Numerical Solution of Vibration of Euler Nanobeams with Different Complicating Effects

Abstract

Nanomaterials are those for which a single unit is sized between 1 and 100 nm. Nanoparticles show very interesting and unique behaviours under thermal, electrical, and magnetic environments. Due to these characteristics, nanostructural members such as nanowires, nanobeams and nanoplates etc. play a significant role in the making of devices for information technology, solar panels, optical and electronics, medical and health care applications etc. Research on nanostructures shows that the classical continuum theories could not capture the small scale effect at nano or micro scales. Various nonlocal theories are developed to capture the size effects at nanoscales, which are helpful to predict the dynamic behaviours of nanosized structures. In various engineering and industrial sectors such as aerospace, nuclear power, automobiles, steel etc., heat and thermal conductivity of materials are important. In this aspect Functionally Graded Materials (FGM) have gained attention of researchers. FGMs are composite materials formed of two or more constituent phases with a continuously variable composition. So combining FGM concept with nanoscaled materials helps to make many advanced devices and equipment such as nanoelectromechanical devices, optoelectronic devices, aerospace equipment etc. Modelling and analysis of nano and functionally graded nanobeams are challenging because experiments at nanoscale are always costly and time dependent. Also capturing the small scale effects considering internal length scale parameters is not easy for nanostructural members. On the other hand, dynamic behaviours of nanobeams are affected by complicating effects like different elastic foundations, magnetic environment, thermal effects, flexoelectric effects etc. The influence of these complicating effects at nanoscale is challenging. Hence investigation of dynamic behaviours of nanobeams placed under different complicating effects is a major concern in the field of vibration of nanobeams. In view of the above, the present thesis investigates the vibration behaviours of Euler nanobeams considering various complicating effects. Also, handling different boundary conditions is a challenge for this type of investigation. In this work, several computationally efficient methods such as RayleighRitz, Differential Quadrature, Adomian Decomposition, Differential Transform, Homotopy Perturbation methods etc., are successfully implemented under different classical boundary conditions like simply supported, clamped, and free. Furthermore, convergence and validation of different numerical methods are discussed in terms of frequency parameters. Different nonlocal theories such as Eringen’s nonlocal theory, nonlocal strain gradient theory, and nonlocal strain gradient theory for piezoelectric nanobeam are used to capture the small scale effects of the vibration of beams. As mentioned above, regarding the complicating effects, nonhomogeneous beam model, Winkler foundation, WinklerPasternak foundations, thermal effects, torsional vibration of functionally graded nanobeam, magnetic effects etc., are taken in action and dynamic behaviours of Euler nanobeams are analysed in systematic manners. Obtained results are compared with existing literature and many new results are reported in terms of figures and tables. The new results obtained through the above mathematical models may serve as benchmarks and those may certainly be used by design engineers and practitioners to validate their experimental work for better design of the related nanostructures.