VIBRATION ANALYSIS OF A CARBON NANOTUBE IN THE PRESENCE OF A GRAPHENE SHEET. A Thesis. Presented to. The Graduate Faculty of The University of Akron

Transcription

1 VIBRATION ANALYSIS OF A CARBON NANOTUBE IN THE PRESENCE OF A GRAPHENE SHEET A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Sunny Rahul Nimmalapalli August, 2016

2 VIBRATION ANALYSIS OF A CARBON NANOTUBE IN THE PRESENCE OF A GRAPHENE SHEET Sunny Rahul Nimmalapalli Thesis Approved: Accepted: Advisor Dr. S.Graham Kelly Department Chair Dr. Sergio Felicelli Committee Member Dr. Alper Buldum Interim Dean of the College Dr. Donald P. Visco Jr. Dean of the Graduate School Dr. Chand K. Midha Date ii

3 ABSTRACT The free and forced vibrations of multiwalled carbon nanotube in the presence of a graphene sheet using the Timoshenko beam model for fixed-fixed end conditions and for various aspect ratio s are studied. The multiwalled carbon nanotube can be studied by considering each of the nested nanotube to be a Timoshenko beam and are coupled through the van der Waals forces. The interaction between the graphene sheet and the outer nanotube are also coupled through the van der Waals forces. The van der Waals interaction energy potential expressed as a function of interlayer spacing can be estimated using Lennard-Jones potential. The change in the equilibrium distance d can effect the natural frequencies of the system, hence calculation of d is necessary. Intertube radial displacement because of the applied external force on the structure at a frequency nearer to the resonant frequency or due to the disturbances within the structure may result in different resonant frequencies and non-coaxial modes. These non-coaxial modes may result in the distortion of the geometry of the structure. Hence even the study of these resonant frequencies and the non co-axial modes is needed. In this thesis, the study of the vibration modes are analyzed for the iii

4 double walled carbon nanotube. Once the problem formulation for the double walled nanotube is obtained, it can be extended to the multi walled carbon nanotube. Modal analysis is performed to study the forced responses of the double walled nanotube system assuming that the graphene sheet has a harmonic excitation. The time response plots for various frequencies and the frequency response plots for various lengths of the carbon nanotube are obtained. iv

5 ACKNOWLEDGEMENTS First and foremost, I would like to express my sincere thanks to my adviser Dr. S.Graham Kelly for his continuous support. His patience in explaining me and letting me learn all that I needed for my research is really commendable. I am really grateful to him for his guidance throughout my Master s program. I would like to thank my thesis committee member: Dr. Alper Buldum for taking time out of his busy schedule to attend my defense. Most importantly, I would like to express my thankfulness to my parents N.V.Sudhaker and A.Sreematha, for their continuous support and love. I would like to thank my brother Nitin Parsa, sister Sushma, better half Anusha and my friends for their support. v

6 TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES viii ix CHAPTER I. INTRODUCTION Motivation Brief view of the properties of carbon nanotubes and graphene Chirality of carbon nanotubes and structural characteristics of graphene sheet Literature review of carbon nanotubes Timoshenko beam theory and Vanderwaal s forces Overview II. MECHANICS OF CARBON NANOTUBES Mechanical properties of carbon nanotubes Strength of carbon nanotube Lennard-Jonnes potential and van der Waals forces Estimation of the equilibrium distance III. PROBLEM FORMULATION vi

7 3.1 Timoshenko Beam Theoretical modeling of the beams Extended Hamilton s principle Non dimensionalization of Timoshenko beam equations IV. FREE VIBRATION RESPONSES Timoshenko beam partial differential equations Frequency response equations Determination of the natural frequencies and mode shapes V. FORCED VIBRATION RESPONSES Modal Analysis Forced response equations Frequency response plots Time response plots VI. CONCLUSIONS AND RECOMMENDATIONS Conclusion Recommendations BIBLIOGRAPHY APPENDICES APPENDIX A. LIST OF SYMBOLS AND THEIR NUMERICAL VALUES USED FOR THE CALCULATION OF THE NATURAL FREQUENCIES APPENDIX B. TRANSFORMATION OF THE EQUATIONS IN CHPATER vii

8 LIST OF TABLES Table Page 2.1 Elastic properties of multiwalled carbon naotubes Natural frequencies of fixed-fixed double walled carbon nanotube for different aspect ratio s viii

9 LIST OF FIGURES Figure Page 1.1 Schematic diagram of a two dimensional graphene sheet[1] Chirality of CNT aarmchair bzigzag [2] Euler Bernoulli and Timoshenko Beam a Schematic view of the vanderwaal bonds between an atom of carbon nanotube and the nearby counterparts in the graphene sheet and b Front view of the graphene sheet-carbon nanotube vanderwaal bonds in cut-off radius zone [3] Graph of Interatomic potential versus distance Double walled carbon nanotube in the presence of a graphene sheet Mode shapes of a double walled carbon nanotube with fixed-fixed end conditions for an aspect ratio of Mode shapes of a double walled carbon nanotube with fixed-fixed end conditions for an aspect ratio of Mode shapes of a double walled carbon nanotube with fixed-fixed end conditions for an aspect ratio of Mode shapes of a double walled carbon nanotube with fixed-fixed end conditions for an aspect ratio of Frequency response plots for fixed-fixed double walled nanotube near x= Frequency response plots for fixed-fixed double walled nanotube near x= ix

10 4.7 Frequency response plots for fixed-fixed double walled nanotube near x= Time response plots of a double walled carbon nanotube with fixedfixed end conditions for ω = Time response plots of a double walled carbon nanotube with fixedfixed end conditions for ω = Time response plots of a double walled carbon nanotube with fixedfixed end conditions for ω = x

