THE INEQUALITY OF VIRTUAL WORK FOR DISSIPATIVE SYSTEMS AND ITS APPLICATIONS

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.