中文摘要：

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C^1连续。将non-Sibsonian插值函数作为三次单纯形Bemstein．Bezier多项式的基坐标，构建了C^1自然邻近插值函数。由于C^1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C^1自然邻近迦辽金法能够用来分析应变梯度弹性理论问题。

英文摘要：

The governing equations of strain gradient elasticity are fourth-order partial differential equations. Galerkin discretization would require C1 continuous shape functions. C^1 natural neighbor interpolant can be realized when embedding non-Sibsonian interpolant in the Bernstein-Beier surface representation of a cubic simplex. Essential boundary conditions are imposed directly in a Galerkin scheme for the strain gradient elasticity because the C^1 interpolant has the interpolating properties for nodal function and nodal gradient values. Boundary layer analysis and infmite plate with a central circular hole subjected to the biaxial tension are analyzed. The numerical solutions agree well with the analytical solutions, which shows that C^1 natural neighbor Galerkin method can be used to analyze the strain gradient elasticity problems.