This study presents the analytical formulation and the finite element solution of fractional order nonlocal plates under both Mindlin and Kirchoff formulations. By employing consistent definitions for fractional-order kinematic relations, the governing equations and the associated boundary conditions are derived based on variational principles. Remarkably, the fractional-order nonlocal model gives rise to a self-adjoint and positive-definite system that accepts a unique solution. Further, owing to the difficulty in obtaining analytical solutions to this fractional-order differ-integral problem, a 2D finite element model for the fractional-order governing equations is presented. Following a thorough validation with benchmark problems, the 2D fractional finite element model is used to study the static as well as the free dynamic response of fractional-order plates subject to various loading and boundary conditions. It is established that the fractional-order nonlocality leads to a reduction in the stiffness of the plate structure thereby increasing the displacements and reducing the natural frequency of vibration of the plates. Further, it is seen that the effect of nonlocality is stronger on the higher modes of vibration when compared to the fundamental mode. These effects of the fractional-order nonlocality are noted irrespective of the nature of the boundary conditions. More specifically, the fractional-order model of nonlocal plates is free from boundary effects that lead to paradoxical predictions such as hardening and absence of nonlocal effects in classical integral approaches to nonlocal elasticity. This consistency in the predictions is a result of the well-posed nature of the fractional-order governing equations that accept a unique solution.

中文翻译：

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非局部板静态和动态分析的分数阶模型

本研究介绍了在 Mindlin 和 Kirchoff 公式下分数阶非局部板的解析公式和有限元解。通过对分数阶运动学关系采用一致的定义，基于变分原理推导出控制方程和相关的边界条件。值得注意的是，分数阶非局部模型产生了一个接受唯一解的自伴随和正定系统。此外，由于难以获得该分数阶微分积分问题的解析解，因此提出了分数阶控制方程的二维有限元模型。在对基准问题进行彻底验证之后，二维分数阶有限元模型用于研究分数阶板在各种载荷和边界条件下的静态和自由动态响应。已确定分数阶非局域性导致板结构刚度降低，从而增加位移并降低板的固有振动频率。此外，可以看出，与基本模式相比，非局域性对更高振动模式的影响更强。无论边界条件的性质如何，都会注意到分数阶非定域性的这些影响。进一步来说，非局部板块的分数阶模型不受边界效应的影响，边界效应会导致矛盾的预测，例如非局部弹性的经典积分方法中的硬化和缺乏非局部效应。预测的这种一致性是分数阶控制方程接受唯一解的适定性质的结果。