Issue 29

A. Infuso et alii, Frattura ed Integrità Strutturale, 29 (2014) 302-312; DOI: 10.3221/IGF-ESIS.29.26 305 where C is the tangent operator stemming from the consistent linearization of the local force vector F loc with respect to the local gap vector g loc : C   F LJ  g n 0 0 0           (11) whereas the term  R /  u related to the differentiation of the rotation matrix with respect to the displacements vector is a third order tensor. This mathematical formulation has been implemented in the Finite Element Analysis Program FEAP [19]. A flow chart of the operations is shown in Tab. 1. Table 1 : Sequence of element operations. N UMERICAL I NVESTIGATION : M ONODIMENSIONAL M ODEL he first step of the study is the understanding of the nonlinear behaviour of a L-J nonlocal system of atoms for different degrees of nonlocality. Let us consider a 1D system of nodes representing atoms equally spaced by a length l 0 along a straight line. A local system would be represented by the case where only local interactions take place between each node and its nearest neighbours (Fig. 4(a)). This is just a mechanical system of nonlinear springs in series, where the removal of a link (failure of one spring) leads to the failure of the entire chain (Fig. 4(b)). (a) (b) Figure 4 : (a) System with only local links. (b) Collapse of the system due to failure of just one link. Long range interactions can be modelled by introducing additional links between a generic node and the nodes located at distances 2  l 0 , 3  l 0 ,… and so forth. Various systems with different degree of nonlocality can be generated by changing the cut-off distance for nonlocal interactions. For comparison purposes, it is useful to introduce a dimensionless number NLI (nonlocality index) synthesizing the topological properties of the system: NLI  max( l i ) l 0 (12) where l i is the length of the i-th link between two connected nodes and l 0 is the length of the local link between two adjacent nodes. Therefore, NLI=1 corresponds to a local system, whereas NLI>1 refers to systems with long range interactions. As a first example, let us consider the uniaxial tension test for a system composed of 5 nodes and NLI=2 (Fig. 5(a)). Four local links are present and three additional links due to long-range interactions are introduced. For a preliminary qualitative analysis of the system, a linearized version would consist in springs with different stiffnesses k i , depending on T LOOP until convergence of the Newton-Raphson scheme CALL to the element subroutine INPUT: X initial nodal coordinates, u nodal displacements COMPUTE: updated coordinates x=X+u rotation matrix R and gap vector g (Eq. 5) gap vector in the local frame g loc (Eq. 7) local force vector F loc and the tangent operator C (Eq. 11) OUTPUT: residual vector p and tangent stiffness matrix K (Eq. 10) END LOOP