On the capability of generalized continuum theories to capture dispersion characteristics at the atomic scale

Abstract Generalized theories of continuum mechanics, such as gradient and nonlocal elasticity, have been widely used to account for the small scale effects on materials’ behavior when dealing with structures at the micro- or nano- scale. It has been demonstrated that these enhanced theories provide better approximations that are closer to experimental observations than classical ones for problems in the field of fracture mechanics, dislocations, and wave propagation. The present work investigates the capability of one-dimensional elastic models -gradient, nonlocal and mixed- to predict the dispersive behavior of traveling waves, in comparison with the Born–Karman model of lattice dynamics. The linear theories adopted herein are limited to Mindlin’s first (grade 2) and second (grade 3) strain gradient theories in elasticity with two and three intrinsic parameters and Eringen’s nonlocal elasticity theory with one and two intrinsic parameters. Mixed models of nonlocal and gradient theories with up to three intrinsic parameters are also considered. More specifically, seven 1D models are considered: one grade 2 elastic bar with micro-inertia, one grade 3 elastic 1D model, three nonlocal elastic bars -two with Helmholtz operator, and one with bi-Helmholtz operator after Lazar et al. (2006) , one mixed nonlocal / grade 2 elastic bar with Helmholtz operator, and the mixed nonlocal model after Challamel et al. (2009) . Only three models, under specific assumptions for their intrinsic parameters, result in matching satisfactorily the dispersion curve of Born–Karman’s atomic model. The rest suffer violation of their fundamental thermodynamic restrictions. This violation is naturally explained by further analyzing the mathematical structure of the obtained dispersion relations, via Pade approximants, whose coefficients are directly related to each model’s intrinsic parameters.