Shaat M, Ghavanloo E, Emam S
Mechanical Engineering Department, Abu Dhabi University, Al Ain, P.O.BOX 1790, United Arab Emirates.
School of Mechanical Engineering, Shiraz University, Shiraz, 71963-16548, Iran.
Sci Rep. 2020 May 14;10(1):7984. doi: 10.1038/s41598-020-64542-y.
A novel micromorphic beam theory that considers the exact shape and size of the beam's microstructure is developed. The new theory complements the beam theories that are based on the classical mechanics by modeling the shape and size of the beam's microstructure. This theory models the beam with a microstructure that has shape and size and exhibits microstrains that are independent of the beam's macroscopic strains. This theory postulates six independent degrees of freedom to describe the axial and transverse displacements and the axial and shear microstrains of the beam. The detailed variational formulation of the beam theory is provided based on the reduced micromorphic model. For the first time, the displacement and microstrain fields of beams with elongated microstructures are developed. In addition, six material constants are defined to fully describe the beam's microscopic and macroscopic stiffnesses, and two length scale parameters are used to capture the beam size effect. A case study of clamped-clamped beams is analytically solved to show the influence of the beam's microstructural stiffness and size on its mechanical deformation. The developed micromorphic beam theory would find many important applications including the mechanics of advanced beams such as meta-, phononic, and photonic beams.
一种考虑梁微观结构精确形状和尺寸的新型微观形态梁理论被提出。新理论通过对梁微观结构的形状和尺寸进行建模，对基于经典力学的梁理论进行了补充。该理论对具有形状和尺寸且表现出与梁宏观应变无关的微观应变的微观结构梁进行建模。该理论假定六个独立的自由度来描述梁的轴向和横向位移以及轴向和剪切微观应变。基于简化的微观形态模型给出了梁理论的详细变分公式。首次推导出具有细长微观结构的梁的位移和微观应变场。此外，定义了六个材料常数来全面描述梁的微观和宏观刚度，并使用两个长度尺度参数来捕捉梁的尺寸效应。对两端固定梁的一个案例进行了解析求解，以展示梁的微观结构刚度和尺寸对其机械变形的影响。所提出的微观形态梁理论将有许多重要应用，包括超材料梁、声子梁和光子梁等先进梁的力学。
