Issue34

R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 80-89; DOI: 10.3221/IGF-ESIS.34.08 82 of continuous materials. This remark suggests to adopt a unified potential-based formulation for the assessment of the interparticle actions, by properly choosing the distance of influence for the transmission of forces. As is typically done in the atomic description of solids where the material can be characterized by a potential energy functional ( )  x , it can be assumed the existence of a similar potential function at the meso or macro-scale. The equilibrium equations, referred to the discrete element i , correspond to the configuration of minimum energy of the system: ( ) 0 tot tot i i i E        x P x x with i i tot tot E xP x x    )( )( (1) where ( ) tot  x and i P are the strain energy and the force applied to the particle i , respectively. By writing the power series expansion of the potential energy function, starting from its equilibrium state identified by the position vector 0 x , the stationary condition of the above functional identifies the equilibrium configuration as follows: 0 0 0 2 2 0 0 0 2 2 ( ) ( ) ( ) tot tot tot E E                 x x x x x x x P K x x P x x x (2) where 0 / tot E    x x 0 at the equilibrium, and the stiffness matrix of the system is indicated with 2 2 / / ij tot i j tot i j E          K x x x x . At the atomic scale, various potential functions have been proposed in order to represent the mechanical behaviour of materials, such as the Lennard-Jones (LJ) potential and the Morse potential [23]. Figure 1 : Scheme of pair-wise interparticle forces. The simplest mechanical model describing the stress-strain behaviour of a continuum linear elastic solid is the generalized Hooke’s law that assumes the force between two infinitely closed material points to be proportional, through proper coefficients, to the strain value estimated in such a small region (Fig. 1). The region of material – representative of a small region (of a continuum solid) lying between two particles i, j – is subjected to the deformation (evaluated with respect to the equilibrium distance e r ) given by: 2 2 1 1 2 2 e e e e e e e e r r r r s s s s r r s s                      (3)