Abstract

Starting from the variational principle of virtual power for the three-dimensional equations of the micropolar theory of elasticity and using generalized series in terms of the plate thickness coordinates a new higher order models of orthotropic micropolar plates have been developed here for the first time. Following carrera unified formulation, the stress and strain tensors, as well as the vectors of displacements and rotation, have been expanded into series in terms of the plate thickness coordinates. Then, all the equations of the micropolar theory of elasticity (including generalized Hooke’s law) have been transformed to the corresponding equations for the coefficients of the series expansion on the plate thickness coordinates. A system of differential equations in terms of the displacements and rotation vectors and natural boundary conditions for the coefficients of the series expansion of the plate thickness coordinates have been obtained in the same way as in the classical theory of elasticity. All equations for the higher order theory of micropolar plates have been developed and presented here. The case of complete linear expansion has been considered in detail and compared with the theories based on shear deformation and Kirchhoff hypothesis. The obtained equations can be used for calculating the stress-strain and for modeling thin walled structures in macro, micro, and nanoscale when taking into account micropolar couple stress and rotation effects.

摘要

首次从微极弹性理论三维方程的虚功率变分原理出发，利用板厚坐标的广义级数，建立了新的正交各向异性微极板高阶模型。遵循卡雷拉统一公式，应力和应变张量以及位移和旋转矢量已根据板厚坐标展开成系列。然后，将微极弹性理论的所有方程（包括广义胡克定律）转化为相应的板厚坐标上的级数展开系数方程。以与经典弹性理论相同的方式获得了关于位移和旋转矢量以及板厚坐标级数展开系数的自然边界条件的微分方程组。微极板高阶理论的所有方程都已在此处开发和介绍。详细考虑了完全线性膨胀的情况，并与基于剪切变形和基尔霍夫假设的理论进行了比较。当考虑到微极耦合应力和旋转效应时，获得的方程可用于计算应力应变和模拟宏观、微观和纳米尺度的薄壁结构。