基于偶应力/应变梯度理论的样条扩展有限元方法

The couple stress theory and the strain gradient theory are two kinds of new and widely used gradient theories, which focus on materials at the micro/nano-scale, and explain the size effects by introducing the material length parameters into the constitutive models. Compared with the conventional continuum mechanics, the convergence and nodal parameters for the couple stress theory and the strain gradient theory are substantially more complicated. Besides, it includes both macro and micro scales, which makes some difficulties in FEM for the couple stress/strain gradient theory. The extended finite element method (XFEM) is a powerful tool for solving the problems with discontinuous or singular fields, which is a great development for conventional FEM. The XFEM is good at overcoming the difficulties in mesh partitions due to the difference in double scales. However, the XFEM is based on the conventional continuum mechanics, and the size effects are not considered in the XFEM. In this project, we will use the spline interpolation method to generalize the XFEM with the couple stress/strain gradient theory for macro-micro scales analysis. The key problems are to construct the enrichment functions for XFEM according to the couple stress/strain gradient theory by using the spline interpolation method. The new spline enrichment functions need to be satisfying the partition of unity and linear independence, so that the new spline XFEM can get rid of extra dofs and overcome those difficulties in XFEM with the couple stress/strain gradient theory.

偶应力/应变梯度理论是刻画材料在细观尺度的应变局部化现象的新理论，其基本方程是在传统连续体力学方程基础上增加含有材料长度参数的高阶方程，在收敛性理论和节点参数都有很大差异，并且同时具有宏观和微观两个尺度，现有的有限元方法用于求解偶应力/应变梯度理论方程存在很大的困难。扩展有限元是在传统有限元的基础上进行了重要改进，适合解决尺度差异造成的有限元网格剖分问题。但目前扩展有限元主要以连续介质模型为基础，还未考虑微缺陷结构引起的尺度效应。本项目拟利用样条插值方法开展基于偶应力/应变梯度理论的样条扩展有限元方法的研究，将扩展有限元方法的优势和偶应力/应变梯度理论相结合，发展宏观-微观相结合的有限元分析方法。关键问题为基于三角形面积坐标插值方法构造用于扩展有限元的样条扩充形函数，满足偶应力/应变梯度理论的要求，使其具有单位分解性和线性无关性，并消除附加自由度，克服扩展有限元的不足。

本项目围绕基于偶应力/应变梯度理论的样条扩展有限元方法开展研究工作，为了研究扩充单元形函数的方法，我们同时结合了近年来发展的比例边界有限元方法。本项目开展了相关比例边界有限元方法的一些理论和算法研究，并结合多元样条插值方法，发挥两种方法的优势，构造了基于样条插值和比例边界有限元方法的包含节点位移和转角自由度的扩展形函数，并满足用于偶应力/应变梯度理论有限元关于单元形函数的连续性和多项式完备阶的要求。其他主要成果还包括：证明了比例边界有限元的高阶多项式完备性；构造了基于离散Kirchhoff薄板理论的允许1-irregular退化的多边形样条薄板单元；构造了两个基于Mindlin板理论的四边形样条单元；构造了求解双调和方程具有超收敛性的非协调四边形样条单元；构造了具有二阶多项式完备性的15节点三棱柱样条单元等，获得了一系列的研究成果。在项目执行期间，在《中国科学: 物理学力学天文学》，《Computer Methods in Applied Mechanics and Engineering》，《European Journal of Mechanics - A/Solids》，《Journal of Computational and Applied Mathematics》等高水平期刊共发表论文15篇，其中SCI检索9篇，EI检索6篇。