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3. Finite element formulations for piezoelectric structures

Hamilton's principle is used to derive the dynamic variational principle []:

(1)

where V is the volume of the piezoelectric structure, ρ is the mass density, {u} is the displacement field and {δ u } its variation, {φ} is the elecctric pontential and {δ φ } its variation. {f } is the volumic force, {ρ_e } the volumic charge density, {t } the vector of applied surface forces on Ω₁ and {σ } the charge density applied on Ω₂. The variational principle is the starting point for all discrete finite element formulations. 3D and shell approximations are detailed below.