Issue 33

C. Gao et alii, Frattura ed Integrità Strutturale, 33 (2015) 471-484; DOI: 10.3221/IGF-ESIS.33.52

experimental observations like Figure 2. In addition, the standard deviations of the associated normal distribution of the equivalent length and diameter will be eliminated in their average values of lognormal distribution under the presupposed condition in Eq. (19) for the lognormal distribution. Probability density function p (x) Logorithmic normal distribution Figure 4 : Illustration of the actual upper and lower boundaries of the equivalent length and diameter of CNT clusters with the lognormal distribution. Since the equivalent length and diameter of CNTs are closely related with the volume fraction of CNTs, l  and D  was assumed to change with the volume fraction. The depending relationship of them on the volume fraction are still unknown, however, they can be evaluated by using the polynomial function interpolation method to numerically fit the real nonlinear curves. Based on the experimental stress-strain curves at four different volume fractions [9], the cubic spline function was used to match these data points, then l  can be expressed as: Radom variable x ( l c or D c ) (nm) x min x max

3    2 1 a v 1 b v f f

 

h v l

1 c v l ) f

(22)

( ) (

l

f

D  with volume fraction ( 0

l  and the change of

D  is also faster than

l  ,

1 f    ) is faster than

Since the change of

D  and

D  can be expressed by a quadric polynomial interpolation function:     4 3 2 2 2 2 D f f f f g D a b c D         

then

(23)

Finally, by substituting Eqs. (20-23) into Eq. (15) and then into Eq. (1) together with Eq. (10), we can get the modified micromechanical model of CNT reinforced metal matrix composites to describe their thermo-viscoplastic flow behaviors:

   

    

1/2

  

   

   

  

2 2 a v D R ( f

2 b v c b v c    v   2 ) ) f f

E G

1

f

 



c 

= ( ) 1-  

 

1

f

f

f

2

( l a v

f 

f

f

m

1

1

1

(24)

    



1 1 p

   



q

  

  

  

  

0   s        

0          

    

ˆ Y

n

(1 )  



 





1  exp ln T 

2 

T

1   

ln

f

ath

   

478