The engineering applications of the innovative materials, such as carbon nanotubes (CNTs) and graphene sheets, constitute a developing branch of the modern science. CNTs, in particular, exhibit extraordinarily mechanical properties capable of making them a reliable material for the above applications. The simulation of the mechanical response of a CNT is effectively conducted by a generalized beam model. This work investigates the response of a CNT, subjected to a static loading, by means of a gradient beam model. Given that the existence of a boundary layer is demonstrated by the analytical solutions of the aforementioned problem, the interior penalty discontinuous Galerkin finite element methods (IPDGFEMs) are crucial to its solution. The hp-version IPDGFEMs are developed in this study for the solution of a static gradient elastic beam in bending, derived from two different equilibrium formulations, for the first time. An a priori error analysis is also performed for the above method, and numerical simulations are then carried out. A comparison is finally drawn between the exact deflection and those of the IPDGFEM and the conforming C²-continuous finite element method (C2CFEM) for a number of discretizations and each length scale that is investigated. The deduced results highlight the suitability, the efficiency, and the accuracy of the investigated models over the already existing ones, and they have considerable significance for the engineering design in the range of micro and nanodimensions.

中文翻译：

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用于静态碳纳米管应用中梯度束的不连续 Galerkin FEM

碳纳米管 (CNT) 和石墨烯片等创新材料的工程应用构成了现代科学的一个发展分支。尤其是 CNT，表现出非凡的机械性能，使其成为上述应用的可靠材料。通过广义梁模型有效地模拟了碳纳米管的机械响应。这项工作通过梯度光束模型研究了 CNT 在静态负载下的响应。鉴于上述问题的解析解证明了边界层的存在，内部惩罚不连续 Galerkin 有限元方法 (IPDGFEM) 对其求解至关重要。惠普_ _本研究首次开发了版本 IPDGFEM，用于求解弯曲中的静态梯度弹性梁，源自两种不同的平衡公式。对上述方法也进行了先验误差分析，然后进行数值模拟。最后在精确偏转与 IPDGFEM 和符合C ² -连续有限元法 (C2CFEM) 的精确偏转之间进行比较，用于研究的多个离散化和每个长度尺度。推导结果突出了所研究模型相对于现有模型的适用性、效率和准确性，对微米和纳米尺寸范围内的工程设计具有重要意义。