A non-classical analytical model for vibration analysis of thin isotropic and FGM plate containing multiple part-through cracks (star shaped) of arbitrary orientation is proposed. A plate containing four concentric cracks of arbitrary orientation in the form of continuous line is considered for analysis. The proposed governing equation is derived based on classical plate theory and modified couple stress theory. Line spring model is modified to accommodate all the crack terms. The application of Berger’s formulation introduces nonlinearities in the governing equation and then the Galerkin’s method is applied for solving final governing equation. Results for fundamental frequencies for different values of crack length, crack orientation, gradient index and material length scale parameters are presented for two different boundary conditions. Furthermore, to study the phenomenon of bending hardening/softening in a cracked plate, the frequency response curves are plotted for the parameters stated above. Based on the outcomes of this study, it can be concluded that stiffness of the plate is severely affected by the presence of multiple cracks and the stiffness goes on decreasing with increase in number of cracks thereby affecting the fundamental frequency.

提出了非均质薄板和FGM板振动分析的非经典分析模型，该板包含多个任意方向的部分贯通裂纹（星形）。考虑将包含四个以连续线形式任意定向的同心裂纹的板进行分析。提出的控制方程是基于经典板理论和改进的偶应力理论推导的。修改了线弹簧模型以适应所有裂纹项。Berger公式的应用在控制方程中引入了非线性，然后将Galerkin方法用于求解最终的控制方程。给出了两种不同边界条件下不同裂纹长度，裂纹取向，梯度指数和材料长度尺度参数的基频结果。此外，为了研究裂纹板弯曲硬化/软化的现象，绘制了上述参数的频率响应曲线。根据这项研究的结果，可以得出结论，存在多个裂纹会严重影响板的刚度，并且随着裂纹数量的增加，刚度会继续降低，从而影响基频。