Mechanics Research Communications 92 (2018) 107–110 Contents lists available at ScienceDirect Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom Discrete systems with geometry-driven evolution: Application to 1D elasticity and granular media A. Battista a,b,c,d , P. D’Avanzo e , M. Laudato a,b,d,∗ a International Research Center M&MoCS, Università degli Studi dell’Aquila, Via Giovanni Gronchi 18 - Zona industriale di Pile, L’Aquila 67100, Italy b Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studi dell’Aquila, Via Vetoio, Coppito, L’Aquila 67100, Italy c Laboratoire des Sciences de l’Ingénieur pour l’ Environnement, Université de La Rochelle, 23 avenue Albert Einstein BP 33060, La Rochelle 17031, France d Research Institute for Mechanics, National Research Lobachevsky State University of Nizhni Novgorod, Russia e Dipartimento di Fisica, Universitá degli studi Federico II, Naples, Italy a r t i c l e i n f o Article history: Received 18 February 2018 Revised 14 July 2018 Accepted 8 August 2018 Available online 11 August 2018 Keywords: Granular Swarm Geometry-driven Dynamics a b s t r a c t In this paper, we present a discrete system characterized by a geometrically-driven evolution law and very light computational costs. Potential applications to the simulation of the nonlinear deformation of extensible strings with null flexural stiffness, as well as to dynamics of granular media, are suggested. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction: The algorithm In this paper, we consider a one-dimensional particularization of the discrete model introduced in [1] and investigated e.g. in [2]. The aim of the model was the approximation of the deformation of (generalized) elastic bodies with discrete particle systems. For gen- eralized bodies, we mean higher gradient deformable media, i.e. continua in which the deformation energy depends on second or higher gradient of the displacement field (see [3–6], for theoreti- cal introductions and recent results). This analysis can be useful in the framework of dynamics and wave propagation phenomena in granular media (see the seminal paper [7]) from both the theoret- ical and the experimental point of view. Here we propose a quick summary of the algorithm, referring to the aforementioned works for more details. Let us consider a discrete system S consisting of N points in the real plane. We call them the ”elements” of S. We introduce an orthogonal reference system (x 1 , x 2 ). In the initial configura- tion C 0 the elements of S are set along the x 1 -axis, equally spaced with unitary step and with the first element in (0,0) and the last element in (N − 1, 0). We consider a set of discrete time steps ∗ Corresponding author at: International Research Center M&MoCS, Università degli Studi dell’Aquila, Via Giovanni Gronchi 18 - Zona industriale di Pile, L’Aquila 67100, Italy. E-mail addresses: antonio.battista@univ-lr.fr (A. Battista), paolo.dav93@outlook.it (P. D’Avanzo), marco.laudato@graduate.univaq.it (M. Laudato). T m = {0, t 1 , . . . , t m , . . .}. Each element of S has an index (i). We call nth neighbors of the element ¯ i the elements N n ( i ) defined by: N n ( i ) := {i ∈ C 0 : ρ [ i, i] = n} where ρ is the Lagrangian distance, i.e. the distance evaluated in C 0 . The elements at the boundary of the discrete system have only one neighbor. To have always the same number of neighbors, we introduce two fictitious elements at the edges, whose evolu- tion will be described later. We select a leader element L with starred index (i ∗ ) whose time-depending position is defined by a prescribed motion M : t m ∈ T m → (x 1 i ∗ , x 2 i ∗ )(t m ) ∈ R 2 The system evolves through a family of sets of virtual configu- rations, each set ending with an actual configuration corresponding to a given time step. We define the first virtual configuration V t 1 0 as the one in which the leader L is positioned in M(t 1 ) and all other elements are where they were in C 0 . Given a virtual configuration V t 1 n , the next virtual configuration V t 1 n+1 is defined as follows: 1. the leader L along with its first (n − 1) − th neighbors are in the same position they had in V t 1 (n−1) ; 2. every one of the n − th neighbors of L undergoes the displace- ment u ′ i (t m ); 3. all others elements are in the same position they had in V t 1 (n−1) . When the virtual step n reaches the value of the maximum La- grangian distance n = n ∗ of L from the boundary in C 0 , we de- https://doi.org/10.1016/j.mechrescom.2018.08.006 0093-6413/© 2018 Elsevier Ltd. All rights reserved.