标题： 基于广义连续体的单维结构的新框架 显示英文标题

标题： A New Framework for Unidimensional Structures Based on Generalised Continua

摘要： 本文介绍了一类从三维高阶弹性框架导出的梁模型。通过引入三个运动场——宏观位移u、微扭曲张量P和三阶张量N，该研究系统地探讨了三种情况：全等的、半全等的和非全等的。这些情况对应于不同层次的运动约束，从经典弹性到一个完全放松的模型。全等情况退化为一个高阶欧拉-伯努利梁模型，而半全等情况则推广了提莫申科梁模型。非全等情况提供了一个统一的框架，自然地结合了位错和位错环。此外，通过增加特定的惩罚系数，全等和半全等模型被证明是来自非全等模型的奇异极限。针对特定情况，如纯拉伸和弯曲，推导出了简化的常微分方程系统，说明了这些模型的实际适用性。结果突显了所提出框架的分层结构及其在捕捉梁状结构中的材料缺陷方面的能力。 显示英文摘要

摘要： The present work introduces a family of beam models derived from a three-dimensional higher-order elasticity framework. By incorporating three kinematic fields - the macroscopic displacement u, the micro-distortion tensor P, and the third-order tensor N - the study systematically explores three regimes: holonomic, semi-holonomic, and non-holonomic. These regimes correspond to varying levels of kinematic constraints, ranging from classical elasticity to a fully relaxed model. The holonomic case reduces to a higher-order Euler--Bernoulli beam model, while the semi-holonomic case generalises the Timoshenko beam model. The non-holonomic case provides a unified framework that naturally incorporates both dislocations and disclinations. Furthermore, the holonomic and semi-holonomic models are shown to emerge as singular limits of the non-holonomic model by increasing specific penalty coefficients. Simplified ordinary differential equation systems are derived for specific cases, such as pure traction and bending, illustrating the practical applicability of the models. The results highlight the hierarchical structure of the proposed framework and its ability to capture material defects in beam-like structures.