Mechanical conservation and balance laws

Abstract

In the previous chapter, we derived kinematic fields to describe the possible deformed configurations of a continuous medium. These fields on their own cannot predict the configuration a body will adopt as a result of a given applied loading. To do so requires a generalization of the laws of mechanics (originally developed for collections of particles) to a continuous medium, together with an application of the laws of thermodynamics. The result is a set of universal conservation and balance laws that apply to all bodies: conservation of mass; balance of linear and angular momentum; thermal equilibrium (zeroth law of thermodynamics); conservation of energy (first law of thermodynamics); second law of thermodynamics. These equations introduce four new important quantities to continuum mechanics. The concept of stress makes its appearance in the derivation of the momentum balance equations. Temperature, internal energy and entropy star in the zeroth, first and second laws, respectively. In this chapter we focus on the mechanical conservation laws (mass and momentum) leaving the thermodynamic laws to the next chapter. Conservation of mass A basic principle of classical mechanics is that mass is a fixed quantity that cannot be formed or destroyed, but only deformed by applied loads. Thus, the total amount of mass in a closed system is conserved. For a system of particles this is a trivial statement that requires no further clarification.