PSI - Issue 6

Yurii Meshcheryakov / Procedia Structural Integrity 6 (2017) 109–114 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

112

4

2 2     u u

 

2

s

.

(8)

(0, 0) [ ( ), ] r      d



 

2

t   

m



Heretheassumptionhasbeenmadethattime intervalof integration ismuch more greater than the duration of correlations themselves. Indeterminationofthefirstdiffusioncoefficient one should take into account for changingthe mean velocity due to fluctuative character of stress field:

(9)

'





s

' d d u          " ( ", ") ( , ) u





 

( ( ), )] r        , u



m

r



0

0

whereaTaylorexpansionhasbeenmade. In the frame of the Mode 2 of motion, thetrajectoryofmesoparticleis determined by fluctuative part of stress σ [ r ( τ ) , τ )] which defines the random way '  

s

   

.

(10)

r

u  

' d d u

( )

" ( ", ")     

m

0

0

In accordance with this equation, in the case of motion of mesoparticle in heterogeneous medium, the position of mesoparticle is determined by two items. Thefirstitemdescribes ameandisplacementofmesoparticle which equals to product of mean particle velocity and time interval between two successive velocity correlations. The second item characterizes the random deviations of mesoparticle.Then the total change of mesoparticle velocity in the second mode of motion M2equals:

1 ( ( )) t u s r d t m t            

.

(11)

0 





Since ( , ) 0 u     , averaging on time interval t  yields:

u s

'        ' d d d " ( ", ") u     t 

  

2



.

(12)

( , ) u   





 

2

t   

m

r



0 0

0

Changingtheorderofintegrationandtakingintoaccountthatexpressionunderintegraldependsonlyondifference ( τ – τ ‖), one obtains:

0  

.

(13)

2

u   

s



  

 

( , ) u d    

(0, 0)

 



2

t   

m

r



Comparison of Eq.(8) and(13) yields:

u   

2 2      u u

1 2

,

(14)

1



 

 

 

 

t

u

t







Thetimeinterval, Δ t , canbetakentoequalto thetime of relaxation of the particle velocity distribution function r  . Then the second diffusion coefficient equals:

2

D

,

(15)

F



2



R

andEq. (14) can be written in the form:

1 1 2 r 

2

D



.

(16)

F



1

u

