Spatial chaos in discrete mechanical systems: elastic linkage and elastic web of links

Postcritical behaviour of discrete and continuous bar structures is relevant in case of imperfection sensitivity in engineering practice, but it also plays an important role in biology when spatial configurations of macromolecules are investigated. Biological filaments, like DNA, (bio)polymers, or tendrils, and engineering structures, like marine cables and drill strings may exhibit complicated spatial patterns. For the investigation and characterization of such spatially complex configurations, chaos theory can provide useful tools. This phenomenon of spatially complex behaviour of structures was lately called spatial chaos, although it still does not have a general definition. Here we derive and organize equilibrium configurations of simple, one and two dimensional discrete mechanical models considering large displacements. The aim of the dissertation is to characterize and more deeply understand spatial chaotic behaviour. The main goal is to give such a definition of spatial chaos which can be fairly easily used for boundary value problems defined on one or more dimensional finite domain. We start with discussing the buckling problem of a simply supported elastic linkage loaded by a follower load. It is shown, that the related initial value problem is area-preserving. Then we investigate a clamped non-linear elastic linkage under general loading and prove, that the related initial value problem is non-dissipative in every case. We prove, that every equilibrium configuration of the clamped non-linear elastic linkage under general loading is uniquely connected to a periodic orbit of the corresponding dynamical system. Based on this observation, we suggest, that a boundary value problem should be called chaotic, if the number of solutions depends exponentially on the extent of the domain, and the exponent—the topological entropy of the corresponding dynamical system—is positive. Bifurcation analysis of the elastic linkage is carried out for the clamped case under various loads and for the simply supported case loaded by a follower force. The buckling problem of a two dimensional structure, the elastic web of links is studied in the second part of the dissertation. The web is supported by a fixed hinge in one case and by a fix hinge and a roller in another case. In both cases, the web is loaded in one direction. The equilibrium configurations are obtained using the simplex algorithm based on piecewise linearisation. Bifurcations of the trivial equilibrium path is analyzed. Finally, the similar behaviour of the elastic web of links and the unbendable, unstretchable rod is pointed out for infinitesimally small displacements.