Issue 51

F. Clementi et alii, Frattura ed Integrità Strutturale, 51 (2020) 313-335; DOI: 10.3221/IGF-ESIS.51.24 320 The reaction force and the relative velocity of the contact are described by the laws of Signorini and Coulomb. In fact, the impenetrability of contact between blocks is represented by the Signorini’s condition (see Fig. 8 (b)), written as: 0 ,        0 ,       0 , n n g r g r        (4) where g is the distance between the bodies and n r is the normal component of the contact force. The same equations can be written for the velocities, considering the normal component n u , in the following way:     0 0 0      ,                0   0 ,   0 ,   0 . n n n n g t at initial time step t g t u r u r          (5) Additionally, the dry friction Coulomb’s law (see Fig. 8 (c)) permits to comprehend the tangential force between blocks and the sliding, as written in the following system: 0,  0                           ,  0 , 0            , T T n T T n T T T u r r u Sticking r r u kr k u Sliding                (6) where  is the friction coefficient. Thus, the bodies exhibit dynamics regulated by the following equation of motion:     , , q dv F q v t dt dI   M (7) where M is the mass matrix,   , , F q v t is the vector of internal and external forces of the system, dt is the Lebesgue measure on  , d v is a differential measure of velocity denoting the acceleration measure and dI is a differential measure of the impulse of contact resultant. It is important to highlight that it is not necessary to manage explicitly the contact events in the time-stepping integration scheme, as in the case of the event-driven scheme. The time subdivision is done on intervals [ti, ti+1] of length h and it is fixed, consequently it is possible to deal with a great number of discontinuities during one-time step, and the contact problem is solved over the range in terms of measures of this interval and not in a point-wise way. Thus, the Eq. (7) can be integrated on each time step, which involves to:     1 1 1 1 , ,  , i t i i i t v v F q v t dt I        M (8) with the velocity 1 i v  is the approximation of the right limit at the time 1 i t  ,     1 1 1 1 1 1 1 1 1 , ,  , , ,  ,                     i i t i i i t t free i t v v F q v t dt I v v F q v t dt                    M M M (9) where free v is the velocity of the bodies in the absence of contacts. Hence, (9) can be rewritten by means of the Delassus operator ఈఉ and the local unknows in this form:   1 ,    ,     ,   0 , i loc free v v W I Contact law I v               (10)