PSI - Issue 79

Pranaw Parihar et al. / Procedia Structural Integrity 79 (2026) 404–412

407

∂ w o ∂ x  ∂ y  ∂ w o

3  ϕ 3  ϕ

u ( x , y , z ) = u o ( x , y ) + z ϕ x ( x , y ) − c 1 z

x +

(3)

v ( x , y , z ) = v o ( x , y ) + z ϕ y ( x , y ) − c 1 z

y +

w ( x , y , z ) = w o ( x , y )

where, c 1 = = plate thickness. Considering small deformation, the strain is related with displacement as, 4 3 h 2 , h

 

  ε

γ   =

  ε (0) + z ε (1) + z 3 ε (3) γ (0) + z 2 γ (2)

(4)

     ∂ϕ x ∂ y

     , γ (0) =     ϕ y + ϕ x +

     , ε (3) = − c 1

     ∂ϕ x ∂ y

∂ 2 w ∂ x 2 ∂ 2 w

    

     ∂ u o ∂ y

∂ϕ x ∂ x ∂ϕ y ∂ y

∂ u o ∂ x ∂ v o ∂ y

∂ϕ x ∂ x ∂ϕ y ∂ y

0

+

    , and γ (2) =

∂ w o ∂ y ∂ w o ∂ x

0

ε (0) =

, ε (1) =

+

∂ y 2

∂ v o ∂ x

∂ϕ y ∂ x

∂ 2 w 0 ∂ x ∂ y

∂ϕ y ∂ x

+

+

+ 2

+

− 3 c 1         . The stress is related with strain as, ϕ y + ∂ w o ∂ y ∂ w o ∂ x ϕ x +

     1 ν

     ,

0

ν

E ( x , z ) 2(1 + ν )   

1 0 0 1    .

E ( x , z ) 1 − ν 2

1

0

D a ( x , z ) =

D ( x , z ) =

(5)

(1 − ν ) 2

0 0

Using Hamilton principle, the weak form for vibration of bi-directional FGM plate can be obtained as,

 Ω

a D ε a d Ω+  Ω

a D a γ a d Ω=  Ω

δ ε T

δ γ T

δ u T m ¨ u d

(6)

Ω

4. XIGA Formulation for Vibration Analysis of Bi-directional FGM Plate

The XIGA displacement approximation is written as,