Towards a unified approach to nonlocal elasticity via fractional-order mechanics
Towards a unified approach to nonlocal elasticity via fractional-order mechanics
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DOI：
10.1016/j.ijmecsci.2020.105992
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发表时间：
2021
影响因子：
7.3
通讯作者：
Sansit Patnaik;Sai Sidhardh;F. Semperlotti
中科院分区：
文献类型：
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作者：
Sansit Patnaik;Sai Sidhardh;F. Semperlotti
本研究提出了一种分数阶连续介质力学方法，允许在单一框架不变框架下结合非局部弹性的选定特征，典型的经典积分和梯度公式。由此产生的广义理论能够捕捉到硬化和软化效应，并且不受在选定的外部载荷和边界条件下经常观察到的不一致的影响。一维连续统的控制方程是由一维晶格的拉格朗日量在远程相互作用下的连续化导出的。这种方法特别适合于突出分数阶算子与介质微观特性之间的联系。该方法还被推广到利用变分原理推导强形式三维连续体的控制方程。正确定的势能，我们分数式公式的特点，总是确保控制方程的适定性。这一方面，结合分数阶算子的微分积分性质，保证了稳定性和捕捉色散的能力，而不需要额外的惯性梯度项。所提出的公式适用于Timoshenko梁或Mindlin板的静态和自由振动分析。通过分数阶有限元方法得到的数值结果表明，分数阶公式能够同时模拟这些细长结构的加筋和软化响应。数值结果为批判性地分析不同分数阶模型参数的物理意义及其对结构单元响应的影响提供了基础。
This study presents a fractional-order continuum mechanics approach that allows combining selected characteristics of nonlocal elasticity, typical of classical integral and gradient formulations, under a single frame-invariant framework. The resulting generalized theory is capable of capturing both stiffening and softening effects and it is not subject to the inconsistencies often observed under selected external loads and boundary conditions. The governing equations of a 1D continuum are derived by continualization of the Lagrangian of a 1D lattice subject to long-range interactions. This approach is particularly well suited to highlight the connection between the fractional-order operators and the microscopic properties of the medium. The approach is also extended to derive, by means of variational principles, the governing equations of a 3D continuum in strong form. The positive definite potential energy, characteristic of our fractional formulation, always ensures well-posed governing equations. This aspect, combined with the differ-integral nature of fractional-order operators, guarantees both stability and the ability to capture dispersion without requiring additional inertia gradient terms. The proposed formulation is applied to the static and free vibration analyses of either Timoshenko beams or Mindlin plates. Numerical results, obtained by a fractional-order finite element method, show that the fractional-order formulation is able to model both stiffening and softening response in these slender structures. The numerical results provide the foundation to critically analyze the physical significance of the different fractional model parameters as well as their effect on the response of the structural elements.