In this paper, we introduce and analyze a lowest-order locking-free weak Galerkin (WG) finite element scheme for the grad-div formulation of linear elasticity problems. The scheme uses linear functions in the interior of mesh elements and constants on edges (2D) or faces (3D), respectively, to approximate the displacement. An $H(div)$-conforming displacement reconstruction operator is employed to modify test functions in the right-hand side of the discrete form, in order to eliminate the dependence of the $Lam\acute{e}$ parameter $\lambda$ in error estimates, i.e., making the scheme locking-free. The method works without requiring $\lambda \|\nabla\cdot \mathbf{u}\|_1$ to be bounded. We prove optimal error estimates, independent of $\lambda$, in both the $H^1$-norm and the $L^2$-norm. Numerical experiments validate that the method is effective and locking-free.

翻译：本文针对线弹性问题的梯度-散度形式，引入并分析了一种最低阶免锁死弱伽辽金（WG）有限元格式。该格式在网格单元内部采用线性函数，在边（二维）或面（三维）上采用常数分别逼近位移。为消除离散形式右端测试函数中对拉梅参数λ的依赖性（即使格式免锁死），本文引入了一个H(div)相容的位移重构算子来修改测试函数。该方法无需要求‖∇·u‖₁有界。我们证明了与λ无关的H¹范数和L²范数下的最优误差估计。数值实验验证了该方法的有效性和免锁死特性。