General shape transformations of thin hyperelastic shells through stress-free differential growth

To address the needs of engineering applications, researchers often wish that the shapes of samples can be precisely controlled. The current work aims to propose a promising approach, i.e., through stress-free differential growth, to realize general shape transformations of thin hyperelastic shells. First, within the finite-strain regime, we formulate the 3D governing equations system for modeling the growth behavior of hyperelastic shells. To facilitate the derivations, it is assumed that the shell sample attains the stress-free state in its current configuration. Subsequently, through series expansions of the unknown variables, we derive the explicit analytical formulas that elucidate the intricate relationships between growth functions and the geometric quantities of general 3D target surfaces. Based on these analytical formulas, we propose a theoretical framework for controlling the shape changes of the shell sample from the reference configuration to a desired target configuration. Notably, our framework accommodates a wide array of geometric mappings, including topology transformation, conformal mapping, and isometry mapping. To promote applications of the theoretical framework, a numerical scheme is further proposed to achieve shape transformations of shell samples between complex surfaces without explicit parametric equations. Both the theoretical framework and the numerical scheme are validated through 3D finite element simulations. The results of the current work can be applied for the design of novel intelligent soft devices, which also reveal the connections between solid mechanics and differential geometry.