This paper presents a novel equilibrium-based approach to the linear homogenization of shear-deformable beams and plates with periodic heterogeneity. The proposed approach leverages the fact that, under equilibrium, the stress resultants and sectional strains in beams and plates vary at most linearly with respect to the axial or in-plane coordinates. Consequently, the displacement fields within a representative volume element (RVE) are composed of rigid-body, constant-strain, and linear-strain deformation modes, which are proportional to the stress resultants at the center of the beam or plate. By enforcing kinematic compatibility and equilibrium conditions on the lateral surfaces of adjacent RVEs, along with energetic equivalence conditions, the local and global equilibrium equations of the RVEs are derived, leading to singular linear equations for the warping matrix and sectional compliance matrix. The proposed method accounts for all potential stiffness couplings, resulting in fully coupled 6 × 6 and 8 × 8 sectional stiffness matrices for periodic beams and plates, respectively. Notably, the approach addresses stiffness coupling between transverse shear and other deformation modes, which are overlooked in other homogenization methods for periodic structures. Additionally, an equivalent minimization formulation is introduced to determine the warping field and sectional compliance matrix, addressing the homogenization problem in a variational manner. Numerical examples demonstrate that the macro-beam and plate models, using the predicted stiffness matrices, provide accurate displacement fields and three-dimensional stress fields within the linear deformation range. The limitations of the proposed method in addressing problems with significant geometric nonlinearities are also highlighted through numerical examples.

中文翻译：

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具有周期性异质性的剪切变形梁和板的均质化：一种基于统一平衡的方法


本文提出了一种新的基于平衡的方法，用于对具有周期性异质性的剪切变形梁和板进行线性均匀化。所提出的方法利用了这样一个事实，即在平衡下，梁和板中的应力合力和截面应变相对于轴向或平面内坐标最多呈线性变化。因此，代表性体积单元 （RVE） 内的位移场由刚体、恒定应变和线性应变变形模式组成，这些模式与梁或板中心的应力合力成正比。通过在相邻 RWE 的侧面强制执行运动学兼容性和平衡条件以及能量等效条件，推导出 RWE 的局部和全局平衡方程，从而得到翘曲矩阵和截面柔度矩阵的奇异线性方程。所提出的方法考虑了所有潜在的刚度耦合，从而分别为周期性梁和板的 6 × 6 和 8 × 8 截面刚度矩阵完全耦合。值得注意的是，该方法解决了横向剪切和其他变形模式之间的刚度耦合问题，而这些模式在其他周期性结构的均质化方法中被忽视了。此外，引入了等效最小化公式来确定翘曲场和截面柔度矩阵，以变分方式解决均质化问题。数值实例表明，宏梁和板模型使用预测的刚度矩阵，在线性变形范围内提供精确的位移场和三维应力场。 通过数值示例，还强调了所提出的方法在解决具有显著几何非线性的问题方面的局限性。