严格收敛的C0类偶应力/应变梯度理论高阶有限元方法

The couple stress/strain gradient theories are theories for microstructures and can describe scale effects. Unlike the homogeneous differential equation of continuum mechanics, the equations of couple stress/strain gradient theory are non-homogeneous differential equations. The strain in conventional continuum elasticity only contains first derivatives of displacements, and the strain of thin plate only contains second derivatives of displacements. However, the first and second derivatives of displacements are simultaneously contained in the finite element in couple stress/strain gradient theory. The patch test can be used to examine the convergence of the element and construct a convergence element. The enhanced patch test of C0 type couple stress/strain gradient theory has test function with 3rd order, however, it is difficult to satisfy these requirements by the conventional finite element method. Up to now, there is not any finite element for C0 couple stress/strain gradient theory can pass the 3rd enhanced patch test. Based on a new arbitrary order Timoshenko beam function, we try to establish finite elements of couple stress/ strain gradient theories which can pass the strict test, and this can solve the problem of finite element for C0 couple stress/strain gradient theory

偶应力/应变梯度理论有限元是描述材料在细观尺度上尺度效应的有限元方法，不同于连续体力学的齐次阶微分方程,其基本方程在传统连续体力学方程基础上增加含材料长度参数的高阶方程，属非齐次阶微分方程。偶应力/应变梯度理论有限元中应变同时有位移的一阶和二阶导数，这两个导数部分都要求节点参数包括位移的函数值和导数值，是一个新的提法。分片检验在有限元方法中即可以用来检验单元的收敛性又能用来指导建立收敛单元。C0类偶应力/应变梯度理论的分片检验函数应是满足平衡方程的三次位移函数，这些要求给单元构造带来了很大困难，目前尚无能通过增强型分片检验的C0偶应力/应变梯度理论的单元。本项目中利用任意阶的Timoshenko梁函数来构建单元函数，建立能通过这一检验的C0单元。这种构造单元的新方法可以解决C0偶应力/应变梯度理论有限元的问题。

现有有限元仿真软件中使用的C0单元，不通过严格的收敛检验。本项目目的在于研究位移和转角独立的C0理论及其收敛性理论，并关注基于此理论的高性能有限元方法。全面审视了C0和C1偶应力理论、C0中厚板理论及其收敛检验，给出了C1和C0理论的增强型分片检验的检验函数，指出现有C0单元存在收敛性问题。依据C0理论和增强型分片检验，建立了高阶杂交应力单元，并建立了自由振动和稳定分析的有限元格式。构造的单元同时通过零剪力分片检验和非零常剪力分片检验。为了通过严格的收敛条件，运用任意阶Timoshenko梁函数来构造杂交应力元的边界位移插值函数，域内应力采用多项式插值，应力插值函数需要满足位移表示的平衡条件，建立了高阶杂交应力单元。自由振动有限元格式中分别采用集中质量矩阵和一致质量矩阵，其中集中质量矩阵采用了精化元方法来改进精度。稳定分析的有限元格式中采用了精化方法改进的几何刚度矩阵。算例表明单元满足严格的增强型分片检验，且收敛速度快，精化元方法显著提高了振动和稳定分析的计算精度,并且通过零剪力分片检验和非零常剪力分片检验，保证严格收敛。这一工作有利于促进有限元软件里的单元扩充和发展。