A Variational Derivation of Stoney-Like Formulas for Self-Stressed Bilayered Plates

Abstract

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.