PSI - Issue 79

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

405

Nonlinear vibration of FGM beams analyzed, showing material gradation and supports strongly a ff ect dynamic behavior Ke et al. (2010). Wei et al. (2012) study about the free vibration of cracked FGM beams under axial load, considering shear deformation and various boundary conditions. The study by Huang et al. (2012) investigated how cracks a ff ect the natural frequencies of FGMs with cracks, considering both 2D and 3D models. Nevertheless, the abrupt interfaces between distinct layers frequently resulted in structural failures such as cracking or delamination Sinha and Kumar (2021). To address these limitations, researchers developed bi-directional FGMs, marking a major advancement in materials science by enabling a continuous and gradual variation in composition and properties across the material’s cross-section. Many researchers have examined the mechanical behaviour of functionally graded plates and shells. Cheng and Batra (2000) study about first-order shear deformation plate theory (FSDT) and third-order shear deformation plate theory (TSDT), relating their behaviour to equivalent homogeneous Kirchho ff plates. Reddy (2000) examined the static response of functionally graded rectangular plates using a TSDT. The Van Do et al. (2017) presents a FEM approach coupled with a novel TSDT is developed to vibration analysis of bidirectional 2D-FGM plates. Hissaria et al. (2023) has done his study on bidirectional FGMs, investigating the vibration analysis, using FEM developed in ABAQUS. Qian et al. (2004) examined meshes in the local Petrov-Galerkin (MLPG) method, combined with the HSDT, to optimise a bidirectional FGM cantilever plate improving natural frequencies, compar ing results with FEM. Isogeometric Analysis (IGA), introduced by Hughes et al. (2005), integrates Computer-Aided Design (CAD) with Finite Element Analysis (FEA) for improved computational accuracy. Lezgy-Nazargah (2015) uses the NURBS-based isogeometric approach applied to bidirectional FGM beams for evaluation of temperature and displacement fields. Be´zier extraction expresses NURBS basis functions as Bernstein polynomials, enabling C 0 con tinuous isogeometric elements Borden et al. (2011) or Singh et al. (2018). XIGA with Be´zier extraction and S-FSDT analyzes cracked FGM plates, capturing shear e ff ects accurately Singh et al. (2019). The literature review indicates a growing need for the development of bidirectional composite FGMs, where the distribution of material properties vary along two principal directions. Such materials can significantly enhance the overall performance of components under complex loading conditions and improve the structural integrity. Numerous theoretical models, such as the FSDT, TSDT and HSDT models, have been developed to describe plate behaviour. Conventional FEM have been widely em ployed for vibration and buckling analysis, but they often require mesh refinement and special treatment to accurately capture discontinuities. XIGA enhances this framework using level sets, enrichment functions, and higher-order B splines for discontinuities. In this manuscript, an extended isogeometric analysis (XIGA) formulation is developed to study the vibration of crack-graded bidirectional plates, using Reddy’s HSDT and appropriate enrichment functions. The study evaluates natural frequencies by varying parameters such as volume fraction coe ffi cients, aspect ratio ( b / h ), and crack length ratio ( d / a ). The influence of di ff erent boundary conditions on the vibrational behaviour of cracked bidirectional FGM plates is also investigated.

2. Bi-directional Functionally Graded Plates

A bi-directional FGM plate having length a ,width b and thickness h as shown in Fig. 1 is considered. The xy -plane represents the mid-surface of plate and z -axis directed upward from it. The plate composed of three di ff erent materials whose volume fractions V 1 , V 2 , and V 3 are varied according to Lezgy-Nazargah (2015).

Table 1: Material properties of the bi-directional FGM plate.

Material properties

Material 1

Material 2

Material 3

E (GPa)

110 0.3

70

380 0.3

0.3

ν

3 )

4500

2700

3900

ρ (kg / m

1 2

1 2

1 2

V 1 = 1 −

, V 2 = 1 −

, V 3 =

n

n x

q

x a

a

1 −

n

q

z h

z h

z h

(1)

+

+

+