河北水利电力学院学报

河北水利电力学院学报 ›› 2021, Vol. 31 ›› Issue (1): 75-80.DOI: 10.16046/j.cnki.issn2096-5680.2021.01.015

• 基础科学研究 • 上一篇    

用张量分析方法推导含偶应力弹性力学有限元理论

孙晓勇１，２，宋兴海２，侯娜娜１，２，付建航２，刘立悦1，２   

  1. 1.河北省数据中心相变热管理技术创新中心，河北省沧州市重庆路1号 061001；
    2.河北水利电力学院 土木工程学院，河北省沧州市重庆路1号 061001

  • 收稿日期:2020-03-25 修回日期:2020-09-11 出版日期:2021-03-31 发布日期:2021-04-30
  • 作者简介:孙晓勇(1989-)，男，天 津 市 人，河 北 水 利 电 力 学 院 助 教,研 究 方 向 为 固 体 力 学｡E-mail:893941282@qq.com
  • 基金资助:
    河北省教育厅教学改革课题（2018GJJG386）

Derivation of the Finite Element Theory of Elasticity with Couple Stress by Tens or Analysis

SUN Xiao-yong1,2,SONG Xing-hai 2,HOU Na-na1,2,FU Jian-hang2,LIU Li-yue1,2   

  1. 1.Technology Innovation Center of Phase Change Thermal Management of Internet Data Center,061001,Cangzhou,Hebei,China;
    2.Hebei University of Water Resources and Electric Engineering,061001,Cangzhou,Hebei,China

  • Received:2020-03-25 Revised:2020-09-11 Online:2021-03-31 Published:2021-04-30

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摘要： 经典弹性力学理论用位移梯度表示无限小变形,不考虑旋转变形,把微元体的旋转视为刚体旋转。含偶应力弹性力学理论将旋转变形以旋转张量表示,微元体旋转和微元体平动位移同量级,而旋转张量和应变张量同量级,旋转张量与旋转矢量一一对应,用旋转矢量的梯度表示旋转变形。含偶应力弹性力学理论本构关系包括应力-应变关系和偶应力-曲率张量关系,用等参变换方法离散单元位移到节点上,从虚功原理出发,增加罚函数项以降低有限元方程对高阶单元的需求,推导了拟解决三维及二维问题的含偶应力弹性线力学有限元理论,可得三维及二维问题中位移、应力、应变等分布情况,对结构进行力学评价。

关键词: 偶应力, 旋转变形, 旋转张量, 张量分析

Abstract: The classical theory of elasticity expresses infinitesimal deformation by displacement gradient.The rotation of the infinitesimal body is regarded as the rotation of the rigid body without considering the rotation deformation.In the theory of elasticity with couple stress,rotation deformation is represented by rotation tensor.The rotation and translational displacement of the micro element are the same order,while the rotation and strain tensors are the same order.The rotation tensor corresponds to the rotation vector one by one,and the gradient of the rotation vector is used to represent the rotation deformation.The constitutive relation of elasticity with couple stress includes stress-strain relation and couple stress curvature tensor relation.Discrete the displacement of element to nodes by using isoparametric transformation method.Based on the principle of virtual work,apenalty function is added to reduce the demand of higherorder elements in the finite element equation.The finite element theory of elastic lines with coupled stresses to solve the plane problem is derived.The distribution of displacement,stress and strain in the problem of plane can be obtained,and the mechanical evaluation of the structure can be carried out.

Key words: ouple stress, rotational deformation, rotation tensor, tensor analysis

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