ESHELBY’S INCLUSION THEORY IN LIGHT OF NOETHER’S THEOREM

In a variational setting describing the mechanics of a hyperelastic body with defects or inhomogeneities, we show how the application of Noether’s theorem allows for obtaining the classical results by Eshelby. The framework is based on modern differential geometry. First, we present Eshelby’s original derivation based on the cut-replace-weld thought experiment. Then, we show how Hamilton’s standard variational procedure “with frozen coordinates”, which Eshelby coupled with the evaluation of the gradient of the energy density, is shown to yield the strong form of Eshelby’s problem. Finally, we demonstrate how Noether’s theorem provides the weak form directly, thereby encompassing both procedures that Eshelby followed in his works. We also pursue a declaredly didactic intent, in that we attempt to provide a presentation that is as self-contained as possible, in a modern differential geometrical setting