Kinematics

The second group of assumptions required by the first-order piezoelectric symmetricthickness model is of kinematic nature and requires deformation of the normals to the midsurface onto normals, not necessarily perpendicular to the mid-surface, and no elongation of these normals during deformation. The first assumption reads:

$$\boldsymbol{\mu}'\_{j} = \frac{1}{2} (\boldsymbol{\mu}'^{b}\_{j} + \boldsymbol{\mu}'^{t}\_{j}) + \frac{\boldsymbol{\mathcal{X}}'\_{3}}{t} (\boldsymbol{\mu}'^{t}\_{j} - \boldsymbol{\mu}'^{b}\_{j}), \qquad \mathbf{x} \in V,\tag{25}$$

where *V* represents the symmetric-thickness domain or such a part of the complex domain. Above, the local displacements *u* can be transformed into the global ones *u* typical for the applied 3D-based approach, with use of the transformation matrix *θ<sup>T</sup>* such that *uj* = *<sup>θ</sup>ijui*. It is obvious that *i* = 1, 2, 3 represent the global directions, while *j* = 1, 2, 3 denote the local directions of which the first two are tangent and the third one is normal to the mid-surface *Sm* of the symmetric-thickness piezoelectric structure. It can be noticed that the above condition is expressed by the top and bottom local displacements, *u<sup>t</sup>* and *u<sup>b</sup>* that can be described with *u<sup>t</sup>* = *u* for *x* ∈ *St* and *u<sup>b</sup>* = *u* for *x* ∈ *Sb*, where *St* and *Sb* are the top and bottom surfaces within the symmetric-thickness structure. Note also that the first and second terms of the right hand-side of the above condition represent the local displacements of the mid-surface and the displacements due to rotation of the normal. The latter displacements change linearly along the normal as *t* is the structure thickness, while *x*3∈ (− *t*2, *t*2) measures the distance from the mid-surface *Sm*.

The second assumption says that the third local displacements of the top and bottom surfaces are the same. Because of this no elongation is possible in this direction, namely:

$$
\mu\_{\clubsuit}^{\prime t} - \mu\_{\clubsuit}^{\prime b} = 0, \qquad \mathbf{x} \in V\_{\prime} \tag{26}
$$

where *V* is defined as in (25). One can express the above condition by the global displacements with use of *u*3= *θi*3*ui* and *ut*3 = *<sup>u</sup>*3 for *x* ∈ *St*, *ub*3 = *<sup>u</sup>*3 for *x* ∈ *Sb*.

### *3.2. Transition Piezoelectric Models*

Three piezoelectric transition models (classical, modified, and enhanced) will be presented. In each of these three models, continuity of the displacement and electric potential fields, on the boundaries between the transition and basic models, is maintained due to three-dimensional approach which lies in application of the three-dimensional displacements and three-dimensional potential for any model of the piezoelectric structure. The hierarchical or first-order models are consistently treated as constrained threedimensional models.

The classical transition model is characterized with the mechanical field of the threedimensional model. Only on the boundary with the first-order model, the plane stress assumption is valid. In the case of the modified transition model, the stress field changes gradually from of three-dimensional one (on the boundary with the three-dimensional domain) to the plane stress one (on the boundary with the first-order model). The same stress field description is valid also for the enhanced transition model.

The enhanced transition model is characterized with the gradually changing assumption of no elongation of the lines perpendicular to the mid-surface of the domain. This assumption changes from its total validity on the boundary with the first-order model to its lack on the boundary with the three-dimensional model. Note that, in the cases of the classical and modified models, this assumption is valid only on the boundary with the first-order model. Apart from this boundary, this assumption does not apply.

## 3.2.1. The Classical Approach Stress State

The mechanical and electric fields within the classical transition piezoelectric model are defined as in the case of the three-dimensional piezoelctricity characterized with the constitutive Equation (16), completed with the definitions (13), (18), and (19).