11 CHAPTER I INTRODUCTION 1.1 Motivation Nanotubes are long, slender fullerenes, having cylindrical walls composed of carbon atoms bonded in a hexagonal shape. Fullerenes [4] are clusters of carbon atoms, closed-cage like structures which led to the discovery of nanostructures such as carbon nanotubes. They take the form of a hollow sphere, ellipsoid, or tube. Carbon nanotubes acquire special properties which are not found in any other compounds. Since their discovery in 1991 by Iijima and coworkers [5], carbon nanotubes have been of special interest. The large length of some carbon nanotubes coupled with a small diameter result in a large aspect ratio. They can be nearly assumed as one dimensional form of fullerenes, making carbon nanotubes, the materials which are expected to possess interesting electrical, mechanical and molecular properties. Carbon nanotubes are cylindrical molecules consisting of a hexagonal arrangement of carbon atoms. They can be described as single or multiple layers of graphene sheets rolled into hollow cylinders. These carbon nanotubes are of two forms, single walled carbon nanotube and multi walled carbon nanotube. A single walled carbon 1

12 nanotube contain a single graphene layer whereas multi walled carbon nanotubes are comprised of several to tens of concentric graphene layers, each one forming a single walled carbon nanotube. Carbon nanotubes are synthesized by three main techniques [5], arc discharge, laser ablation and chemical vapor deposition. The arc discharge method employ a vapor created between the two carbon electrodes, the vapor results in the formation of carbon nanotubes. In laser ablation technique, a high power laser beam is impinged on a certain volume of the stock gasmethane or carbon monoxide containing carbon. Arc discharge method and chemical vapor deposition technique produces both the single walled and multi walled nanotubes, whereas laser ablation produces only single walled nanotubes. Complex experiments were used by researchers to study graphene. But the first crystals of graphene were discovered in 2004,using a simple and ordinary technique. Graphene is the first material to be two dimensional, which gives graphene a unique set of properties. Graphene is stronger that diamond and about 300 times stronger than steel and is used in many leading industries such as auto-motives, electronics, energy storage, communications, sensors, oil production and aerospace. 1.2 Brief view of the properties of carbon nanotubes and graphene Carbon nanotubes characteristics mostly depend on their size. So a effective computational method has to be developed to study the properties of the 2

13 nanotubes. The larger aspect ratio of carbon nanotubes and one dimensional form make them the materials that are expected to have interesting electrical and mechanical properties. Some of the most important properties of the carbon nanotubes are chemical reactivity, electrical conductivity, optical activity and mechanical strength. When the chemical reactivity of a carbon nanotube is compared with that of a graphene sheet, the carbon nanotubes of smaller diameter show increased reactivity. To study the electrical conductivity of a carbon nanotube, its chirality plays an important role. The smaller diameter carbon nanotubes are either semi conducting or metallic. The differences in the conducting properties of the carbon nanotubes are due to the different band structure. Studies show that the optical activity of the nanotube disappears when they become large. Carbon nanotubes have a very high elastic modulus, which make them very suitable for applications in composite materials. Carbon nanotubes are the most compelling materials of all the nanomaterials due to their fundamental properties measurements and potential applications. The physical properties revealed from theoretical and experimental studies have been proven to be quite impressive. Studies show that the specific tensile strength of a single walled nanotube is 100 times that of steel. A cantilevered beam model has been used in an experiment by Wong et al [6], in which multi walled nanotube were bent using an atomic force microscope tip, a Young s modulus of 1.28 ± 0.59 TPa was observed. The maximum tensile strength of a single walled nanotube is close enough to 30 GPa. From all the experiments 3

14 conducted in studying the elastic behavior of nanotubes lead to a conclusion that the diameter and shape of the nanotube is a determining factor in obtaining the elastic modulus. Experiments conducted on multi walled nanotubes by Forr et al[6] proved that the elastic modulus really did not depend on the diameter of the nanotube instead it depended on the amount of disorder in the nanotube walls. However, their work showed that the elastic modulus for single walled nanotube does depend on diameter of nanotube. Another factor that does effect the elastic modulus of nanotube is its thickness. If the nanotube is treated as a solid cylinder then a lower elastic modulus is expected. If the nanotube is treated as a hollow cylinder, then the elastic modulus is higher and thinner the walls of nanotube are higher is its elastic modulus. Other properties of nanotubes such as Poisson s ratio, bulk modulus depend strongly on helicity, diameter, stiffness, flexibility and strength of carbon nanotubes. All these unique properties of nanotubes motivated researchers and led to further study of these materials. The first experiment to observe the elastic properties and strength of graphene sheet were performed by Lee and his co-workers [7]. They found that the graphene exhibits both the non-linear elastic behavior and brittle fracture. The mono layer graphene also exhibits crystallographic characteristics of crack growth. Regular defects that are observed in graphene are vacancies, dislocations and grain boundaries. Dislocations and grain boundaries have higher impact on the mechanical properties of graphene. The plastic behavior and the unique set of properties that are possessed by graphene, because of its two dimensional structure 4

