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}:

${\frac {dU}{dt}}={\dot {Q}}-P_{i}$

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

$dU=-P_{i}dt$

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:

$dU=FdX$

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 $X$ .

Remark:

If $X$ is a scalar $x$, the energy differential is:

$dU=f.dx$

If $X$ is a vector $x_{i}$, the energy differential is:

$dU=f_{i}dx_{i}$

Si $X$ is a tensor $x_{ij}$, the energy differential is:

$dU=f_{ij}dx_{ij}$

Remark:

If a displacement $x$, is considered as thermodynamical variable, then the conjugated variable $f$ has the dimension of a force.

Remark:

One can go from a description using variable $X$ as thermodynamical variable to a description using the conjugated variable $F$ of $X$ as thermodynamical variables by using a Legendre transformation (see section secmaxient and ([#References

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:

$F(X)={\frac {\partial U}{\partial X}}$

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