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 references])). \index{Legendre transformation} The internal energy is then transformed to another function called depending on the case, enthalpy, free energy, etc\dots 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.