PSI - Issue 79

Pranaw Parihar et al. / Procedia Structural Integrity 79 (2026) 404–412

406

Fig. 1: A schematic of bi-directional FGM plate.

where, q and n denote non-negative gradient indices that govern the material composition variation along the x and z directions, respectively.

(a)

(b)

(c)

Fig. 2: Distribution of V 1 , V 2 , and V 3 in a bi-directional FGM plate for n = 2 and q = 5.

From Eq. (1), the material assigned (100% material 3) at the top surface ( z = h / 2) varied to a mixture of materials 1 and 2 at the bottom surface ( z = − h / 2). Besides, at the bottom surface material composition varied from material 1 assigned at the left edge ( x = 0) to material 2 assigned at the right edge ( x = a ). Morever, when q = 0, Eq. (1) reduces to two-phase FGM, and material properties vary from materials 2 to material 3 in thickness direction. The distributions of V 1 , V 2 , and V 3 for n = 2 and q = 5 are illustrated in Fig. 2. Furthermore, the rule of mixtures based on Voigt model is used to obtain the e ff ective material properties of the FGM. The e ff ective Young’s modulus ( E ) and density ( ρ ) can be obtained as,

E ( x , z ) = E 1 V 1 + E 2 V 2 + E 3 V 3 , ρ ( x , z ) = ρ 1 V 1 + ρ 2 V 2 + ρ 3 V 3

(2)

where, E i and ρ i ( i = 1 , 2 , 3) represent the Young’s modulus and density of each constituent material, respectively.

3. Higher-Order Shear Deformation Plate Theory

The displacement field at any point ( x , y , z ) in the plate using Reddy’s HSDT is given as Singh et al. (2018),