Discussion of “Derivation of Mindlin's first and second strain gradient elastic theory via simple lattice and continuum models” by Polyzos and Fotiadis

Abstract

We discuss “Derivation of Mindlin's first and second strain gradient elastic theory via simple lattice and continuum models” by Polyzos and Fotiadis in [Int. J. Solids Struct.49 (2012), pp. 470-480]. Polyzos and Fotiadis derived the equations of motion of Mindlin's first and second strain gradient elasticity theory by continualizing the response of simple mass-and-spring chains with one-neighbor and two-neighbor interactions in addition to the inclusion of distributed mass alongside lumped mass. When passing from the finite difference equations of the discrete lattice to the continualized equations of the corresponding higher-order continuum model, certain inconsistencies in truncating the various Taylor series are noted. By including the missing terms, corrected potential and kinetic energy densities are presented here, after which corrected governing equations are derived by the application of Hamilton's variational principle.