Analytical and computational aspects for torsional statics and dynamics of cylindrical nanostructures based on nonlocal elasticity theory

Abstract

In late 1960s, the fundamental works on nonlocal elasticity was conducted by Kroner, Kunin and Krumshansl where they considered the long range cohesive forces and atomic lattice theory. Later in early 1970s, Eringen and co-workers developed nonlocal elasticity theory in the presence of nonlocality residuals of field such as body force, mass, entropy, internal energy etc. and determined these residuals, along with the constitutive laws, by means of thermodynamic restrictions. This theory states that the stress at a point in a nonlocal elastic continuum not only depends on the local stress at that point also on stress at all other points in the body. Consequently, in the constitutive relation of nonlocal elasticity theory Hooke's law (for local theory) is replaced by the integration which governs the nonlocal material behavior. As the constitutive relation is of the integral form, the subsequent integro-partial differential equations are extremely difficult to solve in terms of displacement field in the nonlocal elasticity. However, after approximately ten years of original development, under certain conditions using Green's function with a certain approximation error, Eringen himself transformed the original integral constitutive relation into differential constitutive form. Later, this differential constitutive form has been extensively used to study the mechanical properties of nanostructures. But the exact nonlocal boundary effects presented by the integration of kernel function in the integral formulations are not perfectly matched or transformed in the differential form which is a weak point of approximation. In order to capture this issue, an integral constitutive relation which was also proposed by Eringen and his associates is utilized to solve boundary value problem. The new contributions and significance of the present model over the analytical nonlocal model is highlighted in terms of static and dynamic torsional analysis of cylindrical nanostructures for boundary value problem. In the Eringen's nonlocal elasticity, Hooke's law is replaced by an integral relation which contains a nonlocal kernel function describing the relative influence of the stress at various locations on the stress at a given location. It is noted that the nonlocal kernel or attenuation function is not complete near the boundary. Thus, the analytical nonlocal model derived by using differential constitutive relation which is a transformation of the original integral constitutive relation provides inconsistent results at the boundary. The present nonlocal finite element model (NL-FEM) where an integral constitutive relation is incorporated exhibits reliable and consistent results in the presence of nonlocal nanoscale effect even at the boundary of a finite domain. In this thesis, one dimensional cylindrical nanostructure is considered to study the mechanical properties of cylindrical nanostructures based on nonlocal elasticity theory. This work is divided into three main parts. In the first part, based on the standard finite element method, a new nonlocal finite element is developed and utilized for rod model to compare the validity of analytical results for boundary value problems. It is shown numerically that the proposed numerical method provides consistent results for arbitrary external load while the existing analytical nonlocal model gives unstable results for some particular case. In the second part, an extensive study is performed for the boundary value problems which contain torsional static and torsional vibration analysis in the presence of small scale effect. A comparison is made between the present results and the existing results in this area. Finally, the third part treats the dynamical problem of torsional propagation in infinite periodically loaded single rod and double rods system. The main results are the effect of nonlocal nanoscale on dispersion relation. Besides, the effects of nonlocal nanoscale on group velocity, phase velocity, escape frequency etc. are also treated with the nonlocal rod model. It is seen that wavenumber has nonlinear variation with wave frequency and the wave propagates dispersively.

Date of Award	2 Oct 2013
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	C W LIM (Supervisor)