Issue 24

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

criteria are proposed: 1. Linear dependence of ij link

k on specific normal force  ij :





 

ij 



  ij 

    ij       0, ij  



ij   k link

ij





 

if

,

,

(38)

peak

min







max

min

   

ij link

k

otherwise

0

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 >  min ) is another parameter having the dimension of specific force and regulating the slope of the  is a maximum value of specific normal force previously achieved from the moment of linking  ;

linking;  max dependence

(  max

  ij link ij

ij peak

k

beginning. 2. Linear dependence of ij link

taking into account specific normal force  ij :

k on plastic work of deformation W ij



W



    ij ij  

ij

ij link        k

,     

if

W W

,



ij

min

W



ij link ij link

,     

k k

if

W W

1,

(39)

     



ij

min

otherwise

0

is normalizing parameter depending on specific normal force  ij :     max min min min max min ij W W W W              

where 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 (per unit of volume). Parameters  min

max W W  

(normally

min ). Plastic work in (39) has to be considered in specific units  in criteria (39)-(40) are assigned (user-defined) values, which 

,  max ,

W

W

and



min

max

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 considered as the sum of increments of plastic work of deformation of both interacting elements/automata: (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). . As a first approximation it can be ij j W W W      i

42