Biological materials-fish scales exhibit ultra-flexibility due to its functionally graded materials. Inspired by the hierarchical gradient structure of fish scales, a new flexible gradient model that can adequately describe the characteristics of the bioinspired hierarchical structures is proposed in this work. To assess the flexibility of the proposed gradient model, a combination of extended finite element method (XFEM) and stabilized discrete shear gap (DSG) is established to analyze the vibration of bioinspired gradient plates with/without cracks. The DSG technique is employed to eliminate the shear locking phenomenon, while the XFEM is used for a mesh-independent modelling of crack. The combined approach is applicable to both moderately thick and thin plates, and is insensitive to mesh distortion. Functionally gradient plates take two types: power law function (Type I) and bioinspired hierarchical mode (Type II). For Type I, the natural frequencies decrease by increasing the gradient factor, i.e., the exponent of the power law. When the gradient factor is larger than one, the improvement of the plate stiffness by material gradient is restricted. For Type II, the natural frequencies are mostly independent of the step smoothing factor, yet quite sensitive to the number of step layers, providing an additional degree of freedom in tailoring the material properties. In addition, the natural frequencies of the bioinspired gradient plate are lower than that of the homogeneous ceramic plate. By using Type II, the stiffness of the plate can be reduced effectively, making the plate prone to deformation, which coincides with the flexible scale design. Therefore, the present study provides an incisive method and instructive guideline for a new era of artificially designed flexible materials inspired by natural (or biological) materials and structures.

生物材料鱼鳞由于其功能分级的材料而具有超柔韧性。受鱼鳞分级梯度结构的启发，本文提出了一种新的柔性梯度模型，该模型可以充分描述生物启发性分级结构的特征。为了评估所提出的梯度模型的灵活性，建立了扩展有限元方法（XFEM）和稳定的离散剪切间隙（DSG）的组合，以分析具有/不具有裂纹的生物启发梯度板的振动。DSG技术用于消除剪切锁定现象，而XFEM用于独立于网格的裂纹建模。组合方法适用于中厚板和薄板，并且对网格变形不敏感。功能梯度板有两种类型：幂律函数（I型）和生物启发式分层模式（II型）。对于类型I，固有频率通过增加梯度因子（即幂定律的指数）而降低。当梯度因子大于1时，限制了材料梯度对板刚度的改善。对于类型II，固有频率大部分与阶跃平滑因子无关，但对阶跃层的数量非常敏感，从而在调整材料属性时提供了额外的自由度。另外，受生物启发的梯度板的固有频率低于均质陶瓷板的固有频率。通过使用II型，可以有效降低钢板的刚度，使钢板易于变形，这与柔性标尺设计相吻合。因此，