  In this paper, we study the problem of shape-programming of incompressible
hyperelastic shells through differential growth. The aim of the current work is
to determine the growth tensor (or growth functions) that can produce the
deformation of a shell to the desired shape. First, a consistent finite-strain
shell theory is introduced. The shell equation system is established from the
3D governing system through a series expansion and truncation approach. Based
on the shell theory, the problem of shape-programming is studied under the
stress-free assumption. For a special case in which the parametric coordinate
curves generate a net of curvature lines on the target surface, the sufficient
condition to ensure the vanishing of the stress components is analyzed, from
which the explicit expression of the growth tensor can be derived. In the
general case, we conduct the variable changes and derive the total growth
tensor by considering a two-step deformation of the shell. With these obtained
results, a general theoretical scheme for shape-programming of thin
hyperelastic shells through differential growth is proposed. To demonstrate the
feasibility and efficiency of the proposed scheme, several nature-inspired
examples are studied. The derived growth tensors in these examples have also
been implemented in the numerical simulations to verify their correctness and
accuracy. The simulation results show that the target shapes of the shell
samples can be recovered completely. The scheme for shape-programming proposed
in the current work is helpful in designing and manufacturing intelligent soft
devices.
