In this paper, we propose a robust low-order stabilization-free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress-hybrid principle. We refer to this approach as the Stress-Hybrid Virtual Element Method (SH-VEM). In this method, the Hellinger$-$Reissner variational principle is adopted, wherein both the equilibrium equations and the strain-displacement relations are variationally enforced. We consider small-strain deformations of linear elastic solids in the compressible and near-incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are $\textit{virtual}$. The stress field, similar to the stress-hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5-parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain-displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH-VEM is free of volumetric and shear locking, and it converges optimally in the $L^2$ norm and energy seminorm of the displacement field, and in the $L^2$ norm of the hydrostatic stress.

翻译：本文针对线弹性问题，提出了一种基于应力-混合原理的四边形网格鲁棒低阶无稳定化虚拟单元法，我们称之为应力-混合虚拟单元法（SH-VEM）。该方法采用Hellinger–Reissner变分原理，对平衡方程和应变-位移关系进行变分强加。我们考虑可压缩与近不可压缩条件下线弹性固体的小变形问题，网格采用四边形单元（包括凸四边形与凹四边形）。单元内位移场由虚拟规范形函数的线性组合逼近。应力场则借鉴Pian和Sumihara的应力-混合有限元法，采用对称张量多项式的线性组合表示。每个单元采用5参数应力场展开，并对畸变四边形应用应力变换方程。在应变-位移关系的变分表述中，通过散度定理将应力系数表达为节点位移的函数，从而得到仅以节点位移为未知量的公式形式。针对线弹性中的多个基准问题给出数值结果，表明SH-VEM无体积闭锁和剪切闭锁，在位移场的L²范数和能量半范数以及静水应力的L²范数下均达到最优收敛阶。