Studies in mechanics have evolved to be the most fundamental tool in the innovation of new materials and advanced technology. Specifically, studies in Continuum Mechanics, has been very influential in producing materials that exhibit various desirable characteristics at the expense of size and weight. Nanotubes are a perfect example of how this study has impacted the evolution in material technology. It has been used as the basis for the explanation of these complex materials as well as provisions for their behavioral effects during machining. The wide use of nanotubes in fragile fields like technical textile supplements, truly depict the stretchy characteristic the combination of CM and nanotubes display. Scores of publications and research reports have been theorized to help explain different aspects of this technology with direct reference to nanotubes. A comparative and comprehensive analysis of a variety of these resources will most probably reveal the most crucial characteristics of this relationship. As a matter of fact, establishing the applications of CM with respect to modeling and simulation, defect formation, torsional buckling, wave propagation, and the elastic modulus of CNTs are objectified.
Keywords: Carbon Nanotubes (CNT), Continuum Mechanics, Modulus, Grapheme sheet, Euler-Bernoulli beam theory, Crystal Structures, Quantum Mechanics.
Carbon nanotubes are technically materials made of cylindrical layers of carbon molecules forming a tube. These materials are formed using different graphite layer configurations that give it the characteristics it exhibits. The divergence in it microstructure for CNT considered is responsible for the diversified fields of application this material exhibits. It is important to note that technological advancements coupled with research have made it possible to alter the micro-configuration of nanotubes to produce lengthier tubes. CNTs can be categorized as either single-walled NTs or double walled NTs based on the molecule layers that form its repeating structure. CNTs exhibit unique material characteristics with respect to strength, tenacity as well as thermal and electrical conductivity. This accounts for its extensive applications in structural composite materials, conductive plastics, and biosensors. Fundamentally, CNTs are a product of micro technological approach in material science through which molecules are awarded perfect orientations to yield particularly predetermined characteristics.
The history of CNTs cannot be accorded to one particular individual as it came to existence on a gradual line of invention. However, it is traced that the concept began through Marinobu Endo, who produced the first complete carbon filaments as he was striving for a Ph. D at the University of Orleans. His concept is, however, not directly linked to the history of CNTs, as it did not give a clear connection with the modern CNTs. The first CNT was developed by Sumio Lijima. The introduction of the high-powered electron microscope by Roger Bacon paved the way for interested scientists to study and produce more NTs as it helped in studying visual impressions of the first NTs. This field of science was advanced with the seriousness it deserved because the involved scientists could not pre-determine the potential of CNTs and the impact it would have in material technology in the future. As a result of this, the first single walled nanotube was made in 1993 by Lijima and Donald Bethune. This drew a transition line for the complete invention of the CNTs despite corresponding inventions by Russian Nanotechnologists. In this context, the tribute was given to these two scientists, who when working together, came up with the single walled CNTs without the slightest idea the impact their invention had for the future.
Continuum mechanics can be defined as a branch of mechanics that deals with materials that exhibit collective characteristics as continuous bands and not as distinguished particles. In normal studies in mechanics, say quantum mechanics, the behavior of materials are explained with respect to a particular nature of matter. Each characteristic depicted by a particular material is attributed to each particle in its structure. However, continuum mechanics views particular materials as rather continuous strand that collectively contribute to the characteristics of the block. As a matter of fact, continuum mechanics explains CNTs with respect to their lengthwise bit formation of compact cylindrical carbon arrangement. Hence, the effects of shear, deformation, torsional buckling, and propagation of waves are viewed with respect to compact blocks.
