| 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 |
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| 引用本文： | 付宝连.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[J].应用数学和力学,2017,38(11):1251-1268. |
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| 作者姓名： | 付宝连 |
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| 作者单位： | 燕山大学 建筑工程与力学学院, 河北 秦皇岛 066004 |
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| 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.
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| 关 键 词： | 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 |
| 收稿时间： | 2017-01-05 |
Variational Principles for Dual and Triple Mixed Variables of Linear Elasticity With Finite Displacements and the Application |
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| Affiliation: | College of Architectural Engineering and Mechanics, Yanshan University,Qinhuangdao, Hebei 066004,P.R.China |
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| 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 fimite 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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| Keywords: | finite displacement theory linear elasticity variational principle with dual (triple) mixed variables principle of virtual (complementary) work with mixed variables generalized variational principle semi-inverse method |
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