Parameter Optimization and Comparison of Different Small-Scale Elasticity Theories for Carbon Nanotubes

During the structural analysis of micro- and nanostructures, classical elasticity theory fails to capture the small-scale size effects that are present in the experimental results. To incorporate the small-scale size effects into the equation of motion, either intermolecular forces over relatively long distances or higher-order derivatives of displacement field need to be considered. Two such theories employing the mentioned models are nonlocal elasticity theory and strain gradient theory, respectively. In this work, nonlocal strain gradient theory, which includes small-scale size effects, is used to model carbon nanotubes with the assumptions of Timoshenko beam theory. Natural frequencies are found with the differential quadrature method, and genetic algorithm is used to determine the nonlocal and length scale parameters present in the theory using molecular dynamic results for different aspect ratios to obtain the best overall fit. Instead of obtaining different small-scale parameters for each mode number, optimization of all modes at once is carried out, which is extended to also include the classical material properties. The results are inclined toward nonlocal elasticity theory for the problem at hand.