Configurational forces and gauge conditions in electromagnetic bodies

Abstract

Material inhomogeneities, defective materials or fractured solids address the notion of configurational force. In electromagnetic deformable bodies this force also depends upon the electromagnetic potentials, which unfortunately are not uniquely defined. In electrostatics the problem of uniqueness is scarcely relevant, as only the scalar potential plays a role. Thus, dielectrics are adequately described in this context, even in the presence of a crack-line. In electrodynamics also the vector potential plays a prominent role and the lack of uniqueness of the electromagnetic potentials cannot be disregarded. The problem of solving this indeterminacy for the quantities of interest appeals to additional conditions, the gauge conditions. Gauge transformations, which leave invariant the Maxwell equations and the balance of momentum in material form, are here examined. In configurational mechanics of electromagnetic solids, a possible gauge dependent quantity is the material momentum, which seems to be related to some extent to supercurrents in deformable superconductors.

On the dynamics of electromagnetic bodies

“…However, care is needed in the constitutive relationships. In fact, even in the trivial case of a vacuum, the consistency of the equations (1)-(3) with (12)- (14) appeals for the introduction of following co-moving fields:…”

“…More specifically,Ȧ ≡ (∂A/∂t)| X . [5,8,13,14]. The equations (21) hold true if the Lagrangian electromagnetic fields that enter these equations are so related to the fields in the current frame…”

“…Alternatively, this balance law can be established independently of the equations of motion, through a non-standard variational procedure. The two different derivations address the same final equation, which is [5,14] Div [(F T (∂ L/∂F)…”

“…With further reference to the equation (43), the argument of the material time derivative {(∂/∂t)| X }, which is denoted hereafter by p M , represents the material momentum. By taking into account the Lagrangian (33) and the results of the previous sections, the electromagnetic Eshelby tensor and the material momentum explicitly read, respectively, [14] b…”