Since the beginning of the 20th century, it is known that the spontaneous bending of heterogeneous bilayered plates correlates with the self-stress due to the contrast in the material properties of the two layers, and that this correlation can be exploited to gauge the internal stress state. Over the last decades, ever-growing device miniaturization has made stress assessment and even stress engineering an area of major technological interest. In this paper, we obtain two effective 2D models accounting for the spontaneous bending of devices comprised of a thin substrate and a much thinner coating by applying a \(\varGamma \)-convergence technique to the standard 3D linear hyperelastic model of a bilayered plate. Our procedure is characterized by the introduction of two distinct smallness parameters plus three independent energy scaling parameters.

自 20 世纪初以来，由于两层材料特性的差异，异质双层板的自发弯曲与自应力相关，并且可以利用这种相关性来测量内应力状态。在过去的几十年里，不断发展的设备小型化已经使压力评估甚至压力工程成为一个主要的技术领域。在本文中，我们获得了两个有效的二维模型，通过应用\(\varGamma \)- 将技术收敛到双层板的标准 3D 线性超弹性模型。我们的程序的特点是引入了两个不同的小参数加上三个独立的能量缩放参数。