Equilibrium Paths of Polygonally Elastic Structures∗

Abstract

Abstract Stability analysis of structures having polygonally elastic material is presented. The polygonal material model can be used for approximating any nonlinearly elastic behavior or originally polygonal behavior. A discrete model is considered, in which perfectly rigid elements are connected to each other by springs that represent material characteristics. The behavior of the springs is governed by uniaxial arbitrary non-decreasing stress-strain polygons with horizontal and vertical jumps. The stability analysis presented is based on potential energy. Since the material law is polygon-like, the related strain energy is a nonsmooth function, for which, since the material is reversible, it is a convex function. Thus, a nonsmoot convex analysis is needed. The stability analysis presented in this paper is global, and related to the total domain of the possible deflections. Equilibrium paths of one-dimensional problems are analyzed. Finally, a visual presentation of the detailed nonsmooth stability analysis is introduced. This paper is the first part of a series of three papers related to nonsmooth stability. The non-smooth nonconvex (dissipative) problems and nonsmooth damage with localization are detailed in Refs. 1 and 2, respectively.