Mechanics of a discrete chain with bi-stable elements

It has become common to model materials supporting several crystallographic phases as elastic continua with non (quasi) convex energy. This peculiar p roperty of the energy originates from the multi-stability of the system at the microlevel associated with the possibility of several energetically equ ivalent arrangements of atoms in crystal lattices. In this paper me study t he simplest prototypical discrete system-a one-dimensional chain with a fin ite number of bi-stable elastic elements. Our main assumption is that the energy of a single spring has two convex we lls separated by a spinodal region where the energy is concave. We neglect this interaction beyond nearest neighbors and explore in some detail a comp licated energy landscape for this mechanical system. in particular we show that under generic loading the chain possesses a large number of metastable configurations which may contain up to one (snap) spring: in the unstable (spinodal) state. As the loading parameters vary, the system undergoes a nu mber of bifurcations and we show that the type of a bifurcation may depend crucially on the details of the concave (spinodal) part of the energy funct ion. In special cases we obtain explicit formulas for the local and global minima and provide a quantitative description of the possible quasi-static evolution paths and of the associated hysteresis. (C) 1999 Elsevier Science Ltd. All rights reserved.