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Perturbation approach to Eringen’s local/non-local constitutive equation with applications to 1-D structures
Meccanica ( IF 2.7 ) Pub Date : 2020-03-16 , DOI: 10.1007/s11012-020-01145-x
Ugurcan Eroglu

Eringen’s two-phase local/non-local constitutive equation is preferred over its full non-local counterpart due to mathematical simplifications it provides. Then again, an integro-differential equation must be solved, which requires rigorous examination of the existence of an exact solution in certain forms. For this purpose, some additional constraints are attained to strain field for the sake of an exact solution which may be in contrast with the balance equations. It is the aim of this study to look for possible approximated solutions in series by a perturbation approach. Indeed, we find that response of structures with non-local constitutive relation may be approximated by a set of local elasticity problems, the existence and uniqueness of which are ensured. The present approach does not require any more conditions than physical boundary conditions, such as constitutive boundary conditions. It is applied to simple one-dimensional structural elements, and numerical evidence on possible convergence of the series expansion is provided. Some structural problems of bars and beams, which may be the simplified models of nanostructures in modern engineering applications, are discussed and solutions to them are given in closed-form.

中文翻译:

用于一维结构的 Eringen 局部/非局部本构方程的微扰方法

由于其提供的数学简化,Eringen 的两相局部/非局部本构方程优于其完整的非局部对应方程。再说一次,必须求解积分微分方程,这需要严格检查某些形式的精确解的存在性。为此,为了得到可能与平衡方程相反的精确解,对应变场进行了一些额外的约束。本研究的目的是通过扰动方法寻找可能的串联近似解。事实上,我们发现具有非局部本构关系的结构的响应可以通过一组局部弹性问题来近似,这些问题的存在性和唯一性是有保证的。本方法不需要比物理边界条件更多的条件,例如本构边界条件。它应用于简单的一维结构元素,并提供了级数展开可能收敛的数值证据。讨论了可能是现代工程应用中纳米结构的简化模型的杆和梁的一些结构问题,并以封闭形式给出了解决方案。
更新日期:2020-03-16
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