1. Introduction#

Mechanical models of thin structures (shells and plates) have reached an extremely advanced stage of development, at least for elastic structures that are homogeneous in thickness. The problem has been known for a very long time and various theories have emerged, generally dedicated to specific problems (thick shells, buckling, etc.). However, a basic model, that of LOVE - KIRCHHOFF, is unanimously accepted in the most common applications. The difficulties lie rather in the numerical calculation of this one because, on the one hand, of the need to correctly approach the surface of the shell (in particular its curvature), and on the other hand, of the high order of the partial differential equations that must be solved (4th order).

In thermal technology, on the other hand, the situation is much less clear and a large number of approaches coexist. In fact, it was only recently that the problem arose with the possibilities (and the necessity) of thermomechanical calculations. The first models neglect conduction parallel to the mean surface to only retain heat transfers in the thickness of the shell. This approach is completely antithetical to that of thin structures where, on the contrary, the small thickness of the structure leads to simplifying hypotheses on the variation in the thickness of the fields of physical quantities.

The most recent models are inspired by mechanical ideas of thin shells relating to the second approach; they can be classified in a very similar order.

  1. Models involving a more or less advanced polynomial development of temperature in thickness [bib2], [bib9], [bib10]. It is essentially the formulation of finite elements.

  2. Models associated with surface to director theories (Surfaces of COSSERAT) [bib5], [bib8]. The director here is the temperature gradient in thickness. The problem with these approaches lies in the law of behavior to be introduced. Consistency with the three-dimensional law leads to choices that can be interpreted as a hypothesis of linear distribution of temperature in thickness. This formalism is therefore in practice similar to previous models (the introduction of several directors identifying with various orders of development of polynomials).

  3. Degenerate finite element models [bib11]: starting from a three-dimensional finite element, the introduction of constraints between the degrees of freedom located on the same normal to the mean surface allows by condensation to deduce a « thermal shell » element. In practice, again, since the base element uses parabolic interpolation according to the thickness, the shell element corresponds to a linear distribution in the thickness.

In parallel with these numerical approaches (1) and (3) or based on a priory assumptions (2), results on the shape of the temperature field of a thin plate and of the problem for which it is a solution were obtained by asymptotic methods [bib3], [bib1].

As for the mechanical model, these make it possible to justify the a priory hypotheses made in thin shell theories, or even to obtain the equations of the shell problem. The results of [bib 1] are mentioned below and will serve as the basis for the proposed model. Let us simply note here that the idea underlying any asymptotic approach is to introduce a small parameter (here the ratio of thickness of the plate to a characteristic dimension of the plate), then having obtained the limit problem when tending to zero from the three-dimensional problem, to approach in applications (where obviously takes a non-zero value) the solution by its limit.

From a practical point of view, the limit obtained for the equations of stationary heat seems to be too « poor » to be of real interest, (we will give an illustration of this in [§2.2.2]). More precisely, the values of \(\varepsilon\) to be reached in order to identify the solution at its limit are very small in the real situations encountered.

This is the reason why we propose in this note to keep the shape of the limit solution (parabolic distribution in the thickness) but to use it as an a prior hypothesis on the three-dimensional solution making it possible to reduce ourselves to a problem posed on the average surface.

We thus have an approximate model of a thin structure converging to the limit model of three-dimensional equations. In this sense, it is « optimal » since a hypothesis of linear distribution in thickness leads to a model not converging towards the limit solution and a model based on a richer development in thickness sees its terms of order greater than two converge to zero when the shell is thin.

The outline of the note is as follows:

  • we start by recalling the equations of the stationary thermal problem for the three-dimensional solid and their expressions in a coordinate system adapted to cases where the solid is a « thin shell »,

  • then, having recalled the results of an asymptotic study of these equations carried out in the case of a plate, a complete description of the proposed model is given,

  • the model is then applied to a certain number of geometries and thermal loads and a comparison is made with respect to analytical solutions or three-dimensional numerical calculations,

  • finally, some indications are given on the numerical aspects of the use of the model in a calculation using surface and linear finite elements.