偶应力/应变梯度理论的精化不协调元方法

项目名称： 偶应力/应变梯度理论的精化不协调元方法

项目编号： No.11202039

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 赵杰

作者单位： 大连海事大学

项目金额： 24万元

中文摘要： 偶应力/应变梯度理论是成功解释尺度效应的连续介质理论，其相应的数值方法是微纳结构研究的必要基础。偶应力/应变梯度理论的势能泛涵同时包含位移的一、二阶导数，建立协调有限单元需满足位移插值函数C1连续。然而，C1协调单元的节点参数含有位移的高阶导数，构造和应用都较为困难。对于目前广泛采用的C0单元，需要通过Lagrange乘子或罚函数来约束独立插值的位移和位移梯度，由此带来额外的计算量和计算结果的不确定性。相对于协调单元，不协调单元放松了单元间的连续条件，可以构造更为灵活的单元函数，便于建立高精度单元。本项目将研究偶应力/应变梯度理论不协调元的收敛准则，提出一类放松单元间连续性要求的变分原理，建立同时满足C0连续（或弱连续）、二次完备和C1弱连续的精化不协调单元。通过对偶应力/应变梯度理论精化不协调元方法的系统研究可以加速推进该理论的研究和工程应用，促进微纳技术的发展。

中文关键词： 偶应力/应变梯度理论；有限元；变分原理；弱连续；分片检验

英文摘要： Couple stress/strain gradient theory is a type of constitutive theory which can successfully explain the size effects and the related numerical method is the necessity of micro/nano structure research. The displacement interpolation function of conforming element should satisfy the requirement of C1 continuity as first and second derivatives of the displacement are involved in potential energy principle of the couple stress/strain gradient theory. C1 conforming elements contain the nodal parameters with high order derivatives, and are complicated to construct and implement. Currently, the most widely used couple stress/strain gradient elements are C0 elements, in which displacements and displacement gradients are interpolated independently and their kinematic constraints are enforced via the penalty or Lagrange multiplier method. Consequently, the computation cost is dramatically increased and the analysis results are varied with the penalty function. Compared with the conforming element methods, it is easier for the nonconforming element methods to establish high-performance elements as they relax the continuity condition more loosely and offer more flexible interpolation algorithms. In this item we will study the convergence criteria of nonconforming elements for couple stress/strain gradient theory and propos

英文关键词： Couple stress/Strain gradient theory；Finite element；variational principle；weak continuity；Patch test