Issue 52

N. Hebbar et alii, Frattura ed Integrità Strutturale, 52 (2020) 230-246; DOI: 10.3221/IGF-ESIS.52.18

B ASIC ASSUMPTIONS

T O

he hypotheses of the present theory are as follows; the origin of the Cartesian coordinate system is taken on the neutral axis of the beam in functionally graded materials; The displacements are small in comparison with the thickness of the beam thus the deformations involved are infinitesimal; Displacements ( u ) in the x direction consist of extension, bending and shear components. K INEMATIC AND CONSTITUTIVE EQUATION n the basis of the assumptions made in the previous section, the displacement field can be presented as follows:



w x t

( , )



0







( ) ( , ) , K f z x t dx 

(4a)

u x z t

u x t

z

( , , )

( , )

a

0



x

( , , ) 0, x z t  

(4b)

  g z x t 





w x z t

w x t

( , , )

( , )

( , ),

(4c)

0

where: u 0 is the axial displacement in the median plane, and t represents the time. In this study, f (z) represents the shape function determining the distribution of transverse shear deformation as follows:

z z  

  

2



h

f z

2

( )







(5)

f z

, ( ) g z

( )

 



h

z

3 4

The deformations associated with displacements in Eqn. (4) are:



f z z 

0 ( ) xz 

0



   

( ) , x x zk f z 





(6)

x

x

xz

where:

2



w



u

 

,

, x

a K A     ,

0

0

0

 

0



( , ) a K x t dx 

k





(7)

x

xz

x

2





x

x

and



g z

( )

0 0  

z 









(8)

g z

, '( ) g z

'( ) ,



z

z

z

The Navier method is used to solve the integrals defined in the equations [64]:

x    





(9)

dx

A

where the coefficient A' is considered according to the type of solution used, in this case via the Navier method. Consequently, A ′ and K a are expressed as follows :

2 1 A - , 

2

 

a K (10) The beam in functionally graded materials obeys Hooke's law, so the behavioral relations can be given as follows:  

234