Asymptotic Thermoelastic Behavior of Flat Plates 647

Introduction. The derivation of the mechanical behavior of a flat plate as the limit behavior of a three-dimensional flat solid whose thickness tends to zero is a well-established theory since the works of Goldenveizer [G] or Ciarlet and Destuynder [CD1], Due attention has been paid to the static response of such plates for various linear or nonlinear mechanical behaviors (Ciarlet-Destuynder [CD1, CD2], Ciarlet [C], Blanchard-Ciarlet [BC]). Much less attention, however, has been devoted to the dynamic response of flat plates from a similar standpoint. The linearly elastic case was considered in Raoult [Rl, R2], but we are not aware of any further work in that direction. The present study is devoted to the dynamic behavior of a three-dimensional linearly thermoelastic flat plate. Specifically, a three-dimensional flat plate with small thickness is submitted to an arbitrary system of initial and loading conditions. The limits of the displacement, stress, and temperature fields as the thickness approaches zero are investigated. Thermoelastic behavior is characterized by a coupling between the mechanical equations of motion and the "energy" equation. The limiting procedure is seriously affected by the presence of the coupling terms. The initial conditions are seen to play an essential role in the analysis. In particular, a change of initial condition generally occurs for the temperature field. A similar phenomenon appears in the homogenization of a thermoelastic composite (Francfort [F]). These concurring results seem to indicate that such shifts in initial data are closely linked to any kind of asymptotic problem for coupled systems. The first section is very short and entirely devoted to notation and basic definitions. In the second section, the problem under investigation is formulated in a mathematical framework. It is then rescaled in the usual manner (Ciarlet-Destuynder [CD1]) so as to obtain a family of problems indexed by the thickness of the plate and defined on a fixed domain.