Molecular dynamics for the finite deformation of random elastic grids

Abstract

Structural foams are widely used as energy absorbing materials for impact protection. The microstructure of these materials is often modeled as a periodic lattice of elastic beams. In previous studies [High strain compression of random open cell foams, MMC2001, San Diego, 2001; Int. J. Solids Struct. 39 (2001) 3599; Détermination de la charge de flambement dans un matériau cellulaire périodique, XVme Congrès Français de Mécanique, Nancy, 2001; Scale effects in high strain compression of periodic open cell foams, ICTAM, 2000], a tetrakaidecahedral foam unit cell geometry was considered. The corresponding macroscopic buckling yield surface, defined as the first instability of the periodic solution, for proportional loading paths was numerically computed using finite element. In addition, for uniaxial compression, the post-buckling response has been computed and exhibits the well-known plateau regime of foam materials. In reality however, imperfections always arise during the foaming process. This leads to a random distribution of the constitutive material at cell edges, which modifies the material response and especially the plateau stress. If the microstructure were modeled as a random network of elastic beams [Eur. J. Mech. A/Solids 11 (1992) 585], the concept of buckling yield surface would not apply, and the stress level of the plateau could not be determined in this way. Some authors have carried out time consuming incremental finite element simulations on these random networks, in order to get the plateau region behavior. In our study, we have applied the molecular dynamics method to this problem. In order to illustrate our approach, we have considered a square network of elastic beams defined by a cross pattern, in which the material description has been randomly defined. The boundary conditions applied on the cell are strain-controlled. The elastic energy of each beam can be expressed in terms of the displacements of its ends. Therefore, total energy can be expressed in terms of the displacements of the lattice nodes U . The problem then is to identify the minimum of the total potential energy of the network under prescribed boundary conditions. The idea herein is to compute the solution as follows: each node is ascribed a virtual mass. Starting from arbitrary nodal displacements, with zero velocity, molecular dynamics are applied to compute system oscillations during a prescribed period of time T. We then determined the moment when the potential is minimized and we iterated the algorithm with nodal displacements corresponding to this minimum at zero velocity. We will begin by showing that this method is able to capture both the first instability of the perfectly periodic lattice (obtained analytically) and the post-buckling response. We will then study the random case, in order to demonstrate the capabilities of this methodology.