Issue 24

S. Psakhie et alii, Frattura ed Integrità Strutturale, 24 (2013) 26-59; DOI: 10.3221/IGF-ESIS.24.04

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 calculated in incremental fashion taking into account the rotation of elements [19, 22]:

of the elements in the pair. The value shear ij l is

shear ij



l

tan V V q      shear g



,

(5)

q

ij

ij

i ij

j ji



t

where is the tangential component of the relative velocity vector ij V  V  are velocity vectors of the centers of mass of the elements),  i 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: for the elements i and j ( ij i V V V      j , i V  and j and  j tan g ij V

shear           shear V t l r q  

,

(6)

q

 

 

ij

ij

ij ij

ij

ji

i j

j i

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) ). 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:  j(i)

K 

K 

q

ij

gear

gear

 

(7)

ij

ji

q

ji

gear ij K K   ij

gear

gear ij  , which is calculated by analogy with shear ij  :

Value of the torque

is defined by bending angle

gear





ij

gear V q     i ij ij



r

q

(8)

ij

j ji



t

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 V t q  

  gear

  gear j i

r

q

(9)

ij

ij

ij

ij

ji

i j

  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 

  gear





where

and

i j

  gear

  gear j i

gear ij K , bending angles



 have opposite signs (in contrast to shear

follows from the definition of torque angles), as reflected in the expression (9).

and

i j

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.

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