1. Introduction

We consider continuous media or heterogeneous structures, whose geometric properties and/or physical vary « rapidly » and periodically in space compared to the scale of loads exerted, here with a rectangular step. They are therefore characterized by a three-dimensional « base cell », also called « representative elementary volume », having In the framework studied here, geometric and material symmetries in relation to planes orthogonal in Cartesian space. We are interested in the « macroscopic » behavior of these structures, to determine the global response, but also we want to return to the « microscopic » scale on a « base cell », to access quantities of interest to this ladder.

This can be multi-perforated plates, fiber composites, made up of a repeating pattern…

The objective is to establish homogenized properties in three-dimensional linear thermal, and, according to the type of heterogeneous structure, cf. [bib4], in linear thermoelasticity three-dimensional, cf. [bib5], or in the linear elasticity of a Love-Kirchhoff plate in membrane-flexure, cf. [bib3]. For this purpose, the « method of averages » is used, justified by asymptotic development, cf. [bib6].

In three-dimensional linear thermics, it is accepted to be able to consider for the heterogeneous structure studied sufficiently slow transients, so as to be able to separate spatial scales on The effect of the time derivative of temperature on the evolution of volume enthalpy macroscopic. The effect of thermal shock cannot therefore be treated in this way. In fact, in one-dimensional stationary heat, under the action of a temperature imposed on On edge \({x=0}\) to \({t=0}\) the solution is written:

\[\]

declaremaThoperatorerfc {erfc}

T (x, t) =T_ {imp}cdoterfcleft (frac {x} {2sqrt {tcdotfrac {K_ {cond}}} {rho_ {cond}}}}}right)

This solution reveals the thermal diffusivity coefficient \(\frac{K_{cond}}{\rho_{cp}}\) which is homogeneous to the square of a distance divided by a time. For example, let’s take ordinary steel with conductivity \(K_{cond} = 35 \frac{W}{m \cdot K}\) and heat capacity \(\rho_{cp} = 3.5 \frac{MJ}{m^3 \cdot K}\), with a length of \(3~cm\); the time derivative of temperature becomes very low (\(0.01~\frac{°C}{s}\)) beyond a duration of \(90~s\). This means that taking Account of a transition whose duration is shorter will cause a solution that will not be able to be represented correctly on a homogenized scale for a base cell of size \(3~cm\).

In thermo-elasticity, a distinction is made between situations:

• periodic heterogeneous three-dimensional solids in the three directions of Cartesian space;

• heterogeneous, slender three-dimensional solids with plane or curved geometry: plate or

  shell, periodic in both directions of the tangential plane. We place ourselves in the case standard, cf. [bib3] and Figure 1-a, where the periodicity step is of the same order as size than the thickness of the plate or shell, and where we consider that this step of periodicity is low compared to the average curvature of the shell, so as to admit a Cartesian metric in the base cell.