15 has made graphene a very important super material for many leading industries and also led researchers to study about the properties of graphene in detail. 1.3 Chirality of carbon nanotubes and structural characteristics of graphene sheet Carbon nanotubes chirality is expalined by its chiral vector C h and chiral angle θ. Chiral vector, defined with the help of a pair of integers n and m and is given as C h = n a 1 + m a 2 [8], where a 1 and a 2 are the unit vectors illustrated in the figure 1.1. This pair of integers n, m make carbon nanotube have three different unique geometries, which are armchair, zigzag and chiral. The representations in terms of chiral vectors for theses configurations are armchairn, n, zigzagn, 0 and chiraln, m. Chiral angles for zigzag and armchair are 0 and 30 respectively. The study of chirality of the nanotubes is important due to the reason that the chirality of nanotube determines it s conductivity, allowing them for a variety of use in electronic switching devices. The integers n and m are helpful in determining the chirality or so called twist of the nanotube. As already said chirality of nanotube affects it s conductivity, in addition it also affects it s density, lattice structure and many other properties. For a single walled nanotube its metallic nature can be determined by these integers, n and m. If n m is divisible by 3, then it s metallic [9]. Otherwise it s semiconducting. And it can also be said that if two thirds of nanotube is metallic then the other one third is semi conducting. Single walled carbon nanotubes are capable of undergoing 5

16 elastic bending deformations, whereas the elastic behavior is less for multi walled nanotubes when compared to the single walled nanotubes. Graphene is an allotrope of carbon, which is in the two dimensional form and is stronger than diamond and steel. Graphene exhibits an excellent thermal conductivity when compared to the other materials, which is really impressive and makes it a unique material because of the unique set of properties it possess because of its two dimensional hexagonal pattern. The carbon-carbon bond length in graphene is nm. Graphene possess a tensile strength of 130 GPa and Young s modulus of 1 TPa [7]. Apart from the excellent properties possessed by graphene it also acquires exceptional electronic properties. All these make graphene a dependable material for many industries such as aerospace, paints and coatings, light/heavy composite materials etc. Figure 1.1: Schematic diagram of a two dimensional graphene sheet[1] 6

17 Figure 1.2: Chirality of CNT aarmchair bzigzag [2] 1.4 Literature review of carbon nanotubes The growth of science and technology has evolved into the era of nanotechnology. As the properties of the nanomaterials mostly depend on their size, providing the researchers significant difficulties in studying the mechanical, electrical properties of these nanomaterials. Also modeling of these nanomaterials have received much interest over the years. The modeling of the carbon nanotubes can be classified into two approaches. One is the atomistic modeling and the other is the continuum mechanics modeling. Atomistic modeling [8] can be done through several techniques such as classical molecular dynamics, tight binding molecular dynamics, density functional 7

18 theory and several other techniques. But the atomistic modeling is not preferred over continuum modeling as the later has the advantage of modeling the nano-structures with large scale over the former. Though the molecular dynamics modeling possess a advantage of considering the surface effect, which results in a system with a standard lattice so that the carbon nanotube is in an unstrained equilibrium position, the huge computational tasks of atomistic molecular dynamics modeling technique makes it less approved over continuum modeling techniques. The practical application of both the atomistic and continuum modeling techniques are limited to systems containing small number of atoms and also to the systems which have short life of around picoseconds to nanoseconds. The other category of modeling is the continuum mechanics modeling [8]. Equivalent continuum modeling is referred as the most efficient method, particularly for the nano-structures with large scale. Ru [8] assumed a elastic shell model to study the buckling behavior of the double walled carbon nanotubes. Li and Chou [8] investigated the Young s modulus, shear modulus and the vibrational behavior of both the single and double walled carbon nanotubes using molecular structural mechanics approach, which is the integration of structural and classical molecular mechanics. Though the equivalent continuum mechanics approach has been efficient and effective, it possess a technical challenge of not addressing the surface effect, which has been noted as the main cause for the size dependence of the nanomaterials. For example, The hollow cross sectional shape of the carbon nanotube would approach to a polygon instead of circle due to the neglection of the 8

19 surface effect. Furthermore surface effect is responsible for inducing unstrained deformation due to the loss of some neighbor atoms for the atoms near the surface. Even the effect of the weak in-layer non-bonded Vanderwaal atomistic interactions among the atoms was never addressed in the equivalent continuum mechanics approach. To deal with both the surface effect, which results in the unstrained equilibrium state and the weak in-layer Vanderwaal interactions, a new approach, atomistic-continuum modeling is developed. This approach integrates the classical molecular dynamics simulation and the effective continuum mechanics based on the structural mechanics. The effective continuum modeling is a link between the molecular mechanics and the continuum mechanics and is derived based on transforming atomic potential energy between two atoms to an equivalent continuum model, such as springs, truss and beams [8]. Tersoff [8] in 1992 conducted calculations of energies based on the deformations of a planar graphite sheet and concluded that the elastic energies of fullerenes and nanotubes can be estimated using the elastic properties of graphite sheet. A continuum shell model was developed by Yakobson et al.1996 [8] after he noticed the unique features of the fullerenes. The static or dynamic mechanical properties of the nanotubes can be analyzed using the continuum shell model, but the model doesn t account for the chirality and forces acting on the individual atoms in carbon nanotubes. So, there is a need for developing a computational technique that evaluates the mechanical responses of the nanosystems at atomistic scale. 9

