PSI - Issue 25

A.Yu. Smolin et al. / Procedia Structural Integrity 25 (2020) 477–485 A.Yu. Smolin et al. / Structural Integrity Procedia 00 (2019) 000 – 000

480

n

   

F

S





ij

ij ij

(5)

ij F 

τ

S



ij

ij

.

To characterize the deformation of the automaton i in its normal interaction with the automaton j , we can use the following dimensionless parameter (normal strain)

ij d q d 

/ 2

(6)

i

 

ij

/ 2

i

In general case, each automaton of a pair represents different material, and the overlap of the pair is distributed between i th and j th automata :

(7)

h

q

q

d

d

2

2

      



 



ij

ij

ji

ij

i

ji

j

where symbol Δ denotes the increment of a parameter per time step Δ t of numerical integration of the motion equation (1). The distribution rule of strain in the pair is intimately associated with the expression for computing the interaction forces of the automata. This expression for central interaction is similar to Hooke's relations for diagonal stress tensor components:

G

(8)

P

G ) (1 2 2 (       

)

ij

ij

i

K

where K is the bulk modulus; G is the shear modulus of the material of i th automaton; and P i is the pressure of the automaton i , which may be computed using Eqs. (3) and (4) at the previous time step or by the predictor-corrector scheme. To determine a parameter characterizing shear deformation in the pair of automata i – j , we start with kinematics formula for free motion of the pair as a rigid body

(9)

ω v v   

r

j

i

ij

ij

where r ij = ( R j − R i ), v i is the translation velocity of the i th automaton centroid; and  ij is the rotational velocity of the pair as a whole (rigid body). If we multiply both sides of Eq. (9) on the left by r ij and neglect rotation about the axis connecting centers of the automata of the pair (i.e. let  ij · r ij = 0 because the rotation about the axis of the pair does not produce a shear deformation), then we get the following formula

( n v v  

)

(10)

ij

j

i

ω



ij

r

.

ij

Besides such rotation of the pair as a whole (defined by the difference in translational velocities of the automata), each automaton rotates with its own rotational velocity  i . The difference between these rotational velocities produces a shear deformation. Thus, the increment of shear deformations of the automata i and j per time step Δ t is defined by the relative tangential displacement at the contact point l ij shear divided by the distance between the automata :       t q q        n ω ω n ω ω l

shear ij

ij

ij

i

ij

ji

ij

j

ji

(11)

    γ γ



ij

ji

r

r

.

ij

ij