Introduction to the Variational Formulation in Mechanics

10Heat Transfer

10.1 Introduction

In this chapter we develop the variational formulation to model the steady‐state heat transfer in rigid continuum media, that is, in bodies where the spatial position of particles remains invariant.

According to the roadmap established in Chapter 3, the first step in the construction of a variational model consists of defining the kinematics for the model, that is, the motion actions that particles can execute. In the heat transfer problem, the temperature is the primary scalar field that characterizes the average kinetic energy of molecules, and so it characterizes the kinematics in this problem. The temperature in a body is constrained to satisfy certain conditions, and therefore it is possible to define the concept of admissible variations of the temperature fields. With this, it is possible to introduce the generalized strain action operator, which in this context is denoted by (in Chapter 3 it was denoted by ), which leads us to the conception of the virtual internal power. From there, the characterization of generalized internal and external forces will follow, and then the application of the Principle of Virtual Power will be exploited to define the concept of equilibrium for the system.

Let us consider a body occupying a bounded and regular ...