20 The classical Euler-Bernoulli beam theory assumes that the neutral beam axis remain straight even after the deformation and neglects the effect of transverse shear deformations, which is the condition that is present only in the slender beams. To overcome this problem Timoshenko-beam model is used. The Timoshenko beam theory is higher order than the Euler-Bernoulli beam theory. Timoshenko beam theory is known to be superior in predicting the transient response of the beam and is more promising for beams with a low aspect ratio [10]. Timoshenko proposed a beam theory which adds the effect of shear and rotation to the Euler-Bernoulli beam. The Timoshenko model can be effectively used for non-slender beams and for high frequency responses in which the shear and rotary effects are not negligible. Kruszewski 1949, Traill-Nash1951 and Collar 1953, Dolph 1954, and Huang 1961 [11]. Kruszewski obtained the first three anti-symmetric modes of a cantilever beam, and three anti-symmetric and symmetric modes of a free-free beam. Traill-Nash and Collar [11] gave a complete theoretical treatment and experimental results for the case of a uniform non slender beams for which the shear and rotary effect are are not negligible. The first part of their paper contained expressions obtained for the frequency equation and mode shapes for six common boundary conditions which are fixed-free, free-free, hinged-free, hinged-hinged, fixed-fixed and fixed-hinged. The experimental results in addition to the numerical results obtained by theoretical models were discussed in the second part. They reported the difference for the first and second natural frequencies between the theoretical and experimental values anticipated by the theoretical models, Euler-Bernoulli and Timoshenko models. 10

21 Huang 1961 [11] independently obtained the frequency expressions for the mode shapes for all six different end conditions. The frequency equations are difficult to solve except for the case of a simply supported beam. The challenge of presenting the roots in a meaningful way still persists even when the roots of the equation are obtained. Kruszewski, Traill-Nash and Collar, and Huang [11] gave expressions for the natural frequencies and mode shapes. They were not able to calculate the response of beams due to initial conditions and external forces and even the knowledge of the orthogonality conditions is required to do so. A essential parameter in Timoshenko beam theory is the shape factor which is also called the shear coefficient or the area reduction factor. The cause of this parameter is due to the reason that the shear is not constant over the cross section. The shape factor is expressed as a function of Poisson s ratio and the frequency of vibration as well as the shape of the cross-section. Generally, the functional dependence on frequency is ignored. Davies 1948, Mindlin and Deresiewicz 1954, Cowper 1966 and Spence and Seldin 1970 [11] suggested methods to calculate the shape factor as a function of the shape of the cross-section and Poisson s ratio. Stephen 1978 showed variation in the shape factor with frequency [11]. The buckling instabilities of a thick elastic beam which is simply supported has been studied by Matsunaga[1996][12]. For a triple Timoshenko beam elastically connected to a Winkler elastic layer, the critical buckling load calculated using the trigonometric method were only slightly differ from the values determined by numerical solution of the characteristic equation. The static stability region of this 11

22 model is slightly influnced by the Winkler layer of stiffness K and the number of Timoshenko beams and the static stability region of the double and triple Timoshenko beam systems is mostly smaller than that for a single Timoshenko beam system[12]. Single beam model assumes that all the concentric tubes of a multi walled carbon nanotube remain coaxial during the vibration and so can be defined by a single deflection curve, whereas the double beam model considers each of the inner and outer nanotubes as two individual Timoshenko beams and thus each tube as its own deflection curve which need not to be necessarily coincident with the deflection curves of the other tubes. Both the Timoshenko beam effect and the double beam effect are compelling when the wavelength is a few times larger than the outer radius of the double walled carbon nanotubes. For smaller diameters of the nanotubes the double beam effect is compelling than for the nanotubes of larger diameters, where as the Timoshenko beam is significant for both the smaller and larger diameters of the nanotube[13]. The carbon atoms in the carbon nanotube are under the effect of Vanderwaal s forces. Lennard-Jones potential can be used to predict the Vanderwaal s interaction energy, expressed as a function of the interlayer spacing between any two carbon atoms. Through direct summation presuming a Lennard-Jones 6-12[14] potential between carbon atoms, the lattice summations of the potential energy have been computed by Girifalco and Lad[14]. The multiple layers of the nanotube are held together through the Vanderwaal s forces. These Vanderwaal s forces can be an attraction force or a repulsion force. The force 12

23 becomes repulsive when the distance between the interacting atoms becomes less than the sum of their contact radii. An elastic shell model is conferred to study the infinitesimal buckling of a double walled nanotube by Ru[8]. The results of the study showed that the Vanderwaal s forces does not increase the critical axial strain. The study by Ru shows that the critical axial strain for a single walled nanotube cannot be increased by inserting an inner nanotube, but this increases the critical axial load due to the increase in the total area of cross section[15]. Continuum mechanics can be used to study the strength and effect of the Vanderwaal s forces on the single and multi walled nanotubes. A study on the free vibrations of a single walled nanotube constrained completely in the vicinity of a graphene sheet by R.D. Firouz-Abadi and A.R. Hosseinian[3] showed that the natural frequencies of the nanotube are effected by the distance between the nanotube and the graphene sheet. The study also shows that their exists a equilibrium distance between the nanotube and the graphene sheet where the effect of the Vanderwaal s forces is completely negligible and if the distance between the nanotube and the graphene sheet is less than this equilibrium distance the natural frequencies of the nanotube increase at an impressive rate. 1.5 Timoshenko beam theory and Vanderwaal s forces The basic definition of vibration can be given as, when a body as its motion described about a reference point. The disturbance, discomfort caused in the system are also known as vibrations. Vibration analysis is used to reduce the equipment downtime 13

