energies. Also, a short review of the finite element method is presented as it applies to the modeling of carbon nanotubes subjected to mechanical loading.
\n\t\t\tThe geometry of a CNT could be described with the pair (n,m), the lattice translational indices, and the bond length a. In general, the diameter d of a CNT is defined using the expression
\n\t\t\t\tFor zigzag CNTs (Fig. 1a), the value of m is zero. In this case, the diameter is simply given by the formula\n\t\t\t\t\t\t
The bond energies between carbon atoms include the stretching Ua, bending Ub and torsional Ut energies. For small distortions from equilibrium, these energies could take the forms:
\n\t\t\t\twhere K, C, and Ct are the bond stretching, bending and torsional stiffnesses.
\n\t\t\t\tr, Θ, and Ф are the stretched position, bending and torsional angles, respectively.
\n\t\t\t\tThe subscript “o” refers to the initial equilibrium configuration.
\n\t\t\t\tAs far as structural mechanics, the linear elastic deformation is assumed to be the combination of axial, bending and torsional deformations. Their corresponding strain energies are expressed as:
\n\t\t\t\twhere EA, EI, and GJ are the axial, bending and torsional stiffnesses; and u, θ and φ are the axial, bending and torsional deformations, respectively.
\n\t\t\t\tCarbon Nanotubes: (a) Zigzag, (b)Armchair.
For linear elastic behavior of 3D space frames, the axial, bending and torsional strain energies can be expressed as in equation (3).
\n\n\t\t\t\tThe bonding between two carbon atoms is modeled by placing a 3D space frame element between them, Fig. 2. When this procedure is repeated throughout the tube, a finite element mesh is obtained with the carbon atoms becoming the nodes in the mesh.
\n\t\t\t\tEach node is assumed to have six degrees of freedom, three translational and three rotational.
\n\t\t\t\tSpace frame element.
The stiffness matrix K relating the degrees of freedom to their corresponding forces and moments at both ends of a 3D frame element is a 12x12 matrix.
\n\t\t\t\tChang and Gao derived closed form expressions for carbon nanotubes subjected to simple tensile loading using molecular mechanics.
\n\t\t\tRepresenting units of zigzag and armchair carbon nanotubes are shown in Fig. 3 with α and β being the internal angles. Equivalent equations to the ones of Chang and Gao will be derived in the next section for the armchair CNT.
\n\t\t\ta) Two units of a zigzag CNT. (b) One unit of an armchair CNT.
We start by expressing the results of Chang and Gao [3] in a more suitable form using the principle of minimum total potential energy. We will split the approach into bond stretching and bond bending deformations.
\n\t\t\tDue to bond stretching, it is easy to verify that\n\t\t\t\t\t\t
We start with the molecular energy expression
\n\t\t\t\twhere F is the axial load applied to a single carbon atom, Δ is the deflection of the end of the tube, Σ1 is a summation over the number of bonds and Σ2 is the summation over the number of atoms with applied loads.
\n\t\t\t\tExpression (5) takes the following form in structural mechanics:
\n\t\t\t\tSince both K and EA/a are conjugate to the axial deformation between carbon atoms in the energy equation, they represent the same axial stiffness. Thus,
\n\t\t\t\tTo determine the tube deformation, we let nu be the number of vertical units of Fig. 3 in a carbon nanotube and nv equal to (2nu–1). The deflection at the end of the tube due to axial bond deformations can be expressed as:
\n\t\t\t\tDue to bond bending, the total potential energy is written as:
\n\t\t\t\tLet nx be the number of units along the circumference and nv the number of vertical units. Then, for an infinite cylinder with no end effects,i.e., all units have the same deformation, Π will be equal to:
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t
Minimizing the total potential energy with respect to α gives
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t
Since cosβ = -cos(π/2n)cos(α/2) and cosφ = tan(α/2)/tan(β), we get dβ/dα = cosφ/2, where φ is the torsion angle between the planes of adjacent units of an armchair nanotube (Fig. 3).
