On the fundamental equations of unsteady anisothermal viscoelastic piezoelectromagnetism

Abstract

The derivation of the piezoelectric equations requires only the Maxwell equations of electrostatics and the momentum equation with electric forces, together with the constitutive equations derived from the first principle of thermodynamics. In the non-isothermal case, the inclusion of thermal conduction requires also the second principle of thermodynamics and then the equation of energy becomes necessary. Furthermore, in the viscoelastic case, the viscous stresses modify the momentum and energy equations. The sequence outlined above provides the simplest derivation of the fundamental equation of anisothermal viscoelastic piezoelectricity including thermal and viscous dissipation. The equations of piezomagnetism are similar to those of piezoelectricity, replacing the electric by magnetic fields. The coupling of the unsteady electric and magnetic fields through electromagnetic waves interacts with both piezoelectricity and piezomagnetism leading to piezoelectromagnetism. In the general case of unsteady anisothermal piezoelectromagnetism, the energy, momentum and Maxwell equations specify the temperature, displacement vector, electric field and magnetic field. The coefficients involve constitutive and diffusion tensors specifying the properties of matter, generally anisotropic, such as crystals and orthotropic plates. They can also specify the properties of amorphous substances corresponding to the simplest isotropic case.