Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59; DOI: 10.3221/IGF-ESIS.24.04 37 forces ( ij ji      and ij ji     ) in the pair i-j . In view of the need for implementation of Newton’s third law, the current values of element interaction forces in (10) are calculated on the basis of the following proportion: ij ji ji ij ij ij ij ji ji ij ij ij q q r q q r                    (28) In considered two-dimensional problem statement scaling of specific forces  ij and  ij and stress i zz  has peculiarities for approximations of plane strain and plane stress state. In the first case these variables are scaled using the expressions (26) for i zz  and (27) for specific forces. In the second case the special iterative procedure proposed by Wilkins [34] is adopted to particle-based approach (see Appendix C). By analogy with the case of elastic problem, law of scaling of resistance of the pair i-j to bending has to be defined extra. In the present model this was done by analogy with scaling of shear resistance force  ij :           2 gear gear ij ij i gear gear ji ji j gear gear ij ji ji ij ji gear ij ij K K M K K M K q K q q K r                   (29) Rheological properties of material of discrete element i are defined through assigning constitutive relation   int int i i     (when applied to movable cellular automaton, it is called as “mechanical response function of automaton/element” [22- 25]). Current value of int i  could be calculated incrementally using known values of int i  after solution of elastic problem at the considered time step ( n +1) and at the end the previous time step n :         int int 1 int int 1 3 final i i i i n n n n i G          (30) where   int 1 i n   is stress intensity value, which results from solution of elastic problem at the current time step ( n +1);   int final i n  is stress intensity value at the end of the previous time step n (after realization of stress return procedure if necessary). Note, that following the idea of Wilkins’ algorithm the value of   int 1 i n   remains unchanged throughout the stress return procedure (this is applied both to central and shear strains of discrete elements i and j in the pair i - j ). Calculation of current values of element volume and square of area of interaction of the pair In addition to forces  ij , and  ij , acting on the surface of discrete element i , important constituents of the expression (16) for the components of average stress tensor are element volume  i and squares of areas of interaction of the element with neighbours S ij . The current values of these variables can be found using the average strain tensor components i   in the volume of the discrete element i . The values of components i   can be found directly in terms of pair strains  i(j) and  i(j) of element (see Appendix A) or through the components of the average stress tensor i   . In the last case, the specific relations between i   with i   are determined by considered rheological model of the medium. In the framework of above described two-dimensional model of elastic-plastic interaction of discrete elements with von Mises criterion of plasticity these relations should be presented in hypoelastic form: