Size-dependent free vibration analysis of infinite nanotubes using elasticity theory

Abstract

Exact elasticity theory is employed to study the (two-dimensional) free vibration of nanoscale cylindrical tubes in the presence of free surface energy. Use is made of the Gurtin-Murdoch surface elasticity model to incorporate the surface stress terms into the pertinent boundary conditions. Some numerical examples are provided to depict the influence of the surface energy, and particularly the inner radius size of the nanocylinder, on the natural frequencies of the system. The results indicate a stronger influence of surface effects for both smaller values of the outer to inner radius ratio and higher modes of vibration.

Exact Size-Dependent Elasticity Solution for Free Radial Vibration of Nano-Sphere

“…Two different sets of surface properties corresponding to the crystallographic directions [100] (denoted by "SA") and [111] (denoted by "SB") in aluminum are used in the calculation. The corresponding elastic constants are given as: 5 Surface A "SA": E s = −8.95(N/m), Surface B "SB": E s = 6.08(N/m). Classical elasticity theory, in which surface effects don't appear or E s = 0(N/m), are denoted by Surface C "SC".…”

“…Due to radial vibration analysis, the displacement field was only expressed as a function of the r coordinate. The solution of the scalar Helmholtz equation (5), which lead to the field expansion of compressional waves, was written as: 24 ϕ(r, ω) = a(ω)j 0 (αr) + b(ω)y 0 (αr);…”

“…Recently, due to the increased application of nanoscale structures like the nano-beam, 5 nano-bar, 6 and nano-plate 7 in technological systems, there has been an increase in demand for understanding the behavior of small-sized materials and structures. One of the procedures for studying the characteristics nano-sized materials and devices is "atomic simulation".…”

“…8 Another procedure is "size-depended elasticity theories". 5 In particular, several size-dependent elasticity theories are used to study very small-sized structures. One of the size-dependent elasticity theories is the "nonlocal elasticity theory."…”

Thermal and magnetic effects on the vibration of a cracked nanobeam embedded in an elastic medium

“…If θ = 0, i.e., there is no influence of change in temperature, we then return to the constitutive relation for nonlocal elasticity. It should be noted that Young's modulus of some types of nanomaterials, for example CNT, is insensitive to temperature changes in the tube at temperatures less than nearly 1100 K, but it decreases at higher temperatures [Hsieh et al 2006]. In what follows, we will use the constitutive relation for nonlocal thermoelasticity to derive governing equations of motion.…”