Issue 24

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

      

2 G K G G K G K G G K     2 2 2 i xx i i i i i yy i



i    

i 



i

xx

mean

2



i    

i 



i

yy

mean

2

(31a)

i

i

i

i xy     

i xy

i  

2        i i  xz yz G

0

i zz      

 

Plane strain

0

(31b)

G K 

2

i

i 



Plane stress

i

i

 

zz

mean

G K

2

i

i

Here  are stress increments calculated after solving the elastic task at the current time step. Calculated values i   are used to define current volume of the discrete element i :      0 1 1 1 i i i i i xx yy zz          i  

(32)

0 i  is “initial” volume of the element (in undeformed state).

where

Determination of current value of S ij

is more complicated problem due to possible significant element shape distortion

under large loads. The following approximation to estimate the value of S ij is suggested. It is based on definition of local values of strain tensor components at the area of interaction (contact area) of considered pair of discrete elements i and j (hereinafter denote this tensor as ij   ) by linear interpolation of corresponding components of average strain tensors ( i   and j   ) to the central point of this area:

i 

j





q

q





ji

ij

ij





(33)



r

ij

ij

  tensor are transformed from the laboratory system of coordinates to the instantaneous local

Components of

ij x x

coordinate system X  Y  of the considered pair (Fig.7): ij 

ij

ij zz  thus defined in local







. Components

and

 

   



coordinate system are used for calculation of the current value of square of the area of pair interaction:     0 1 1 ij ij ij ij x x zz S S       

(34)

Here 0 ij S is the initial value of square corresponding to the pair of undeformed elements i and j .

Figure 7 : Instantaneous coordinate system X  Y  associated with the current spatial orientation of the pair i-j .

38