2 2 y x z t t t u div t u x y z t ρ ρ ɺɺ 6 where

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{ } 2 2 y x z T T T u DIV T u x y z t ρ ρ = + + = = ɺɺ (6) Where DIV is divergence of a dyadic. ρ is density of the piezoelectric medium. Whereas the electric behavior is described by Maxwell’s Equation considering that piezoelectric media are insulating (no free volume charge). { } 0 DIV D = (7) Equations (1)-(7) constitute a complete set of differential equations which can be solved with appropriate mechanical (displacements and forces) and electrical (potential and charge) boundary conditions. [1] The dynamic equations of a piezoelectric continuum can be derived from the Hamilton principle, in which the Lagrangian and the virtual work are properly adapted to include the electrical contributions, as well as the mechanical ones. The potential energy density of a piezoelectric material includes contributions from the strain energy and from the electrostatic energy. [5], [6] { } { } { } { } 1 2 T T H S T E D = (8) Similarly, the virtual work density { } { } t W u F δ δ δφσ = (9) Where { } F is the external force and σ is the electric charge. From (8), (9), into the Hamilton principle. [7] { } { } { } { } { } [ ] { } { } [ ] { } { } { } { } { } { } { } { } { } 1 2 0 t t E t t t V t t S b t S S t c S u u S c S S e E E e S dV E E u P u P dS u P dS Q ρ δ δ δ δ δ ε δ δ δ δφσ δφ = − + + + + + + ɺɺ (10) In the finite element formulation, the displacement field { } u and the electric potential φ over an element are related to the corresponding node values { } i u and { } i φ by the mean of the shape functions { } u N , { } N φ . { } [ ] { } u i u N u = (11) { } i N φ φ φ = (12) And therefore, the strain field { } S and the electric field { } E are related to the nodal displacements and potential by the shape functions derivatives [ ] u B and B φ defined by: { } [ ] { } [ ][ ] { } [ ] { } u i u i S D u D N u B u = = = (13) { } { } { } i i E N B φ φ φ φ φ = −∇ = −∇ = − (14) Substituting expressions (11) – (14), into the variation principle (10), yields: { } [ ] [ ] { } { } [ ] [ ] { } { } [ ] [ ] { } { } [ ] [ ] { } { } { } { } [ ] { } { } [ ] { } { } [ ] { } { } { } 1 2 0 t t i u u V t t E i u u i V t t i u i V t t t i u i V t t S i i V t t i u b V t t t t i u S i u c S t t t t i i S u N N dV u u B c B dV u u B e B dV B e B dV u B B dV u N P dV u N P dS u N P N dS N Q φ φ φ φ φ φ δ ρ δ δ φ δφ δφ ε φ δ δ δ δφ σ δφ = − + + + + ɺɺ (15) Which must be verified for any arbitrary variation of the displacements { } i u δ and electrical potential { } i δφ compatible with the essential boundary conditions. For an element, (15), can be written under the form: [ ] { } [ ] { } { } { } i uu i u i i M u K u K f φ φ + + = ɺɺ (16) { } { } { } u i i i K u K g φ φφ φ + = (17) With [ ] [ ] [ ] t u u V M N N dV ρ = (18) [ ] [ ] [ ] t E uu u u V K B c B dV = (19) [ ] [ ] t t u u V K B e B dV φ φ = (20) t S V K B B dV φφ φ φ ε = − (21)
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