Thick rods are employed in nanotechnology to build modern electromechanical systems. Design and optimization of such structures can be carried out by nonlocal continuum mechanics which is computationally convenient when compared to atomistic strategies. Bishop’s kinematics is able to describe small-scale thick rods if a proper mathematical model of nonlocal elasticity is formulated to capture size effects. In all papers on the matter, nonlocal contributions are evaluated by replacing Eringen’s integral convolution with the consequent (but not equivalent) differential equation governed by Helmholtz’s differential operator. As notorious in integral equation theory, this replacement is possible for convolutions, defined in unbounded domains, governed by averaging kernels which are Green’s functions of differential operators. Indeed, Eringen himself, in order to study nonlocal problems defined in unbounded domains, such as screw dislocations and wave propagation, suggested to replace integro-differential equations with differential conditions. A different scenario appears in Bishop rod mechanics where nonlocal integral convolutions are defined in bounded structural domains, so that Eringen’s nonlocal differential equation has to be supplemented with additional boundary conditions. The objective is achieved by formulating the governing nonlocal equations by a proper variational statement. The new methodology provides an amendment of previous contributions in the literature and is illustrated by investigating the elastostatic behavior of simple structural schemes. Exact solutions of Bishop rods are evaluated in terms of nonlocal parameter and cross section gyration radius. Both hardening and softening structural responses are predictable with a suitable tuning of the parameters.

中文翻译：

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Bishop非局部棒的一致变分公式

纳米技术中使用粗棒来构建现代机电系统。这种结构的设计和优化可以通过非局部连续介质力学来进行，这与原子策略相比在计算上是方便的。如果制定了适当的非局部弹性数学模型来捕获尺寸影响，Bishop的运动学就能够描述小尺寸的粗棒。在有关此问题的所有论文中，通过用由亥姆霍兹的微分算子控制的结果（但不是等效的）微分方程代替Eringen的积分卷积来评估非局部贡献。在积分方程理论中臭名昭著，这种替换对于在无界域中定义的卷积是可行的，该卷积由平均内核控制，这些内核是微分算子的格林函数。的确，艾林根本人 为了研究在无界域中定义的非局部问题，例如螺旋位错和波传播，建议用微分条件代替积分微分方程。Bishop杆力学中出现了另一种情况，其中在有界结构域中定义了非局部积分卷积，因此必须用其他边界条件来补充Eringen的非局部微分方程。通过用适当的变分表述公式化控制非局部方程来实现此目的。新方法提供了文献中先前贡献的修正，并通过研究简单结构方案的弹性静力行为得到了说明。根据非局部参数和截面回转半径评估Bishop杆的精确解。