Introduction to Mathematical Physics/Energy in continuous media/Introduction

From Wikibooks, open books for an open world
< Introduction to Mathematical Physics‎ | Energy in continuous media
Jump to navigation Jump to search

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 ( term) to system's thermodynamical properties ( term). When modelizing a system some "thermodynamical" variables are chosen. They can be scalars , vectors , tensors , \dots Differential can be naturally expressed using those thermodynamical variables by using a relation that can be symbolically written:

where 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

. In general it is looked for expressing as a function of .

Remark:

If is a scalar , the energy differential is:

If is a vector , the energy differential is:

Si is a tensor , the energy differential is:

Remark:

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

Remark:

One can go from a description using variable as thermodynamical variable to a description using the conjugated variable of 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 } \index{internal energy} as a function of thermodynamical variables . Relation is obtained by differentiating with respect to , symbolically:

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