Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59 ; DOI: 10.3221/IGF-ESIS.24.04 58 Here upper stroke marks scaled values of stresses. Note that original expressions in [34] bind scaling of diagonal stresses to correction (adjustment) of the values of plastic increments of strain tensor and volume change during the time step:   , pl f          . Expressions (C3) are obtained by simple mathematical transformation of original relationships into the notation in terms of stresses:   , , , pl zz zz zz f s s          . 5. The total increment of Z-component of strain tensor (  zz ) is calculated:     2 zz zz mean mean el pl pl n n zz zz zz zz s s G K                    (C4) where el zz   is elastic component of strain increment, mean   is mean stress value after stress scaling procedure (note that in contrast to 3D problem the procedure of stress correction in plane stress approximation is not accompanied by the persistence of mean stress because its value after the solution of elastic problem is overestimated [34]), index “ n ” indicates the value of corresponding variable referred to the end of the previous time step n . Implementation of this procedure within the framework of discrete element concept is concerned with scaling of element response forces  ij and  ij to satisfy the expressions (C3) for average stress tensor components in the volume of the element i . By the analogy with elastic problem the expressions for correction of  ij and  ij are obtained by direct reformulation of relationships for   and   in (C3):       , 1 2 i zz i i i pl i ij ij zz i zz i zz i ij ij i s G s M s M b M                      (C5) where , i pl zz   is irreversible (plastic) increment of Z-component of average strain tensor in the volume of the element i . It is easy to demonstrate that scaling of element/automaton response forces  ij and  ij in accordance with (C5) rigorously satisfies the expressions (C3) for i xx  , i yy  and i xy  . Note that in the case of multiple iterations of the procedure of stress/force scaling the relationships used have more cumbersome form [34], however their matter is the same.