Issue 24

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

interacting pairs of discrete elements (15’) and the need to distribute relative displacement of elements in the pair the expressions for specific interaction forces can be written as follows:

   

  

2            pre pre i G

cur

i 

ij 

ij 

ij 

ij 



G

2

1

 

 

i

mean

i j

K

i

                                2 2 1 j cur pre pre ji ji ji ji j j i j j i ij j i i j G G K d d r       

(21)

j



mean

2

2

and

                        2 2 cur pre pre cur pre pre ij  ij  ij  ij  i ji ji ji ji j i j ij V t j i G G d d       



 

j i

(22)

2

2

 

 

shear

i j

j i

Here, relations for calculating the central and tangential interaction forces are written in incremental fashion (in hypoelastic form); “ cur ” and “ pre ” upper indexes mark values of specific reaction forces at the current step of integrating the equations of motion of discrete elements (or cellular automata); mean stress increments i mean   and j mean   are taken from previous time step or determined with use of predictor-corrector modification of a numerical scheme. Equations (21)-(22) are first solved for strain increments   i j   ,   j i   ,   i j   and   j i   . Found values of strain increments are then substituted in (21)-(22) to calculate current values of specific forces cur ij  and cur ij  . The above relations have a general (three-dimensional) form. In considered two-dimensional problem statement the approximations of plane stress (  zz =0) or plane strain (  zz =0) state are used. In the model under consideration, this means that the average stress tensor components i xz  and i yz  are equal to zero. A component i zz  is calculated as follows:         , 0 0, 2 2 i i i i zz i xx yy zz i i i zz zz i i i i mean Plane strain G K G K Plane stress                     (23) is a Poisson ratio. Resistance of the pair i-j to bending (Fig.3) is not included in the expressions (21)-(22). So, the form of the dependence   gear gear ij ij K  has to be assigned extra. In the present model this is done by analogy with (22): where  i



gear V t  

G

ij

  gear                     gear j i i i j i j ij j ij gear gear K q S G ij ji ij ij i G G q q G q K q   

ji

(24)

 

i j gear



ji

Testing of the proposed model of element interaction by the example of elastic wave propagation through ensemble of linked elements showed that use of shear modulus G as an elastic modulus of resistance to bending in (24) provides the best agreement of simulation results with analytical solutions and results of simulation by finite difference method. Relations (21)-(24) make it possible to simulate fragments of a locally isotropic linear elastic medium by an ensemble of linked discrete elements (or movable cellular automata). Elastic-plastic medium An important advantage of the proposed approach to building many-body interaction of discrete elements is a capability to realize various models of elasticity and plasticity within the framework of DEM/MCA. In particular, a model of plastic flow (incremental plasticity) with von Mises criterion of plasticity was implemented to simulate deformation of locally isotropic elastic-plastic medium.

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