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耗散系统的虚功变分不等式及其应用

冯晔 李杰

冯晔, 李杰. 耗散系统的虚功变分不等式及其应用. 力学学报, 2023, 55(4): 1-8 doi: 10.6052/0459-1879-23-006
引用本文: 冯晔, 李杰. 耗散系统的虚功变分不等式及其应用. 力学学报, 2023, 55(4): 1-8 doi: 10.6052/0459-1879-23-006
Feng Ye, Li Jie. The inequality of virtual work for dissipative systems and its applications. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(4): 1-8 doi: 10.6052/0459-1879-23-006
Citation: Feng Ye, Li Jie. The inequality of virtual work for dissipative systems and its applications. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(4): 1-8 doi: 10.6052/0459-1879-23-006

耗散系统的虚功变分不等式及其应用

doi: 10.6052/0459-1879-23-006
基金项目: 国家自然科学基金(51538010)和上海市科委高层次专家支持专项(22YS1400700)资助项目
详细信息
    通讯作者:

    李杰, 教授, 主要研究方向为随机力学、工程可靠性理论. E-mail: lijie@tongji.edu.cn

  • 中图分类号: O34

THE INEQUALITY OF VIRTUAL WORK FOR DISSIPATIVE SYSTEMS AND ITS APPLICATIONS

  • 摘要: 对于保守系统, 能量变分原理为推导力学系统控制方程提供了简洁的途径. 对于耗散系统, 控制方程的建立往往需要引入经验的或理性的假定, 增大了建模的难度. 针对耗散系统, 引入系统局部稳定的概念, 并在此基础上, 提出一类虚功变分不等式. 这一不等式事实上揭示了耗散系统的一类虚功不等原理. 该原理的物理含义为: 使系统状态稳定的必要条件是, 在该状态附近所有可能的虚拟路径上系统释放的势能不大于系统耗散的能量. 研究表明: 仅需结合虚功不等原理和能量守恒原理, 即可导出准静态系统力学状态量的全部控制方程. 作为应用, 文章重新讨论了塑性力学, 结合虚功不等原理与能量守恒原理, 导出经典塑性力学的全部控制方程, 并证明了经典的最大塑性耗散原理可以作为虚功不等原理的推论导出; 同时, 以Mohr-Coulomb强度准则为例, 讨论了虚功不等原理在强度理论中的应用, 说明基于应力的强度准则可以是基于能量的稳定性准则的推论. 上述例子说明了虚功不等原理的广泛适用性和在建立耗散系统控制方程中的有效性.

     

  • 图  1  一维系统稳定性示意图

    Figure  1.  Schematic diagram of the stability of a one-dimensional system

    图  2  矩形板单轴受压问题

    Figure  2.  Uniaxial compression of a rectangular plate

    图  3  线性黏聚律和对应的势能

    Figure  3.  Linear cohesive law and corresponding potential

  • [1] 钱伟长. 变分法及有限元. 北京: 科学出版社, 1980

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出版历程
  • 收稿日期:  2023-01-03
  • 录用日期:  2023-03-11
  • 网络出版日期:  2023-03-12

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