Major Simplifications in a Current Linear Model for the Motion of a Thermoelastic Plate

A dynamic model for a thin thermoelastic plate proposed by Lagnese and Lions in 1988 [1] has been used recently by several authors (e.g., [2]—[5]) to study existence and stability of solutions to initial/boundary-value problems. Simple, systematic orderof-magnitude arguments show that it is consistent to neglect several terms appearing in the governing differential equations that couple a temperature moment to the average vertical displacement. Further, because the time scale on which the temperature adjusts itself to the strain rate contribution to the energy equation is quite small compared with the longest (isothermal) period of free vibration of the plate, the energy equation can be solved for the temperature in terms of derivatives of the vertical displacement and hence the system reduced to a single equation, only slightly more complicated than the classical (Kirchhoff) equation of motion. Among other things, it is shown that the temperature has a cubic rather than a linear variation through the thickness. Finally, another order-of-magnitude estimate for a clamped aluminum plate of one meter radius and 1mm thickness shows that thermal damping acting alone takes on the order of 200 cycles of vibration to halve the initial amplitude. 1. The equations of linear thermoelasticity. Let xQ, a = 1,2, and z denote Cartesian coordinates in an inertial reference frame and let a homogeneous, isotropic, thermoelastic plate-like body occupy the closure of the open set Q x where fl, the midplane of the plate, is a connected region of the xQ-plane with piecewise smooth boundary dfl and '2H is the plate's thickness. Then, from the standard references by Boley and Weiner [6] or Carlson [7], the linearized governing three-dimensional equations in Cartesian tensor notation comprise the equations of motion, &af3,a Tf3,z f(3 = ; (1*1) Ta,a + + / = pil, (1.2) Received November 10, 1997. 1991 Mathematics Subject Classification. Primary 35B20, 73B30. ©1999 Brown University 673