偶应力/应变梯度理论的精化不协调元方法

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 propose a class of variational principles which relax the continuity conditions. Based on the proposed variational principles, a series of refined nonconforming elements which satisfies C0 continuity (or weak continuity), quadratic completeness and weak C1 continuity will be developed for couple stress/strain gradient theory. The systematic research of the refined nonconforming element method for couple stress/strain gradient theory will accelerate the theory research process, and furthermore improve the development of micro/nano technology.

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。

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。本项目基于精化不协调元方法，研究偶应力/应变梯度理论不协调元的收敛准则。目前已经建立的偶应力/应变梯度单元都只分别考虑满足C0连续或C1连续。我们的方法可建立同时满足C0连续（或弱连续）和C1连续的偶应力/应变梯度精化不协调单元。传统轴对称单元只能满足C0连续，本项目提出了一类放松轴对称单元间连续条件的变分原理，利用已知的单元位移函数, 根据建立的变分原理，只修改对应常曲率部分, 即可得到满足C1弱连续的轴对称单元。在此基础上，建立了轴对称四边形单元（ACQ12+ADKQ）,其中ACQ12满足C0连续且具有二次完备性,用于计算位移的一阶导数，ADKQ满足本文建立的C1弱连续条件,用于计算位移的二阶导数。数值结果表明该单元有良好的收敛性并适用于微观结构的分析。