\n\t\t\t\tThe vertical deformation of the tube can be expressed as:
\n\t\t\t\twhere Ho,H are the initial and current heights of the tube, respectively. Differentiating Eq. (10) with respect to α\n\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t
Solving for Δα by minimizing the total potential energy, we get
\n\t\t\t\tSubstituting Δα in Eq. (10) above, we get the following expression for the vertical deflection of an infinitely long CNT (due to bond bending only)
\n\t\t\t\tIn the next Section, we will derive an equivalent expression in terms of the material properties of structural mechanics. This will allow us to deduce a relation between EI/a and C.
\n\t\t\tLet O and A be two atoms on the CNT with coordinates (Rcos(-θ/2), Rsin(-θ/2), 0 ) and ( Rcos(-π/n+θ/2), Rsin(-π/n+θ/2), a sin(α/2) ), respectively. The angle θ is given by 2R2(1-cosθ)=a2.
\n\t\t\t\tAlso, let u be the unit vector from O to A expressed as:
\n\t\t\t\twhere i, j and k are the unit vectors in the Cartesian coordinate system.
\n\t\t\t\tFor an infinite cylinder and due to symmetry, the radial and tangential rotations at O and A are zero.
\n\t\t\t\tIn addition, due to multiple symmetries (for n=4,8,12,16,..), we assume the displacement and rotation fields at O and A take the following forms:
\n\t\t\t\tand
\n\t\t\t\twhere ( δro, δrA ) and ( θo, θA ) are the displacements and rotations vectors at O and A respectively. In this work, small displacements and rotations are assumed. Without loss of generality, the tensile load at each carbon atom is assigned a value of one.
\n\t\t\t\tFollowing a similar procedure to Kasti [1], one can show that θ is equal to zero.
\n\t\t\t\tAnd, the only force in the inclined member OA in Fig. 3 is the vertical force F and the moment is (1/2)F.a.sin(α/2).
\n\t\t\t\tDue to multiple symmetries (for n=4,8,12,16,..), and after some simplifications, the axial deflection at the end of the whole CNT is given by:
\n\t\t\t\tand the corresponding elastic modulus Ys\n\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\tZigzag CNTs under simple tension\n\t\t\t\t
\n\t\t\t\tThe corresponding axial deformation at the end of a zigzag CNT under simple tension is given by [1]:
\n\t\t\t\t\n\t\t\t\t\tZigzag CNTs under simple torsion\n\t\t\t\t
\n\t\t\t\tThe tangential deformation at the end of a zigzag CNT unit under simple torsion is given by following expression for ΔB [2]:
\n\t\t\t\tComparing the molecular mechanics expression Eq. (12) with the structural mechanics Eq. (16), we obtain
\n\t\t\tThus, in general, the bending stiffness to be used in the structural mechanics varies with the bond bending stiffness C and torsional angle φ.
\n\t\t\tFor long CNT tubes with large diameters, cos2(φ) → 1 and EI/a → C/2, which is the same result derived by Kasti [1] for the particular case of a graphene sheet.
\n\t\t\tTo validate the closed form solution Eq. (16), we compared the axial deformation of member OA (Fig. 3) and the change in radius to the results from a finite element model in ABAQUS [14]. The results are shown in Tables 1 and 2 below for C=1.42 nN.nm.rad-2. The accuracy obtained is excellent.
\n\t\t\t\n\t\t\t\t\t\t\tLattice Translational Index, n\n\t\t\t\t\t\t | \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMolecular/Structural Mechanics\n\t\t\t\t\t\t | \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tABAQUS\n\t\t\t\t\t\t | \n\t\t\t\t\t
4 | \n\t\t\t\t\t\t5.9844 | \n\t\t\t\t\t\t5.9849 | \n\t\t\t\t\t
8 | \n\t\t\t\t\t\t5.9477 | \n\t\t\t\t\t\t5.9470 | \n\t\t\t\t\t
12 | \n\t\t\t\t\t\t5.9358 | \n\t\t\t\t\t\t5.9358 | \n\t\t\t\t\t
16 | \n\t\t\t\t\t\t5.9310 | \n\t\t\t\t\t\t5.9306 | \n\t\t\t\t\t
Vertical deformation of member OA (x10-4).
