PSI - Issue 79

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

409

simulated by taking at least (28 × 28) control points. Fig. 3(b) shows the plot of first four vibration mode shapes obtained using HSDT-XIGA.

(a)

(b)

Fig. 3: (a) A square FGM plate having centre crack (b) First four vibration mode shapes.

Table 2: First four NNFs of a square FGM plate ( n = 0 . 2 and q = 0) obtained by HSDT-XIGA

Method

Number of Control Points

Modes

1

2

3

4

HSDT-XIGA

20 × 20 24 × 24 28 × 28 32 × 32 36 × 36 40 × 40

5.288 5.286 5.300 5.299 5.298 5.297 5.259

13.71 13.74 13.70 13.71 13.70 13.69 13.53

13.93 13.93 13.94 13.94 13.94 13.94 13.78

22.25 22.25 22.25 22.25 22.25 22.25 21.99

Reference Huang et al. (2012)

5.2. Parametric Study

A comprehensive parametric study is carried out to quantify the e ff ects of volume fraction coe ffi cients ( n , q ), aspect ratio ( b / h ), crack length ratio ( d / b ) and boundary conditions. A square BDFGM plate with centre crack, as illustrated in Fig. 3a, is considered. Two di ff erent cases are analysed, i.e., case I: ( n = 2, q = 0.2) and case II: ( n = 2, q = 5). For both cases, first five lowest NNFs are computed for di ff erent aspect ratio ( b / h = 5, 20, 50, 100) with a fixed crack length ratios ( d / b = 0.3), as presented in Fig. 4(a). Fig. 4 shows that for both FGM cases, the NNFs increase with an increase in the aspect ratio ( b / h ). This is due to the fact that thin plates experience smaller shear deformation as compared to thick plates, leading to a higher sti ff ness-to-mass ratio and consequently higher natural frequencies. Figure. 4(b) shows the plot of five di ff erent modes for (Case I, b / h = 50, d / b = 0.3) obtained using HSDT-XIGA. Next,