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应变梯度弹性理论的控制方程是位移场的四阶偏微分方程,Galerkin离散要求形函数C1连续。将non-Sibsonian插值函数作为三次单纯形Bernstein-Bézier多项式的基坐标,构建了C1自然邻近插值函数。由于C1形函数对结点函数值和梯度值的插值特性,本质边界条件可以直接施加。具体算例包括双材料系统的边界层分析和中心圆孔无限大板承受双轴拉伸时位移和应力分布的分析,数值解与理论解吻合得较好,表明C1自然邻近迦辽金法能够用来分析应变梯度弹性理论问题。
The governing equation of strain-gradient elasticity theory is the fourth-order partial differential equation of displacement field. Galerkin discretization requires that the shape function C1 be continuous. The non-Sibsonian interpolation function is taken as the base coordinate of the cubic simplex Bernstein-Bézier polynomial, and a C1 natural neighbor interpolation function is constructed. Due to the interpolating nature of the C1-shaped function on node function values and gradient values, the essential boundary conditions can be applied directly. Specific examples include the analysis of the boundary layer of bimaterial system and the analysis of the displacement and stress distribution under biaxial tension in the infinite plate of the central circular hole. The numerical solution is in good agreement with the theoretical solution, which shows that C1 is naturally close to the Galerkin method Used to analyze the theory of strain gradient elasticity.