Introduction to Mathematical Physics/Energy in continuous media/Introduction

The first principle of thermodynamics (see section secpremierprinci) allows to bind the internal energy variation to the internal strains power \index{strains}:

if the heat flow is assumed to be zero, the internal energy variation is:

This relation allows to bind mechanical strains ($$P_i$$ term) to system's thermodynamical properties ($$dU$$ term). When modelizing a system some "thermodynamical" variables $$X$$ are chosen. They can be scalars $$x$$, vectors $$x_i$$, tensors $$x_{ij}$$, \dots Differential $$dU$$ can be naturally expressed using those thermodynamical variables $$X$$ by using a relation that can be symbolically written:

where $$F$$ is the conjugated\footnote{ This is the same duality relation noticed between strains and speeds and their gradients when dealing with powers.} thermodynamical variable of variable $$X$$. In general it is looked for expressing $$F$$ as a function of $$X$$.

The next step is, using physical arguments, to find an {\bf expression of the internal energy $$U(X)$$} \index{internal energy} as a function of thermodynamical variables $$X$$. Relation $$F(X)$$ is obtained by differentiating $$U$$ with respect to $$X$$, symbolically:

In this chapter several examples of this modelization approach are presented.