PSI - Issue 26

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

38 4

1. The displacements are small in comparison with the plate thickness, and, therefore, strains involved are infinitesimal. 2. The transverse displacement w includes two components of bending (w b ), and shear (w s ). The bending and shear parts are functions of coordinates x, y and t only, and the stretching part is a function x, y, t and z. 3. The in-plane displacements (u and v) in coordinates x and y are divided into extension, bending and shear parts. It is shown that the in-plane displacements are functions of x, y, t and z in which the bending parts are alike to those presented by CPT, and shear parts of that are in conjunction with the hyperbolic variations of shear strains.

4. General theory

According to the higher-order shear deformation plate theory (HSDPT), the axial and transverse displacements of the plate fields can be defined by Merdaci et al. (2019) are given by:

w

w





b

s

( , , , )

u x y t z ( , , ) −

( )

u x y z t

f z

=

+

0

x

x





w

w





(2)

b

s

( , , , )

v x y t z ( , , ) −

( )

v x y z t

f z

=

+

0

y

y

 +



( , , , )

( , , )

( , , )

w x y z t

w x y t

w x y t

=

b

s

In which t represents the time, u 0 and v 0 signify the displacement functions of the middle surfaces of the plate. Also f (z) is the representative shape function that denotes the distribution of transverse shear stress or strain along the plate thickness. In this study, we have

2

2

5z 4z 

  

df z 5 5z ( )

(3)

( )

and g z f z ( ) '( ) = =

f z

1 = − 

= −

2

2

4

dz 4

3h

h



The strains associated with the displacements in Eq. (2) are

0 x 0 y y = +             xy           xy xy k             ε ε γ k 0 ε ε γ b x b y b x    

s     x yz           s k ,

s         yz s xz γ γ

γ γ

(4)

z k f z k ( ) +

g z

ε 0 =

( )

,

=

y

z

xz  

s xy     k  



   

   

 

2

2

w

w





   

   

u



b

s

−

0

2

2

x

x





               =                 s s yz s xz s w   γ γ y w x ,

x

0 x 0 y 0 xy                =      ε ε γ

b x b y b                  xy       k k k ,

s     x

k k k

2

2

(5)

v

w

w





s y               s xy     = ,

0

b

s

,

= −

2

2

y

y

y





0 u v y x  +        0

2

2

w

w





  

  

  

b

s

2

2

−

x y

x y

 

 

The stress-strain relations for a linear elastic plate and isotropic, are written in the following of

x         y xy τ  

 

11 σ Q Q 0 ε σ Q Q 0 ε     = 12

x yz       zx τ τ           y ,

yz           zx γ γ

44 Q 0

  

  

(6)

=

  

  

12

22

0 Q

55

0 0 Q

γ

xy  

66