Continuum Mechanics Applications in Carbon Nanotubes
The derivation of continuum theories from inter-atomic interactions has been an approach utilized by scores of researchers. Marino Arroyo and Ted Belytschko, in the report, Continuum mechanics modeling, and simulation of CNTs highlights the integration of inter-atomic interactions with the exploration of crystalline behaviors exhibited by NTs (Marino & Ted, 2005). Fundamentally; this resource prompts the crystalline behavior and concepts in NTs. The main ideology bases the similarity in material strength exhibited by crystalline materials in comparative grounds with NTs. This study introduces the concepts of finite crystal elasticity and Cauchy-Born rule as fundamental principles in the explanation of crystalline film and NTs. This study takes particular interest in the strong crystalline network coupled with the hollow cylindrical geometry that are a characteristic of NTs and formulates a conceptual analysis with respect to crystalline solids that have similar characteristics.
Particularly, cases of deformation and failure that are exhibited by crystalline solids and technically explained by plasticity as well as fracture points when certain amounts of stress are applied in-plane with the material is used as the main comparative aspect. It is believed that fracture and failure in nanotubes are as a result of relaxation in deformed sections of the NT. Interestingly, the study exploits evidence of fractured nanotubes as a result of tension. With respect to the flexibility of three-dimensional structures depicted by multidimensional NTs, the report highlights how stress and tension are distributed and counteracted by the rest of the crystalline structure of the nanotube. Furthermore, this is tabled as the main principle behind the application of NTs in devices and sensors. The approach employed in this analysis is based on the concepts of CM in rationalization of mechanics of materials with respect to irreducible material characteristic. As a matter of fact, this has seen through the production of nanotubes with thin elastic shells, non-elastic beams, and non-local elastic beams.
This is demonstrated by the Cauchy-Born rule used in expressing deformation parameters of the continuum. The homogeneity expressed in molecular theories help in the formulation of bond parameters in NTs, which ultimately help in the characterization of the deformation gradient. Essentially, behavior of molecular provisions, help in the determination of continuum studies in NTs. However, the question is raised on the extent to which the theories address atomic sizes, relative atomic spacing, and the initial assumption of finite systems.
Defects in any engineering material have tremendous effects on its applications. There has been growing concerns displayed in the form of studies and publications about the various defects exhibited by CNTs. The study of these defects are important as they reveal special optimum limits and operational parameters of these materials, as a matter of fact, CNTs are ranked as one of the most influential materials that drive technological advancements. The publication, Continuum field model of defect formation in carbon nanotubes, strives to explain the importance of studying the defects in CNTs and the various forms of defects that are characteristic of these materials (Zhiling, Prasad, Pradeep, Satish, & Boris, 2005). As a matter of fact, tribute is given to the several researchers who have taken interest in studying this particular field of CNTs. The report uses several calculations retrieved from reviewed sources that depict energy and bond relations with respect to various forms of deformations.
Specifically, the report takes special interest in the SW (single-walled) defect and creates an analysis with respect to dislocations and disclinations. As portrayed by the modeling and simulation mechanisms, CNTs can be viewed as crystalline structures. Therefore, it is a logical fact that dislocations would be a conventional deformation characteristic along its crystalline layers. In the analysis, a grapheme sheet is considered with assumptions of normal operational conditions and the absence of buckling. Fundamentally, the total energy in the grapheme sheet is considered after which this is determined after a deflection from which the gradient is used to gauge how favorable the deformation is to the original characteristics of the grapheme sheet. Deflections in the CNTs are discussed in this report with respect to the initial illustration demonstrated by the grapheme sheet. Fundamentally, as grapheme has carbon atoms as its building blocks, it is assumed that the CNT can be viewed as a rolled up grapheme. Through this, the possible effects of deformations are viewed on energy impact basis. The following is the energy equation that is used to determine the value of deformation on the grapheme sheet.
W = WE + WD + WI,
Where, WE is the stored elastic strain energy of the grapheme sheet, WD is the self-energy of the nucleated edge dislocation dipoles, and WI is the interaction energy between the applied stresses and the dislocation dipole. Therefore the change in energy that represents the effect of deformation on the characteristics of the grapheme is;
(∆) W = WD + WI.
