The F\"oppl-von K\'arm\'an equations of elastic plates with initial stress

TL;DR

This paper derives F"oppl-von K"arm"an equations for initially stressed elastic plates, enabling analysis of shape morphing driven by initial stress distributions without predefined target metrics, with applications in engineering.

Contribution

It provides a formal asymptotic derivation of F"oppl-von K"arm"an equations incorporating initial stresses, allowing for shape control without preset target metrics.

Findings

Derived equations enable analysis of stress-induced shape changes.

Explicit solutions illustrate how initial stress drives spontaneous curvature.

Framework applicable to engineering fields like soft robotics and 4D printing.

Abstract

Initially stressed plates are widely used in modern fabrication techniques, such as additive manufacturing and UV lithography, for their tunable morphology by application of external stimuli. In this work, we propose a formal asymptotic derivation of the F\"{o}ppl-von K\'{a}rm\'{a}n equations for an elastic plate with initial stresses, using the constitutive theory of nonlinear elastic solids with initial stresses under the assumptions of incompressibility and material isotropy. Compared to existing works, our approach allows to determine the morphological transitions of the elastic plate without prescribing the underlying target metric of the unstressed state of the elastic body. We explicitly solve the derived FvK equations in some physical problems of engineering interest, discussing how the initial stress distribution drives the emergence of spontaneous curvatures within the deformed plate. The proposed mathematical framework can be used to tailor shape on demand, with applications in several engineering fields ranging from soft robotics to 4D printing.