The force method and displacement method on the basis of the reciprocal theorem play an important role in the field of structural mechanics and have been successfully applied in structural mechanics. However, it is interestingly found that the unexpected paradox exists when the authors attempt to apply it to problems of deformations of strain gradient beams. The reciprocal relation between higher order stresses and higher order strains within the framework of linear elastic strain gradient elasticity is proposed with a view toward studying the physical nature of this paradoxical phenomenon, and it is then used to prove the updated reciprocal theorem. At the same time, the reciprocal theorem of any gradients of any second-order symmetric stress tensors and their corresponding gradients of displacements are derived according to the proposed reciprocal relation. The results show that the essential reason for the failure of the conventional reciprocal theorem is that the effect of higher order surface forces and surface stresses that are produced by strain gradients contributes to the reciprocal work. When the strain gradients work-conjugating to stress gradients are considered, they satisfy the local reciprocal relation that cannot be degenerated to the conventional reciprocal theorem in the form of body forces and inertial forces. The theory developed in this paper may have an increasingly profound effect on continuum mechanics and is expected to be a helpful tool for the mechanics of cellular structures homogenized by strain gradient elasticity.

中文翻译：

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传统的倒数定理在应变梯度弹性中是否失效？


基于倒数定理的力法和位移法在结构力学领域发挥着重要作用，并在结构力学中得到了成功的应用。然而，有趣的是，当作者试图将其应用于应变梯度梁的变形问题时，却出现了意想不到的悖论。为了研究这一矛盾现象的物理本质，提出了线弹性应变梯度弹性框架内高阶应力与高阶应变之间的倒数关系，并用于证明更新的倒数定理。同时，根据所提出的倒数关系，推导了任意二阶对称应力张量的任意梯度及其对应的位移梯度的倒数定理。结果表明，传统倒易定理失效的根本原因是应变梯度产生的高阶表面力和表面应力的作用对倒易功做出了贡献。当考虑应变梯度与应力梯度共轭时，它们满足局部倒易关系，该关系不能退化为体力和惯性力形式的常规倒易定理。本文提出的理论可能会对连续介质力学产生日益深远的影响，并有望成为应变梯度弹性均匀化多孔结构力学的有用工具。