Wave Propagation in Timoshenko–Ehrenfest Nanobeam: A Mixture Unified Gradient Theory

A size-dependent elasticity theory, founded on variationally consistent formulations, is developed to analyze the wave propagation in nanosized beams. The mixture unified gradient theory of elasticity, integrating the stress gradient theory, the strain gradient model, and the traditional elasticity theory, is invoked to realize the size effects at the ultra-small scale. Compatible with the kinematics of the Timoshenko–Ehrenfest beam, a stationary variational framework is established. The boundary-value problem of dynamic equilibrium along with the constitutive model is appropriately integrated into a single function. Various generalized elasticity theories of gradient type are restored as particular cases of the developed mixture unified gradient theory. The flexural wave propagation is formulated within the context of the introduced size-dependent elasticity theory and the propagation characteristics of flexural waves are analytically addressed. The phase velocity of propagating waves in carbon nanotubes (CNTs) is inversely reconstructed and compared with the numerical simulation results. A viable approach to inversely determine the characteristic length-scale parameters associated with the generalized continuum theory is proposed. A comprehensive numerical study is performed to demonstrate the wave dispersion features in a Timoshenko–Ehrenfest nanobeam. Based on the presented wave propagation response and ensuing numerical illustrations, the original benchmark for numerical analysis is detected.