PSI - Issue 79

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

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(a)

(b)

Fig. 4: (a) Normalized natural frequencies for bi-directional FGM plate with centre crack (b) Vibration mode shapes for SSSS boundary condition

5.3. E ff ect of Plate Thickness and Aspect Ratio

As the thickness ratio increases (i.e., the plate becomes thinner), the bending sti ff ness of the plate decreases. Thin ner plates exhibit greater bending flexibility but reduced shear deformation e ff ects, leading to higher non-dimensional frequencies as shown in Fig. 4 and discussed in Section 4.2.

5.4. E ff ect of Crack Length

Fig. 5(a) shows the influence of crack length investigated at a constant thickness ratio of b / h = 50. As the crack length increases from d / b = 0 . 5 to d / b = 0 . 7, the natural frequency decreases. A longer crack reduces the plate sti ff ness and increases the local flexibility near the crack tip. This reduction in frequency is more pronounced in thicker plates, where the crack has a stronger e ff ect on the overall sti ff ness.

5.5. E ff ect of Crack Orientation

Fig. 5(b) present the orientation of the central crack angle influence the normalized natural frequency for both simply supported (SSSS) and fully clamped (CCCC) plates, with d / b = 0 . 5 and b / h = 50. The crack angle increases from 0 ◦ to 45 ◦ , the natural frequency decreases and reaches its minimum value around θ = 45 ◦ for both cases it a ff ects both the longitudinal and transverse sti ff ness paths simultaneously, which causes the maximum reduction in sti ff ness and, hence, the lowest vibration frequency. When the crack orientation exceeds 45 ◦ , the natural frequency begins to rise again and reaches a maximum around 90 ◦ . At this orientation, the bending sti ff ness in the dominant vibration direction is partially recovered, and the frequency increases again. The V-shaped trend show how the crack orientation is directly linked with the principal vibration direction. The di ff erent boundary conditions influence the natural frequency of the plate as shown in Fig. 5. The vibration frequency is proportional to the square root of the ratio of sti ff ness to mass, i.e., f ∝ k m . Since the mass ( m ) remains constant, any decrease in sti ff ness ( k ) leads to a corresponding reduction in the natural frequency ( f ). The fully clamped case (CCCC) provides the highest sti ff ness and frequency because all edges are completely restrained against motion and rotation. The simply supported condition (SSSS) allows edge rotation, which slightly lowers the sti ff ness and results in a lower frequency compared to the clamped case. In the CFCF case, two edges are free, resulting 5.6. Influence of Boundary Conditions