Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59; DOI: 10.3221/IGF-ESIS.24.04 55 radius-vector to corresponding axis of laboratory system of coordinates. The volume integral in (A5) can be rewritten as a surface integral by applying the Gauss divergence theorem: 1 1 1 i i i i i i i S r e e d x e n d x n dS r                         , (A6) where S i is the surface of element i ; n  is the unit outward normal to infinitesimal surface area dS . The element i is loaded by forces acting at discrete interaction surfaces S ij (10). So, its total surface S i can be represented as a set of S ij , and surface integral can be replaced by the following sum: , , 1 1 1 cos cos i i N i ij ij ij ij j i i S e x n dS S q               , (A7) where  ij,  is defined as shown in the Fig.5; N i is total number of interaction surfaces ( 1 i N ij i j S S    ). In accordance with (A7) the sum in the second contribution in the right part of (A2) is: 2 , 1 1 cos 1 i N i i ij ij ij x xx xx j i S q e e        . (A8) So, the expression (A2) can be rewritten in the form: 2 2 1 i i i i i xx i xx mean xx i G G K               (A9) which is Hooke's law for average stress and strain tensor components i xx  and i xx  . Corresponding expressions for other components of average stress tensor can be derived in analogous fashion:     2 , , , 1 1 2 , , , 1 1 2 , 1 1 sin cos sin 1 2 sin cos sin 2 1 2 1 sin 2 1 i i i i i N N i yy ij ij ij ij x ij ij ij ij x ij x j j i N N i ij ij ij x ij ij ij x ij x i j i j j j i N i i i i mean ij ij ij x i yy j i i S q S q G S q S q G G S q G K K                                                         i i mean yy i          (A10)     2 , , , 1 1 2 , , , 1 1 , 1 cos sin cos 1 2 cos sin cos 2 1 1 cos sin i i i i N N i xy ij ij ij ij x ij x ij ij ij ij x j j i N N i ij ij ij x ij x ij ij ij x i j i j j j i i i mean ij ij ij x ij i i S q S q G S q S q G S q K                                                                    , 1 2 2 1 2 i N x j i i i i i i i xy xy mean i xy xy i G G e G K                   (A11) Relations (A9)-(A11) are derived on the basis of assumption that total surface S i of the element i is “occupied” by interacting neighbours j . However, if a part of surface S i of automaton/element i is free (i.e. neighbouring elements j “occupy” only part of S i ), corresponding unoccupied surfaces S ik can be formally considered as surfaces of interaction with virtual neighbours having zero stiffness. In that case specific forces at unoccupied surfaces S ik has to be assigned as zero:  ij =  ij =0 (this doesn’t hold true for corresponding pair strains:  i ( j )  0 and  i ( j )  0). Therefore equalities (A9)-(A11) are correct for discrete elements with partially free surface as well. In accordance with above mentioned the number of real