Introduction to the Variational Formulation in Mechanics

Part IVOther Problems in Physics

In this part of the book we will address the modeling of other problems typically encountered in the physics of continua through the use of the variational framework. In Chapter 10 we will concentrate attention on steady‐state heat transfer in rigid bodies. This domain of physics has usually been tackled by employing modeling techniques based on local (strong) forms, that is, making use of partial differential equations (Euler–Lagrange equations). By exploiting the general concepts developed in Chapter 3, we will build a variational model for this problem, making possible the analysis of several aspects such as constitutive equations, reactive forces, and kinematical constraints. Moreover, the connection to traditional minimization problems will be established, in line with that presented in Chapter 4. In particular, regarding the heat transfer problem in rigid bodies, the work by P. Podio‐Guidugli deserves special mention [242], where the Principle of Virtual Power is developed in parallel for both the duality between strain rate and stress, and the duality between temperature and entropy. These duality pairings in fact deliver power as outcome. In our case, we will utilize a more classical variational approach in which the duality is understood as a result of pairing temperature and heat flux, which renders power per unit temperature.

In Chapter 11 we will examine the modeling of incompressible fluid flow. As with the heat transfer problem, ...