Although the passively adjusting and actively tuning of pure longitudinal (primary (P-)) and pure transverse (secondary or shear (S-)) waves band structures in periodically laminated piezoelectric composites have been studied, the actively tuning of coupled elastic waves (such as P-SV, P-SH, SV-SH, and P-SV-SH waves), particularly as the coupling of wave modes is attributed to the material anisotropy, in these phononic crystals remains an untouched topic. This paper presents the analytical matrix method for solving the dispersion characteristics of coupled elastic waves along the thickness direction in periodically multilayered piezoelectric composites consisting of arbitrarily anisotropic materials and applied by four kinds of electrical boundaries. By switching among these four electrical boundaries—the electric-open, the external capacitance, the electric-short, and the external feedback control—and by altering the capacitance/gain coefficient in cases of the external capacitance/feedback-voltage boundaries, the tunability of the band properties of the coupled elastic waves along layering thickness in the concerned phononic multilayered crystals are investigated. First, the state space formalism is introduced to describe the three-dimensional elastodynamics of arbitrarily anisotropic elastic and piezoelectric layers. Second, based on the traveling wave solutions to the state vectors of all constituent layers in the unit cell, the transfer matrix method is used to derive the dispersion equation of characteristic coupled elastic waves in the whole periodically laminated anisotropic piezoelectric composites. Finally, the numerical examples are provided to demonstrate the dispersion properties of the coupled elastic waves, with their dependence on the anisotropy of piezoelectric constituent layers being emphasized. The influences of the electrical boundaries and the electrode thickness on the band structures of various kinds of coupled elastic waves are also studied through numerical examples. One main finding is that the frequencies corresponding to (with the dimensionless characteristic wavenumber) are not always the demarcation between pass-bands and stop-bands for coupled elastic waves, although they are definitely the demarcation for pure P- and S-waves. The other main finding is that the coupled elastic waves are more sensitive to, if they are affected by, the electrical boundaries than the pure P- and S-wave modes, so that higher tunability efficiency should be achieved if coupled elastic waves instead of pure waves are exploited.
On the quasistatic effective elastic moduli for elastic waves in three-dimensional phononic crystals
