A discrete model for elasticity with microstructure

Abstract

In the mid-1960s, Mindlin introduced his theory of strain gradient elasticity (SGE), along with one of its most widely recognized simplified versions, known as SGE-Form II. These enhanced theories were motivated by the need to capture dynamic phenomena that classical elasticity fails to address. Although theoretically elegant, the inclusion of numerous material constants and internal length scale parameters makes the practical application of both SGE and SGE-Form II challenging—even for one-dimensional (1D) problems. A practical approach for identifying these parameters involves validating the theories through lattice models, where both the microstructure and macrostructure are materially and geometrically well-defined. Using a simple 1D lattice model with nearest and next-nearest neighbor spring connections and distributed mass, Polyzos and Fotiadis (Int. J. Solids Struct. 49, 470–480, 2012) were the first to successfully validate Mindlin’s SGE-Form II and to interpret the intrinsic parameters introduced by the theory. Despite significant efforts, no lattice model has yet been shown to fully replicate the behavior described by the general SGE theory, even in 1D cases. This gap is addressed in the present work through the use of a simple 1D lattice model and the implementation of a continualization process similar to that employed by Polyzos and Fotiadis (2012).