Self-sustained eversion or inversion of a thermally responsive torus

When a cylindrical rod is placed on a flat and hot surface with a constant temperature, it can reach a steady state after certain time. In the steady state, though the temperature field inside the rod is inhomogeneous, it does not change with time. The inhomogeneous temperature change in the rod may induce inhomogeneous thermal expansion in it. Recent experiments have determined that if the rod is slightly curved, the inhomogeneous thermal expansion in the rod can drive its continuous and self-sustained rolling on a hot surface. It has been further shown that if the rod is bent to a closed torus and placed on a hot surface, the torus everts or inverts continuously due to the cross-coupling between the thermal field and the cyclic rotation. Such cyclic eversion or inversion of a torus can be regarded as a zero-elastic-energy mode because both the elastic energy and the shape of the torus remain unchanged during the rotation. In this article, we develop a coupled mechanics theory to model the continuous self-sustained eversion or inversion of a viscoelastic torus on a hot surface. We hope our modeling will inspire more novel designs of elastic motors being capable of zero-energy mode motion and help to quantitatively predict their performance.