Issue 29

A. Infuso et alii, Frattura ed Integrità Strutturale, 29 (2014) 302-312; DOI: 10.3221/IGF-ESIS.29.26 306 the distance between the connected nodes. Due to the symmetry of the problem, only half model can be analyzed (Fig. 5(b)). Figure 5 : (a) Complete 5 nodes NLI=2 system. (b) Simplified configuration due to the symmetry. For a given set of stiffnesses, it is possible to compute analytically the relation between the displacement  of the whole system and the displacement  i of each single spring, satisfying the equilibrium and the congruence:  1  k 2  2  k 4 k 1  k 2  2  k 4    2  k 1 k 1  k 2  2  k 4    3            (13) The relations between the spring forces and the total imposed displacement  are: F 1  k 1   1  k 1  k 2  2  k 4 k 1  k 2  2  k 4   F 2  k 2   2  k 2  k 1 k 1  k 2  2  k 4   F 3  k 3   F 4  k 4  2   2  k 4  2  k 1 k 1  k 2  2  k 4              (14) These analytical results provide some qualitative information on the solution of the fully nonlinear system. In general, comparing  1 and  2 in Eq.(13), we expect  1 >  2 , which is a main difference with respect to a purely local system where all the local springs experience the same elongation. Moreover, if a spring is removed, then the system has still an equilibrium configuration, whereas a purely local system would fail. The solution of the fully nonlinear problem is discussed in the next subsections. Monodimensional model without defects: the effect of nonlocality and system size Let us consider the discrete system in Fig. 6 with NLI=1, 2 or 3. In the local system characterized by springs in series, the forces and the elongations are the same for all the springs. Each spring presents an increasing branch until the achievement of the maximum force and then a softening, according to the L-J constitutive equation (9). In case of NLI=2, the situation is much more complex. Due to the symmetry, it is sufficient to comment on the behaviour of the springs 1, 2, 5 and 6. The forces in such springs vs. the total imposed displacement  are shown in Fig. 7(a). The positions of the nodes of each spring vs.  are illustrated in Fig. 7(b). From these diagrams we note that springs (a) (b)