PSI - Issue 26

132 Merazi et al / Structural Integrity Procedia 00 (2020) 000 – 000 • The transverse normal stress σ z is negligible in comparison with in-plane stresses σ x and σ y . • The axial displacement u in x-direction consists of extension, bending, and shear components. • The bending component b u and b v are assumed to be similar to the displacements given by the classical plate theory. Therefore, the expression for b u and b v can be given as • The shear components s u and s v gives rise, in conjunction with s w , to the hyperbolic variation of shear strains , and hence to shear stresses τ , τ through the thickness of the plate in such a way that shear stresses τ , τ are zero at the top and bottom faces of the plate. Consequently, the expression for s u and s v can be given as Merazi Mohamed et al. / Procedia Structural Integrity 26 (2020) 129–138 4





(5 )

,

v

y f z w ns 

= − ( )

u

x f z w ns 

s

= − ( )

s

s

s

Where

h z d

  

 −  

2 1

3





   + ns

  

  

  

(

)

(6)

2

f z

= + + z d

h

h

ns ( )

2 1 3

sec

tanh

ns

2

2.3 Constitutive relations

Based on the assumptions made in the preceding section, the displacement field as:

w

w





b

s

u(x, y, z ) u (x, t) z − =

f (z ) ns

−

ns

0

ns

x

x





w

w





b

s

f (z ) ns

v(x, y, z ) v (x, t) z − =

−

(7)

ns

0

ns

y

y





w(x, y, z ) w (x, y) w (x, y) s b ns + =

The strains associated with the displacements in Eq. (7) are

2

2

  

  

  

  

u

w

w







0 = + − z

b

s

ε

ns f z ) (

+

−

x

ns

2

2

x



x

x





2

2

  

  

  

  

v

w

w







0 = + − z

b

s

ε

ns f z ) (

+

−

y

ns

2

2

x



y

y





(8)

s yz ns w γ g z ) x ( =      ( ) = 1 − ′ ( ) ′ ( ) = ( )         s xz ns w    γ g z ) ( = x

(9)