Three-dimensional (3D) constitutive equations of piezoelectric (PZ) plates and shells are considered for inverse linear and electrostrictive (quadratic) piezoeffects. Prestressed multilayer PZ shells reinforced with metal including the case of uneven thickness polarization are studied. Asymptotic and variational methods to solve the governing differential equations of PZ shells are considered. Concentrations of electrical and mechanical fields near structure imperfections and external local loading are investigated. The electrothermoviscoelastic heating of PZ shells is considered at harmonic excitation. From numerical analysis and the experimental data of energy dissipation and the temperature behavior of PZ shell the conditions of optimal transformation of electric energy into mechanical deformations are defined. Thus, the geometrical parameters and working frequencies are determined with due account of dielectric relaxation processes. The following nonlinear phenomena are studied: acoustoelectronic wave amplification; electron injection into metalized polar dielectric; resonance growth by 5–20 times of internal electrical field strength in the PZ shells and plates; and autothermostabilization of ferroelectric resonators. For a better understanding of R.D. Mindlin’s gradient theory of polarization in view of electron processes in thin metal-dielectric-metal structures, use was made of solid state physics interpretations as well as experimental data. High concentration of mechanical stresses and temperature and electrical fields near structure defects (first of all, near boundary between various materials) defines the main properties of polar dielectrics. An unknown domain of electrode rough surface influence was estimated, and as result an uneven polarization distribution was found. A theory of nonlinear autowave systems with energy dissipation was used in a physical model of the electrothermal fracture of dielectrics (contacting with metal electrodes), and as a result a nondestructive testing method to study the microstructure defect formation has been suggested.

This is a review article on a few special topics in piezoelectricity: gradient and nonlocal theories, fully dynamic theory with Maxwell equations, piezoelectric semiconductors, and motions of rotating piezoelectric bodies. They all require some extension of the classical theory of piezoelectricity. They are relatively new, more advanced, and growing subjects with applications or potential applications in various electromechanical devices. The article contains 209 references. (In memory of Raymond D. Mindlin (1906–1987)).

This is a review article on the mechanics of small-amplitude motions superposed on finite biasing or initial fields in an electroelastic body. It begins with a summary of the nonlinear theory of electroelasticity, which is the theoretical foundation for the theory of small fields superposed on a bias. This theory, obtained by different approaches, and the development of structural theories for electroelastic beams, plates, and shells under biasing fields are discussed. Applications of these theories for small fields superposed on biasing fields in the buckling of thin electroelastic structures, frequency stability of piezoelectric resonators for time-keeping and telecommunication, acoustic wave sensors based on frequency shifts due to biasing fields, characterization of nonlinear electroelastic materials by propagation of small-amplitude waves in electroelastic bodies under biasing fields, and electrostrictive ceramics which operate under a biasing electric field are reviewed. A summary of some current and possible future research topics in this field is given. The article contains 166 references.