This study presents the framework to perform a stability analysis of nonlocal solids whose response is formulated according to the fractional-order continuum theory. In this formulation, space fractional-order operators are used to capture the nonlocal response of the medium by introducing nonlocal kinematic relations. First, we use the geometrically nonlinear fractional-order kinematic relations within an energy-based approach to establish the Lagrange-Dirichlet stability criteria for fractional-order nonlocal structures. This energy-based approach to nonlocal structural stability is possible due to a positive-definite and thermodynamically consistent definition of deformation energy enabled by the fractional-order kinematic formulation. Then, the Rayleigh-Ritz coefficient for the critical load is derived for linear buckling conditions. The fractional-order formulation is finally used to determine critical buckling loads of slender nonlocal beams and plates using a dedicated fractional-order finite element solver. Results establish that, in contrast to existing studies, the effect of nonlocal interactions is observed on both the material and the geometric stiffness, when using the fractional-order kinematics approach. We support these observations quantitatively with the help of case studies focusing on the critical buckling response of fractional-order nonlocal slender structures, and qualitatively via direct comparison of the fractional-order approach with the classical nonlocal approaches.

中文翻译：

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分数阶结构稳定性：细长结构临界荷载的公式化及应用

本研究提出了对非局部固体进行稳定性分析的框架，这些固体的响应是根据分数阶连续介质理论制定的。在这个公式中，空间分数阶算子用于通过引入非局部运动学关系来捕获介质的非局部响应。首先，我们在基于能量的方法中使用几何非线性分数阶运动学关系来建立分数阶非局部结构的拉格朗日-狄利克雷稳定性标准。这种基于能量的非局部结构稳定性方法是可能的，因为由分数阶运动学公式实现的变形能的正定和热力学一致定义。然后，推导出线性屈曲条件下临界载荷的 Rayleigh-Ritz 系数。分数阶公式最终用于使用专用分数阶有限元求解器确定细长非局部梁和板的临界屈曲载荷。结果表明，与现有研究相比，当使用分数阶运动学方法时，可以观察到非局部相互作用对材料和几何刚度的影响。我们在关注分数阶非局部细长结构的临界屈曲响应的案例研究的帮助下定量地支持这些观察，并通过分数阶方法与经典非局部方法的直接比较来定性地支持这些观察。结果表明，与现有研究相比，当使用分数阶运动学方法时，可以观察到非局部相互作用对材料和几何刚度的影响。我们在关注分数阶非局部细长结构的临界屈曲响应的案例研究的帮助下定量地支持这些观察，并通过分数阶方法与经典非局部方法的直接比较来定性地支持这些观察。结果表明，与现有研究相比，当使用分数阶运动学方法时，可以观察到非局部相互作用对材料和几何刚度的影响。我们在关注分数阶非局部细长结构的临界屈曲响应的案例研究的帮助下定量地支持这些观察，并通过分数阶方法与经典非局部方法的直接比较来定性地支持这些观察。