Issue34

R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 80-89; DOI: 10.3221/IGF-ESIS.34.08 83 where 0 0 , i j e i j r r     x x x x are the distances between the particle centers in a generic state and at the equilibrium state, respectively, whereas , e s s are the corresponding effective distances counterparts (Fig. 2) defined as 2 ( ) / 2 2 2 i j s r d d r r          , 2 2 e e s r r     . In order to account for the no-penetration condition, the maximum co-penetration depth 1 2 r         is expressed as a fraction  of the averaged radius r of the particles. Eq. (3) is written by taking into account the large displacements effect, whereas the maximum co-penetration distance enables to avoid the complete penetration condition between particles. Figure 2 : (a) A couple of particle at the equilibrium distance. (b) Maximum co-penetration corresponding to 0 ( ) F r r    . (c) Generic particles relative position with geometrical parameters. By assuming the linear stress-strain relationship, the following expression for the force acting along the line joining the particle centers can be written: 2 infl infl 1 ' if 2 2 2 ( ) 0 if 2 2 i j e e ij e e i j d d s s s s A E s r s s F s d d s r                                      (3) where ij A and 0 '( ) ( ) E s c s E   are the cross-section and the elastic modulus of a bar element that represents the material connecting the particles , i j . The Young modulus '( ) E s is assumed to depend on s in order to take into account the no-penetration condition: it increases when 0 s  (i.e. ( ) c s  if 0 s  ), whereas ( ) 1 c s  when s increases. The assumption of the no-penetration condition ( 0 s  or 0 2 w    ) entails an unlimited compressive strength of the particles when they are in contact, up to the limit 0 s  (i.e. 2 w   or 0 r r  ), w being the particles overlapping and 0 r being the minimum allowed distance between particles. The force-potential and the corresponding stiffness, according to Eq. (3), can be written as follows: 3 2 2 ( ) 1 1 ( ) ' 2 6 e ij e e e s s s s A E s s s s                     , 2 2 2 ( ) ( ) 1 ( ) ' e ij e e s s d s K s A E ds s s                       (4) In Eq. (3) the influence radius infl r indicates the maximum interacting distance between two particles, i.e. it is the maximum distance beyond which two particles are not interacting. This assumption is suitable to represent long distance forces, such as in continuum materials, since the internal actions between the material points act also if such points are not in direct reciprocal contact (non-local interaction). In Fig. 3a the distance dependence of the potential, of the force and of the stiffness are displayed for the linear elastic particles interaction related to the representative equilibrium distance / 1 e i s d  , while the same functions are reported in Fig. 3b for the case of no-penetration constraint. In order to correctly represent the elastic behaviour of the material described through bar elements connecting the particles, a suitable choice of the cross section area ij A of such elements must be done. Since the bar arrangement in the