Attila Kocsis

Conservative spatial chaos of buckled elastic linkages

Buckling of an elastic linkage under general loading is investigated. We show that buckling is related to an initial value problem, which is always a conservative, area-preserving mapping, even if the original static problem is nonconservative. In some special cases, we construct the global bifurcation diagrams, and argue that their complicated structure is a consequence of spatial chaos. We characterize spatial chaos by the associated initial value problems topological entropy, which turns out to be related to the number of buckled configurations.

Buckling Analysis of the Discrete Planar Cosserat Rod

In this paper, a discrete model of the planar Cosserat rod is presented. Based on the calculus of variations, the equilibrium equations of the model are derived for potential forces and hyperelastic material. Buckling of the structure under axial loading is thoroughly studied assuming linear elasticity. Dimensionless stiffness parameters are introduced, and analytical solutions are given for the critical loads and the corresponding buckled shapes of the model. Classification of the axially loaded structure is accomplished based on the number, sign, and physical admissibility of its buckling loads. It is revealed that the model can possess several buckling modes under tension.

SPATIALLY CHAOTIC BIFURCATIONS OF AN ELASTIC WEB OF LINKS

Spatially chaotic bifurcations of an elastic web of links are investigated. We numerically construct the global bifurcation diagrams uniquely describing the buckled states, and show that the exponential growth of the number of equilibrium branches with the size of the web indicates spatial chaos. The types of bifurcations from the trivial equilibrium branch are also determined, and we show that cusp catastrophes of any order can appear. This observation relates the buckling of the elastic web of links to the buckling of rods with finite shear and infinite bending and normal stiffness.

On the Foundation of a Generalized Nonlocal Extensible Shear Beam Model from Discrete Interactions

In this paper a generalized discrete elastica model including bending, normal and shear interactions is developed. Nonlinear static analysis of the discrete model is accomplished, its buckling and post-buckling behavior are thoroughly studied. It is revealed that based on what finite strain theory is used, the discrete model yields a generalized (extensible) Engesser elastica, or a generalized (extensible) Haringx elastica. The local continuum counterparts of these models are also obtained. Then nonlocal models are developed from the introduced flexural, extensible, shearable discrete systems using a continualization technique. Analytical and numerical solutions are given for the discrete and nonlocal models, and it is shown that the scale effects of the discrete models are well captured by the continualized nonlocal models.

Buckling under conservative and nonconservative load

Since the first invention of chaos theory, it has been found to play a very important role in many different fields, ranging from physics through biology to engineering, among others. We deal with a phenomenon called spatial chaos which is a special form of spatial complexity, when the governing equations are reminiscent of a chaotic dynamical system, but the role of time is taken over by a spatial coordinate (e.g. arc-length). Many examples of spatial chaos have been addressed recently in general mathematical studies, in fluid dynamics, in the case of buckling of elastic rods or linkages. It also plays an important part in biology where biological filaments – like DNA, (bio)polymers, or tendrils – may exhibit complicated spatial patterns. It has been shown that the elastic linkage provides both a mathematical discretization of Euler’s buckling problem and a mechanical discretization of a continuous rod.The discrete problem is in the state of spatial chaos: it has much more complicated equilibrium shapes than has the continuous Euler-problem. The reason of this is that the governing equations of the continuous problem coincide with a non-chaotic initial value problem, the mathematical pendulum, while the equations of the linkage are the same as the well-known chaotic map, the standard map. We deal with the buckling problem of a cantilever under a quite general set of loads which can be either conservative or non-conservative. We assume that the material behavior can be nonlinear and the rod can be non-prismatic. Using a discrete model, an elastic linkage we show that the static stability is related to a chaotic map, which is conservative both in case of conservative or non-conservative loads. It proves that conservative spatial chaos is not a unique feature of conservative buckling problems. We detail some special examples and construct their global bifurcation diagrams.