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 ( 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 .


If is a scalar , the energy differential is:

If is a vector , the energy differential is:

Si is a tensor , the energy differential is:


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


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.