New weak form with reduced boundary parameters for facilitating the numerical investigation of first-order linear elastic microstructures

The article delves into the challenges posed by the size-dependent deformation of micro- and nano-scale structures, which classical elasticity theory fails to address. It explores various non-classical continuum theories, including strain gradient, micropolar, and nonlocal elasticity theories, which incorporate additional deformation measures and length scale parameters. The focus is on the modified couple stress theory (M-CST), which simplifies these complexities by reducing the number of length-related constants to one. The article introduces a new weak form for the numerical analysis of micro-beams, derived from the M-CST and non-shear beam theory, which simplifies the boundary conditions and governing equations. This approach leads to the development of a novel two-node couple stress beam element, which is validated through numerical examples and comparisons with existing models. The results demonstrate the element's accuracy and efficiency in capturing size and shear effects, making it a valuable tool for the analysis of microstructures under various loading and support conditions. The article also discusses the limitations of previous models and the advantages of the proposed approach, providing a comprehensive overview of the current state and future directions in the numerical analysis of microstructures.