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基于Hellinger-Reissner变分原理的应变梯度杂交元设计
引用本文:李雷,吴长春,谢水生.基于Hellinger-Reissner变分原理的应变梯度杂交元设计[J].力学学报,2005,37(3):301-306.
作者姓名:李雷  吴长春  谢水生
作者单位:北京有色金属研究总院,有色金属材料加工与制备国家重点实验室, 100088
基金项目:国家自然科学基金资助项目(10172078,50374014)
摘    要:从一般的偶应力理论出发,基于Hellinger-Reissner变分原理,通过对有限元 离散体系的位移试解引入非协调位移函数,得到了偶应力理论下有限元离散系统的能量相容 条件,并由此建立了应变梯度杂交元的应力函数优化条件. 根据该优化条件,构造了一 个C0类的平面4节点梯度杂交元,数值结果表明,该单元对可压缩和不可压缩状态的 梯度材料均可给出合理的数值结果,再现材料的尺度效应.

关 键 词:应变梯度  变分原理  杂交元  偶应力理论  设计  优化条件  数值结果  位移函数  相容条件  离散系统  应力函数  梯度材料  压缩状态  尺度效应  有限元  非协调  可压缩  平面
文章编号:0459-1879(2005)03-0301-06
修稿时间:2004年4月8日

Optimization method of hybrid element stress function for strain gradient theory based on Hellinger-Reissner principle
Li Lei,Wu Changchun,Xie Shuisheng.Optimization method of hybrid element stress function for strain gradient theory based on Hellinger-Reissner principle[J].chinese journal of theoretical and applied mechanics,2005,37(3):301-306.
Authors:Li Lei  Wu Changchun  Xie Shuisheng
Abstract:Recent experiments have shown that materials will display strong scale effect when the scale of non-uniform plastic deformation field associated their intrinsic length scale is on the order of microns. In order to explain such scale effect phenomena, Fleck and Hutchinson developed a couple stress theory of strain gradient plasticity based on the reduced couple stress theory, which incorporates the rotation gradient of deformation into constitutive model, and introduces a material characteristic length parameter related to the rotation gradient. Theoretical predictions agree well with the micro-torsion and micro-bending experiments. In the finite element implementation of Fleck-Hutchinson couple stress plasticity, the higher order nature of theory requires that both the displacement and its first-order derivatives to be continuous across the adjacent elements' boundaries. Noticed that the micro-rotation *********, an independent kinematic quantity with no direct dependence on displacement u, is introduced in the general couple stress theory. This enables the C0-continuous element to be developed based on the general couple stress theory. Fitting within the framework of general couple stress theory, the energy consistency condition of the discrete finite element system for couple stress strain gradient theory is derived by introduction of incompatible displacement trial functions. Furthermore, the optimization condition of stress trial functions for hybrid element of strain gradient theory is constructed based the energy consistency condition. A 4-node C0 kind hybrid element is designed in terms of the optimization condition. Numerical tests show that the scale effects can be reflected with the element designed in the paper and reliable results is delivered both for compressible and incompressible materials.
Keywords:finite element method  hybrid element  strain gradient theory  couple stress theory  scale effect
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