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Integral and differential approaches to Eringen's nonlocal elasticity models accounting for boundary effects with applications to beams in bending
ZAMM - Journal of Applied Mathematics and Mechanics ( IF 2.3 ) Pub Date : 2021-02-02 , DOI: 10.1002/zamm.202000152
Aurora Angela Pisano 1 , Paolo Fuschi 1 , Castrenze Polizzotto 2
Affiliation  

The Eringen's fully nonlocal elasticity model is known to lead to ill-posed boundary-value problems and to suffer some boundary effects arising from particle interactions impeded by the body's boundary surface. An enhanced model is derived from the original fully nonlocal one by the addition of a regularizing non-homogeneous local phase which accounts for boundary effects and which leads to a Fredholm integral equation of the second kind, hence to well-posed boundary-value problems, without paradoxes, nor other drawbacks. The enhanced integral model applied to a beam in bending proves to be equivalent to a sixth order differential equation with variable coefficients, with extra nonlocality boundary conditions here also derived. Both the integral approach and the differential one lead to a same unique solution of the small-scale beam problem. An efficient numerical algorithm is presented in which the sixth order differential equation with variable coefficients is reduced to one of the second order, which is addressed by a finite difference method. The proposed theory is applied to a set of engineering beam problems, for each of which the inherent size effects are reported and graphically illustrated. The influence of the length scale parameter upon the beam's response is highlighted by means of a function δ ( λ ) representing the normalized maximum deflection of the beam as a function of the length scale parameter. It is shown that the enhanced model always predicts softening size effects no matter the boundary and loading conditions, and that the related response function δ ( λ ) generally exhibits a waved pattern with positive slopes first, then negative, as the length scale parameter increases, with a limit asymptotic behavior like an atomic lattice model. A comparison with other theories is also presented together with possible future developments.

中文翻译:

考虑边界效应的 Eringen 非局部弹性模型的积分和微分方法应用于弯曲梁

众所周知,Eringen 的完全非局部弹性模型会导致不适定的边界值问题,并且会遭受由物体边界表面阻碍的粒子相互作用引起的一些边界效应。通过添加正则化的非均匀局部相位,从原始的完全非局部的模型中衍生出一个增强模型,该相位解释了边界效应,并导致了第二类 Fredholm 积分方程,从而产生了适定的边界值问题,没有悖论,也没有其他缺点。应用于弯曲梁的增强积分模型证明等效于具有可变系数的六阶微分方程,此处还导出了额外的非局部边界条件。积分方法和微分方法都导致小规模梁问题的相同唯一解决方案。提出了一种有效的数值算法,其中将具有可变系数的六阶微分方程简化为二阶之一,并通过有限差分方法解决。所提出的理论应用于一组工程梁问题,每个问题的固有尺寸效应都被报告并以图形方式说明。长度尺度参数对梁响应的影响通过函数突出显示 所提出的理论应用于一组工程梁问题,每个问题的固有尺寸效应都被报告并以图形方式说明。长度尺度参数对梁响应的影响通过函数突出显示 所提出的理论应用于一组工程梁问题,每个问题的固有尺寸效应都被报告并以图形方式说明。长度尺度参数对梁响应的影响通过函数突出显示 δ ( λ ) 表示作为长度尺度参数的函数的梁的归一化最大偏转。结果表明,无论边界和加载条件如何,增强模型始终可以预测软化尺寸效应,并且相关的响应函数 δ ( λ ) 随着长度尺度参数的增加,通常首先呈现出具有正斜率波浪模式,然后是负斜率,具有像原子晶格模型一样的极限渐近行为。还介绍了与其他理论的比较以及可能的未来发展。
更新日期:2021-02-02
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