A Nonlinear Theory of Prestressed Elastic Stick-and-Spring Structures

TL;DR

This paper develops a nonlinear theoretical framework for prestressed elastic stick-and-spring structures, applicable to nanostructures like graphene, incorporating geometrical nonlinearity and prestress effects with analytical and numerical solutions.

Contribution

It introduces a nonlinear theory combining exact strain measures with linear stress response, accounting for prestress states, and provides analytical and numerical solutions for complex structures.

Findings

The theory accurately models nanostructures such as graphene and nanotubes.

Analytical solutions are derived for simple cases, numerical methods for complex structures.

The approach captures the effects of prestress and nonlinear geometry on structural behavior.

Abstract

The discrete modeling of a large class of mechanical structures can be based on a stick-and-spring concept. We here present a stick-and-spring theory with potential application to the statics and the dynamics of such nanostructures as graphene, carbon nanotubes, viral capsids, and others. A key feature of our theory is its geometrical nonlinearity: we combine exactly defined strain measures with a general linear stress response; another, rarely found feature is a careful account of prestress states. A linear version is firstly proposed, where attention is restricted to study small displacements from an unstressed reference placement. Next, a theory linearized about a prestressed (preloaded or not) placement is displayed, which is based on a careful analysis of the tangent stiffness operator and its two parts, the elastic and prestress stiffness operators. Finally, two examples are proposed and solved; when an analytical solution is of prohibitive complication, numerical solutions are given, by the use of a specifically implemented `stick-and-spring' code.