STABILITY OF EQUILIBRIUM STATES OF NONLINEAR STRUCTURES AND CHAOS PHENOMENON.

Abstract

The “classical” chaos of deterministic systems is characteristic for the motion of dynamical systems. Recently, some attempts were made to find static analogies of chaos [Thompson & Virgin, 1988; Naschie & Athel, 1989; Naschie, 1989]. However, this was considered for structures in specific artificial conditions (for example, infinitely long bars with sinusoidal geometric imperfections) transferring de facto the boundary value problem (which always describes static deformation of structures) into an initial value problem characteristic for problems of motion. In this article, chaotic (unpredictable) behavior is described for a usual (not special) nonlinear structure in statics, which is governed, naturally, by a boundary value problem in a finite interval of the argument. The behavior of this structure (geometrically nonlinear plate), which is an example of the class of static chaotic structures, is investigated by a new geometrical approach called the “deformation map.” The presented results are one of the first steps in the chapter of chaos in statics, and therefore the link between “classical” and static chaos needs further investigations.