Size-Dependent Nonlinear Analysis of Microbeams: A Corotational Finite Element Formulations based on Modified Strain Gradient and Couple Stress Theories

This study develops a novel nonlinear finite element framework for microbeams that synergistically integrates modified strain gradient theory (MSGT) and modified couple stress theory (MCST) with corotational formulations. This unique integration enables the accurate modeling of size-dependent effects and geometric nonlinearity in both Euler-Bernoulli and Timoshenko beam models. A key advancement is our robust computational approach, which achieves unprecedented solution stability and efficiency even under extreme deformations, overcoming critical convergence challenges that have limited previous simulations. Numerical results reveal a strong multiscale coupling between geometric nonlinearity and material length scales, directly impacting MEMS reliability predictions. These findings provide critical guidelines for MEMS design, particularly in micro-actuators and sensors where classical continuum theories significantly underestimate stiffness. The proposed formulation bridges a key gap in computational microbeam mechanics by simultaneously addressing geometric nonlinearity, shear deformation, and size effects, offering a rigorously validated tool for optimizing performance in real-world microscale applications.