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

Abstract

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. In Section 3, global relations of balance for open systems are studied, and are set into analogy to the results of Section 2. An open system is represented by a non-material control volume, the surface of which moves at a velocity different from the velocity of the material particles instantaneously located on that surface, such that a flow of mass takes place. Supplies of mass and linear momentum due to this flow of mass are shown to be analogous to the supplies introduced in Section 2. The theoretically as well as practically important special case of a rigid body that experiences a surface growth is exemplarily treated. Section 4 deals with extended relations of jump for systems with a variable mass. Relations of jump are needed, when certain entities suffer considerable changes across some region of transition. This region of transition is replaced by an equivalent singular surface, for which relations of jump are formulated by including additional non-classical surface supply terms, such as surface supply of mass and linear momentum. Other surface supply terms are derived as mathematical consequences of the latter. As an example for the formulations presented in Sections 3 and 4, the problem of a chain heaped up on a table, the hanging part of the chain being set into motion, is considered in Section 5. It is believed that the corresponding formulation can explain some seemingly controversial results from the literature.

It should be mentioned that our emphasis lies on a rational treatment of the topics under consideration. While our methodology has many important predecessors, but appears to be novel in the systematic manner here presented, we do not intend to give a historical review on the topic due to the limited space available. For the latter, the reader is referred to reviews by Mikhailov (1975), historical presentations to be found, e.g., in the important works of Eke (1998) and Cveticanin (1998), as well as to a review by Irschik and Holl (2004) on balance of mass and momentum for systems with a variable mass. (An extended review by the latter authors concerning balance of moment of momentum and kinetic energy for variable mass systems is being prepared since and hoped to be finished soon). Last but not least, the reader is referred to the other chapters of the present book.