| 有限位移理论线弹性动力学二类和三类混合变量的最小势作用量原理和驻值余作用量原理及其应用 |
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| 引用本文： | 付宝连. 有限位移理论线弹性动力学二类和三类混合变量的最小势作用量原理和驻值余作用量原理及其应用[J]. 应用数学和力学, 2017, 38(12): 1359-1376. DOI: 10.21656/1000-0887.380005 |
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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： | 两个新的概念，即势作用量的概念和余作用量的概念被引入弹性动力学变分原理中.根据势作用量的概念，最小作用量原理（即Hamilton原理）被改称为最小势作用量原理.根据余作用量的概念，首次提出了驻值余作用量原理.考虑边界条件的变化并应用有限位移理论的功的互等定理，导出了以位移和应力为变分变量的二类混合变量的最小势作用量原理及驻值余作用量原理.应用应变势能密度与应力余能密度的关系式于上述二类混合变量作用量原理，导出了以位移、应力和应变为变分变量的三类混合变量的相关作用量原理.最后，应用拉氏乘子法给出了广义势作用量原理及广义余作用量原理，并且应用大挠度梁二类混合变量最小势作用量原理计算了一悬臂梁的受迫振动.
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| 关 键 词： | 势作用量 余作用量 有限位移理论的功的互等定理 二类（三类）混合变量最小势作用量原理 二类（三类）混合变量驻值余作用量原理 |
| 收稿时间： | 2017-01-05 |
Principles of Minimum Potential Action and Stationary Complementary Action With Dual and Triple Mixed Variables for Linear Elastodynamics of Finite Displacement Theory 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: | 2 new concepts, potential action and complementary action, were first introduced into the variational principles for linear elastodynamics. On the basis of the concept of potential action, the principle of minimum action (Hamilton’s principle) was renamed to the principle of minimum potential action. In terms of the concept of complementary action, the principle of stationary complementary action was proposed for the first time. Next, the principles of minimum potential action and stationary complementary action with dual mixed variables of displacement and stress were derived in view of the boundary condition changes by means of the reciprocal theorem. And then, through the application of the relations between the strain energy density and the complementary energy density to the above 2 principles with dual mixed variables, the principles of potential action and complementary action with triple mixed variables of displacement, stress and strain were derived. Finally, the generalized principles of potential action and complementary action were given with the Lagrange multiplier method, in the meantime, the principle of minimum potential action with dual mixed variables of large deflection beams was applied to the calculation of a bending cantilever beam under forced vibration. |
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