论文标题
一种基于丽兹的有限元方法，用于非局部弹性的分数边界值问题
A Ritz-based Finite Element Method for a Fractional-Order Boundary Value Problem of Nonlocal Elasticity
论文作者
论文摘要
我们介绍了Euler-Bernoulli梁的分数非局部连续体模型的分析公式和有限元解。使用一致的定义对分数阶运动关系，管理方程式和相关边界条件是基于变异原理得出的。值得注意的是，分数非局部模型产生了一个自我接合和正定系统，该系统接受了独特的解决方案。此外，由于难以获得解决此边界价值问题的分析解决方案，因此提出了分数阶阶程方程的有限元模型。在与基准问题进行彻底验证之后，使用分数有限元模型（F-FEM）来研究经受各种载荷和边界条件的Euler-Bernoulli光束的非本地响应。这里将使用分数阶正定系统，以通过经典的非局部弹性方法来解决非局部光束获得的一些矛盾的结果。尽管在1D Euler-Bernoulli梁的上下文中呈现，但FEM公式非常通用，并且可以扩展到任何一般的分数边界值问题的解决方案。
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.
