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.
We dedicate this work to the memory of
our maestro Professor Gaetano Giaquinta (Catania, Italy,
1945–2016), who first taught us Noether's theorem and showed
us its unifying beauty.