PSI - Issue 2_B

Chyanbin Hwu et al. / Procedia Structural Integrity 2 (2016) 1327–1334 Hwu and Yeh / Structural Integrity Procedia 00 (2016) 000–000

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Table 1. Poisson’s ratio of carbon nanotubes.

z θ ν

z θ ν

Tube type

Armchair (12,12) Armchair (24,24) Armchair (36,36)

0.2067 0.2047 0.2043 0.2086 0.2054 0.2047

0.2055 0.2044 0.2042 0.2070 0.2056 0.2045

Zigzag (20,0) Zigzag (36,0) Zigzag(52,0)

(2) Determine the position of each atom of RVE in the undeformed state, which may be expressed as ( cos , sin , ) i i i i R R z θ θ = r for the i th atom of RVE. Here, R is the radius of carbon nanotubes, i θ and i z denote, respectively, the angular and longitudinal position of the atom i . (3) Apply a suitable deformation field to RVE, and calculate the position of each atom in the deformed state, ( ) ( ) ( ) ( cos( ), sin( ), ) i i i i i i i z R u R u z u θ θ θ θ ′ = + + + r . Under the assumption of linear elastic fracture mechanics, the displacements ( ) i u θ and ( ) i z u near the crack tip associated with strain intensity factor I S and II S can be expressed as (Yeh and Hwu, 2016)

S

2 S

1 1 1 1

ν ν ν ν

+ − + −

( ) i

I f r x i

II f r x i

u

( , ) θ

( , ), θ

=

+

I

II

i

i

θ

R

R

(3a)

S

( ) i

I S f r I y i

II f r y i

u

( , ) θ

( , ), θ

=

+

II

z

i

i

2

where

r

i θ

θ

2(1 ) ν

 −  + 

  

I f r x i

2 2sin , i

( , ) θ

cos

=

+

i

i

2

2 1

2

π

ν

r

i θ

i θ

4

 

  

I f r y i

2

( , ) θ

sin

2cos

=

−

i

i

2 2 1 π

2

ν

 +

(3b)

r

i θ

i θ

4

 

  

II f r x i

2

( , ) θ

sin

2cos

,

=

+

i

i

2 2 1 π

2

ν

 +

r

i θ

θ

2(1 ) ν

 −  + 

  

II f r y i

2 2sin . i

( , ) θ

cos

= −

−

i

i

2

2 1

2

π

ν

ij ∆  between any two atoms and angle change

ijk α ∆ between any three atoms

(4) Calculate the distance change

by

(4a)

, ′ ′ ∆ = − ∆ = −   

,

ijk ijk ijk α α α

ij

ij

ij

where