Issue34

R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 80-89; DOI: 10.3221/IGF-ESIS.34.08 84 3D space depends on the influence radius infl r and on the particles distribution (the bar connecting two particles exists only if the particles are within their influence distance), we can assume infl 0, ( ) ij ij A a r A   , where the reference area is 2 0, ( ) / 8 ij i j A d d     , i.e. the cross section of the cylindrical region of material placed between two particles with diameter equal to the mean value of the particles diameters. The function infl ( ) a r is evaluated by best fitting procedure once the particles arrangement in the space and their influence distance are assumed. Dimensionless effective distance between particles surface, s / d i Interparticles force, F contact stiffness, K Force potential,  F ( s )  ( s ) (a) K ( s ) s e / d i = 1 1.0 0.0 Dimensionless effective distance between particles surface, s / d i Interparticles force, F contact stiffness, K Force potential,  F ( s )  ( s ) (b) K ( s ) s e / d i = 1 1.0 0.0 Figure 3 : Force potential and corresponding forces and stiffness according to the linear elastic behaviour for an equilibrium distance / 1 e i s d  , (a) without and (b) with the no-penetration condition. The above potential formulation can also be adopted for granular materials, where the internal forces act only when the particles are in reciprocal contact and their motion make them approach. In these cases, an influence distance can be adopted ( infl / ( / 2) i r d lower than or equal to about 2), i.e. the interaction of particles occurs only when they are in contact under compression condition, while no forces exist when ' 0 s  . Further, a tangential force can be assumed to exist between the colliding particles when ' 0 s  (Ref. [22]):   , , ( ) sign( ) min , ( ) t rel t rel md T w F w        u u (5) where  is the viscosity coefficient, md  is the dynamic friction coefficient of the materials, , t rel u is the relative tangential velocity between the two particles in contact. Note that Eq. (5) provides a tangential force as the minimum value between the dynamic friction and the viscosity action. The present authors have verified that the function infl ( ) a r , for a given particles arrangement, is almost independent of the particles arrangement in the space and depends on infl r only [24]. On the other hand, the Poisson’s coefficient tends to 1/3 as the influence radius increases, irrespective of the particles layout used to represent the domain of the problem of interest. When compact materials have to be simulated, a suitable choice of the influence radius value is infl / ( / 2) 4 i r d  (corresponding to the correction function value equal to about infl ( ) 0.4 a r  for both cubic and tetrahedral arrangements), i.e. the interaction forces exist for neighbor particles even if they are not in direct contact. Granular materials can be simulated by assuming infl / ( / 2) 2 i r d  and infl ( ) 1.0 a r  , i.e. the interaction of particles exists only when they are in reciprocal compressive contact, while no forces are present when ' 0 s  . It is worth noting that the particle-boundary simulation can be tackled in the same way as the particle-particle interaction. SPH property of the potential force approach Smoothing Particle Hydrodynamic (SPH) approximation of a function and of its n -th derivative at a point i x in the region  can be written as follows [13]: