We establish a new H\(^2\)-Korn’s inequality and its discrete analog, which greatly simplify the construction of nonconforming elements for a linear strain gradient elastic model. The Specht triangle (Specht in Int J Numer Methods Eng 28:705–715, 1988) and the NZT tetrahedron (Wang et al. in Numer Math 106:335–347, 2007) are analyzed as two typical representatives for robust nonconforming elements in the sense that the rate of convergence is independent of the small material parameter. We construct the regularized interpolation operators and the enriching operators for both elements, and prove the error estimates under minimal smoothness assumption on the solution. Numerical results for the smooth solution, and the solution with boundary layer are consistent with the corresponding theoretical prediction.

我们建立了一个新的 H \(^2\) -Korn 不等式及其离散模拟，这极大地简化了线性应变梯度弹性模型的不合格单元的构建。Specht 三角形 (Specht in Int J Numer Methods Eng 28:705–715, 1988) 和 NZT 四面体 (Wang et al. in Numer Math 106:335–347, 2007) 作为稳健不合格元素的两个典型代表被分析收敛速度与小的材料参数无关。我们为两个元素构造了正则化插值算子和丰富算子，并证明了在最小平滑度假设下对解的误差估计。光滑解和带边界层解的数值结果与相应的理论预测一致。