偶应力理论的径向点插值无网格法研究

STUDY ON THE RADIAL POINT INTERPOLATION MESHLESS METHOD FOR COUPLE STRESS THEORY

偶应力理论引入了位移二阶导数和相应的高阶边界条件,在有限元实施过程中,需要实现位移函数C1连续,这对传统有限元方法来说是一个苛刻的要求。无网格法能够实现位移函数的高阶导数连续,该文基于应变梯度偶应力理论下的虚功原理,导出了其无网格实施方法。无网格法的形函数一般不具有插值特性,本质边界条件施加困难,为克服这一困难,采用了带有多项式基的径向点插值法,构造的形状函数具有插值特性,可直接施加本质边界条件。数值结果表明,该方法数值结果稳定、精度高。

Displacement＇s second order derivatives and the higher order boundary conditions are introduced in the couple stress theory.In its finite element implementation,displacements＇ shape functions are required to be C1 continuous across the adjacent elements＇ boundaries for the couple stress theory,which is a rigorous requirement for the traditional finite element method.A meshless method can realize the higher order continuity of displacement shape functions.Based on the virtual wok principle,the meshless implementation process for the couple stress theory is derived.Generally,the shape functions of the meshless method do not possess the merits of interpolation functions,and the essential boundary conditions cannot be exerted directly.In order to overcome this defect,the radial point interpolation method coupled with the polynomials is adopted to develop the shape functions.The resulted shape functions have the merits of interpolation characteristics and the essential boundary condition can be exerted directly.Numerical results show that the developed meshless method can deliver stable and precise results.