Yan Shang

In this work, a stabilized second‐order virtual element method is proposed for two‐dimensional elasticity problems. Relying solely on low‐order degrees of freedom, a Euclidean orthogonal projection in the degree of freedom space is constructed using a least‐squares approach. Leveraging this projection, a streamlined serendipity virtual element scheme is established that eliminates redundant moment degrees of freedom within the standard virtual element, achieving a significant reduction in implementation complexity while preserving high accuracy. To validate the performance of the new element, classical numerical examples are presented, with a special emphasis on bending problems of various beam types solved on an extremely coarse mesh—a demanding capability that is scarcely reported in the existing literature on the virtual element method. The proposed element demonstrates superior accuracy in displacement and stress fields, achieving optimal convergence rates in both the norm and energy semi‐norm. Numerical results indicate that the new element model exhibits remarkable robustness against mesh distortion and inherent geometric adaptability, outperforming existing standard finite element and virtual element models while delivering highly competitive computational accuracy relative to hybrid virtual element models, thereby establishing itself as a reliable and efficient computational tool for continuum solid mechanics.

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.

Flexoelectricity, an electromechanical coupling phenomenon induced by strain gradients, finds plate structures to be particularly suitable due to their inherent susceptibility to developing such gradients. However, practical exploitation requires efficient numerical tools for modeling geometrically complex thin‐walled structures. This work presents a novel 4‐node quadrilateral Trefftz element for analyzing flexoelectric thin plates within consistent couple stress theory (CCST). To accomplish this, the non‐classical flexoelectric Kirchhoff plate model is first established based on CCST, and Trefftz functions satisfying the governing equations are systematically derived. These Trefftz functions subsequently serve as basis functions for constructing the displacement field, and generalized conforming conditions at collocation points are used to link them to nodal degrees of freedom. The performance of the proposed element is rigorously evaluated through benchmark problems involving static bending and free vibration analyses. Numerical results are compared against analytical solutions or three‐dimensional finite element simulations. It is shown that the element achieves high precision in capturing the size‐dependent flexoelectric responses, which are prominent at the micro/nano‐scale, and exhibits superior robustness against mesh distortion. This establishes the method as an efficient numerical tool for the design of flexoelectric plates, such as those used in micro‐sensors and energy harvesters.

Purpose: This paper aims to extend the recent unsymmetric 8-node, 24-degree-of-freedom hexahedral solid element US-ATFH8 to simulate the compressible and nearly incompressible behaviors associated with growth-induced deformations in soft materials. Design/methodology/approach: An effective total Lagrangian algorithm is designed to extend the element US-ATFH8 to the geometrically nonlinear analysis in which different test and analytical trial functions (ATFs) are employed. Then, the volumetric growth theory of biological tissue is embedded into the novel element US-ATFH8 to simulate the growth-induced deformations in soft materials. Findings: The numerical examples, including cubes, cantilever beams, rods, tubes, petals and growth-induced deformations in skin and ventricles, fully prove that the element possesses high accuracy and a fast convergence rate in modeling growth behaviors. It can still perform well even when other conventional finite element methods cannot work. Originality/value: It is again proved that the unsymmetric finite element with ATFs is still suitable for the constitutive model of growth-induced deformation.

The cover image is based on the article Corotational Unsymmetric Membrane Element Formulation for Geometric Nonlinear Analysis of Flexoelectric Solids Within the Consistent Couple Stress Theory by Yan Shang et al., https://doi.org/10.1002/nme.70045. image

In this work, the corotational (CR) penalty membrane element formulation for geometric nonlinear analysis of flexoelectric solids with large displacement but small deformation is proposed. To achieve this, the two‐dimensional couple stress‐based flexoelectric model is first extended to the CR form where the effect of rigid body rotation is excluded and, accordingly, the curvature that governs the electromechanical behavior of flexoelectricity is defined as the derivatives of the elastic rotation in the CR configuration. Next, a quadrilateral 8‐node element is constructed based on the CR method. In the development, with the use of the penalty function method, the independent nodal rotation degrees of freedom are utilized to approximate the elastic rotation field, ensuring the higher‐order continuity requirement for the displacement test function in a weak sense. In addition, to fully improve the element performance, the trial function of CR force‐stress is formulated based on the polynomial force‐stress functions, which are continually updated with reference to the latest deformed configuration. As demonstrated by the numerical benchmark examples, the new element can efficiently and accurately predict the geometric nonlinear electromechanical response of slender flexoelectric structures and capture the size effect very well.