PSI - Issue 45

Oscar Zi Shao Ong et al. / Procedia Structural Integrity 45 (2023) 140–147 O.Z.S. Ong et al. / Structural Integrity Procedia 00 (2019) 000 – 000

143

4

with the following inertia and stiffness terms,

/2

h

   z









2 z z dz ,

1

I I I





1

2

3

/2

h



      11 21 v z v z           21 22 21 v z v z 12 12 12 E z     11 v z E z v z v z E z  21  

/2

h









 

2 z z dz ,

1

A A A



1

2

3

1

/2

h



/2

h









2 z z dz ,

1

B B B



(13)

1

2

3

1

/2

h



/2

h











2 z z dz ,

1

C C C



1

2

3

1

/2

h



/2

h

  









2 z z dz ,

1

D D D G z 

1

2

3

12

/2

h



and non-dimensional parameters (star signs are removed for simplicity)

A

x

y

3 

1 

2 

x a

y

b b a h

M

3,

m

*

*

*

* 0

* 0

*

* 2 

*

* ,       , t ,

,

,

,

,

,

,

,

,

x

y

x

y

t

M

0

0















1 

3 

4

b

a

b

h

h

h

I

ab

1, I b m

1

,

m

2

2

I

A

B

I

I

1 Ah

2 Ah

1 Bh

2 B h

*

*

*

*

*

*

*

*

*

,

,

,

,

,

,

,

,

,

(14)

I

I

I

A

A

A

B

B

B

3

3

3



















1

2

1

2

3

1

2

3

1

2

3

2

I

1, I h m

A

A

A

A

A

A

I

h

1,

3,

3,

3,

3,

3,

3,

1,

m

m

m

m

m

m

m

m

2

2

C

D

1 Ch

2 C h

1 Dh

2 D h

*

*

*

*

*

*

,

,

,

,

,

,

C

C

C

D

D

D

3

3













1

2

3

1

2

3

A

A

A

A

A

A

3,

3,

3,

3,

3,

3,

m

m

m

m

m

m

where I 1, m and A 3, m are the I 1 and A 3 for a non-CNT reinforced isotropic plate. 3. Solution

To solve Eqs. (10), (11) and (12), the modal decomposition method is utilised. Expanding the deformations of a FG CNT reinforced plate with simply supported boundary conditions in the axial, lateral and transverse directions results in the following:

q x 

w y 

  

sin                   sin sin

  

1 1    1 1 1 1 N M q w N M q w N M q w      





    , q w t  , 1

, , x y t

sin

1 



a

b

q x 

w y 

     

     

    , q w t  , 2





, , x y t

sin

2 



(15)

a

b

q x 

w y 

    . q w t  , 3





, , x y t

sin

3 



a

b

Substituting Eq. (15) into the equation of motions developed in the previous section, and using the decomposition method for the modes gives the natural frequencies of the FG CNT reinforced plate with mass imperfections. 4. Results and discussion To help analyse the system as shown in Fig. 1, the following dimensions and material properties are selected for the numerical solutions of the FG CNT reinforced Poly[(m-phenylenevinylene)-co-(2,5-dioctoxy-p phenylenevinylene)] (PmPV) (matrix) plate unless stated otherwise: / 1, b a  / 50, b h  2.1GPa, m E  3 1160kg/m , m   0.34 m v  , 11 5.6466TPa, CNT E  22 7.08TPa, CNT E  12 1.9445TPa, CNT G  3 1400kg/m , CNT   and 0.175 CNT v  (Huang et al. 2021).