A Treatise on the Equations of Balance and on the Jump Relations in Continuum Mechanics

Abstract

In this Lecture, we are concerned with the range of applicability of the balance relations in continuum mechanics. These relations have been already introduced in Lecture 1 of the present book as an important foundation of the dynamics and control of structures and machines, Belyaev (2004). In the present Lecture, we start with the general form of the relations of balance, which we afterwards specialise to the equations of balance of mass, momentum and energy (the first law of thermodynamics) and to the balance relation of entropy (the second law of thermodynamics), the latter in the form of the Clausius-Duhem inequality. We also discuss the frequently used equations of balance of kinetic energy (the law of power), and we point out the consequences of inserting the latter into the first and second law of thermodynamics, leading to the balance relations for the internal energy and to the Clausius-Planck inequality, respectively. We lay special emphasis on reviewing the continuity conditions that must be satisfied in order that the global and local forms of the relations of balance do hold. When a surface of discontinuity, a so-called singular surface, is present, across which some entity shows different values when approaching from the two sides of the surface, jump relations are needed in order to connect the local forms of the relations of balance at the two sides. We summarise and extend a recent formulation, which allows to connect the local equations of balance of mass, momentum, energy, kinetic energy and internal energy in a consistent manner, Irschik (2003). In the latter reference, it has been shown that surface growth terms must be introduced for the sake of consistency, and relations between these surface growth terms have been derived. In the present Lecture, these results are extended with respect to the second law of thermodynamics, and with respect to the resolution of a contradictory result in the literature on the jump relations of energy and of internal energy.