Kinematic Assumptions

In this classical transition model, the kinematic assumption of the first-order mechanical theory consisting in deformation of the normals to the mid-surface into normals during deformation and the lack of elongation of these normals are valid on the boundary *R* between the transition and first-order theories, i.e.,

$$\begin{aligned} \boldsymbol{u}\_{\boldsymbol{j}}^{\prime} &= \frac{1}{2} (\boldsymbol{u}\_{\boldsymbol{j}}^{\prime b} + \boldsymbol{u}\_{\boldsymbol{j}}^{\prime t}) + \frac{\boldsymbol{x}\_{3}^{\prime}}{t} (\boldsymbol{u}\_{\boldsymbol{j}}^{\prime t} - \boldsymbol{u}\_{\boldsymbol{j}}^{\prime b}), \quad \mathbf{x} \in \nabla \cap \boldsymbol{R} \equiv \boldsymbol{R} \\\ 0 &= \boldsymbol{u}\_{\boldsymbol{j}}^{\prime t} - \boldsymbol{u}\_{\boldsymbol{j}}^{\prime b}, \quad \mathbf{x} \in \nabla \cap \boldsymbol{R} \equiv \boldsymbol{R}, \end{aligned} \tag{27}$$

where *V* = *V* ∪ *∂V*, *∂V* ≡ *S*. As in the case of the first-order model, the above conditions can be expressed by the global displacements with use of the same relations as before. Note that the mid-surface within the transition domain may only be defined on the boundary *R* between the first-order and transition models.

#### 3.2.2. the Modified Transition Model

## The Assumed Stresses

The second approach is based on the transition character of the plane stress assumption. This assumption is valid on the boundary with the first-order piezoelectric model. It does not hold, however, on the boundary with the hierarchical higher-order or threedimensional piezoelectric model, i.e., three-dimensional stress state is present on this boundary. Between these two boundaries, the stress state is intermediate, namely:

$$\sigma\_{33} = D \left\{ \nu \varepsilon\_{11} + \nu \varepsilon\_{22} + (1 - \nu) \left\langle n \varepsilon\_{33} + (1 - n) \left[ \frac{-\nu}{1 - \nu} (\varepsilon\_{11} + \varepsilon\_{22}) + \frac{\mathcal{C}\_{33}}{D(1 - \nu)} E\_3 \right] \right\rangle \right\} - \mathcal{C}\_{33} E\_3. \tag{28}$$

Above, the function *α* = *α*(*Sm*) ∈ 0, <sup>1</sup>, with *Sm* being the mid-surface of the thick- or thin-walled part of the transition domain, is a blending function equal to 1 at the boundary with the three-dimensional model, and 0 at the boundary with the first-order model. Consequently, the definition (28) becomes identical with (20) for *α* = 1, and with (22) for *α* = 0.

Taking the above equation into account in the third row of the first relation (16) leads to: *σ* = *σ*1 − *σ*2, where the first component equals:

$$
\sigma\_1 = \begin{bmatrix} D\_{11} & D\_{12} & D\_{13} & 0 & 0 & 0 \\ D\_{21} & D\_{22} & D\_{23} & 0 & 0 & 0 \\ D\_{31} & D\_{23} & D\_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & D\_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & D\_{55} & 0 \\ 0 & 0 & 0 & 0 & 0 & D\_{66} \end{bmatrix} \begin{bmatrix} \varepsilon\_{11} \\ \varepsilon\_{22} \\ \kappa \varepsilon\_{33} + (1 - \kappa) \frac{-\upsilon}{1 - \upsilon} (\varepsilon\_{11} + \varepsilon\_{22}) \\ \varepsilon\_{12} \\ \varepsilon\_{23} \\ \varepsilon\_{33} \end{bmatrix},\tag{29}
$$

while the second one is equal to:

$$
\sigma\_2 = \begin{bmatrix}
0 & 0 & \mathbf{C}\_{13} - (1 - \alpha) \frac{\mathbf{v}}{1 - \mathbf{v}} \mathbf{C}\_{33} \\
0 & 0 & \mathbf{C}\_{23} - (1 - \alpha) \frac{\mathbf{v}}{1 - \mathbf{v}} \mathbf{C}\_{33} \\
0 & 0 & \alpha \mathbf{C}\_{33} \\
0 & 0 & 0 \\
0 & \mathbf{C}\_{52} & 0 \\
\mathbf{C}\_{61} & 0 & 0
\end{bmatrix} \begin{bmatrix} E\_1 \\ E\_2 \\ E\_3 \end{bmatrix}.\tag{30}
$$

In addition, the second equation (16) can be divided into two components, i.e., *d* = *d*1 + *d*2, where the first component is:

$$d\_1 = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & \mathcal{C}\_{61} \\ 0 & 0 & 0 & 0 & 0 & \mathcal{C}\_{52} & 0 \\ \mathcal{C}\_{13} - (1 - a) \frac{\nu}{1 - \nu} \mathcal{C}\_{33} & \mathcal{C}\_{23} - (1 - a) \frac{\nu}{1 - \nu} \mathcal{C}\_{33} & a \mathcal{C}\_{33} & 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} \varepsilon\_{11} \\ \varepsilon\_{22} \\ \varepsilon\_{33} \\ \varepsilon\_{12} \\ \varepsilon\_{23} \\ \varepsilon\_{33} \end{bmatrix}, \tag{31}$$

while the second component reads

$$d\_2 = \begin{bmatrix} \gamma\_{11} & 0 & 0 \\ 0 & \gamma\_{22} & 0 \\ 0 & 0 & \gamma\_{33} + (1 - a) \frac{C\_{33}^2}{D(1 - \nu)} \end{bmatrix} \begin{bmatrix} E\_1 \\ E\_2 \\ E\_3 \end{bmatrix}. \tag{32}$$

Note that, when *α* = 1, the relations (29)–(32) transform into Equations (16)–(19), while, for *α* = 0, into Equations (23) and (24).

## The Assumed Kinematics

As it comes to the kinematic assumptions within the modified transition element, they are the same as in the case of the classical elements, i.e., the conditions (27) are valid on the boundary *R* between the transition and first-order models.

#### 3.2.3. The Enhanced Transition Model
