A finite element formulation for out of plane behavior of curved beams in nanotechnology

Nowadays, manufacturing less error including, lighter and better nanotechnological materials with the help of the nanotechnology which enables people to produce advanced structures in terms of physical, chemical and electrical characteristics is considered as one of primary reasons of being centre of attention of nanotechnology. Nanomechanics that examine functional and elastic characteristics of nanoscale systems by making use of force-displacement relations is one of the most important fields of nanotechnology. In nano-devices and nano-machines, one of the structural elements that is used as widespread is nanobeams. Knowing mechanical behavior of these nano-sized beam elements is important for the improvement of nano-sized machines. In the modelling of nanobeams, nonlocal theory of elasticity is used instead of classical theory of elasticity. Because of the fact that definition of classical theory of elasticity is independent of the size effect, there can be some problems in investigations which are related material behaviors of nanostructures. As the dimensions become smaller, the size of granular distance or lattice parameter will have a significant effect on the material behavior. In this situation, intermoleculer gap between molecules becomes crucial and the results obtained with local theory of elasticity are not acceptable. Thus, in micro-scaled and nano-scaled structures, nonlocal theory of elasticity that takes into consideration the small-scale effect is used. In addition, superiority of the nonlocal theory of elasticity does not only consider the small scale-effect in the static, dynamic and buckling behaviours of nanomaterials but also it is easier and cheaper than other methods. In the literature survey, it is observed that while most of studies are usually made by using local theory of elasticity, there are a few studies are benefited from nonlocal theory of elasticity. In most of these studies, one or several effects such as shear deformation, axial and shear forces were ruled out. Most of these studies are dealt with solutions of static problems of straight nanobeams. In the problems, nonlocal effect of bending moments are taken into account and shear deformation effect has been excluded. The studies that are used nonlocal elasticity theory have included nonlocal effect of bending moments and excluded shear force. In the most of studies that are investigated dynamic problems of straight nanobeams .shear deformation and rotatory inertia effects are neglected. The studies that are used nonlocal elasticity theory have included nonlocal effect of bending moments and excluded rotary inertia. In the literature, there are a few studies related to curved beam. A few articles on nonlocal beam theory were dealt with curved nanobeams and there are no studies related with out of plane behaviors of curved planar beams. In the literature, very few studies on the finite element formulation of nanobeams is available. Almost all of these studies are coupled with straight nanobeams. In these studies, it is observed that approximate functions are put account for displacement. In this research, a finite element formulation based on the exact analytical solution of governing equations for out of plane static and dynamic behaviour of curved nanobeams with varying curvature and cross-section. The effects of shear deformation, bending and torsional rotary inertias are taken into account. In the formulation of the stiffness and mass matrices, the exact displacement functions are used. The results of this study are compared with those given in the literature. In the first chapter, the importance of the nanotechnology and design approaches of nanostructures in modern engineering applications are discussed. The originality of this study is emphasized. In the second chapter, studies dealing with the static and dynamic problems of nanobeams are given. In the literature survey, it is observed that while most of the studies investigates the subject by using local theory of elasticity, there are a few studies using the nonlocal theory of elasticity. In most of these studies, one or more several effects such as axial extension, shear deformation, and the nonlocal effects of these forces are not considered. There is only a few articles studying the curved nanobeams. There are scarcely studies about the out of plane behaviors of curved planar nanobeams. In the literature, very few studies on the finite element formulation of nanobeams are available. Almost all of them deal with straight nanobeams. In these studies, it is observed that some approximate functions are used as the displacement functions. In the third chapter, constitutive equations of Eringen's nonlocal theory of elasticity are rewritten in cylindrical coordinates and applied into the classical beam equations. These equations are written for the beams with varying curvature and cross-section. In the equations, nonlocal effects of forces and moments were taken into consideration. The exact analytical solution of equations governing the out of plane behavior of curved planar beams are obtained by using the initial value method. The main advantage of this method