PSI - Issue 26

Merazi Mohamed et al. / Procedia Structural Integrity 26 (2020) 129–138 Merazi et al. / Structural Integrity Procedia 00 (2020) 000 – 000

131

3

Where n is the power law index which takes the value greater or equal to zero and d is the distance of neutral surface from the mid-surface. (Suresh and Mortensen, 1998). Thus, using Eq. (1), the material non-homogeneous properties of FG plate, as a function of thickness coordinate, become

n

+

2 1

  

  

,

(2)

ns h E z E E z d = + ( ) M CM

+

C M CM E E E = −

Where M E and C E are the corresponding properties of the metal and ceramic, respectively. In the present work, we assume that the elasticity modules E are described by Eq. (2), while Poisson’s ratio  , is considered to be constant across the thickness. The variation of Young’s modulus in the thickness direction of the P -FGM plate, which shows that the Young’s modulus changes rapidly near the lowest surface for 1  n and increases quickly near the top surface for 1  n . The position of the neutral surface of the FG plate is determined to satisfy the first moment with respect to Young’s modulus being zero as follows:

h

/ 2

(

)

(3)



( ) ms ms E z z d dz −

0

=

ms

h

/ 2

−

The position of neutral surface can be obtained as

h

/ 2



( ) ms E z z dz ms

ms

(4)

d

h

− =

/ 2

h

/ 2



( ) ms E z dz

ms

h

/ 2

−

2.2 Basal hypothesis

Consider a plate of total thickness h and composed of functionally graded material through the thickness (Fig. 2). It is assumed that the material is isotropic and grading is assumed to be only through the thickness.

Fig. 2. Functionally graded materials (FGM) plate model.

The assumptions of the present theory are as follows: • The origin of the Cartesian coordinate system is taken at the neutral surface of the FG plate. • The displacements are small in comparison with the height of the plate and, therefore, strains involved are infinitesimal. • The transverse displacement w includes two components of bending b w , and shear s w . These components are functions of coordinates x, y only.