Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59 ; DOI: 10.3221/IGF-ESIS.24.04 38 2 2 2 2 2 2 2 0 i i i xx i i xx mean i i i i yy i i i i yy mean i i i i xy i xy i i i xz yz G K G G K G K G G K G                                         (31a) 0 2 2 i zz i i i i zz mean i i Plane strain G K Plane stress G K              (31b) Here i    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          (32) where 0 i  is “initial” volume of the element (in undeformed state). 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 ji ij ij ij q q r         (33) Components of ij   tensor are transformed from the laboratory system of coordinates to the instantaneous local coordinate system X  Y  of the considered pair (Fig.7): ij ij         . Components ij x x    and ij zz  thus defined in local 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 .