24 and maintenance cost of the unit by detecting the experimental faults when applied in industries. Due to this, the study of vibration analysis is important and even to get some knowledge on how to avoid any resonances caused in the system. The study of elastic models such as elastic beams has earned a lot of attention. Many experiments have been conducted to study the deflections of elastic beams. The shear deformable beams, Timoshenko beams are often considered better than the shear indeformable beams, Euler-Bernoulli beams. Timoshenko beams account for both the rotational and shear deflections. Timoshenko beam theory is also known as thick beam theory and Euler-Bernoulli beam is also known as thin beam theory. Investigation of carbon nanotubes modes of transverse vibration by modeling them as Timoshenko beams, which contain a system of coupled equations accounting for both the shear and rotational deflections. Multi walled carbon nanotubes have several graphene sheets rolled over to form a hollow cylinder.weak van der Waals forces hold together each layer of carbon nanotubes. The van der Waals energy is minimized, when the carbon nanotube is in its fully retracted position. Equlibrium distance for which the carbon nanotube and the graphene sheet will be in equilibrium is given by minimization of the van der waals forces. 1.6 Overview In this thesis, the free and forced vibrations of the nanotubes in the vicinity of a graphene sheet are examined with the fixed-fixed end conditions. In chapter 1 14

25 Figure 1.3: Euler Bernoulli and Timoshenko Beam the complete background of carbon nanotubes and the graphene sheet are studied. Chapter 2 deals with the brief view of the properties of nanotubes and the problem formulation. To explain in detail, chapter 2 comprises of mechanical properties of the nanotubes and calculation of the equilibrium distance by minimization of the van der Waals forces about a arbitrary distance d and developing the equations of motion governing the multiple carbon nanotubes, as Timoshenko beams, using the extended Hamilton s principle. In chapter 3, the equations that are developed in the chapter 2 are extended to the carbon nanotubes and graphene sheet system. Later in chapter 3 these equations are non dimensionalized and the boundary conditions are applied to obtain the free vibrations of the system. Using these set of coupled equations, a Matlab code is 15

26 written to find the roots, which are the natural frequencies of the system. A analytical solution for the free vibrations of the carbon nanotube is obtained for these set of partial differential equations. Initially the formulation is done for a double walled carbon nanotube but can be extended for multiple walled carbon nanotube. In chapter 4 the forced vibratons of the system are examined. Similar to the free vibrations analytical solution, even a analytical solution for the forced vibrations can also be obtained. The variation in the applied force also varies the forced responses. To explain this, different responses of the nanotube and the graphene sheet system are studied by applying various forces such as sinusoidal and a linearly varying force are examined in this thesis. 16

27 CHAPTER II MECHANICS OF CARBON NANOTUBES 2.1 Mechanical properties of carbon nanotubes The prediction of electrical properties of the carbon nanotube needs subtle theoretical analysis, whereas the mechanical properties of the nanotubes can be foreseen based on several features that a nanotube possess such as strength of the carbon-carbon bond, uniform arrangement within a graphene sheet and seamless folding into hollow cylinders. In general, mechanical properties of a solid material depend upon the spatial arrangement of the atoms in the structure and its interatomic forces. The specific arrangement of the carbon atoms in the nanotube allows the nanotube to have a stiffness of diamond. Carbon nanotubes exhibit very high stiffness to an axial load or a bending of small amplitude. Due to the hollow shell like structure the carbon nanotubes especially single walled nanotubes are susceptible to buckling, kink forming and may even collapse at larger strains. The yield strength of a nanotube explicitly depends on the nanotube helicity. The mechanical strength of the multi walled carbon nanotubes is concealed by the poor load transfer from the outer tube of the nanotube to the core of the multi walled nanotube. The utilization of the nanotubes in the nanoscale 17

28 devices or composite materials due to their high stiffness makes it necessary to have a broad knowledge about the mechanical properties of these materials, thus leading to the necessity for conducting research in this area[5]. Apart from the exceptional mechanical properties possessed by the carbon nanotubes, they also possess excellent electrical and thermal properties. Specific combinations of the integers n and m indicate the conductivity of the nanotube. Absorbed resistivity of the single walled carbon nanotube ropes was of the order of 10 4 Ω.cm at 27 C. Single walled carbon nanotubes of ultra small size below 20K temperature have shown to display superconductivity. Therefore it is expected that the nanotube reinforcements in polymeric materials may improve the thermomechanical and thermal properties of the composites[5]. 2.2 Strength of carbon nanotube The higher elastic modulus of the carbon nanotubes makes them perfect materials for the composites reinforcement. It is well known fact that the mechanical properties of a any solid depend on its interatomic forces and the spatial arrangement of the atoms. In materials like carbon nanotubes, the stiffness is directly related to the stiffness of the bonds and can be expressed as, Y k r where k is the spring constant of the bond and r is the interatomic distance. The values of k varies much for the different bonds [16]. It s quite evident from all the 18

29 studies done on the strength of the carbon nanotubes, the higher strengths for the carbon nanotubes is due to the ultimate stiffness they possess. So a detailed study about the stiffness or more properly called Young Modulus of the carbon nanotubes is required. As reported by Irudayum [17], when multiwalled nanotubes of different chiralities are considered, Young s modulus demonstrated a slight dependence on the diameter and the chirality of the nanotube. In general The Young s modulus of a material as defined by Hooke s law is the ratio of the normal stress σ to normal strain ɛ with in the elastic limit. E = σ ɛ. Even the Young s modulus can be defined in the conventional way as the second derivative of the energy U with respect to the applied strain ɛ divided by the equilibrium volume. E = 1 2 U V 0 ɛ 2 ɛ=0 2.1 How rapidly the energy grows as the system is distorted out of its equilibrium is measured by the second derivative. This equation 2.1 provides an ambiguity when a single walled carbon nanotube is considered, as the thickness possessed by the single walled nanotube is very less, leading to an insignificant value for the volume of the nanotube. A number of studies had used the computer simulation to study the properties of the nanotubes. Early in 1993 Overney et al.[18] calculated a Young s 19

