PSI - Issue 26

Merdaci Slimane et al. / Procedia Structural Integrity 26 (2020) 35–45 Slimane et al. / Structural Integrity Procedia 00 (2020) 000 – 000

39

5

where (σ x , σ y , τ xy , τ yz , τ yx ) and (ε x , ε y , Ɣ xy , Ɣ yz , Ɣ zx ) are the stress and strain components, respectively. Using the material properties defined in Eq. (6), the stiffness coefficients, Q ij , can be expressed as

ν E z

E z

( )

( )

( )

E z 1 ν −

(7)

,

,

,

11 Q Q

Q

Q Q Q

= =

=

= = =

(

)

22

12

44

55

66

2

2

2 1 ν +

1 ν −

5. Governing equations The Hamilton’s principle is used herein to derive the equations of motion. The principle can be stated in analytical from as Reddy (2002).

T

(8)

0 δU δK dt 0 ( ) − = 

Where δ U: variation of strain energy; δ K: variation of kinetic energy. The variation of strain energy of the plate is calculated by

h 2 /

(9a)

 

 

 

δU

σ δ ε σ δ ε τ δ γ τ δ γ τ δ γ dA dz + + + +

=

x x y y xy xy yz yz xz xz A h 2 / −

0 A δU N δ ε N δ ε N δ ε M δ k M δ k M δ k M δ k M δ k M δ k S δ γ S δ γ dA   = + + + + + + + + + +    (9b) Where “ A ” is the top surface, and stress resultants N, M, and S are defined by 0 0 b b x x b b y y b b xy xy s s s s s s s s s s x x y y xy xy x x y y xy xy yz yz xz xz

      

      

x N N N M M M M M M , , , , , , y b x b y s s

1           f z ( )

xy

h 2 /

h 2 /

(

)

(

)

(

)

(10)





b

s

s

σ σ τ

z dz S S , ,

xz yz τ τ g z dz , ( ) .

x y xy , ,

=

=

xy

xz yz

h 2 /

h 2 /

−

−

s

x

y

xy

The variation of kinetic energy of the plate can be written as

h 2 /

(11)

 

δK

ρ z u δu vδv w δw dAdz ( )( ) + +

=

A

h 2 /

−

Where dot-superscript convention indicates the differentiation with respect to the time variable t; and (I 1 , I 2 , I 3 , I 4 , I 5 , I 6 ) are mass inertias defined as

h 2 /

(12)

(

)

(

)

− = 

2

2

1 z z f z zf z f z , , , ( ), ( ), ( ) ( ) ρ z dz

1 2 3 4 5 6 I I I I I I , , , , ,

h 2 /

By substituting Eq. (4) into Eq. (6) and integrating through the thickness of the plate, the stress resultants are given as

s ε M A D D k S A γ M B D H k ,             = =                     b s b s s s s s s N A B B

(13)