A novel class of model constitutive laws in nonlinear elasticity: Construction via Loewner theory

Using a solitonic connection, we show that the class of infinitesimal Bäcklund transformations originally introduced by Loewner in 1952 in a gasodynamic context results in physically interesting nonlinear model constitutive laws. We obtain laws previously used to model a variety of hard and soft nonlinear elastic responses. A natural extension of the latter leads to a novel class of model constitutive laws where the stress and strain are given parametrically in terms of elliptic functions. Such models allow a change in the concavity of the stress-strain law. Such behavior can be observed in the compression of polycrystalline materials or in the unloading regimes of superelastic nickel-titanium.