This study presents the analytical and finite element formulation of a geometrically nonlinear and fractional-order nonlocal model of an Euler–Bernoulli beam. The finite nonlocal strains in the Euler–Bernoulli beam are obtained from a frame-invariant and dimensionally consistent fractional-order (nonlocal) continuum formulation. The finite fractional strain theory provides a positive definite formulation that results in a mathematically well-posed formulation which is consistent across loading and boundary conditions. The governing equations and the corresponding boundary conditions of the geometrically nonlinear and nonlocal Euler–Bernoulli beam are obtained using variational principles. Further, a nonlinear finite element model for the fractional-order system is developed in order to achieve the numerical solution of the integro-differential nonlinear governing equations. Following a thorough validation with benchmark problems, the fractional finite element model (f-FEM) is used to study the geometrically nonlinear response of a nonlocal beam subject to various loading and boundary conditions. Although presented in the context of a 1D beam, this nonlinear f-FEM formulation can be extended to higher dimensional fractional-order boundary value problems.

这项研究提出了Euler–Bernoulli光束的几何非线性和分数阶非局部模型的解析和有限元公式。Euler–Bernoulli梁中的有限非局部应变是从不变框架且尺寸一致的分数阶（非局部）连续谱公式获得的。有限分数应变理论提供了一个正定公式，该公式得出了数学上合理的公式，该公式在载荷和边界条件下均保持一致。利用变分原理获得了几何非线性和非局部Euler–Bernoulli梁的控制方程和相应的边界条件。进一步，为了实现积分微分非线性控制方程的数值解，建立了分数阶系统的非线性有限元模型。在对基准问题进行了彻底验证之后，分数有限元模型（f-FEM）用于研究非局部梁在各种载荷和边界条件下的几何非线性响应。尽管在一维光束的背景下提出，但这种非线性f-FEM公式可以扩展到更高维的分数阶边值问题。