We study the dynamic behavior of a thermoelastic diffusion beam with rotational inertia and second sound, clamped at one end and free to move between two stops at the other. The contact with the stops is modeled with the normal compliance condition. The system, recently derived by Aouadi (2015), describes the behavior of thermoelastic diffusion thin plates under Cattaneo’s law for heat and mass diffusion transmission to remove the physical paradox of infinite propagation speeds of the classical Fourier’s and Fick’s laws. The system of equations is a coupling of a hyperbolic equation with four parabolic equations. It poses some new mathematical and numerical difficulties due to the lack of regularity and the nonlinear boundary conditions. The exponential stability of the solutions to the contact problem is obtained in the presence of rotational inertia thanks to a structural damping term. We propose a finite element approximation and we prove that the associated discrete energy decays to zero. Finally, we give an error estimate assuming extra regularity on the solution and we present some results of our numerical experiments.

我们研究了具有旋转惯性和第二种声音的热弹性扩散束的动力学行为，该声束的一端被夹紧，而另一端则在两个止动件之间自由移动。与止动件的接触采用正常的顺应性条件进行建模。该系统最近由Aouadi（2015）推导，描述了在Cattaneo定律下热弹性扩散薄板的传热和传质扩散行为，以消除经典傅里叶定律和Fick定律的无限传播速度的物理悖论。方程组是双曲型方程与四个抛物线方程的耦合。由于缺乏规则性和非线性边界条件，它带来了一些新的数学和数值困难。由于存在结构阻尼项，在存在转动惯量的情况下，可获得接触问题解决方案的指数稳定性。我们提出了一个有限元逼近，并证明了相关的离散能量衰减到零。最后，我们给出了一个误差估计，并假设解决方案具有额外的规律性，并给出了数值实验的一些结果。