Hierarchy of beam models for lattice core sandwich structures

TL;DR

This paper develops a hierarchy of beam models for lattice core sandwich structures, linking discrete lattices to continuum beams, and compares their effectiveness in modeling different load conditions.

Contribution

It introduces a novel non-classical micropolar beam model derived from a discrete-to-continuum transformation, and compares multiple beam models for lattice core structures.

Findings

The 4th-order Timoshenko beam models stretching-dominated lattices effectively.

The 6th-order micropolar model captures bending-dominated behaviors.

The 1-D micropolar model accurately reproduces 2-D lattice responses without boundary effects.

Abstract

A discrete-to-continuum transformation to model 2-D discrete lattices as energetically equivalent 1-D continuum beams is developed. The study is initiated in a classical setting but results in a non-classical two-scale micropolar beam model via a novel link within a unit cell between the second-order macrorotation-gradient and the micropolar antisymmetric shear deformation. The shear deformable micropolar beam is reduced to a couple-stress and two classical lattice beam models by successive approximations. The stiffness parameters for all models are given by the micropolar constitutive matrix. The four models are compared by studying stretching- and bending-dominated lattice core sandwich beams under various loads and boundary conditions. A classical 4th-order Timoshenko beam is an apt first choice for stretching-dominated beams, whereas the 6th-order micropolar model works for bending-dominated beams as well. The 6th-order couple-stress beam is often too stiff near point loads and boundaries. It is shown that the 1-D micropolar model leads to the exact 2-D lattice response in the absence of boundary effects even when the length of the 1-D beam (macrostructure) equals that of the 2-D unit cell (microstructure), that is, when L=l.