30 modulus of 1500 GPa which was similar to that of graphite. The difficulties in handling the single walled nanotubes made the measurements of Young s modulus of the nanotubes take longer time. Young s modulus of 1 TPa was observed by Salvetat et al.[18] using their AFM method for smaller diameters of the single walled nanotubes by bending methods. Shear slippage of the individual nanotubes dominated the mechanical properties of the nanotube of larger diameters in a bundle. Yu et al.[18] were able to measure the tensile properties of the nanotubes and observed Young s Modulus to be in the range of TPa and strengths between 10 and 52 GPa. Failure of the nanotubes occurred at a maximum strain of 5.3% giving a toughness of approximately 770 J m. The shear strength required to pull the nanotube from the bundle depends on the shear modulus. The shear modulus of the nanotube is sensitive to the tube diameter and chirality and is comparable to those of diamond. Shear modulus of a material based on the theory of elasticity can be given by G = T L θj, where T is the torque acting on the nanotube, L is the length of the nanotube, θ is the torsional angle and J is the cross sectional polar inertia of the carbon nanotube. Poisson s ratio of the nanotube is the mechanical property that measures the tube expansion radially when subjected to negative or positive axial strain. The relation between the Poisson s ratio and the Young s modulus is useful to estimate the Poisson 20

31 Table 2.1: Elastic properties of multiwalled carbon naotubes Nanotube no. N Inner radius r ni nm Outer radius r oi nm Young s modulus E TPa Shear modulus G TPa Poisson s ratio υ ratio. The expression that relates the Poisson ratio and Young s modulus is υ = 1 G E 2 G A table 2.1 showing the elastic properties of multiwalled carbon nanotubes by Jian Ping Lu[15] is shown above. 2.3 Lennard-Jonnes potential and van der Waals forces The carbon atoms in the nanotube and the graphene sheet are subjected to the van der Waals forces which can be attractive or repulsive. When the carbon nanotube is in certain retracted position, it is expected to have certain kinetic energy, in this position the van der Waals energy is minimized. The van der Waals interaction energy potential, as a function of the interatomic interaction between adjacent nanotubes, estimated by the Lennard-Jones potential is given by, φd = A 1 d 6 0 a 6 2 d 1 12 d

32 In the equation 2.2 d represents the interatomic distance, a represents the carboncarbon bond length which is nm, d 0 and A are respectively, van der Waals distance and energy constant which are taken to be 2.7 nm and J m Estimation of the equilibrium distance The Lennard-Jones potential can also be expressed as [ σ 12 V LJ = 4ɛ r ] 6 σ r 2.3 where V LJ is the van der Waals force between the carbon nanotube and the graphene sheet, in this thesis. r is the distance between the two atoms. σ is the finite distance at which the inter particle potential is zero and ɛ is the depth of the potential well. For distances greater than 2.5σ i.e, σ = 0.34 nm, the Lennard-Jones potential greatly goes to zero and thus the van der Waals forces over this distance are less active and hence can be neglected. The van der Waals forces between the two atoms can be obtained by differentiating the equation 2.2 with respect to r, namely F LJ = dv LJ dr [ 13 ] 7 σ σ = 4ɛ 12 6 r r 2.4 where, F LJ is the van der Waals force. In this thesis, a system of carbon nanotube and the graphene sheet are considered. Initially it is assumed that the carbon nanotube is at a distance of w above the graphene sheet. As the distance w between the carbon nanotube and the graphene sheet varies, the van der Waals forces between them also varies and to 22

33 satisfy the equilibrium condition the carbon nanotubes static deformation shape changes. Due to the strong non-linear nature of the van der Waals forces interactions, an iterative procedure is to be devised to find the equilibrium state of the carbon nanotube near the graphene sheet. For this system of carbon nanotube and graphene sheet an equilibrium distance of d is assumed. It is assumed that at this equilibrium distance the carbon nanotube has minimal or no deflections. The arrangement of the atoms in the carbon nanotube and the graphene sheet are as shown in the 2.1. The van der Waals force between any two atoms of the carbon nanotube and the graphene sheet is given by the Lennard Jones potential. Girfalco and Lad obtained the value for d from the condition that at the equilibrium V, equilibrium = All the van der Waals forces between the atoms are summed up. Initially the carbon nanotube is divided into 12 sections each of length 0.18 nm and even it is assumed that the graphene sheet length is 2.16 nm, which upon diving into section leads to each single section of length 0.18 nm. This way of approach can later be generalized to any number of atoms and any length of both the carbon nanotube and graphene sheet. It is assumed that each carbon atom of the nanotube is in contact with the each atom of graphene sheet because of the attractive or repulsive van der Waals force. The front view of the graphene sheet and the carbon nanotube carbon atoms in interaction is as shown in the figure 2.1 b. The summation of the van der Waals forces between number of carbon atoms represented by a in the nanotube and the 23

34 Figure 2.1: a Schematic view of the vanderwaal bonds between an atom of carbon nanotube and the nearby counterparts in the graphene sheet and b Front view of the graphene sheet-carbon nanotube vanderwaal bonds in cut-off radius zone [3] number of carbon atoms represented by b in the graphene sheet is, a=n b=n V LJmn 2.6 a=1 b=1 where a and b can range from 1 to n. In this thesis, both the carbon nanotube and the graphene sheet are assumed to have 12 carbon atoms. So the equation 2.6 for the system considered in this thesis can be written as, a=12 b=12 a=1 b=1 V LJmn 2.7 At the equilibrium position the van der Waals energy is minimized. A Matlab code is written to minimize the van der Waals forces with respect to the equilibrium distance d. Now this equation 2.7 is given to the Matlab code and the Matlab returns the minimized value for that equation 2.7, which comes out to be 3.1 Å and is shown in figure 2.2, Due to different radii and the external conditions such as different end conditions, the intertube van der Waals forces get coupled with the individual tube 24

