A new eigenvalue problem solver for thermo‐mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method

Abstract

In this study, size‐dependent thermo‐mechanical vibration analysis of nanobeams is examined. Size‐dependent dynamic equations are obtained by implementing Hamilton's principle based on Timoshenko beam theory and then combined with stress equation of nonlocal elasticity theory. The separation of variables total method and finite element formulation is utilized to solve the eigenvalue problem. Local and nonlocal stiffness and mass matrices are firstly derived by using a weighted residual method for the finite element analysis. The accuracy of the finite element solution is demonstrated by comparisons with the earlier studies. Then, nondimensional frequencies of nanobeams with different boundary conditions based on a nonlocal finite element method are presented for vibration analysis that cannot be analytically solved under different parameters. It is aimed to emphasize the importance of the nonlocal finite element method in the size‐dependent vibration behavior of nanobeams which form different components of nano‐electro‐mechanical systems.

A Group Theoretic Analysis of Mutual Interactions of Heat and Mass Transfer in a Thermally Slip Semi-Infinite Domain

“…The use of a Williamson fluid with many physical special effects resulted in highly non-linear PDEs, making an exact solution unattainable at this time. When an exact answer appears to be impossible, we always look for a numerical solution [37,38]. Researchers in this subject, in particular, used a variety of approaches to report numerical solutions.…”

Yerel olmayan elastisite teorisi kullanılarak basit mesnetli Timoshenko nanokirişlerinin çökmeleri üzerinde kesit etkisi

“…Consequently, the constitutive equations of nonlocal elasticity for different stress components can be written as follows [27,28]:…”

“…On the other hand, an innovative nonlocal finite element formulation has been presented to investigate the effects of thermal environment and elastic foundation on the free vibration behavior of nonlocal shear deformable nanobeams [27,28]. There are also thesis studies that include nonlocal static and dynamic analysis results for Timoshenko nanobeams [29,30].…”

“…There are also thesis studies that include nonlocal static and dynamic analysis results for Timoshenko nanobeams [29,30]. With the Rayleigh theory, which formulates the shear effects in a simpler way, it can be stated that the size-dependent free vibration of atomic scaled structures has been explained [28,31]. Eptaimeros et al [32] performed the free vibration of nanobeams resting on Winkler foundation by using nonlocal integral approach and finite element analysis.…”

Nonlocal Free Vibration of Embedded Short-Fiber-Reinforced Nano-/Micro-Rods with Deformable Boundary Conditions

“…These parameters, generally called small-scale parameters, make it possible to investigate the size effect. Researchers working on the analysis of various very small-scale nano- and micro-structures have used these size effect theories, such as doublet mechanics theory [ 29 , 30 , 31 , 32 , 33 ], modified couple stress theory [ 34 , 35 , 36 , 37 , 38 , 39 ], nonlocal elasticity theory [ 1 , 40 , 41 , 42 , 43 , 44 , 45 ], nonlocal strain gradient theory [ 46 , 47 , 48 , 49 ] and strain gradient theory [ 50 , 51 , 52 , 53 ]. Researchers use a variety of techniques to analyze an element or structure, such as a rod, beam, plate, frame, etc., whether at the macro, nano, or micro level.…”