On the Lagrangian electrostatics of elastic solids

The equilibrium equations and the Maxwell equations of electrostatics are derived here through a variational procedure. All the treatment is expounded in the reference configuration of the deformed body and the electromagnetic quantities and equations of interest are expressed in Lagrangian form. The total stress tensor, which includes the electrical effects, emerges in a natural way in this framework. Along with this stress, an additional electromechanical tensor is introduced, the Eshelby stress tensor. This tensor, which plays a relevant role in the presence of inhomogeneities or material defects, addresses remarkable integral identities that are valuable in boundary value problems. In this framework, a Lagrangian Electrostatics emerges as a self-consistent topic that parallels the classical electrostatics. Eventually, simple constitutive relationships are discussed.