Modified Strain Gradient–Based Finite-Element Approach for the Analysis of Size-Dependent Higher-Order Microshells

Abstract This study developed a comprehensive framework for size-dependent higher-order microshells by incorporating the modified strain gradient continuum theory (MSGT). The microshell formulations integrate higher-order kinematics and explicitly account for size-reduction effects via material length scale parameters (MLSPs) within the MSGT framework, which are systematically embedded in the governing equations using Hamilton’s principle. Validation studies confirmed the accuracy and reliability of the proposed finite-element formulation through comparisons with prior literature. The approach was applied to microshells of various geometries and boundary conditions to assess the influence of MLSPs, edge constraints, and shell dimensions on the natural frequencies. For illustrative purposes, free vibration problems were solved using a novel finite-element formulation, with discretization tailored to capture strain gradient effects in the equations of motion. Numerical examples demonstrated that incorporating strain gradients increases shell stiffness and nondimensional natural frequencies, and the effects are most significant in smaller microshells. Results compared against those of modified couple stress theory (MCST) and classical elasticity theory (CT) indicated that both represent special cases of the MSGT formulation achievable through specific parameter selections. Notably, the rigidity and vibration response of the shells analyzed via MSGT exceeded those obtained from MCST and CT, and all models converged as size effects decreased (for higher dimensionless length scale parameter values). The developed MSGT finite-element allows a unified treatment of both classical and couple stress models, providing an effective tool for microscale structural analysis. The findings highlight that size effects are most prominent for low values of dimensionless length scale parameters, resulting in significantly increased natural frequencies and improved mechanical stability in microstructured shell systems.