Static and dynamic problems of nonlocal beam theory in nanotechnology

Today, the device features develop at a great pace, while shrinking the dimensions with the same speed. At nanometer scale, materials have new and excellent physical properties. By designing sequence of atoms and molecules to reveal artificial materials with exceptional properties, and design nano-scale devices and systems with them is a new technology called nanotechnology. One of the most important fundamental areas of nanotechnology is nanomechanics. Subjects of the study are the relations between force and displacement analysis of nano scale systems and to examine the functional and elastic characteristics of these systems. After the discovery of carbon nanotubes, extensive researches have been conducted in nanomechanics. Recently, the interest for the nano-electromechanical systems (NEMS) has increased considerably. In these studies, static and dynamic behavior of nanotubes and nanobeams has been investigated. In the analysis, three main methods have been developed and used: Atomistic modelling. Hybrid atomistic-continuum mechanics and Continuum mechanics. In the atomistic modeling method, in general, conventional molecular dynamics and the density functional theory are used. Interatomic potential energy is directly applied into the continuum analysis in the hybrid atomistic-continuum mechanics. Carbon nanotubes are considered as continuous and homogeneous macro-structures in the theory of continuum mechanics. The material properties of the micro-structures, as the lattice spaces between carbon atoms are neglected. The main reason of limited applicability of classical or local continuum theory to nano-scaled systems is that the assumption of continuous media is not appropriate for modeling the discrete structure of the material due to the lattice spacing between atoms. In other words, the material properties are size dependent at nanoscale and so that the small length scale effect should be taken into consideration while determining the mechanical behavior of a nano-material. The small scale effect becomes very important at the order of nanometers. Thus, in the studies of nano structures, the scale effects in these materials started to be taken into account at the implementation of nonlocal continuum mechanics. Literature survey shows that most of the studies are mainly focused on static and dynamic analysis of nanobeams. A few of these studies have used the nonlocal elasticity theory, and only the bending moment is considered as a nonlocal effect. The studies dealing with the static problems have taken only the effect of bending into account, and the shear deformation effect has been neglected. Rotatory inertia and shear deformation effects are often neglected in the studies investigating the problems of vibration. Also, the solutions are obtained approximately by using various numerical methods. In this study, a curved planar nanobeam having variable curvature and a variable cross-section will be discussed by using the equations of the nonlocal elasticity theory. The nonlocal constitutive equations of Eringen are arranged in cylindrical coordinate and implemented into the classical beam equations. Thus, the equations for nonlocal beams with varying curvature under varying loads are obtained. Using these equations, both static and dynamic problems of planar curved nanobeams are solved. The nonlocal effects of the three forces as well as the three moments are considered while deriving the equations. The governing equations for in-plane static problems of a nanobeam with varying curvature and cross-section bearing distributed loads are presented. The nonlocal effects of moments and forces are considered in the equations. In addition, the axial extension and shear deformation effects are considered in the analysis. This is the first study which includes the nonlocal effects of axial and shear forces in the formulation. The resulting differential equations are solved analytically using the method of initial values. Superiority of the initial value method is that high order statical indeterminacy adds no extra difficulty to the solution. The solution can be obtained for any boundary condition. The fundamental matrix of beams with different geometries and cross-sections are obtained analytically. Thus, in any case of beam axis and cross-sections, the deformation, slope and stress resultants can be defined analytically along the beam axis. The effects of the parameters such as small scale parameter, opening angle, slenderness ratio on the static behavior of the beam are also studied. In order to assess the versatility of the method, several examples with different geometries, loading and boundary conditions are solved. The results are obtained for different cases considering or neglecting the effects of axial extension, shear deformation and rotatory inertia. Various problems are solved and the exact analytical equations of the displacements, rotation and the stress resultants are obtained. The effects of nanoscale parameter and variation of geometric parameters on the static behavior of a circular nanobeam analyzed and discussed through the proposed method. The slenderness ratio of the beam λ=(R θ_t)⁄√(I⁄A) is changed from 20 to 150 and opening angle of the beam (θ_t) is taken as between 10° and 180°. Small scale parameter (R⁄γ) is considered to change from 1 to 10 and γ is taken as 1.56 nm. Poisson's ratio and Young's modulus is taken as υ=0.3 and E=1 TPa, respectively. However, in this study the results do not depend on these values, since they are given as ratio of the results of local and nonlocal theories. The equations of motion are derived by means of d'Alembert principle. The solutions of dynamic problems of in-plane vibrations considering axial extension, shear deformation and rotatory inertia effects are presented. In the solution process, initial value method is used. Exact solution are only available for constant cross section, circular curvature nanobeams. Natural frequencies are obtained for different vibration problems. In addition, using these natural frequencies mode shapes are obtained and shown in figures. The effects of parameters such as small scale parameter, opening angle and slenderness ratio to the dynamic behaviour of beams are investigated and examples are presented. Additionally, curved beams with varying curvature and cross-section are modelled as if they consist of a lot of beams with constant cross section and circular curvature. And using same solution method the result are obtained. The results are obtained for different cases considering or neglecting the effects of axial extension, shear deformation and rotatory inertia. The objective of this study is to show that the nonlocal elasticity is a better approach for static and dynamic analysis of nanobeams. Instead of using classical beam theory, using nonlocal elasticity theory reveals the nonlocal effects which is significant to understand the mechanical behavior of nanobeams. A new size-dependent general beam theory is presented within the framework of Eringen's nonlocal elasticity theory for static behavior of curved nanobeams. Nonlocal constitutive equations are implemented in the classical beam equations. Axial extension and shear deformation effects and their size-dependent effects along with the size-dependent effects of bending moment are incorporated in the analytical model. Initial value method is used for the exact solution and the results are obtained analytically. Other modeling techniques are suffering from some shortcomings. Atomistic approach is incapable of modeling complex atomic structures and computationally expensive. On the other hand, classical continuum approach gives relatively simple formulations but it is inadequate for modeling because of the size-free deficiency. In most of the studies, nanobeams are assumed to be perfectly straight beams, but they may also be fabricated curved to be used as sensors, resonators for nanotechnology applications. Motivated by this fact, the nonlocal beam equations are obtained and an exact solution is developed for the static problems of planar curved nanobeams. Also, for engineering applications, it may be possible to develop an exact nonlocal beam finite element. Advances in nanoscience and nanotechnology shaped the modern world in the last decade. Theory, modeling and simulation have played a critical role in these advances. With the solution method proposed herein, it would be very helpful in design and fabrication of curved beam components in MEMS and NEMS applications, particularly those whose main duties are to transfer securely the applied forces. It is expected that the results obtained from the present study are got to instruct the engineering design of nano devices.