Analytical solutions of out-of-plane static and dynamic problems of curved nanobeams using nonlocal elasticity theory

Nanobeams are widely used as a structural element for nano-devices and nano- machines. The development of nano-sized machines depends on proper understanding of mechanical behavior of these nano-sized beam elements. The mechanical properties of nanobeams can be measured by conventional experimental methods, however, these techniques are expensive and should be improved correspond to nano scale. Considering the experimental limitations, researchers focused on better mathematical models that comprise the effects of small length scales such as lattice spacing between atoms, grain size etc. on material behavior in micro and nano scale. Computational strategies, such as molecular dynamics simulations are adopted to analyze mechanical behavior of nano-sized structures. These simulations enable comparable investigations of dynamics of nano materials to experiments, and introduce detailed information on interatomic interactions of nano materials and molecular complexes. However, it could become too complex to simulate nanostructures, and these simulations are bounded by the computational limits. On the other hand, high-order continuum modeling is less computationally expensive than the former approach and provides relatively simple formulations. The nonlocal theory of elasticity, initiated by Eringen and Edelen takes the small-scale effect into account. This theory states that the stress at a given reference point of a body is a function of the strain field at every point in the body; hence, the theory takes the long-range forces between atoms and the scale effect into account in the formulation. Small length scales such as lattice spacing between atoms, surface properties, grain size etc. need to be considered when applying any continuum model. 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. The small length scale becomes very important in the order of nanometers. Thus, nonlocal continuum mechanics approach is adopted for studying the mechanical behavior of nanobeams. In this study, the out-of-plane static and dynamic behavior of a curved planar nanobeam having variable curvature and cross-section is investigated. The nonlocal constitutive equations of Eringen are arranged in cylindrical coordinate and implemented into the classical beam equations. Therefore, governing differential equations of nonlocal beam theory is obtained for curved beams with non-uniform cross-section. Using these equations, out-of-plane static and dynamic problems of planar curved nanobeams can be studied. The nonlocal effects of both force and moments are considered in the equations. This is the first study in the literature that considers the nonlocal effects of shear force along with the effects of shear deformation in the formulations. The governing differential equations are obtained for curved beams and analytical exact solutions for out-of-plane static problems are obtained by using the initial value method. Superiority of the initial value method is that high order statically indeterminacy adds no extra difficulty to the solution. The exact solution can be obtained for any boundary condition. The fundamental matrix of beams with different geometries and cross-sections can be obtained analytically. Thus, the deformation, slope and stress resultants can be defined analytically along the beam axis. In order to understand the effects of different parameters such as small scale parameter, opening angle of the beam and slenderness ratio etc. on the mechanics of the beam, several examples with different geometries, loading and boundary conditions are solved. In the static analysis, a fully clamped circular beam with non-uniform cross-section and non-uniform distributed load is considered as an example. The effects of the small scale parameter and the other beam characteristics, i.e. opening angle of the beam, the slenderness ratio on the transverse displacements, rotation angle and force resultants are investigated. It is observed that the result of the nonlocal theory always larger than that by the classical model for small sized beams. The size of the beam is determined by small scale parameter which is the ratio of the nanobeam radius to the additional length scale parameter. Thus, the small scale parameter has a relevant importance in calculation of displacement. For smaller values of slenderness ratio, i.e. for thick beams, the effect of small scale parameter on the displacement ratio, i.e. the ratio of the displacement results obtained from nonlocal theory and local theory is more important. The difference between the solution of both theories increases considerably with lower values of small scale parameter for any value of slenderness ratio. A uniform change in displacement ratio is observed at larger values of small scale parameter for different values of slenderness ratio, i.e. the effect is almost the same for both theories. Displacement ratio at the midspan increases with the decreasing opening angle for different values of slenderness ratio. At larger values of opening angle, the displacement ratio remains almost the constant at unity for different values of slenderness ratio. By decreasing the beam length, i.e. for smaller opening angle, size effect is much more important. Consequently, the transverse displacement of the nonlocal beam changes more in comparison to the classic one. Therefore, the size effect must be taken into account for smaller beams. The effects of slenderness ratio - opening angle, small scale parameter - opening angle and small scale parameter - slenderness ratio on the rotation angles and force resultants are also studied. Ratio of rotation angle and also the ratio of moments are not affected by the slenderness ratio. The results for shear forces is equal to unity, meaning the both theories give the same results. The equations for out-of-plane free vibrations of a curved beam with non-uniform cross section are derived by means of d'Alembert principle. The effects of shear deformation, bending and torsional rotatory inertia are considered for the out-of-plane vibrations. In the solution process, initial value method is used. Exact solution is only available for uniform circular beams. The vibration of a curved beam with varying curvature and cross-section is investigated by dividing the beam into a number of circular uniform beam elements. Similar approach is taken in the solution process. Natural frequencies are obtained for different curved beams. The effect of small scale parameter, opening angle of the beam and slenderness ratio on the dynamic behavior of the beams are presented with these examples. It is observed that size effect is more significant when the small-scale parameter is small. It is observed that the ratio of the natural frequencies predicted by nonlocal and classical models becomes more noticeable for both lower values of slenderness ratio and opening angle of the nanobeam. It is expected that the valuable results obtained from the presented continuous model would be very useful for dynamic analyses of beam-like components such as nanosensors or nanoactuators whose discrete models have high computational and labor costs. The literature has shown that, almost all the studies on nonlocal beam theory has discussed the subject in the context of straight nanobeams. There are very limited number of papers on the curved nanobeams and most of them neglect the effect of shear deformation. Numerical and approximate solution methods are used in these studies. Numerical examples for nonlocal curved beam models are insufficient. The present work will be helpful in the analysis and design of curved nanobeams with various combinations of loadings, geometry and boundary conditions. The nonlocal theory and the equations presented in this study forms the basis for the study of out- of-plane static and dynamic analysis of curved 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. The results of the examples presented in this study will give insight to the interplay between the geometry of the beam and the stress distribution and the effect of the nonlocal behavior on static and dynamic behavior of curved nonlocal beams. Considering scale effect, shear deformation and rotatory inertia in the exact solutions of the governing equilibrium equations presented here, the results can be considered as a reference for nonlocal theories of curved nanobeams.