Modeling Size Effects in Elasticity With <scp> <i>C</i> ¹ </scp> Virtual Elements: A Two‐Dimensional Formulation for Modified Couple Stress Theory

ABSTRACT To accurately capture the size effects commonly observed in micro‐ and nano‐scale mechanics, many higher‐order continuum theories, such as strain gradient theory (SGT) and couple stress theory (CST), have been proposed, placing higher continuity requirements on the elements. However, traditional finite element methods encounter inherent difficulties in constructing elements that satisfy C 1 continuity conditions, primarily due to the complexity of shape function construction. In contrast, the virtual element method (VEM), owing to its exceptional adaptability to arbitrary element geometries and its unique advantages in constructing elements with arbitrary‐order continuity, provides a promising solution to this challenge. This work innovatively develops a C 1 continuous VEM for the two‐dimensional modified couple stress theory (MCST). This is achieved mainly via three steps. First, a comprehensive mathematical framework is established, including the construction of the virtual element space, the definition of projection operators, and a rigorous proof of the method's optimal convergence in the L 2 norm. Second, addressing engineering application needs, a detailed numerical implementation procedure is provided, outlining critical steps such as stiffness matrix assembly and stabilization term handling. Finally, the method's superior performance in computational accuracy, mesh adaptivity, and convergence rate is systematically validated through numerical examples, demonstrating its significant potential for applications in the analysis of micro‐ and nano‐scale structures.