A variational principle for the equations of piezoelectromagnetism in elastic dielectric crystals

In piezoelectric or ferroelectric crystals, which are often employed in ultrasonics applications, the acoustic field of small amplitude is governed by the equations of elasticity of infinitesimal displacement gradients and the electromagnetic (EM) field is governed by Maxwell’s equations. The interactions between the mechanical and electromagnetic fields are mainly due to the couplings in the constitutive equations. The governing equations consisting of field equations and constitutive equations are called the equations of piezoelectromagnetism. In a dielectric crystal of volume V bounded by a surface S which separates V from an outer vacuum V′, the kinetic-energy density and electric enthalpy density are defined. By introducing these density functions in a variational principle and by requiring the independent variations of the mechanical displacement, the scalar and vector potentials of the EM field, it is shown that the equations of piezoelectromagnetism and the appropriate jump conditions are obtained. The present variational principle reduces to those for Maxwell’s equations, the equations of elasticity, and the equations of piezoelectricity, respectively.