应变梯度理论自然邻近混合伽辽金法

应变梯度理论考虑了位移二阶梯度对应变能密度函数的贡献，在本构关系中引入了与材料微结构特征尺寸相关的参数，可以唯象地解释尺度效应现象。基于约束变分原理，把位移与位移一阶梯度作为独立场变量，使用拉格朗日乘子法引入二者的协调关系，放松对试探函数连续性与完备性的要求，建立了二维及三维问题的应变梯度理论自然邻近混合伽辽金法。通过算例对方法的计算性能进行了考查，结果表明，该方法具有良好的数值精度，能够模拟材料力学性能的尺度效应。

Mixed natural neighbour Galerkin method for the strain gradient elasticity

Based on the constrained variational principle,the mixed natural neighbour Galerkin method for 2D and 3D strain gradient elasticity is proposed.The displacements and the first-order gradients of displacements are chosen to be independent nodal degrees of freedom,and the geometrical constraints between them are enforced by using Lagrange multipliers,which enables the relaxation of the continuity and consistency requirements.All the nodal unknowns including the Lagrange multipliers are approximated by the non-Sibsonian interpolation,which is easy to evaluate and can simplify the imposition of essential boundary condition.The validity and accuracy of the proposed method are investigated through numerical examples.The results show that size effects are observed when the length of deformation field and the characteristic length of the material are comparable,and excellent agreements with analytical solutions are obtained.