Cauchy-Maxwell equations: A unified field theory for coupled electromagnetism and elasticity

Abstract

A conformal gauge theory is used to describe and unify myriad electromechanical and magnetomechanical coupling effects observed in solid continua. Using a space-time pseudo-Riemannian metric in a finite-deformation setup and exploiting the local conformal symmetry of the Lagrangian, we derive Cauchy-Maxwell (CM) equations that seamlessly combine, for the first time, Cauchy's elasto-dynamic equations with Maxwell's equations for electromagnetism. Maxwell's equations for vacuum are recoverable from our model, which in itself also constitutes a new derivation of these equations. With deformation gradient and material velocity coupled in the Lagrange density, various pseudo-forces appear in the Euler-Lagrange equations. These forces, not identifiable through classical continuum mechanics, may have significance under specific geometric or loading conditions. As a limited illustration on how the CM equations work, we carry out semi-analytical studies, viz. on an infinite body subject to isochoric deformation and a finite membrane under both tensile and transverse loading, considering piezoelectricity and piezomagnetism. Our results show that under specific loading frequencies and tension, electric and magnetic potentials may increase rapidly in some regions of the membrane. This may have significance in future studies on efficient energy harvesting.