Issue 29

A. Infuso et alii, Frattura ed Integrità Strutturale, 29 (2014) 302-312; DOI: 10.3221/IGF-ESIS.29.26 309 (a) (b) Figure 10 : Global force-displacement response. (a) The effect of NLI ( h  4  l 0 ). (b) The effect of the system size (NLI=2). N UMERICAL I NVESTIGATION : B IDIMENSIONAL M ODEL Computational complexity s is the monodimensional case, a bidimensional system can be modelled as a collection of nodes and links, thus forming a grid in the plane. An hexagonal disposition of nodes is adopted to simulate materials like graphene. Also in this case the NLI can be defined as the maximum cut-off radius for long range interactions and the length of the link for pure local connections. The numerical construction of the finite element meshes for numerical simulations requires the definition of the list of nodes and the element connectivity matrix. For the latter, a pre-processor written in MATLAB ® has been developed where, centering in the i-th node of the grid, all the other nodes within the cut-off radius defined by the product NLI  l 0 are identified and spring elements are inserted. Clearly, by increasing NLI and the size of the system, the computational complexity drastically increases. As an illustrative example, let us consider square grids with l 0 =3.5Å and with lateral size 14x14 Å, 31x31 Å, and 70x70 Å, respectively. For each grid, the NLI can range from 1 up to a maximum value NLI max , given by the longest distance between two nodes belonging to the system. For the three system sizes considered, we have NLI max  6,12 and 29. The maximum number of links in case of NLI max is equal to 600 for the smallest system, 4,851 for the intermediate system, and 126,756 for the largest one. In dimensionless form, Fig. 11, the ratio between the number of links and the maximum value has a nonlinear dependency with the ratio between NLI and NLI max . The deviation from linearity observed for all the cases when NLI is approaching NLI max is due to the finite size of the system. Figure 11 : Number of links vs. NLI in dimensionless form, for 3 systems with different sizes. A