有限位移理论线弹性力学二类和三类混合变量的变分原理及其应用

付宝连

doi:10.21656/1000-0887.380004

有限位移理论线弹性力学二类和三类混合变量的变分原理及其应用

doi: 10.21656/1000-0887.380004

付宝连

燕山大学建筑工程与力学学院, 河北秦皇岛 066004

详细信息

作者简介:
付宝连（1934—），男，教授（E-mail: ysufubaolian@163.com）.

中图分类号: O343
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出版历程
- 收稿日期: 2017-01-05
- 修回日期: 2017-03-10
- 刊出日期: 2017-11-15

Variational Principles for Dual and Triple Mixed Variables of Linear Elasticity With Finite Displacements and the Application

FU Bao-lian

College of Architectural Engineering and Mechanics, Yanshan University,Qinhuangdao, Hebei 066004,P.R.China

摘要

摘要: 提出了有限位移理论线弹性力学二类混合变量和三类混合变量的变分原理.考虑已知边界条件的变化并应用有限位移理论的功的互等定理，在导出上述两类变分原理的过程中起到了关键作用和桥梁作用.首先，考虑已知位移边界条件的变化和应用功的互等定理，导出了二类混合变量的最小势能原理.用类似的方法，导出了二类混合变量的驻值余能原理.应用应变能密度和应力余能密度的关系式于上述两个变分原理，得到三类混合变量的变分原理.然后，给出了二类和三类混合变量的虚功原理和虚余功原理.同时，应用拉氏乘子法导出了广义变分原理.以一个算例说明了在某些情况下拉氏乘子法会失效，介绍了构成广义变分原理泛函的半逆法.最后，应用二类混合变量最小势能原理计算了一大挠度悬臂梁的弯曲.
- 有限位移理论 /
- 线弹性力学 /
- 二（三）类混合变量的变分原理 /
- 混合变量的虚（余）功原理 /
- 广义变分原理 /
- 半逆法
Abstract: Variational principles for dual and triple mixed variables of linear elasticity with finite displacements were proposed. Considering the variation of prescribed boundary conditions and using the reciprocal theorem of finite displacements played the key and bridging roles in derivation of the above variational principles. First, in view of the variation of the prescribed geometrical boundary conditions and based on the reciprocal theorem, the principle of minimum potential energy with dual mixed variables was derived. In a similar way, the principle of stationary complementary energy with dual mixed variables was also given. Then the relation between the strain energy density and the complementary energy density was applied to the above 2 principles, and the variational principle with triple mixed variables was deduced. In turn, the principles of virtual work and virtual complementary work with dual and triple mixed variables were directly given. Meantime, the generalized variational principles were derived with the Lagrangian multiplier method. Through an example the Lagrangian multiplier method in certain cases was proved to be ineffective. The semiinverse method for construction of the functionals for generalized variational principles was also introduced. Finally, a cantilever beam with large deflection was calculated by means of the principle of minimum potential energy for dual mixed variables.

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