Mechanics of a discrete chain with bi-stable elements

Abstract

It has become common to model materials supporting several crystallographic phases as elastic continua with non (quasi) convex energy. This peculiar property of the energy originates from the multi-stability of the system at the microlevel associated with the possibility of several energetically equivalent arrangements of atoms in crystal lattices. In this paper we study the simplest prototypical discrete systemÐa one-dimensional chain with a ®nite number of bi-stable elastic elements.Our main assumption is that the energy of a single spring has two convex wells separated by a spinodal region where the energy is concave. We neglect the interaction beyond nearest neighbors and explore in some detail a complicated energy landscape for this mechanical system. In particular we show that under generic loading the chain possesses a large number of metastable con®gurations which may contain up to one (snap) spring in the unstable (spinodal) state. As the loading parameters vary, the system undergoes a number 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 function. 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. #

A scalable multiscale LATIN method adapted to nonsmooth discrete media

“…In these so-called mesoscopic scales, systems, sometimes still discrete as the dislocations, can interact to supply the plastic behaviour of the single-crystal [4,5]. Without resorting always to well identified scales, a discrete model can help to report exotic behaviour as metastable states in phase transformations [6,7]. Some authors tried to propose a discrete-to-continuum bridging [8,9], with the aim of carrying out a discrete structural zoom on zones of large deformations or defects, as fracture or buckling in the nanotubes [10,11].…”

Macro- and micro-cracking in one-dimensional elasticity

“…The problem of a discrete chain with N bi-stable springs was studied in Puglisi and Truskinovsky (2000), and we can use some of their results. Obviously, a necessary condition for equilibrium is the equality of`f orces'' acting inside di erent springs,…”

“…The main minors of this matrix can be calculated explicitly (see Puglisi and Truskinovsky (2000)), and we obtain the condition of stability in the form…”

Coarse-graining and renormalization of atomistic binding relations and universal macroscopic cohesive behavior

“…These materials often develop fine microstructure in response to imposed deformations. Truskinovsky et al [3,8,4], and Braides et al [2], have pioneered the application of these methods to fracture. However, the full relaxation of a cohesive potential yields the trivial result that the effective cohesive potential is identically equal to zero.…”