Two- and three-dimensional elastic networks with rigid junctions: modeling within the theory of micropolar shells and solids

Victor A Eremeyev

doi:10.1007/s00707-019-02527-3

Abstract

For two- and three-dimensional elastic structures made of families of flexible elastic fibers undergoing finite deformations, we propose homogenized models within the micropolar elasticity. Here we restrict ourselves to networks with rigid connections between fibers. In other words, we assume that the fibers keep their orthogonality during deformation. Starting from a fiber as the basic structured element modeled by the Cosserat curve beam model, we get 2D and 3D semi-discrete models. These models consist of systems of ordinary differential equations describing the statics of a collection of fibers with certain geometrical constraints. Using a specific homogenization technique, we introduce two- and three-dimensional equivalent continuum models which correspond to the six-parameter shell model and the micropolar continuum, respectively. We call two models equivalent if their approximations coincide with each other up to certain accuracy. The two- and three-dimensional constitutive equations of the networks are derived and discussed within the micropolar continua theory.

Highlights

Lattice beam structures are widely used in civil, mechanical, and aerospace engineering; see, e.g., [17,32,57]
We introduce an averaged continuum model that is a deformable material surface with particular material properties described within the framework of the six-parameter shell theory [13,49,50]
Comparing discrete models for a 2D network and shell in Sect. 4, we present the constitutive equations for the shell which is equivalent in certain sense to the considered network

Summary

Introduction

Lattice beam structures are widely used in civil, mechanical, and aerospace engineering; see, e.g., [17,32,57]. 3, we consider a 2D elastic network similar to a fishnet made of orthogonal flexible fibers. The case of an elastic network with inextensible fibers was considered in [23] For this net, we introduce an averaged continuum model that is a deformable material surface with particular material properties described within the framework of the six-parameter shell theory [13,49,50]. Comparing discrete models for a 2D network and shell, we present the constitutive equations for the shell which is equivalent in certain sense to the considered network. As was mentioned in [26], homogenization is one of the main sources for derivation of the constitutive equations of micropolar solids; see, e.g., [6,18,19,31,35,69] and the references therein

Cosserat curve as a fiber model

Discrete beam lattice

Continuous beam lattice shell: micropolar shell

Conclusions