The energy spectrum of a twisted flexible string under elastic relaxation

A full investigation of the energy spectrum of a twisted flexible string under elastic relaxation is presented and discussed in detail for the first time. New polynomial expressions for critical energy states are derived and the whole spectrum of critical states (minima, maxima and inflexion points) of the elastic energy is found and discussed in relation to superhelicity and elastic characteristics of the string. The study is carried out in the context of the theory of linear elasticity and the thin rod approximation. The relaxation mechanism is studied by using conservation of linking difference (by the formula Delta Lk=Wr+ Delta Tw). We show how specific geometric quantities, such as pitch angle, writhe and twist contributions, as well as physical quantities, such as torsional and bending energy, depend on superhelicity (given by the specific linking difference) and elastic characteristics of the string (given by bending and torsional rigidity). These quantities, expressed per unit length, are examined and compared at each critical energy state. Starting from a supertwisted configuration, we show that the string relaxes (by twist reduction) through two different intermediate helical states (which correspond to different local minima), to reach the lowest minimum energy state in a supercoiled configuration. The case of a generic kink formation (and consequent passage through an inflexional configuration) is then examined and new expressions for the energy change in the vicinity of the inflexion point are derived.