Fusion of Nonlinear Elasticity with Galilean Electromagnetism

Herein we take a first step towards merging nonlinear elasticity with the two non-relativistic Galilean-covariant limits of electromagnetism, namely the electric limit and the magnetic limit, the results of which we call Galilean electroelasticity and Galilean magnetoelasticity, respectively. Using the first law of thermodynamics for dynamical adiabatic processes, we derive, for systems (with zero free-charge and free-current densities) which undergo such processes, the internal energy density function and its associated constitutive equations in Galilean electroelasticity and magnetoelasticity, respectively. Each of the two internal energy density functions (per unit reference volume) thus obtained agrees with one of the two total energy density functions introduced by Dorfmann and Ogden in their work on electro-elastostatics and magneto-elastostatics, respectively. For linear polarizable and magnetizable dielectrics, Galilean-invariant expressions of the Maxwell stress are obtained for the electric limit and for the magnetic limit, respectively.