Issue34

R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 80-89; DOI: 10.3221/IGF-ESIS.34.08 85 ( ) ( ) ( ) ( ) ( ) i i i f f d f W d             x x x x x x x , ( ) ( ) ( ) ( ) ( ) n n i i f f W d       x x x x (6) where ( ) i   x x is the Dirac delta function that is numerically approximated by adopting a proper decreasing function ( ) i W  x x of the distance i  x x ( W is also called kernel or smoothing function), and h is the smoothing length (or support dimension). The numerical evaluation of Eq. (6 1 ) is expressed as follows: ( ) ( ) ( , ) k i k k i k k f f m W h      x x x x (7) where the index k spans all the particles over the neighbors of that placed at i x because of the compact support of the kernel function. The parameters , , k k k m  x are the position, mass and density of the generic particle k , respectively, and ( ) k f x is the function value at k x . By using a SPH approximation for the material’s stress field and inserting it in the dynamic equilibrium equation, , 0 ij j i i x b       , the following expression can be obtained: , ( ) ( ) ( , ) 0 p k ij k j k p p p i p p i k p m V V W h V x V b                  x x x   , 1, 2, 3 i  (8) where p V represents the volume of the particle p and / k k k m V   . By using the potential formulation presented above, we get: , , [ ] 2 ( ) ( ) 0, ( ) [ ] ( ) ( ) ( ) ( , ) ( ) ( , ) ( ) ' 1 2 ij k j k p p k ij k j k p k p k p e kp k p e kp mk p k i kp k p k p e kp e kp W h V V W h s s a h A E V V q V V s s                                        x x x x x x x x x x  (9) where ( ) i kp q is the i -th component of the versor connecting the particles k and p , and [ ] k p indicates the generic particle k falling within the neighbor of particle p . D YNAMIC EQUILIBRIUM EQUATIONS he dynamic governing equations of the discretized problem are given by: i d e b       Mx F F F F 0  or T  Mx F  (10) where M is the mass matrix, x  is the vector of the particles center acceleration (the rotation degrees of freedom are neglected since particles with small rotational inertia are assumed). Further, , , i d e F F F , b F are the internal force vector (see previous sections), the damping force vector, the external force vector and the vector of actions due to the collision of the particle with the elastic boundaries, respectively. The numerical integration of the motion equations can explicitly be done for each particle, once the force vector acting on it is known. Explicit time integration methods, even if conditionally stable, are usually adopted in particles simulations to avoid the use of large matrices such as the stiffness one. T