\n\t\t\t\t\t\t\tLattice Translational Index, n\n\t\t\t\t\t\t | \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMolecular/Structural Mechanics\n\t\t\t\t\t\t | \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tABAQUS\n\t\t\t\t\t\t | \n\t\t\t\t\t
4 | \n\t\t\t\t\t\t-1.3538 | \n\t\t\t\t\t\t-1.3539 | \n\t\t\t\t\t
8 | \n\t\t\t\t\t\t-2.6402 | \n\t\t\t\t\t\t-2.6401 | \n\t\t\t\t\t
12 | \n\t\t\t\t\t\t-3.9383 | \n\t\t\t\t\t\t-3.9383 | \n\t\t\t\t\t
16 | \n\t\t\t\t\t\t-5.2403 | \n\t\t\t\t\t\t-5.2402 | \n\t\t\t\t\t
Change in radius (x10-3).
\n\t\t\t\tZigzag CNTs under simple tension\n\t\t\t
\n\t\t\tFor zigzag CNT under simple tension, the equivalent stiffness is given by [1]:
\n\t\t\tThus, in general, the bending stiffness to be used in the structural mechanics varies with the bond bending stiffness C, torsional angle φ and γ.
\n\t\t\tFor long CNT tubes with large diameters, cosφ → 1, γ→0 and EI/a → C/2, which is the same result for the particular case of a graphene sheet.
\n\t\t\t\n\t\t\t\tZigzag CNTs under simple torsion\n\t\t\t
\n\t\t\tFor a zigzag CNTs under simple torsion, the corresponding stiffness is given by [2]:
\n\t\t\tThus, in general, the bending stiffness to be used in structural mechanics varies with the bond bending stiffness C and lateral translational index n.
\n\t\t\tFor long CNT tubes with large diameters, n→ ∞, \n\t\t\t\t\t
In Sections 4 and 5, a closed form expression was developed for the deformation of infinitely long armchair CNT under simple tension. It included the axial and bending stiffnesses of 3D frame elements. In this Section, we study the effect of the torsional stiffness of 3D space frames.
\n\t\t\tSimilar work to Kasti [1] will show that the torsional stiffness does not enter the expression for the deformation of an infinitely long armchair carbon nanotube under simple tension.
\n\t\t\t\tWhen the axial, bending and torsional deformations are combined, we obtain the following formula:
\n\t\t\t\tTo validate the closed form expression of Eq. (24), we compared the vertical deformation of member OA (Fig. 3) and the change in radius to the results from a finite element model in ABAQUS. The results are shown in Tables 3 and 4 below for K=652nN.nm-1 and C=1.42 nN.nm.rad-2. The accuracy obtained is excellent.
\n\t\t\t\t\n\t\t\t\t\t\t\t\tLattice Translational Index, n\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tMolecular/Structural Mechanics\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tABAQUS\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
4 | \n\t\t\t\t\t\t\t1.7686 | \n\t\t\t\t\t\t\t1.7686 | \n\t\t\t\t\t\t
8 | \n\t\t\t\t\t\t\t1.7500 | \n\t\t\t\t\t\t\t1.7500 | \n\t\t\t\t\t\t
12 | \n\t\t\t\t\t\t\t1.7461 | \n\t\t\t\t\t\t\t1.7461 | \n\t\t\t\t\t\t
16 | \n\t\t\t\t\t\t\t1.7446 | \n\t\t\t\t\t\t\t1.7446 | \n\t\t\t\t\t\t
Vertical deformation of member OA (x10-3).
\n\t\t\t\t\t\t\t\tLattice Translational Index, n\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tMolecular/Structural Mechanics\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tABAQUS\n\t\t\t\t\t\t\t | \n\t\t\t\t\t\t
4 | \n\t\t\t\t\t\t\t-0.53131 | \n\t\t\t\t\t\t\t-0.53136 | \n\t\t\t\t\t\t
8 | \n\t\t\t\t\t\t\t-0.96001 | \n\t\t\t\t\t\t\t-0.96001 | \n\t\t\t\t\t\t
12 | \n\t\t\t\t\t\t\t-1.4088 | \n\t\t\t\t\t\t\t-1.4088 | \n\t\t\t\t\t\t
16 | \n\t\t\t\t\t\t\t-1.8634 | \n\t\t\t\t\t\t\t-1.8634 | \n\t\t\t\t\t\t
Change in radius (x10-3).
The contribution of each of the bond stiffnesses (axial, bending and torsion) to the total vertical deformation of an armchair carbon nanotube is shown in the following example.
\n\t\t\t\tTwo long carbon nanotubes (40 armchair carbon units) with lattice translational indices “n” equal to 4 and 16, respectively, are modeled using MSC/Nastran [15]. The tubes are supported at the bottom and subjected to tensile loading at the top.
\n\t\t\t\tThe resulting vertical deformations are compared to the closed form solution of Eq. (24), as shown in Fig. 4. In spite of the difference in boundary conditions between the closed form solution and the finite element modeling, the errors in the results are less than 3%.
\n\t\t\t\t\n\t\t\t\t\tZigzag CNTs under simple tension\n\t\t\t\t
\n\t\t\t\tGoing through the same manipulations as for an armchair CNT, the vertical deformation at the end of a zigzag CNT under simple tension that accounts for bending and torsional deformations can be expressed as [1]:
\n\t\t\t\twhere F is the load applied at a carbon atom, n\n\t\t\t\t\t\n\t\t\t\t\t\tu\n\t\t\t\t\t is the number of vertical units of Fig. 3 and n\n\t\t\t\t\t\n\t\t\t\t\t\tv\n\t\t\t\t\t = (2n\n\t\t\t\t\t\n\t\t\t\t\t\tu\n\t\t\t\t\t –1). However, in this case, γ takes on the following expression:
\n\t\t\t\tVertical deformations of armchair carbon nanotubes with lattice translational indices of 4 and 16, respectively. Three cases are considered: 1) bending stiffness only, 2) bending + torsion and 3) axial+bending+torsional.
\n\t\t\t\t\tZigzag CNTs under simple torsion\n\t\t\t\t
\n\t\t\t\tWhen the torsional stiffness of 3D frame elements is included, in addition to the axial and bending deformations, the tangential deformation ΔB of a zigzag CNT under simple torsion takes the following form [2]:
\n\t\t\t\twhere N and D are 6x1 vectors function of n, the lattice translational index, and a, the bond length. KN and KD are 6x1 vectors of structural stiffnesses. These vectors could be expressed as:
\n\t\t\t\t\n\t\t\t\t\tKN = [ (EI/L)3 (EI/L)2(GJ/L) (EA/L)(GJ/L)2 (EA/L)(EI/L)(GJ/L) (EI/L)(GJ/L)2 (EA/L)(EI/L)2]T\n\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\tKD = [ (EI/L)4 (GJ/L)(EI/L)3 (EA/L)(GJ/L)2(EI/L) (EA/L)(EI/L)2(GJ/L) (EI/L)2(GJ/L)2 (EA/L)(EI/L)3]T\n\t\t\t\t
\n\t\t\t\t\n\t\t\t\t\tN = [ N1 N2 N3 N4 N5 N6 ]T,D = [ D1 D2 D3 D4 D5 D6]T
\n\t\t\t\tFor example, for n=4,
\n\t\t\t\t\n\t\t\t\t\tN = [ -21.1721 -24.53 -0.0015 -0.0969 -0.4074 -0.0832 ]T\n\t\t\t\t
\n\t\t\t\tD = [ 0. –2.3062x103 -0.4408 -76.1719 497.0279 -62.4643 ]T
\t\n\t\t\t\tOne point worth mentioning is that when the torsional stiffness is neglected, i.e.\n\t\t\t\t\t\t
\n\t\t\t\t\t\n\t\t\t\t\t\t
Thus, in this case of negligible torsional stiffness, the axial and bending deformations are decoupled.
\n\t\t\tSimilar to the previous work by Kasti [1], an elastic modulus Ys (Tpa.nm) could be defined that doesn’t include the thickness of CNTs, and is expressed as:
\n\t\t\t\twhere Ft (equal to 2nF with F=1) is the total load applied, L and R are the length and radius of the tube, respectively.
