Analytical study on growth-induced axisymmetric deformations and shape-control of circular hyperelastic plates

Abstract

Growth of soft material plates is commonly observed in nature. However, the relations between growth fields and shape changes of the plate samples remain poorly understood. The current work aims to derive some analytical results for the growth-induced axisymmetric deformations of a circular hyperelastic plate. The problem of shape-control of the circular plate through differential growth will also be studied. First, starting from the 3D governing system, we formulate the 2D vector plate equation system through a series expansion and truncation approach. Then, the plate equations are solved by using the regular perturbation method. Some analytical (asymptotic) solutions are derived for both the out-of-plane bending and in-plane flat deformations of the plate, which can fit the numerical results quite well. By comparing the energy densities, the preferred deformation styles of the plate corresponding to any given growth functions can be determined, based on which the combined bending-flat deformation of the plate is also studied. Furthermore, by solving an inverse problem, we derived the explicit formula for shape-control of thin circular hyperelastic plates. With this formula, most of the 3D axisymmetric configurations can be generated by selecting proper growth functions in the plates. The results obtained in the current work can not only reveal the mechanisms of growth-induced deformations of soft material samples, but also have wide potential applications in the manufacture of intelligent soft devices.