Challamel N, Wang C M
Université Européenne de Bretagne, INSA de Rennes-LGCGM, 20, avenue des Buttes de Coësmes, 35043 Rennes cedex, France.
Nanotechnology. 2008 Aug 27;19(34):345703. doi: 10.1088/0957-4484/19/34/345703. Epub 2008 Jul 16.
Non-local continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with microstructures or nanostructures. This paper presents some simplified non-local elastic beam models, for the bending analyses of small scale rods. Integral-type or gradient non-local models abandon the classical assumption of locality, and admit that stress depends not only on the strain value at that point but also on the strain values of all points on the body. There is a paradox still unresolved at this stage: some bending solutions of integral-based non-local elastic beams have been found to be identical to the classical (local) solution, i.e. the small scale effect is not present at all. One example is the Euler-Bernoulli cantilever nanobeam model with a point load which has application in microelectromechanical systems and nanoelectromechanical systems as an actuator. In this paper, it will be shown that this paradox may be overcome with a gradient elastic model as well as an integral non-local elastic model that is based on combining the local and the non-local curvatures in the constitutive elastic relation. The latter model comprises the classical gradient model and Eringen's integral model, and its application produces small length scale terms in the non-local elastic cantilever beam solution.
非局部连续介质力学使人们能够考虑在处理微观结构或纳米结构时变得显著的小长度尺度效应。本文提出了一些简化的非局部弹性梁模型，用于小尺度杆的弯曲分析。积分型或梯度非局部模型摒弃了经典的局部性假设，并承认应力不仅取决于该点处的应变值，还取决于物体上所有点的应变值。在现阶段仍有一个未解决的悖论：基于积分的非局部弹性梁的一些弯曲解已被发现与经典（局部）解相同，即完全不存在小尺度效应。一个例子是带有点载荷的欧拉 - 伯努利悬臂纳米梁模型，它在微机电系统和纳米机电系统中作为致动器有应用。在本文中，将表明这个悖论可以通过梯度弹性模型以及基于在本构弹性关系中结合局部和非局部曲率的积分非局部弹性模型来克服。后一种模型包括经典梯度模型和埃林根的积分模型，其应用在非局部弹性悬臂梁解中产生小长度尺度项。
