Issue 24

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

  ij 

  ij 

ij   

pair

pair

ij 

i i A P    ji 





A P

 

ji

j j

(15  )

  ij 

  ij 

pair

pair

ij 

ij 

  







ji

ji

The equality (15  ) provides the basis for calculation of contributions of elements i and j (  i(j) and  j(i) ,  i(j) and  j(i) ) in the total pair strains  ij and  ij . Note that although the right part of the expression (14) formally confirms to notation of element interaction in conventional DE models (1) [16,19,32], their fundamental distinction consists in many-body form of central interaction of discrete elements in the proposed model. It is seen from (14)-(15) that an important problem in building many-particle interaction is definition of local value of pressure ( P i ) in the volume of discrete element. Authors propose to use an approach to calculation of pressure P i (or, what is the same – of mean stress) in the volume of the element i that is based on the computation of components of average stress tensor in the volume of the element [19, 33]. In terms of central ( ij n F ) and tangential ( ij t F ) interaction forces the component i   of average stress tensor in the volume of element i can be written as follows [19, 24, 25]:       1 1 i N i ij ij ij ij ij ij ij j i S q n n t                    , (16) where  ,  = x , y,z (XYZ is a laboratory system of coordinates),  i is a current value of the volume of element i ,   ij n   and   ij t   are projections of unit-normal and unit-tangential vectors onto the X-axis of lab coordinates. In considered two-dimensional problem statement this expression can be rewritten as follows:

N

1

i

1 j i    1 j i  i N

q F 

 

i 

ij

ij

cos cos 

ij 

cos sin 

ij 







F













ij

n

ij

t

ij

,

,

,

,



(16  )

,

1

 

 

ij 

cos cos 

ij 

ij 

cos sin 

ij 





q S









ij ij

ij

ij

,

,

,

,



where  ,  = x , y ( XY is a plane of motion),  ij,  is an angle between the line connecting mass centres of interacting elements i and j and axis  of laboratory system of coordinates (Fig.5), sign «  » implies use «+» in case of  = x and use «–» in case of  = y . Components i xz  and i yz  are identically zero. Definition of i zz  depends on constitutive equations of considered medium. Note that values of i xy  and i yx  coincide only in static equilibrium state of ensemble of discrete elements, while they can slightly differ at the stage of establishing static equilibrium. Therefore their mean value (   2 i i xy yx    ) is used in the proposed model (hereinafter it is called as i xy  ).

Figure 5 : An example of definition of angle  ij,  between line connecting mass centers of discrete elements in the pair i-j and  -axis of laboratory system of coordinates (  = X is considered here). The center of coordinate system is translated to the mass center of the element i.

33