Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59 ; DOI: 10.3221/IGF-ESIS.24.04 30 where symbol  hereinafter indicates increment of corresponding parameter during one time step  t ,  ij is central strain of the pair i-j , variables  i(j) and  j(i) are central strains of discrete elements i and j in the pair (in the general case  i(j)  j(i) ). Tangential interaction is determined by the relative shear displacement shear ij l of the elements in the pair. The value shear ij l is calculated in incremental fashion taking into account the rotation of elements [19, 22]: tan shear ij shear g ij ij i ij j ji l V V q q t         , (5) where tan g ij V is the tangential component of the relative velocity vector ij V  for the elements i and j ( ij j i V V V      , i V  and j V  are velocity vectors of the centers of mass of the elements),  i and  j are angular velocities of the elements (in considered two-dimensional approximation they actually are signed scalars). Note that the accounting of angular velocities in (5) is required for tracking the rotation of the plane of interaction [19, 22]. As in the case of the central interaction, the contribution of the elements i and j in the shear deformation of the pair i-j can be different. By analogy with the central strains (4) the shear angles of the discrete elements i and j in the pair i-j are introduced:     shear shear ij ij ij ij ij ji j i i j l V t r q q             , (6) where  ij is shear angle for the pair i-j , variables  i(j) and  j(i) are shear angles of discrete elements i and j in the pair (in the general case  i(j)  j(i) ). Rotation of discrete elements can lead to “bending” of linked interacting pairs (Fig.3). This type of relative motion of surfaces of elements i and j is not taken into account by expression (5). Nevertheless such type of relative motion of “anchor surfaces” must be accompanied by appearance of special moment of resistance force. So, the special torque gear K  directed against pair bending (Fig.3) and having opposite signs for two elements of the pair is introduced: ij gear gear ij ji ji q K K q     (7) Value of the torque gear gear ij ij K K   is defined by bending angle gear ij  , which is calculated by analogy with shear ij  : gear ij gear ij ij i ij j ji r V q q t         (8) By analogy with the central (4) and shear (6) strains the bending angles of the discrete elements i and j in the pair i-j are introduced:     gear gear gear gear ij ij ij ij ji j i i j r V t q q           (9) where   gear i j   and   gear j i   are bending angles of discrete elements i and j in the pair (in the general case     gear gear j i i j    ). As follows from the definition of torque gear ij K , bending angles   gear i j  and   gear j i  have opposite signs (in contrast to shear angles), as reflected in the expression (9). Figure 3 : An example of bending of the pair of linked elements i and j for the simple case:  i =   j , q ij = q ji and tan g ij V = 0.