\n\t\t\t\tFor a finite length cylinder, the elastic modulus obtained from the closed form expressions of Eq. (28) and ABAQUS are compared in Table 5 for a lattice translational index of 4 and variable tube length. The following values of stiffnesses were used:
\n\t\t\t\t\n\t\t\t\t\tEA/a = 652 nN.nm-1, EI/a = 0.875 nN.nm.rad-2 and GJ/a = 0.278 nN.nm.rad-2.\n\t\t\t\t
\n\t\t\t\tThe closed form results compare very well with the values from ABAQUS.
\n\t\t\t\tLatticeTranslationalIndex, n | \n\t\t\t\t\t\t\tnv=40,60(Eq. 28) | \n\t\t\t\t\t\t\tnv=40(ABAQUS) | \n\t\t\t\t\t\t\tnv=60(ABAQUS) | \n\t\t\t\t\t\t
4 | \n\t\t\t\t\t\t\t358.12 | \n\t\t\t\t\t\t\t358.284 | \n\t\t\t\t\t\t\t358.307 | \n\t\t\t\t\t\t
Elastic Modulus (nN.nm-1).
\n\t\t\t\t\tZigzag CNTs under simple torsion\n\t\t\t\t
\n\t\t\t\tSimilar to the definition of an elastic modulus Ys (Tpa.nm) that doesn’t include the thickness of CNTs [1], an elastic shear modulus Gs (Tpa.nm) could be defined as [2]:
\n\t\t\t\twhere T is the torsional moment applied to the tube, F is the tangential load applied to a single carbon atom which can be taken as unity, L and R are the length and radius of the tube, respectively.
\n\t\t\tRelations between the structural bending stiffness EI/a and the molecular bond bending stiffness C for carbon nanotubes were derived for the cases of simple tension and torsion. In addition, expressions for the deformations and “Young’s moduli” of these nanotubes were presented that account for the axial, bending and torsional effects.
\n\t\tThe effect of temperature and combined loading on composite plate is one of the primary life limiting factors of a bridge engineering application. This chapter will consider the structural evaluation of the localized effect in the bridge engineering. The application of bridge engineering can be found in a structural bridge deck panel. Ray studied the fiber-matrix debonding by applying the thermal shock of thermal fatigue, taking into account the conditioning time. He performed a three-point bending test on glass fiber reinforced with unsaturated polyester and epoxy resin composites in which it exposed to 75
The stress and strain relationship is varied through the laminate thickness, as indicated in Eq. (1):
The general bending equation of rectangular plate is as below:
By taking into account the temperature effect, the mechanical and thermal bending moments are:
where
And,
It can be assumed that all layers have
Lamina of arbitrary of principal material direction.
The normal deflection distribution is derived based on the solution of classical laminate plate theory using Navier equation. Navier solution assumed that the boundary condition is simply supported from all edges under the effect of temperature
where.
And,
By integrating Eq. (7) with respect to (x) and (y), the temperature distribution through the plate thickness is:
The thermal bending moment is defined as in the following:
where
The general solution of normal deflection for simply supported boundary condition from all edges is:
Substitute Eq. (12) and Eq. (10) into Eq. (6), the solution of bending deflection is illustrated in the following equation:
where
The theory of classical laminate plate of Levy solution is used to derive the solution of normal deflection. The Levy solution assumed that the variation of the bending deflection should be along the x-axis. Levy solution can be used on any type of boundary condition which gives flexibility on any type of loading such as
where
By integrating Eq. (15) with respect to (x), the temperature distribution through the plate thickness is:
Ignore the variation of thermal bending moment and normal deflection along y-axis, Eq. (6) will be:
As mentioned earlier, the thermal bending moment is varied along x-axis, as below:
where
The solution of normal bending deflection is as below:
To find
Substitute Eq. (19) and Eq. (22) into Eq. (18) to obtain the particular solution of bending deflection along x-axis,
where
To find
The solution of Eq. (25) is as below:
Substitute Eq. (26) into Eq. (25), to obtain the homogeneous solution of Eq. (25) along x- and y-directions:
Substitute Eq. (27) and Eq. (23) into Eq. (21) to obtain the general solution of normal bending deflection, as indicated below:
The simply supported boundary conditions from all edges are assumed and the constants
where
In this chapter, the finite element discretization is carried out by using ANSYS Ver. 18.2. (SHELL 132) element is used to mesh the composite laminate plate. SHELL 132 is defined by eight nodes having six degrees of freedom at each node to calculate the central normal deflection. In the simulation analysis, the central point of laminate plate is used to calculate the normal deflection. Always the convergence test is needed to determine the size of elements in which the value of normal bending deflection settles down. Finite element analysis of convergence curve defines the relationship between the grid interval and the analysis accuracy. Four types of combined loading is used such as: (temperature affect only
25.07 | 40 | 50 | 60 | 70 | 80 | |
---|---|---|---|---|---|---|
19.933 | 30.4038 | 37.41988 | 44.435 | 51.452 | 58.468 | |
19.933 | 30.4038 | 37.41988 | 44.435 | 51.452 | 58.468 | |
3.0896 | 3.81746 | 4.53322 | 5.5793 | 7.25302 | 10.3612 | |
0.3835 | 0.35098 | 0.32915 | 0.30732 | 0.2855 | 0.26366 | |
1.07675 | 1.33379 | 1.5878 | 1.9614 | 2.5648 | 3.70468 | |
1464.18 | 1686.48 | 1835.4 | 1984.32 | 2133.24 | 2282.16 | |
25.746 E-6 | 21.6044 E-6 | 18.5098 E-6 | 15.3005 E-6 | 12.0234 E-6 | 8.70307 E-6 | |
25.746 E-6 | 21.6044 E-6 | 18.5098 E-6 | 15.3005 E-6 | 12.0234 E-6 | 8.70307 E-6 | |
10.5844 E-6 | 7.932 E-6 | 6.9852 E-6 | 6.3374 E-6 | 5.8663 E-6 | 5.5082 E-6 | |
0.4533 | 0.622 | 0.735 | 0.848 | 0.961 | 1.074 | |
0.4533 | 0.622 | 0.735 | 0.848 | 0.961 | 1.074 | |
0.2174 | 0.2626 | 0.30068 | 0.3553 | 0.43418 | 0.55808 | |
768.139 | 780.8944 | 787.7133 | 793.5087 | 798.495 | 802.8304 |
Mechanical and thermal properties of the simulated materials.
Figures 2 and 3 show the verification test of normal bending deflection using Levy and Navier solutions, taking into consideration ANSYS 18.2 results. The normal bending deflection decreased with the increasing of plate aspect ratio because of the increasing in plate bending stiffness under the temperature effect
Normal bending deflection varying with laminate plate aspect ratio under temperature effect 60C∘.
Normal bending deflection varying with laminate plate aspect ratio under temperature effect −15C∘.
Figure 4 shows the convergence test of normal bending deflection with total degrees of freedom for different fiber volume fractions using ANSYS software. The normal central deflection decrease with the increasing of fiber volume fraction under the effect of temperature
Convergence test of normal deflection and total degrees of freedom under the effect of temperature ΔT, bending moment Mo, and in-plane force Nxx.
Table 2 shows the analytic and simulation verification results of bending deflection under combined loadings for fiber volume fraction
Deflection | Levy method results | ANSYS 18.2 results | Percentage error ( |
---|---|---|---|
0.1853e-3 | 0.1880e-3 | 1.748 | |
−0.1777e-3 | −0.1882e-3 | 5.536 | |
0.7704e-5 | 0.7108 e-5 | 7.736 | |
−0.9859e-5 | −0.9365e-5 | 5.010 |
Analytic and simulation verification of bending deflection under combined loading.
