Issue 29

A. Infuso et alii, Frattura ed Integrità Strutturale, 29 (2014) 302-312; DOI: 10.3221/IGF-ESIS.29.26 304 The gap vector between the nodes in the local reference system is obtained by the product between the rotation matrix R and the gap vector g : g loc  Rg (7) The constitutive law used here stems from a generalization of the Lennard-Jones (L-J) potentials: P LJ  4  r 0 g n       r 0 g n              (8) where  is the depth of the potential well, r 0 is the distance at which the potential between two neighbouring atoms is equal to zero,  ,  are the parameters defining the shape of the L-J potential. Keeping the exponents  ,  as free parameters was suggested in [14-16] to bypass the rapid decay of the law after the equilibrium point and being able compute non-bonded interactions between nodes also at longer distances, or for coarse-grained representations. The corresponding displacement-force relation can be obtained by differentiating the L-J potential with respect to the normal opening displacement g n : F LJ ( g n )   P LJ ( g n )   d d ( g n ) P LJ ( g n )   4   r 0  g n  1       r 0  g n  1           (9) In the proposed formula, whose graphical trend is depicted in Fig. 2(a), the first term in brackets represents the repulsive force contribution between atoms (Pauli repulsion), while the second one describe the long-range attractive force (van der Waals force). Different force-displacement curves are shown in Fig. 2(b) by varying the parameters of the model. In the numerical simulations discussed in the next sections, the selected parameters are   13 and   2 . (a) (b) Figure 2 : (a) Lennard-Jones potential law: attractive and repulsive branches. (b) Different force-displacement forces by varying and  : the red curve refers to L-J potential in Fig. 2(a). The blue curve is the modified law used in this study. The force in Eq. (9) directed along the normal vector n is collected in the local force vector, F loc  (F LJ , 0) T . Starting from the Principle of Virtual Work where each element contribution is related to the scalar product between the force vector and the gap vector as for a classical interface element [17,18], its virtual variation and fully consistent linearization have to be performed in order to derive the residual vector p and the stiffness matrix K for the application of the Newton- Raphson iterative procedure. After a lengthy calculation, which is here omitted for the sake of brevity, we obtain:                               CRL u RLu Lu u RCRL Lu u RC u RLu K CRL RL K K KK F Lu u R RL p T TT T T T TT geom T T mat geom mat loc T (10)