Lattice-Based Nonlocal Elastic Structural Models

This chapter is devoted to lattice-based nonlocal approaches in relation with elastic microstructured elements. Nonlocal continuous approaches are shown to be relevant for capturing length scale effects in discrete structural mechanics models. The chapter contains three complementary parts. In the first part, axial lattices, as already studied by Lagrange during the XVIIIth century are investigated, both for statics and dynamics problems. This discrete model is also called the Born-Klattice model with direct neighbouring interactions. Exact solutions are presented for general boundary conditions. A nonlocal elastic rod model is then constructed from the lattice difference equations. The nonlocal model is similar to the nonlocal model proposed by Eringen in 1983 that is based on a stress gradient approach, although the small length scale of the nonlocal model may differ from statics to dynamics applications. This part is closed with a discussion on generalized lattices with direct and indirect neighbouring interactions and their possible nonlocal modelling. The second part of this study deals with lattice beam elements called Hencky-Bar-Chain models, due to the fact that the discrete beam model was introduced by Hencky in 1920. Exact solutions are presented for general boundary conditions in both statics and dynamics settings. A nonlocal elastic Euler-Bernoulli beam model is then developed from the lattice difference equations. The nonlocal model is similar to a stress gradient nonlocal Euler-Bernoulli beam, where the nonlocality is of the Eringen type, although the length scale of the nonlocal model may also differ for statics or dynamics applications. The last part is devoted to lattice plates as introduced by Wifi et al. in 1988, and El Naschie in 1990, in connection with the finite difference formulation of Kirchhoff-Love plate models. Exact solutions for lattice plate statics and dynamics problems are presented for the Navier-type boundary conditions. A nonlocal elastic Kirchhoff-Love plate model is then derived based on the difference equations of the lattice plate. The microstructure-based nonlocal model slightly differs from an Eringen stress gradient Kirchhoff-Love plate model. The methodology followed from fundamental lattice microstructures shows that new nonlocal structural elements may be built from physical discrete structural approaches. The nonlocal models derived herein are closely related to the lattice microstructure assumed at the discrete level. It is expected that alternative nonlocal models may be achieved for some other microstructures.