## 3 Transition Models For Piezoelectricity
### Basic Models of Linear Piezoelectricity
#### 3.1.2 First-Order Piezoelectric Model  
Stress State  
In the case when the mechanical field of the symmetric-thickness thick- or thin-walled member is linearly constrained through the thickness, the plane stress assumption of the first-order shell model have to be introduced into the three-dimensional piezoelectric constitutive relations. This means that the third row equation of the first constitutive relation (16), namely:  
\[\sigma_{33}=\frac{E}{(1+\
u)(1-2\
u)}[\
u\epsilon_{11}+\
u\epsilon_{22}+(1-\
u )\epsilon_{33}]-C_{33}E_{3}, \tag{20}\]  
has to be combined with the assumption \(\sigma_{33}=0\), so as to give  
\[\epsilon_{33}=-\frac{\
u}{1-\
u}(\epsilon_{11}+\epsilon_{22})+\frac{(1+\
u)(1 -2\
u)C_{33}}{E(1-\
u)}E_{3}. \tag{21}\]  
Now, the condition (20) can be rewritten in the following alternative identity-equation form  
\[\sigma_{33}=\frac{E}{(1+\
u)(1-2\
u)}\left\{\
u\epsilon_{11}+\
u\epsilon_{22}+ (1-\
u)\left[\frac{-\
u}{1-\
u}(\epsilon_{11}+\epsilon_{22})+\frac{(1+\
u)(1 -2\
u)C_{33}}{E(1-\
u)}E_{3}\right]\right\}-C_{33}E_{3}=0. \tag{22}\]  
Substitution of the relation (21) into the first relation (16), arithmetic manipulations on the terms containing \(E_{3}\) (with use of isotropic definitions of the non-zero terms of \(D_{ij}\), \(i,j=1,2,\ldots,6\) from (13)) lead to:  
\[\left[ \begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{23}\\ \sigma_{31}\end{array} \right]=\left[ \begin{array}{c}D_{11}&D_{12}&D_{13}&0& 0&0\\ D_{21}&D_{22}&D_{23}&0&0&0\\ D_{31}&D_{23}&D_{33}&0&0&0\\ 0&0&0&D_{44}&0&0\\ 0&0&0&D_{55}&0\\ 0&0&0&0&D_{66}\end{array} \right]\left[ \begin{array}{c}\epsilon_{11}\\ \epsilon_{22}\\ \frac{-\
u}{1-\
u}(\epsilon_{11}+\epsilon_{22})\\ \epsilon_{12}\\ \epsilon_{23}\\ \epsilon_{31}\end{array} \right]-\left[ \begin{array}{c}0&0&C_{13}-\frac{\
u} {1-\
u}C_{33}\\ 0&0&C_{23}-\frac{\
u}{1-\
u}C_{33}\\ 0&0&0\\ 0&0&0\\ 0&C_{52}&0\\ C_{61}&0&0\end{array} \right]\left[ \begin{array}{c}E_{1}\\ E_{2}\\ E_{3}\end{array} \right]. \tag{23}\]  
The alternative form of the above equation can be obtained after arithmetic manipulations on the terms containing \(\epsilon_{11}\) and \(\epsilon_{22}\). This leads to the change of the constitutive constants of the matrix \(\mathbf{D}\), with zero contributions of its third row and third column. This alternative form requires either retaining \(\epsilon_{33}\) as the third term of the strain vector or removal of: the third row and the third column in \(\mathbf{D}\), the third row in \(\mathbf{\varepsilon}\), and the third row in \(\mathbf{C}\). Taking the plane stress assumption into account in the second constitutive relation (16) needs: introduction of (21) into this relation, arithmetic manipulations on the terms containing \(E_{3}\), and summation of the contributions from the terms containing \(\epsilon_{11}\) and \(\epsilon_{22}\). The latter operation leads to the change of the piezoelectric constants in \(\mathbf{C}^{T}\), including zero contributions of the third column of this matrix. All these lead to:  
\[\left[ \begin{array}{c}d_{1}\\ d_{2}\\ d_{3}\end{array} \right]=\left[ \begin{array}{cccccc}0&0&0&0&C_{61}\\ 0&0&0&C_{52}&0\\ C_{13}-\frac{\
u}{1-\
u}C_{33}&C_{23}-\frac{\
u}{1-\
u}C_{33}&0&0&0&0\\ \end{array} \right]\left[ \begin{array}{c}\epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{12}\\ \epsilon_{23}\\ \epsilon_{31}\end{array} \right]+\left[ \begin{array}{ccc}\gamma_{11}&0&0\\ 0&\gamma_{22}&0\\ 0&0&\gamma_{33}+\frac{C_{33}^{2}}{D(1-\
u)}\end{array} \right]\left[ \begin{array} []{c}E_{1}\\ E_{2}\\ E_{3}\end{array} \right]. \tag{24}\]  
Note that the third column of \(\mathbf{C}^{T}\) and the third row of \(\mathbf{\varepsilon}\) can be removed from the above relation. The alternative form of (24) does not need any changes in \(\mathbf{C}^{T}\) from (16) but requires replacement of \(\epsilon_{33}\) with \(\frac{-\
u}{1-\
u}(\epsilon_{11}+\epsilon_{22})\), as in the Equation (23).