A rational treatment of the relations of balance for mechanical systems with a time-variable mass and other non-classical supplies

摘要

This contribution intends to present a rational methodology for mechanical systems with a variable mass, represented by a supply of mass. Special emphasis is given to the relations of balance and jump for such systems. In these relations, we also allow for other types of additional, non-classical supplies, e.g., supplies of linear and angular momentum. In doing so, we aim at completing and substantially extending formulations laid down in the famous article by Truesdell and Toupin (1960), who stated local relations of balance of mass and linear momentum in the presence of sources of mass, and, among other formulations with relevance to the present article, gave fundamental formulations for the case that a flow of mass through the surface of the system is present in the global relations of balance. Our presentation is organized as follows: We remain in the framework of non-relativistic mechanics, referring to a common inertial frame. Throughout the Chapter, we formulate our relations in the Euler or spatial description, in which every entity is understood as a function of the instantaneous place of the material particles under consideration, and of time. In Section 1, the general equation of balance is stated and is applied to the model of a single mass point with a variable mass. This general equation is specified for the fundamental relations of balance of mass, linear momentum, angular momentum and total energy first. The variable mass is associated with a supply of mass. Afterwards, as mathematical consequences of the fundamental statements, we derive the statements of balance of moment of momentum, intrinsic spin, kinetic energy and internal energy for the single mass point. As a rational procedure for formulating the additional, non-classical supplies that are present in the relations of balance, we assume that the single mass point is gaining or losing differential masses by means of continuous impacts, which are again studied in the framework of the general equation of balance. The outcomes of this procedure include a Seeliger-Meschersky type additional supply of linear momentum. In Section 2, theorems on balance of mass, linear momentum, moment of momentum and kinetic energy for deformable bodies of finite extension with a variable mass are presented. Among these, the first two can be considered as fundamental, while balance of moment of momentum and kinetic energy are derived from balance of mass and linear momentum as mathematical consequences. The supply of mass is associated with distributed sources of mass attached to the material particles, which we call material sources of mass. Both global and local relations of balance are considered, including global and local non-classical supplies of mass and linear momentum. The supplies of moment of momentum and kinetic energy follow as mathematical consequences. A Seeliger-Meschersky type local model for the non-classical supply of linear momentum is presented. Due to limited space, the fundamental relations of balance of angular momentum and total energy for bodies of finite extension are not considered. However, useful global relations concerning the notion of center-of-mass are given, introducing the notions of center-of-mass linear momentum and relative linear momentum, center-of-mass moment of momentum and relative moment of momentum, as well as center-of-mass kinetic energy and relative kinetic energy. Our relations extend some formulations that are well-known for bodies in the absence of a supply of mass. The corresponding relations of balance again follow as mathematical consequences of the fundamental ones, including non-classical supply terms related to the non-classical supplies of mass and linear momentum.