A nonlinear theory of prestressed elastic stick-and-spring structures

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.