35 deformations of the carbon nanotubes. A study by Hertel, Walkup and Avouris[13] investigated that the non coincidence of the deflected tubes and the interlayer radial displacements affects the overall mechanical behavior of the multiwalled nanotube. Figure 2.2: Graph of Interatomic potential versus distance 25

36 CHAPTER III PROBLEM FORMULATION 3.1 Timoshenko Beam A beam is a structural element that has the ability to withstand load essentially by resisting against bending. Beams are used in building different kinds of engineering structures, such as automobile frames, machine frames and many other mechanical and structural systems. So the vibration analysis of these beams is necessary, especially when the response needed to be determined accurately in order avoid any resonances caused by internal and external forces. Two models, Timoshenko beam model and the Euler-Bernoulli beam model are mainly used to analyze the responses of the beams. The former is also known as a shear deformable model, whereas the later is known as shear indeformable model. The drawback of neglecting the shear deformations in the Euler-Bernoulli beam model leads to the much more promising theory, known as Timoshenko beam model. A Timoshenko beam model includes both the rotary inertia of cross section of the beam and shear deformation effects. Timoshenko beam theory also known as the thick beam theory depends on its material properties such as density ρ, Young s modulus E and Shear modulus G and the geometric properties such as Length L, size, shape, cross 26

37 L L Outer nanotube Inner nanotube van Vanderwaals Waals ineteractions interactions Graphene sheet Figure 3.1: Double walled carbon nanotube in the presence of a graphene sheet sectional area A, and the mass moment of inertia I. In this thesis all the properties ρ, E, G, A and I are assumed to be positive. To study the response of multiwalled carbon nanotube in the vicinity of a graphene sheet, a differential element of two beams connected through elastic layers cross section, as shown in the figure is considered. Let m 1 x and m 2 x be the mass per unit length at a distance x for both the beams and the A 1 x and A 2 x be the cross sectional areas. I 1 x, I 2 x, J 1 x and J 2 x be the area and mass moments of inertia for each beam about an axis normal to the plane of motion and passing through mass centers of the differential elements respectively. Let w 1 x and w 2 x be the deflection of outer and inner beams. The total deflection w 1 x in the outer beam is due to the presence of graphene sheet in its 27

38 vicinity and also due to being apart from the equilibrium configuration. The total deflection of each beam will contain two parts, one being deflection caused due to bending and the other is the deflection caused due to shear. Therefore the slope of the deflection curve at any point x is given by, w 1 x, t w 2 x, t = ψ 1 x, t + β 1 x, t 3.1 = ψ 2 x, t + β 2 x, t 3.2 where ψ 1 x, t and ψ 2 x, t are the angles of rotation due to bending, β 1 x, t and β 2 x, t are the angle of distortion due to the shear for respective beams. Both the linear and angular deflections are assumed to be small. The bending moments M 1 x, t and M 2 x, t are related to the bending deformation by, M 1 x, t = E 1 I 1 x ψ 1x, t M 2 x, t = E 2 I 2 x ψ 2x, t deformation by, Even the shearing forces Q 1 x, t and Q 2 x, t are related to the shearing Q 1 x, t = k 1G 1 A 1 xβ 1 x, t 3.5 Q 2 x, t = k 2G 2 A 2 xβ 2 x, t 3.6 where k is the numerical factor depending on the shape of cross section of the beam and for a beam with circular cross section, k is, k = 61 + υ1 + m υ1 + m υm

39 where υ is the Poisson s ratio and m is the numerical factor given by the ratio of inner diameterd i to outer diameterd o. m = D i D o 3.2 Theoretical modeling of the beams The total kinetic energy of the double beam due to rotation and translation is, [ [ ] T t = 1 2 ] 2 ] L w 1 x, t w 2 x, t m 1 x + m 2 x[ dx 2 0 t t [ [ ] ] 2 ] L ψ 1 x, t ψ 2 x, t J 1 x + J 2 x[ dx t t 0 where the mass moment of inertia s are related to the area moment of inertia s by, J 1 x = ρi 1 x = m 1x A 1 x I 1x = k 2 1xm 1 x 3.9 J 2 x = ρi 2 x = m 2x A 2 x I 1x = k 2 2xm 2 x 3.10 where ρ is the mass density, k 1 x and k 2 x are the radii of gyration about the neutral axis. The total variation of kinetic energy equation 3.8 is written as, δt = L 0 w 1 m 1 t δ w 1 w 2 + m 2 t t δ w 2 dx+ t L k1m 2 ψ 1 1 t δ ψ 1 + k 2 ψ 2 t 2m 2 t δ 0 ψ 2 t dx 3.11 For small-deflection linear vibration, the van der Waals pressure at any point between two tubes should be a linear function of the jump in deflection at that point. Thus, the interaction pressure per unit axial length is given by p i = cw i+1 w i 29

