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}: Template:IMP/eq if the heat flow is assumed to be zero, the internal energy variation is: Template:IMP/eq 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: Template:IMP/eq 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 $X$ . Template:IMP/rem Template:IMP/rem Template:IMP/rem 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: Template:IMP/eq In this chapter several examples of this modelization approach are presented.