The report creates a comparative analysis of its deformation principle with existing atomic simulations. Specifically, the two simulations used are the Armchair carbon nanotube and the zigzag carbon nanotube. Each of these is compared with corresponding results from the grapheme deformation tests. In each case, a grapheme sheet is rolled or created in a particular configuration that emulates the structure of the atomistic simulation chosen. However, the research established that the critical strain for a CNT is smaller in comparison with the corresponding grapheme configuration. Conclusively, the report displays the importance of atomistic simulations in the study of microscopic mechanics and interpretation of phenomena as deformations in CNTs.
As the study of CNTs as complete experimental units is virtually impossible, classical molecular dynamics has been constantly used to describe the behavior of CNTs. Particularly; studies have revealed that CNTs exhibit torsional instability (Wang, Quek, & Varadan, 2006). As a result of this, divergent models have been established to demonstrate torsional buckling of CNTs. A report highlighting the results of experiments on torsional buckling of carbon nanotubes have suggested the use of Brenner potential energy concepts as well as simple explicit expression in the determination of strain and instability in single-walled CNTs. This research report discusses various methodologies that have been employed by various analysts in the determination of torsional buckling in CNTs. As a matter of fact, the proposal provides numerous alternatives specific to particular CNTs. Additionally, the Euler-Bernoulli beam theory coupled with the finite element method have been employed.
However, there have been constant conflict with regards to the efficiency and accuracy of these systems, as a result of this; newer systems that address the functional limitations of these systems are in the development process. For instance, the research paper tables a hybrid continuum mechanics and molecular mechanics model as an example of a conclusive model that resolves these arising limitations. Additionally, citing different sources, the paper tables different concepts highlighted from divergent studies relating to torsional buckling of CNTs. Coupling of axial deformation and torsion, application of elastic shell model in the calculation of torsional strain are functional concepts that are used in this explanation. Specifically, continuum mechanics has been a fundamental key in understanding torsion, in materials, with respect to CNTs, a shaft can be considered where the shaft represents a CNT molecule.
The graph above depicts the relationship between the twist angle that a shaft (mentioned above) is subjected to and the corresponding values of strain energy. Various relationships in Continuum mechanics help in the calculation of the strain developed on the shaft from each angle of twist. Different values are gathered after which they are plotted to get the graph above. As a result of the increasing interest in the establishment of torsional behavior with respect to different CNT models, there exist numerous commercial software that aid in the determination of buckling in molecular dynamic simulations. Donnell and Kromm models are used to predict the twist angle given consecutive values of strain energy. The shaft is used in the research report demonstrating how a cylindrical molecular crystal of CNT would behave with constant angular application of torsion.
The study of wave distribution in materials is a fundamental experimental process as far as study and classifications of materials are concerned. As a matter of fact, wave propagation determines the limit that a specific material can withstand while maintaining its original properties. As CNTs are rod like, wave propagation can be studied in the same way a rod’s vibration and its distribution lengthwise is studied (Wang, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, 2005). A journal on Wave propagation in carbon nanotubes through non-local continuum mechanics gives an account of the behavior of CNTs with respect to wave formation and propagation. The two models used in the study of wave propagation in NTs are elastic Euler-Bernoulli and Timoshenko beam models. However, it is important to note that the concepts used in both models can be interchanged successively; this means that one model can be modified to function like the other. There are specific frequency ranges for each version of CNTs measurable through these models.
This report provides comparative results of these two models with respect to wave parameters such as phase velocities, wave numbers, and frequency ranges. The terahertz frequency range was adopted as a test set after which Timoshenko beam model demonstrated better results. For instance, it provided more accurate wave solution at higher frequencies while its counterpart model, exhibited over-estimated wave phase velocity at the corresponding frequencies. The report describes nonlocal continuum models of CNTs as well as their principles of operation to aid in the study of wave propagation in CNTs. It is further demonstrated in the journal that the phase velocity decreases as scale coefficient and wave number increases. However, the research takes particular interest in small-scale effects, which limits its principles to micro study of wave propagation in CNTs. The models discussed are advantageous as they produce the wavelengths. Through these wavelengths, it is possible to estimate the amplitudes and the overall behavior of nanotubes at macroscopic levels. The report accounts for both the efficiency and limitations of the two models used, furthermore; it creates a platform for the appreciation of how significant small-scale effects are on the general properties of the nanotube.