40 where c is the intertube interaction coefficient per unit length between two tubes, which can be estimated by [19] c i = 3202R i 0.16d where R i is the inner radius and d is the equilibrium distance calculated in the chapter 2. In the present thesis a double walled nanotube in the vicinity of a graphene sheet is considered. The interaction between the outer carbon nanotube and the graphene sheet is assumed to be similar to that of interaction between the outer and inner carbon nanotubes. The deflection w 1 x, t of the outer carbon nanotube is due to the van der Waals interaction between the graphene sheet and the outer carbon nanotube. The van der Waals pressure between the graphene sheet and the outer nanotube is expressed as, p 1 = c 0 w 1 w 0 where w 0 x, t is the deflection of the graphene sheet and the van der Waals pressure between the outer nanotube and the inner tube is expressed as, p 2 = c 0 w 2 w 1 where w 1 and w 2 are the deflections of the outer and inner carbon nanotubes and c 0 and c 1 are the respective intertube interaction coefficients. 30

41 sheet is, V t = 1 2 δv = The total potential energy of the double beam in the presence of the graphene L 0 [ ] 2 [ ] 2 ψ 1 x, t ψ 2 x, t E 1 I 1 + E 2 I 2 dx L 0 k 1G 1 A 1 β 2 1x, t + k 2G 2 A 2 β 2 2x, t L 0 c 0 w 1 w 0 2 dx The variation of potential energy can be written as, L 0 L ψ 1 E 1 I 1 δ ψ 1 ψ 2 + E 2 I 2 δ ψ 2 dx L + k 1G w 1 1 A 1 0 ψ w 1 1δ t ψ 1 + k 2G w 2 2 A 2 w 2 2 δ t ψ 2 dx + L 0 c 0 w 1 w 0 δw 1 w 0 dx + L The virtual work done due to the conservative forces is given by, δw nc t = L 0 f 1 x, tδw 1 x, tdx + where f 1 x, t and f 2 x, t are the force densities. 0 L 0 0 dx c 1 w 1 w 2 2 dx 3.13 c 1 w 1 w 2 δw 1 w 2 dx 3.14 f 2 x, tδw 2 x, tdx Extended Hamilton s principle The Hamilton s principle states that the dynamics of a system is determined by a variational problem, the Lagrangian which contains all information regarding the system and the forces acting on it. When non-conservative forces exist, it is 31

42 necessary to apply the extended Hamilton s principle. The extended Hamilton s principle requires kinetic energy T, potential energy V and the non-conservative work W nc and is given by equation The virtual work of the non-conservative forces, when calculated, cannot depend upon the variations of the time derivatives of the generalized coordinates. It can only depend upon the variations of the generalized coordinates [20]. t2 t 1 δt δv + δw nc dt = and δw 1 0, t = 0, δw 2 0, t = 0, δψ 1 0, t = 0 and δψ 2 0, t = 0 at t= t 1 and t 2. Substituting these into expression t2 L w 1 w 1 w 2 w 2 δt δv + δw nc dt = [m 1 δ + m 2 δ t 1 t 1 0 t t t t +k1m 2 ψ 1 ψ 1 1 δ + k 22m ψ 2 ψ 2 2 δ t t t t ψ 1 ψ 1 ψ 2 ψ 2 E 1 I 1 δ E 2 I 2 δ k 1G w 1 1 A 1 ψ w 1 1 δ t ψ 1 k 2G w 2 2 A 2 ψ w 2 2 δ t ψ 2 t2 c 0 w 1 w 0 δw 1 w 0 c 1 w 1 w 2 δw 1 w 2 ] dx dt = There are four dependent variables w 1, w 2, ψ1 and ψ 2 here. Rearranging the integrands will result in four partial differential equations which are coupled to each other. The entire transformation from equation through 3.17 to 3.21 is discussed in the Appendix B. The equations involve bending, shear, rotational and translation 32

43 motion terms. The equations behave like the Euler-Bernoulli equation if the rotational and shear terms are small and negligible. [ ] k 1G 1 A 1 w 1 ψ 1 [ ψ 1 E 1 I 1 2 w 1 m 1 + c t 2 0 w 1 w 0 + c 1 w 1 w 2 = ] k 2G 2 A 2 w 2 ψ 2 E 2 I 2 ψ 2 + k 1G 1 A 1 w 1 ψ 1 + k 2G w 2 2 A 2 ψ 2 2 w 2 m 2 c t 2 1 w 1 w 2 = k 2 1m 1 2 ψ 1 t 2 = k 2 2m 2 2 ψ 2 t 2 = Different boundary conditions are applicable to beam depending on its end conditions. The boundary conditions for the fixed-fixed case are, w j 0, t = 0 ψ j 0, t = w j L, t = 0 ψ j L, t = where j=1,2 In the case of a simply supported beam, ψ j 0, t w j 0, t = 0 M0, t = E j I j = ψ j L, t w j L, t = 0 ML, t = E j I j = where j=1,2 For cantilever beam, the boundary conditions at the clamped end are w j 0, t = 0 ψ j 0, t =

44 At the free end of the beam, the deflection and the rotation are not zero, ψ j L, t ML, t = E j I j = Q j L, t = k jg w j L, t j A j ψ j L, t = where j=1,2. The general form in which the equations 3.18 through 3.21 can be represented by is, [k w j G j A j j ] 2 w j m j + c t 2 j 1 w j w j 1 + c j w j w j+1 = ψ j E j I j + k jg w j j A j ψ j kj 2 2 ψ j m j = t 2 where j=1,2,... Let v j = k G j A j 3.31 w j [v j ] 2 w j j m j + c t 2 j 1 w j w j 1 + c j w j w j+1 = ψ j w j E j I j + v j j kj 2 2 ψ j m j = t 2 where j=1,2, Non dimensionalization of Timoshenko beam equations Non dimensionalization of the equations is performed to simplify the expressions. The advantage of using the non dimensionalization technique is that it lets one know which are the parameters that combined, helps in determining the physical behavior of the solution obtained for that respective system. In this thesis the non 34