Formulations of elastodynamic equations for anisotropic multiphase porous piezoelectric media based on global energy conservation

Abstract

Multiphase porous piezoelectric media are essential for advanced transducers and smart sensors. Existing theories typically postulate Newton's second law for each phase or rely on phenomenological Hamiltonian constructions. The former forces \emph{ad hoc} virtual-mass tensors to describe interphase inertia, while the latter provides no intrinsic safeguard against thermodynamic inconsistency when piezoelectric and multiphase couplings are superposed. In this work, we establish a linear dynamic and constitutive theory for anisotropic multiphase porous piezoelectric media from global energy conservation (GEC). From an abstract energy density functional, Taylor expansion and symmetry constraints derive the standard kinetic and potential energy densities and electric enthalpy, rather than assuming them a priori. Localization of the GEC integral yields the multiphase momentum equations, Gauss's law, the coupled constitutive relations, and the boundary conditions as mathematical corollaries, without invoking Newton's law or Hamilton's principle. The framework eliminates virtual-mass parameters entirely: interphase inertial coupling emerges organically from the off-diagonal kinetic-energy coefficients $ρ_{ij}^{αβ}$. Because all coefficients derive from a single smooth potential, Schwarz's theorem automatically guarantees Maxwell reciprocity and full thermodynamic self-consistency. The formulations agree with those from Hamilton's principle and reduce exactly to Biot's poroelastic theory and Tiersten's single-phase piezoelectric theory in the respective limits. Finally, linear plane-wave analysis produces a generalized Christoffel eigenvalue equation, and numerical phase-velocity calculations for water-saturated porous PZT-2 illustrate the modal structures and reveal strongly directional electromechanical coupling.