Elastic modulus is essentially a constant that relates stress and strain of a material. It provides a parametric provision for the measurement of strength of various materials. Strength is a parameter exhibited in divergent proportions of different CNT structures. These materials could also be classified with respect to their elastic modulus. In a continuum analysis that incorporated inter-atomic potentials, the elastic modulus of single-wall nanotubes was studied, and a theory that related essential aspects of CNTs was established (Zhang, Huang, Geubelle, Klein, & Hwang, 2002). The numerous studies considered by this report unveil important concepts that relate thermal and electrical properties of CNTs.
As a matter of fact, single-wall carbon nanotubes can be used as field effect transistors owing to inter-changeability of its electrical properties. The journal is a report that highlights various studies done by various researchers on the elastic modulus of carbon nanotubes. Also mentioned is the use of transmission electron microscopy (TEM) as a study tool for the establishment of the Young’s modulus of carbon nanotubes. Fundamentally, the report distinguishes the use of experimental means of determination of the elastic modulus of CNTs from extensive atomistic studies. In this category, studies used incorporated the concept of inter-atomic potentials that included the summation of pair-wise harmonic potentials, Keating potential, and effects as a result of both bond stretching and bond angle changes.
The study narrows down to inter-atomic potentials for carbon through which the inter-atomic distances are established coupled with repulsive and attractive forces between two identical carbon atoms. Fundamentally, atoms are analyzed during the formation of bonds with adjacent carbon atoms as well as surface reactions with other elements. The study notes that the single-walled carbon nanotubes have hexagonal atomic structure, a factor that makes it possible to calculate various atomic parameters such as its atomic packing factor, atomic density, and the number of atoms per unit cell. In this context, it is easier to visualize the arrangement of atoms it the microstructure of CNTs. In fact, this gives a breakthrough of visualizing continuum mechanics in particulate forms. Furthermore, the research journal tables a nanoscale continuum theory for materials with non-Centro-symmetric atomic structure. Through this provision, the study incorporates an allowance for dislocations that normally distort the irregular arrangement of crystals. It makes it possible to obtain the elastic modulus of nanotubes irrespective of the modifications present in its atomic structure.
CNTs have attracted considerable interest in research work as depicted by the various literature tabled in the theories above. As a matter of fact, studies in the numerous aspects of their properties and behaviors during machining have furthered the field of research and subjects like material science. Additionally, the ideas presented above depicts that material selection is a rather complex process that not only involves the testing of the right parameter, but also involves the selection of the most appropriate methodology for testing. It is important to note that each parameter addressed in the resources can be studied in more than one method. This ensures that researchers could engage their research work in distinct workshops and come up with contrasting results. With respect to how nanotubes are evaluated microscopically, continuum mechanics is very influential in explaining test concepts as well as behavioral analysis of these and more complex materials. It can be said that the use of CM in the explanation of material behavior adopts the bottom-up approach of problem solving.
Marino, A., & Ted, B. (2005). Continuum mechanics modelling and simulation of carbon nanotubes.
Wang, Q. (2005). Wave propagation in carbon nanotubes via nonlocal continuum mechanics. Retrieved 03 7, 2014, from Journal of Applied Physics: http://dx.doi.org/10.1063/1.2141648
Wang, Q., Quek, S., & Varadan, K. (2006, 12 25). Torsional buckling of carbon nanotubes. Retrieved 3 7, 2014, from ScienceDirect: www.sciencedirect.com
Zhang, P., Huang, Y., Geubelle, P., Klein, P., & Hwang, C. (2002). The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials. International Journal of Solids and Structures , 3893-3906.
Zhiling, L., Prasad, D., Pradeep, S., Satish, N., & Boris, Y. (2005). Continuum field model of defect formation in carbon nanotubes. Journal of Applied Physics , 1-8.