We present the analytical formulation and the finite element solution of a fractional-order nonlocal continuum model of a Euler-Bernoulli beam. Employing consistent definitions for the 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 accepting a unique solution. Further, owing to the difficulty in obtaining analytical solutions to this boundary value problem, a finite element model for the fractional-order governing equations is presented. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the nonlocal response of a Euler-Bernoulli beam subjected to various loading and boundary conditions. The fractional-order positive definite system will be used here to address some paradoxical results obtained for nonlocal beams through classical integral approaches to nonlocal elasticity. Although presented in the context of a 1D Euler-Bernoulli beam, the f-FEM formulation is very general and could be extended to the solution of any general fractional-order boundary value problem.

中文翻译：

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非局部弹性分数阶边值问题的一种基于 Ritz 的有限元方法

我们提出了欧拉-伯努利梁的分数阶非局部连续介质模型的解析公式和有限元解。对分数阶运动学关系采用一致的定义，控制方程和相关的边界条件是基于变分原理推导出来的。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，由于难以获得该边值问题的解析解，提出了分数阶控制方程的有限元模型。在对基准问题进行彻底验证之后，分数有限元模型 (f-FEM) 用于研究 Euler-Bernoulli 梁在各种载荷和边界条件下的非局部响应。这里将使用分数阶正定系统来解决通过非局部弹性的经典积分方法获得的非局部梁的一些矛盾结果。尽管在一维 Euler-Bernoulli 梁的上下文中提出，f-FEM 公式非常通用，可以扩展到任何一般分数阶边界值问题的解决方案。