Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59 ; DOI: 10.3221/IGF-ESIS.24.04 42 criteria are proposed: 1. Linear dependence of ij link k on specific normal force  ij :           min max min , , 0, 0 ij ij ij link ij ij ij peak ij link k if k otherwise                          (38) where  ij is a current value of specific central force in the pair i-j ;  ij is an increment of specific central force during one time step  t ;  min is the minimum (threshold) value of normal pressure in the pair (negative specific normal force) for linking;  max (  max >  min ) is another parameter having the dimension of specific force and regulating the slope of the dependence   ij link ij k  ; ij peak  is a maximum value of specific normal force previously achieved from the moment of linking beginning. 2. Linear dependence of ij link k on plastic work of deformation W ij taking into account specific normal force  ij :     min min , , 1, , 0 ij ij link ij ij ij link ij ij ij link W k if W W W k if W W k otherwise                               (39) where W  is normalizing parameter depending on specific normal force  ij :     max min min min max min ij W W W W               (40) Here  W ij is an increment of plastic work of deformation of the pair i-j during one time step  t ; min W  is the value of plastic work to make pair totally linked at threshold normal pressure  min ; max W  is the value of plastic work to make pair totally linked at normal pressure  max (normally max min W W    ). Plastic work in (39) has to be considered in specific units (per unit of volume). Parameters  min ,  max , min W  and max W  in criteria (39)-(40) are assigned (user-defined) values, which have to be determined for each pair of materials filling elements i and j . One of the problems with the use of the criterion (39) is the calculation of  W ij . As a first approximation it can be considered as the sum of increments of plastic work of deformation of both interacting elements/automata: ij i j W W W      (41) A specific expression to calculate increment of plastic work of deformation of the element is determined by the applied model of plasticity. In the case of the above model of plasticity with von Mises criterion the value of  W i can be calculated as follows:             2 2 int int int int 1 1 int int 1 0.5 1 2 3 3 i i i i n n total elast n n i i i i i i i n n W A A G G                                (42) where total i A  and elast i A  are increments of total work of deformation and its elastic part correspondingly, n is a number of time step. In a general case criterion of pair linking has more complex form as it takes into account combined influence of central force, plastic work and other parameters of pair interaction including strain rate and/or relaxation times. Note that during time-distributed process of pair linking the condition of bond breaking in a pair can be achieved. Therefore, under certain conditions, local processes of linking and unlinking can proceed in parallel ( unlinked  linked transition).