Issue 29

A. Infuso et alii, Frattura ed Integrità Strutturale, 29 (2014) 302-312; DOI: 10.3221/IGF-ESIS.29.26 303 finite element analysis programme FEAP. The tensile and compressive responses of 1D and 2D discrete systems of molecules with or without flaws are numerically simulated, by varying the range of nonlocal interactions defined by a novel nonlocality index NLI inspired by graph and network theories. For each system, the distribution of the internal forces at different load levels is analyzed in relation to the topological properties of the underlying network, in order to understand the force redistribution mechanisms in the presence of defects. F INITE E LEMENT F ORMULATION onsidering a discrete system composed of atoms or molecules subjected by nonlinear interactions governed by generalized Lennard-Jones potentials, graph theory can be efficiently invoked for their characterization. Atoms can be topologically represented by nodes and the nonlinear interactions between them can be modelled by a set of nonlinear springs. In this framework, each link can be implemented as a finite element defined by two nodes whose coordinates in the initial global undeformed reference system are: X  (X 1 ,Y 1 , X 2 ,Y 2 ) T (1) Each i -th node ( i =1,2) has two translational degrees of freedoms in case of 2D problems, u i and v i , collected in the element displacement vector u : u  (u 1 ,v 1 ,u 2 ,v 2 ) T (2) The nodal coordinates vector in the updated configuration is given by: x  X  u (3) A local reference system is now introduced, since the constitutive relation is usually defined in this frame. Hence, the normal vector n pointing from node 1' to node 2' and the vector t perpendicular to the link are introduced, see Fig. 1. The origin of the local reference system is in the node 1'. Such a frame is rotated by an angle  with respect to the axis of the global reference system. The following rotation matrix R can be introduced: R  cos  sin   sin  cos        (4) Figure 1 : Reference coordinate systems for the large displacements analysis. The gap vector g collecting the relative opening and sliding displacements between the nodes 1' and 2' is given by the product between a matrix operator L and the position vector x containing the coordinates of the element nodes in the updated configuration: g  Lx (5) where the matrix L takes the form: L   1 0 1 0 0  1 0 1       (6) C