Large deformation of a punctured disc with the target morphology

Thin sheets widely exist in nature. Studying their fascinating morphologies in three-dimensional space is not only a fundamental scientific issue but of significant practical applications. Utilizing differential manifolds based on the Riemannian metric, this paper addresses the large deformation of a sheet. First, with the polar coordinates, the two-dimensional manifold of a punctured disc and the metric tensor induced by a temperature field are described, providing a geometric interpretation of the thermal deformation mechanism. Next, the equilibrium equations and the boundary conditions for the large deformation of a target cone are derived. Finally, given the temperature field, the deformation and the residual stress are solved. The temperature distribution is also obtained for the stress-free target cone. The results indicate that both the temperature field and the target morphology significantly impact the deformation and the residual stress. This work is a straightforward extension of the customary in-plane thermal deformation theory to the two-dimensional manifold in the three-dimensional Euclidean space, and hence provides the fundamental theory and the solution technique for the finer design of thin sheets.