# Mathematical formulations for 3D linear piezoelectricity

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## Proposed enhanced Euler-Bernoulli model

This section presents an original coupled Euler-Bernoulli model where the electromechanical constitutive coefficients account for the influence of the cross-warping and the induced electric potential. To this end I adopted a deductive approach based on the Hellinger- Reissner mixed variational formulation presented in Section 2.2.3. By exploiting the peculiar geometry of a laminated beam, a priori hypotheses on the distribution of the mechanical displacement, the electric potential, the mechanical stress and the electric displacement are introduced.

### Geometry and notation

A beam is defined as a slender body B having two dimensions relatively small with respect to the third one. Taking A as a straight-line parameterized by the abscissa x referred to an origin o ∈ A, and e1 a unit vector parallel to A, the generic point p of a straight-axis beam B is located by p = o+xe1 + d, d ⊥ A (3.2) where d is an arbitrary vector orthogonal to A. The set where S(¯x) ≡ {p ∈ B : p = pa + d}, (3.3) for a fixed pa = o+¯xe1, is the beam cross-section at the axial point pa. The straight line A is the beam axis.
Laminated beams are composed by stacking up n rectangular cross-section layers. Let C = {o, e1, e2, e3} be a Cartesian reference system with the e3-axis aligned to the stacking direction. The corresponding coordinates are denoted by (x, y, z). The rectangular domain corresponding to the part of the cross-section S(¯x) occupied by the i-th layer is denoted by Si(¯x). It is regarded as the Cartesian product of a thickness-segment Ti of length hi and a width-segment Wi of length ai. Moreover, n local reference frames Ci = {oi, e1, e2, e3}, with oi = o+ ¯zie3, are introduced and the corresponding local coordinates are denoted by (x, y, zi) where zi = z − ¯zi (see Figure 3.1). The local origins are chosen so as to satisfy Z Ti zidz = 0.

#### 1D mixed formulation with conditions on transverse stresses

With the conditions (3.5-3.9), the distributions of the three-dimensional state fields (u, ϕ, T, D) are uniquely determined by the fields {u(x), w(x), V, σα,i(x, y), ζα,i(x, y),Di(x, y)} , (3.10).
where the electric voltage V is independent of x and y because the surfaces of the transducers are completely covered by a single electrode. Here and henceforth, the mute indices α and i are intended to vary from 1 to 2 and from 1 to n, respectively.
The substitution of the hypotheses (3.5), (3.6), (3.7), and (3.9) into the functional (2.18) leads to the following beam mixed functional Θbeam[u, w, V, σα,i, ζα,i,Di] =0.

Euler-Lagrange equations of the mixed functional

The Euler-Lagrange equations of the variational problem are obtained by imposing the stationary conditions on the mixed functional ΘΛ beam for all the admissible beam state fields and for all the admissible Lagrange multiplier fields of ΘΛ beam with respect to (u, w, V, σα,i, ζα,i, Di) and (λ, μ) are null. These conditions produce the following equations. Beam balance equations and boundary conditions (Stationary conditions with respect to kinematical fields).
The stationary conditions with respect to the generalized kinematic variables (u, w, V ) corresponds to the beam balance equations bN (x) + N01 (x) = 0 b0M (x) − bT (x) + M00 1 (x) = 0 R q(x)dx − ¯Q = 0.

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Formulas for the constitutive coefficients

Let us consider the particular layer arrangement of a piezoelectric sandwich shown in Figure 2.2. For this configuration, simple formulas for the beam constitutive parameters appearing in Eqns. (3.34) are found. The case of a beam having piezoelectric and elastic layers with different widths is considered explicitly. For a sandwich beam with thin piezoelectric layers, simplified expressions are derived.
Complete expressions For a piezoelectric sandwich beam, because of the material and geometric symmetry, beam extension and bending are decoupled and the stiffness matrices ˜k 11 and ˜k12 defined in equations (3.26) are diagonal (C11 = C12 = 0). Moreover, due to the electric interconnection of the two piezoelectric layers, the first term of the coupling vector ˜eχd is null, and an imposed electric potential V induces only a bending moment with a vanishing force resultant. Hence, the constitutive equations (3.34) for the bending moment M1 and the electric charge per unit line q, being uncoupled from the extensional  problem, assume the simplified form ∙ M1 q  ¸ = ∙0

1 Introduction
1.1 Motivations
1.2 Objectives
1.3 Outline
2 Piezoelectricity
2.1 Piezoelectric materials
2.2 Mathematical formulations for 3D linear piezoelectricity
2.3 Piezoelectric transducers
2.4 Piezoelectric laminates
3 Beam models of piezoelectric laminates
3.1 Introduction and literature review
3.2 Proposed enhanced Euler-Bernoullimodel
3.3 Standard Euler-Bernoullimodels
3.4 3D Finite Elements
3.5 Numerical comparisons
3.6 Conclusions
4 Numerical Methods for Modal Analysis of Stepped Piezoelectric Beams
4.1 Introduction and literature review
4.2 Problemformulation
4.3 Last Energy Norm(LEN)method
4.4 Galerkinmethods
4.5 Reduced ordermodelling
4.6 Case study and comparisons
5 Passive vibration control through distributed piezoelectric transducers and electric networks
5.1 Introduction and literature review
5.2 Beams with piezoelectric transducers shunted with passive electric networks
5.3 Single shunted piezoelectric transducer
5.4 Distributed wave absorbers for infinite beams
5.5 Modal control with second-order networks
5.6 Conclusions
6 Experiments
6.1 Technical details about the experimental set-ups
6.2 Strain analysis of a beam with piezoelectric transducers
6.3 Experimental modal analysis of stepped piezoelectric beams
6.4 Passive vibration with second-order networks
7 Conclusions
7.1 Summary
7.2 Original contributions
7.3 Suggestions for future works
7.4 Publications
Bibliography
A Constitutive coefficients
A.1 Constitutive coefficients for plane-stress and uniaxial-stress conditions .
A.2 Numerical Values

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