耗散系统的虚功变分不等式及其应用

冯晔; 李杰

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

冯晔,
李杰^,

同济大学土木工程学院, 上海 200092

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
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- 被引次数: 0
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

THE INEQUALITY OF VIRTUAL WORK FOR DISSIPATIVE SYSTEMS AND ITS APPLICATIONS

Feng Ye,
Li Jie^,

College of Civil Engineering, Tongji University, Shanghai 200092, China

摘要

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.
- 变分原理 /
- 变分不等式 /
- 虚功 /
- 塑性力学 /
- Mohr-Coulomb强度准则
Abstract: For conservative systems, energy variational principles provide a concise way to derive the governing equations for mechanical systems. For a dissipative system, the establishment of governing equations often requires the introduction of empirical or rational assumptions, which increases the difficulty of modeling. For dissipative systems, this paper introduces the concept of local stability of mechanical systems. Based on this stability concept, we propose a variational inequality of virtual work. This inequality in fact reveals a virtual work principle for dissipative systems. The physical meaning of this principle is that the necessary condition for the system state to be stable is that the potential energy released by the system on all possible virtual paths near the state is not greater than the energy that may be dissipated in the system. This study shows that it is sufficient to only use the principle of inequality of virtual work together with the sound principle of conservation of energy to derive all the governing equations for the mechanical state variables of a quasi-static system. As an application, this paper revisits plastic mechanics and derives all the governing equations of classical plastic mechanics by combining the principle of inequality of virtual work and the principle of conservation of energy. We prove that the classical principle of maximum plastic dissipation can be derived as a corollary of the principle of inequality of virtual work. Meanwhile, this paper revisits the Mohr-Coulomb strength criterion as an example to show the application of the principle of inequality of virtual work in the strength theory. It is shown that the stress-based strength criterion can be a corollary of the energy-based criterion for system stability. The above examples illustrate the wide applicability of the principle of inequality of virtual work and verify its effectiveness in establishing the governing equations of dissipative systems.
- variational principle /
- variational inequality /
- virtual work /
- plastic mechanics /
- Mohr-Coulomb criterion
注释:

1) ¹推导使用了如下事实：若对于任意可正可负的$\delta u $以及任意的积分区域$\mathcal{B} $都有$\int_{\mathcal{B}}^{{}}{f\left( u \right)\delta u\operatorname{d}V}\geqslant 0 $, 则必须有$f\left( u \right)=0 $. 证明: 取无穷小积分区域，此时上述积分不等式等价于$f\left( u \right)\delta u\geqslant 0 $. 令$\delta u>0 $, 则从$f\left( u \right)\delta u\geqslant 0 $可知$f\left( u \right)\geqslant 0 $; 同理, 令$\delta u<0 $, 则须有$f\left( u \right)\leqslant 0 $. 综上所述, 必须有$f\left( u \right)=0 $.

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图 1 一维系统稳定性示意图

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

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图 2 矩形板单轴受压问题

Figure 2. Uniaxial compression of a rectangular plate

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图 3 线性黏聚律和对应的势能

Figure 3. Linear cohesive law and corresponding potential

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