Shape-programming of hyperelastic plates through differential growth: an analytical approach

In this work, we study the plane-strain deformations of hyperelastic plates induced by differential growth, aiming to derive some analytical formulas for 2D shape-programming of hyperelastic plates. First, we present a plate equation system with the growth functions incorporated, which is derived from the 3D governing system through a series expansion and truncation approach. By proposing a novel analytical method, the plate equation system is solved explicitly. The obtained solutions can reveal the dependence of the current configurations of the hyperelastic plates on the differential growth fields. By solving an inverse problem, some analytical formulas are obtained, which can be used to identify the growth functions for generating arbitrary 2D geometrical shapes of the hyperelastic plates. To demonstrate the efficiency of these formulas, some representative examples are studied, which show good consistency with the numerical simulations. The obtained analytical formulas have wide potential applications in the design of intelligent soft devices.