Issue 24

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

Note that in the “conventional” models of the interaction of discrete elements do not take into account “bending” of the pair of discrete elements. This is perfectly valid in describing the contact interaction of unlinked elements. However, when considering linked pairs, accounting of this effect may be crucial. It follows from (5), (6) and (9) that strains of elements i and j in the pair i-j differ from each other. The rule of strain distribution in the pair is inseparably linked with the expression for element interaction forces and will be discussed in the following Section. By analogy with strains the forces of central ( ij n F ) and tangential ( ij F  ) interaction of discrete elements i and j will be considered in reduced (specific) units: ij n ij ij ij t ij ij F S F S         (10) discrete elements. Thus, in the case of granular material the initial value of S ij shape of the elements i and j and by their mutual orientation. For example, 0 0 ij S  for the elements having a disk (2D case) or spherical (3D case) shape. With increase of compression the value of contact area grows up as a function of h ij :   ij ij S S h  , 0 ij h  . This kind of dependence is determined by the geometrical characteristics of both elements and their mutual orientation. At the same time, under the simulation of consolidated material the parameter 0 ij S is obviously finite and determined by the ratio of sizes of the elements i and j and by features of the local packing of discrete elements in the neighborhood of the pair i-j . As a rule, the 0 ij S is defined to minimize the volume of voids in simulated solid (note that in bonded-particle model [19] this is realized through introduction of the special interfacial zone between two bonded ( linked ) elements and by assigning its geometrical characteristics). Peculiarities of the procedure of assigning the initial area of plane of element interaction lead to the fact that in most models of consolidated solids a simplified form of discrete elements is used (disks or spheres). In two-dimensional approximation discrete elements are considered as disks with diameters d i in the plane of motion and the same value of height h for all of them. Note that in the case of a regular packing of elements with the same size d i = d the value 0 ij S is determined by type of packing. For example, for a close packing 0 3 ij S dh  , for a square package 0 ij S dh  (Fig.4). in the undeformed state ( 0 ij S ) is defined by where  ij and  ij are normal and shear stresses (specific forces) correspondingly, S ij is an area of the plane of interaction of elements (contact area). Note that the definition of S ij is different for the description of consolidated and granular materials by ensemble of

(a) (b) Figure 4 : Examples of regular packing of elements in 2D problem statement: close packing (a) and square packing (b) . Change of S ij during deformation of the pair of discrete elements i and j is determined by approximations of applied model of element interaction. In particular, at small values of pair strain 2 10 ij    (typical strain range for brittle materials) the value of S ij can rely constant ( 0 ij ij S S  ).However, at larger pair strains the value of S ij has to be considered as a function of the deformation process in the pair i - j .

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