R. Brighenti et alii, Frattura ed Integrità Strutturale, 34 (2015) 80-89; DOI: 10.3221/IGF-ESIS.34.08 86 In the present particle approach, the explicit leapfrog method (the Verlet integration scheme) is adopted [25], which allows a satisfactory numerical stability, provides time-reversibility and preservation of the symplectic form on the phase space. The word leapfrog recalls that even derivatives of position are known at on-step points, whereas odd derivatives are known at mid-step points. The particle acceleration at time step n , the velocity at time step 1/ 2 n  and the position of particle i at time step 1 n  are given by: , , , / i n T i n i m  x F  , , 1/2 , 1/2 , i n i n i n t       x x x    ,   , 1 , , 1/2 , i n i n i n i n t t         x x x x   (11) where , 1/2 , , 1 ( ) / i n i n i n t      x x x    is the mean velocity vector across the time step 1 n n   . In order to determine stable results, an upper limit of the time step size / t m K     in the DEM approach is generally obtained from the natural oscillating period of a couple of particles: in the above expression, the minimum ratio / m K among all the particles constituting the discretized solid must be taken into account, and 1   is an appropriate constant to be introduced. N UMERICAL APPLICATIONS Granular material cutting simulations his example considers the cutting of a granular material that resembles the fracture phenomenon in solids. A vertical flat blade with a constant horizontal velocity equal to 10 mm s -1 [26] drives particles that move, leading to some failure lines in the body. An initial volume of particles (Fig. 4) under the gravity action is assumed to be placed inside a box, and the vertical blade is placed at the left side position. The simulation consists in moving the blade, causing the material inside the box to redistribute. The performed dynamic analysis is aimed at determining the particles configurations at different time instants. The material of the particles (assumed with a mass density equal to 855 kgm -3 ) is supposed to have an elastic modulus equal to 6 2.76 10 E Pa   and a negligible tensile strength (or, equivalently, no cohesion). The force potential is used to describe the particles interaction by adopting an influence distance of the particles equal to infl r d  , and the box and the blade are supposed to have elastic modulus equal to 8 3.0 10 E Pa   . The particles volume is modeled through about 1162 spheres with diameter size equal to 20 mm, initially arranged in a cubic fashion. The geometry of the system is depicted in Fig. 4. The integration time step is taken equal to 0.2 ms t   and the final blade displacement is equal to 200mm, with an analysis duration equal to 2.0 s. The damping coefficient for the dynamic analyses is assumed equal to 0.20 d   , and the two coefficients of dynamic friction (between particle and particle, and between particle and planes) are supposed to be 0.10 d   . The obtained arrangement of the particles shows a behaviour similar to that of the soil cutting case analysed in Ref. [26]. Note that the blade in the present study is shorter than that in the literature test, causing a little difference in the disposition of the particles near the blade. Neglecting last aspect, the shape assumed by the particles representing the granular material in the present study is similar to the configuration presented in Ref. [26]. In Fig. 4e, f, the fracture lines arising in the material can be recognized, i.e. the cutting phenomenon can be thought as a crack appearance event inside a non-compact material. Fracture of a brittle concrete beam under a dynamic impact This example examines the fracture behaviour of a compact material. This analysis is performed with the same potential approach described in the theoretical Sections and used also for the previous numerical example. In the present numerical test, a simply supported plain concrete beam under impact load is taken into account [27]. The system is characterized by the reference size 0.3 b m  , whereas the mechanical properties of the materials are as follows: elastic modulus 10 3 10 E Pa   , Poisson’s ratio 0.18   , tensile strength 6 2.7 10 t f Pa   and mass density equal to 2300 kgm -3 for the concrete material, whereas the falling hammer is supposed to be made of steel (with elastic modulus 11 2 10 E Pa   , mass density equal to 8000 kgm -3 ). The system is modelled with about 2200 particles with cubic arrangement and radius equal to 35mm each. The impact velocity of the steel mass is assumed to be equal to T