PSI - Issue 2_B

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

1330

4

′

′ = = −

j ′

′

r r r

r r r

,

,

= = −





ij

ij

j

i

ij

ij

i

′ ′ ⋅ ′ ′

(4b)

ji r r r r ji ⋅

ji r r r r ji

jk

jk

α ′

1

1

−

−

cos

,

cos

.

=

=

α

ijk

ijk

jk

jk

e U of RVE. In the present study, the Modified Morse potential is used and

(5) Calculate the potential energy

1 2

, , i j k U D e = − ∑ [1 e e

( − ∆ β

) 2



2 ) [1 ( k + ∆ + ∆ ( k α

4 ) ],

(5a)

]

α

ij

ijk

s

ijk

θ

where the parameters for carbon nanotubes are

-1 26.25 nm ,

D k

2 0.9 nN nm/rad , ⋅ 0.6031 nN nm, ⋅

=

β

=

e

(5b)

-4 0.754 rad .

k

=

=

s

θ

2 0.9 nN nm/rad ⋅

2 1.42 nN nm/rad k θ = ⋅

k

θ =

Note that k θ was corrected to be

instead of

shown in eqn.(64b)

of (Hwu and Yeh, 2014).

II K by differentiation of energy release rate G with respect to

I K and

(6) Calculate the stress intensity factors strain intensity factors I S and II S , i.e.,

dU

G

G

∂

∂

G

, e

=

K

K

where

(6a)

,

,

=

=

I

II

tda

1 c S ∂

2 c S ∂

I

II

and t is the thickness of the specimen, the coefficients 1 c and 2 c are constants to adjust the equivalency of the relation (6a) and have been obtained to be (Yeh and Hwu, 2016)

2

1

c

c

,

.

=

=

(6b)

1

2

1

1

−

+

ν

ν

The equivalence of potential energy and elastic strain energy has been assumed for the calculation of energy release rate.

- I I K S ,

- II II K S , and determine the fracture toughness by the zero

(7) Plot the generalized stress-strain diagrams,

slope of the curve.

3. Verification of RVE

Conventionally, the strain energy release rate G is approximated by e U stands for the total strain energy of the entire cracked specimen. The selection of circular RVE in eq. (1) and the use of near tip solution (3) can avoid the complicated computational procedure for obtaining the displacement field of entire cracked specimen by traditional finite element approach. In this paper, the strain energy within RVE is calculated based upon the near tip solutions, and the difference of strain energy within RVE is used to stand for the difference of total strain energy. Although the energy change e U ∆ calculated by this way has been proved to be equivalent to the one by the traditional way (Yeh and Hwu, 2016), it still looks strange to employ the near tip solution (3a) to the entire region of RVE set in step (1) since some part of RVE is not that near the crack tip. For example, if 0.2 ν = , Eq.(1) will provide us a circular region whose radius r 0 is 2.18 a a η = for mode I and 0.70 a a η = for mode II. To / ( ) e G U t a = ∆ ∆ in which