As mentioned in Introduction section, Levy and Navier solutions are used to describe the theory of bending deflection by taking into consideration the use of simply supported boundary condition from all edges. ANSYS software is used in the convergence test. The bending deflection value when
α1, α2 | thermal expansion coefficient in longitudinal and lateral directions, 1/C∘. |
ΔT | gradient uniform temperature, C∘. |
A1, A2 | bending moment due to temperature, N.m/C∘. |
Mxx,Myy, and Mxy | bending and twist moments, N.m. |
Qij | reduced stiffness elements, N/m2. |
w0 | midplane deflection along z-direction. |
zk, zk+1 | upper and lower lamina surface coordinates along z-direction, m. |
a, b | length of large and small spans of rectangular plate (m). |
m, n | double trigonometric of Furrier series. |
N | total number of layers. |
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",metaTitle:"Conflicts of Interest Policy",metaDescription:"As an Open Access publisher, IntechOpen is dedicated to maintaining the highest ethical standards and principles in publishing. In addition, IntechOpen promotes the highest standards of integrity and ethical behavior in scientific research and peer-review.",metaKeywords:null,canonicalURL:"/page/conflicts-of-interest-policy",contentRaw:'[{"type":"htmlEditorComponent","content":"In each instance of a possible Conflict of Interest, IntechOpen aims to disclose the situation in as transparent a way as possible in order to allow readers to judge whether a particular potential Conflict of Interest has influenced the Work of any individual Author, Editor, or Reviewer. IntechOpen takes all possible Conflicts of Interest into account during the review process and ensures maximum transparency in implementing its policies.
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\n\nA Conflict of Interest is a situation in which a person's professional judgment may be influenced by a range of factors, including financial gain, material interest, or some other personal or professional interest. For IntechOpen as a publisher, it is essential that all possible Conflicts of Interest are avoided. Each contributor, whether an Author, Editor, or Reviewer, who suspects they may have a Conflict of Interest, is obliged to declare that concern in order to make the publisher and the readership aware of any potential influence on the work being undertaken.
\n\nA Conflict of Interest can be identified at different phases of the publishing process.
\n\nIntechOpen requires:
\n\nCONFLICT OF INTEREST - AUTHOR
\n\nAll Authors are obliged to declare every existing or potential Conflict of Interest, including financial or personal factors, as well as any relationship which could influence their scientific work. Authors must declare Conflicts of Interest at the time of manuscript submission, although they may exceptionally do so at any point during manuscript review. For jointly prepared manuscripts, the corresponding Author is obliged to declare potential Conflicts of Interest of any other Authors who have contributed to the manuscript.
\n\nCONFLICT OF INTEREST – ACADEMIC EDITOR
\n\nEditors can also have Conflicts of Interest. Editors are expected to maintain the highest standards of conduct, which are outlined in our Best Practice Guidelines (templates for Best Practice Guidelines). Among other obligations, it is essential that Editors make transparent declarations of any possible Conflicts of Interest that they might have.
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I am also a member of the team in charge for the supervision of Ph.D. students in the fields of development of silicon based planar waveguide sensor devices, study of inelastic electron tunnelling in planar tunnelling nanostructures for sensing applications and development of organotellurium(IV) compounds for semiconductor applications. I am a specialist in data analysis techniques and nanosurface structure. I have served as the editor for many books, been a member of the editorial board in science journals, have published many papers and hold many patents.",institutionString:null,institution:{name:"Sheffield Hallam University",country:{name:"United Kingdom"}}},{id:"54525",title:"Prof.",name:"Abdul Latif",middleName:null,surname:"Ahmad",slug:"abdul-latif-ahmad",fullName:"Abdul Latif Ahmad",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"20567",title:"Prof.",name:"Ado",middleName:null,surname:"Jorio",slug:"ado-jorio",fullName:"Ado Jorio",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Universidade Federal de Minas Gerais",country:{name:"Brazil"}}},{id:"47940",title:"Dr.",name:"Alberto",middleName:null,surname:"Mantovani",slug:"alberto-mantovani",fullName:"Alberto Mantovani",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"12392",title:"Mr.",name:"Alex",middleName:null,surname:"Lazinica",slug:"alex-lazinica",fullName:"Alex Lazinica",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/12392/images/7282_n.png",biography:"Alex Lazinica is the founder and CEO of IntechOpen. 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