is obtaining the solution easily even in the presence of high static indeterminancy. In the fourth chapter, the exact analytical solution of governing equations of out of plane static behavior of curved nano beams are used for obtaining the stiffness matrix. The beam with varying curvature and cross-section is considered and the effect of shear deformation is included in the equations. The stiffness matrix is obtained for each element and global stiffness matrix is obtained by assembling the joint nodes of adjacent elements. For out of plane dynamic problems, the mass matrix is also formulated by considering the effects of bending and torsional inertia moments. Moreover, both the stiffness and the mass matrices are also obtained by using Hamilton's principle, and it is shown that the results are the same with those obtained by the direct method. To acquire natural frequencies of curved nanobeam, using both stiffness and mass matrices eigenvalue problem is solved. Thanks to developed finite element formulation, problems that have different loading and boundary conditions can be easily solved. In the fifth chapter, some out of plane static and dynamic problems of curved nanobeams are solved by using the developed finite element formulation. In the calculations, the effects of bending and torsional rotary inertia with shear deformation are included. Obtained results are compared with analytical studies. In the static analysis, under clamped-free boundary conditions, a circular nanobeam having uniform cross-section and constant curvature is examined as an example. In examples, force and moments, in normal and tangential directions, are applied from the free end of the nanobeam, respectively. Transverse displacements and rotation angles are calculated. Obtained results are compared with analytical results. In the finite element modeling of problems, one element is used. It is shown that obtained results from both analytical and finite element formulation are identical. In the dynamic analysis, a clamped - clamped circular beam with uniform cross-section and curvature is examined. Dimensionless natural frequency values are acquired with the changing of small scale parameterand the number of elements. In the first example, changing of first natural frequency value with small scale parameter is studied. It is shown that nondimensional natural frequency values are converged to the values that are acquired with local theory of elasticity for increasing small scale parameter values. In the next example, small scale parameter is taken as 2 nm. Changing of the first nondimensional natural frequency with the number of elements which is used in the analysis is examined. It is shown that the value does not change after ten elements. In addition, the effect of nonlocal parameter on nondimensional natural frequency values is investigated for the slenderness ratio of the beam is 10 and 150 and opening angle of the beam (θ_t) is taken as 60°, 120° and 180°. The nonlocal parameter is changed from 0 to 2. Nondimensional natural frequency values regarding first and second mode are examined. It is shown that nonlocal parameter is gone up, nondimensional natural frequency values are decreased. If the slenderness ratio of the beam is gone up, dimensionless natural frequencies are increased for the nanobeams that have same opening angle of the beam. There is no study to compare the obtained results in the literature. In the sixth chapter, the results and conclusions obtained in this study is discussed. In this research, a new finite element formulation has been improved for out of plane static and dynamic behaviours of curved planar nanobeams by taking advantage of nonlocal theory of elasticity. In equations which take account of effects of small scale, shear deformation and torsional rotatory inertia, besides nonlocal effects of bending, torsional moments and shear force. In the formulation, stiffness and mass matrices have made up. Displacements that are acquired from exact analytical solution is utilized in stiffness and mass matrices. Besides, stiffness and mass matrices have constituted by using Hamilton principle that is variational approach. Stiffness and mass matrices obtained with direct method and Hamilton principle are same. Velocity gradient theory as kinetic energy formulation is used. The theory includes micro and macro structure effects is used the first time. The formulation which comprise of exact analytical solution as shape function in the literature is the first. In addition, this is the first study that is investigated out of plane static and dynamic behaviours of planar curved nanobeam with finite element formulation. It is obvious that this new finite element formulation has made a contribution to be done studies with respect to out of plane static and dynamic behaviours of curved nanobeams. It is thought that close results by using less number of finite elements have been acquired inasmuch as displacement fields are procured from exact analytical solution. In addition, it is expected that analysis of complicated nanobeam structures can be simplified with that solution. Thus, it will be possible that complicated nanosystem designs can be truer, more confidential and achieved in shorter time.