Introduction to Mathematical Physics/Energy in continuous media/Introduction

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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_idx_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.