Several beam and plate models have been recently developed in the literature to accommodate for size-dependence. These are usually obtained starting from a generalized continuum theory (such as the couple-stress, strain-gradient or non-local theory or their modifications) and then deducing the governing equations through Hamilton’s principle and ingenuous kinematical assumptions. This approach, originated by Kirchhoff, usually fails to reproduce the dispersion features of the equivalent 3D theory. Besides, it produces a variety of models, in dependence of the different assumptions, such as Kirchhoff’s or Mindlin’s. In contrast, in this paper we adopt asymptotic reduction: moving from the couple-stress linear theory of elasticity with micro-inertia, we deduce new models for elongation and flexural deformation of microstructured plates. The resulting models are consistent, in the sense that they reproduce the dispersion features of the corresponding 3D body. Also, models are unique, for they may only differ by the order of the approximation. We find that microstructure especially affects inertia terms, which can be hardly captured by a-priori kinematical assumptions. For static flexural deformations, our results match those already obtained assuming plane cross-sections within the modified couple-stress theory. In fact, we show that couple-stress, reduced couple-stress and strain gradient theories all lead to equivalent results. Higher order models are also given, that describe the near first-cut-off behaviour and account for thickness deformations in the spirit of Timoshenko.

文献中最近开发了几种梁和板模型，以适应尺寸依赖性。这些通常是从广义连续论（例如耦合应力，应变梯度或非局部理论或它们的修改）开始，然后通过汉密尔顿原理和固有的运动学假设推导控制方程而获得的。这种由基尔霍夫（Kirchhoff）提出的方法通常无法重现等效3D理论的色散特征。此外，它还根据不同的假设（例如基尔霍夫或明德林的假设）生成了各种模型。相反，在本文中，我们采用渐近归约：从偶应力线性理论出发考虑到微惯性的弹性，我们推导了微结构板的伸长和弯曲变形的新模型。从某种意义上说，所得模型是一致的，它们可以再现相应3D实体的色散特征。而且，模型是唯一的，因为它们可能仅在近似顺序上有所不同。我们发现，微观结构特别影响惯性项，而先验运动学假设几乎无法捕获惯性项。对于静态挠曲变形，我们的结果与修改的耦合应力理论中假设平面横截面的情况已经获得的结果相匹配。实际上，我们证明了偶应力，减小的偶应力和应变梯度理论都可以得出相同的结果。还给出了高阶模型，