The virtual element method (VEM) allows discretization of the problem domain with polygons in 2D. The polygons can have an arbitrary number of sides and can be concave or convex. These features, among others, are attractive for meshing complex geometries. VEM applied to linear elasticity problems is now well established. Nonlinear problems involving plasticity and hyperelasticity have also been explored by researchers using VEM. Clearly, techniques for extending the method to nonlinear problems are attractive. In this work a novel first order consistent virtual element method is applied within a static co-rotational framework. To the author's knowledge this has not appeared before in the literature with virtual elements. The formulation allows for large displacements and large rotations in a small strain setting. For some problems avoiding the complexity of finite strains, and alternative stress measures, is warranted. Furthermore, small strain plasticity is easily incorporated. The basic method, VEM specific implementation details for co-rotation, and representative benchmark problems are illustrated. Consequently, this research demonstrates that the co-rotational VEM formulation successfully solves certain classes of nonlinear solid mechanics problems. The work concludes with a discussion of results for the current formulation and future research directions.

翻译：虚拟单元方法（VEM）允许在二维问题中使用多边形进行域离散化，这些多边形可以具有任意数量的边，且可为凹形或凸形。此类特性（包括其他优势）使其在复杂几何体网格划分中极具吸引力。目前，VEM在线弹性问题中的应用已相当成熟，研究者也尝试将其拓展至涉及塑性和超弹性的非线性问题。显然，发展适用于非线性问题的VEM扩展技术具有重要意义。本文首次在静态协同旋转框架下应用一种一阶一致虚拟单元方法（据作者所知，此前文献中尚未出现虚拟单元与此框架的结合）。该方法在小应变设定下允许大位移和大转动，从而避免某些问题中有限应变及其替代应力度量的复杂性，且可便捷地纳入小应变塑性模型。文中阐述了基本方法、协同旋转的VEM特定实现细节及代表性基准问题。研究表明，协同旋转VEM公式能够成功求解特定类别的非线性固体力学问题。最后，本文讨论了当前公式的计算结果及未来研究方向。