Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59; DOI: 10.3221/IGF-ESIS.24.04

Organization of the step of numerical integration of motion equations Thus, numerical algorithm realizing the proposed model of elastic-plastic interaction of discrete element (or movable cellular automaton) i with neighbours j includes the following main stages: i. Calculation of  ij ,  ij and gear ij K at the current time step ( n +1) in elastic approximation according to Eqs. (21)-(24). Calculation of corresponding values of i xx  , i yy  , i xy  , i zz  or i zz s , i mean  , int i  and int i  with use of expressions (16)- (18), (23), (30). ii. Calculation of current values of i xx  , i yy  , i xy  , i zz  and int i  with use of expressions (31). iii. Examination of exceeding the yield surface by element i stress state controlling parameter int i  . Calculation of the coefficient M i if necessary. iv. Carrying out the procedure of stress returning to yield surface for element i with use of expressions (26)-(27),(29), i.e. calculation of ij   , ij   and   gear ij K  . Calculation of scaled values of average stress tensor components and invariants. (32)-(34) vii. Integration of Newton-Euler equations of motion of discrete elements (or movable cellular automata). The equations of motion (1) of discrete elements are numerically integrated with the use of the explicit integration scheme (for example, with use of well-known velocity Verlet algorithm) [35] modified by introducing a predictor cycles for estimation of i   at the current step. The need of the numerical scheme modification is caused by the fact that specific forces of the central interaction of discrete elements  ij at the current time step are computed using mean stress i mean  (21). At the same time these forces are used to define mean stress according to (16). To solve this problem of numerical integration the predictor cycles are used for the calculation of the elastic task. At the initial predictor cycle increment of mean stress is calculated from the previous time step:     1 final final i i i mean mean mean n n        . Value of i mean   such calculated is used in (21) to get estimated values of  ij and corresponding estimated value of   1 pred i mean n   at the current step. The obtained value is used for the calculation of     1 pred final i i i mean mean mean n n        at the next predictor cycle. The number of predictor cycles is defined by assigned tolerance of mean stress change during one cycle (normally the solution converges very fast and 2 predictor cycles are enough). M ODELING FRACTURE AND “ HEALING ” WITH PARTICLE - BASED FORMALISM ne of the main advantages of particle-based methods in mechanics is the feasibility of direct simulation of fracture (including multiple fracture) of material and bonding of fragments through changing the state of a pair of particles (“linked” pair  “unlinked” pair, Fig. 1). The criterion for pair state switching is normally the ultimate value of interaction force or the ultimate value of relative displacement [19]. The developed approach to the description of interaction of movable cellular automata (or discrete elements) in the many-particle approximation makes it possible to apply various multiparametric ”force” fracture criteria (Huber-Mises, Drucker-Prager, etc) as element-element bond fracture criteria. The model to calculate debonding of the pair (linked  unlinked transition) It’s well known that crack formation is a fundamentally brittle and extremely localized phenomenon. Fracture is connected with rupture of interatomic bonds and spatial diversity of atomic layers. That is realized under the influence of local, rather than the average stress. Thus, the used criterion of pair bond breakage between linked elements i and j must be expressed in terms of local variables of the interaction of elements, in particular, through the forces  ij and  ij acting in a pair. In this regard, the following approach to implement parametric fracture criteria as criteria of pair bond breaking ( linked  unlinked transition of the state of the pair) is suggested. In the framework of classical formalism of discrete elements pair bond breakage occurs on the surface of their interface (at the area of interaction of the pair, in other words, O v. Calculation of forces and torques of element interaction. vi. Calculation of new values of automaton volume  i and squares of areas of pair interaction S ij with use of expressions

39