## 3 Establishment of 2D Higher-Order Theory and Equations of Thermo-PS Thin-Film Structures
### Theoretical Solutions  
The relevant zeroth-order divergence equations, which are sufficient to predict extensional deformation-dependent electromechanical fields, can be obtained from the theory established in the previous section  
\[ \begin{split} T^{(0)}_{11,1}+T^{(0)}_{13,3}+F^{(0)}_{1}=2h\rho \tilde{u}^{(0)}_{1},\ T^{(0)}_{31,1}+T^{(0)}_{33,3}+F^{(0)}_{3}=2h\rho\tilde{u}^{ (0)}_{3},\\ D^{(0)}_{1,1}+D^{(0)}_{3,3}+D^{(0)}=2 hq\left(\Delta p^{(0)}-\Delta n^{ (0)}\right),\ J^{n(0)}_{1,1}+J^{n(0)}_{3,3}+J^{n(0)}=2 hq\Delta\dot{n}^{(0)},\\ f^{p(0)}_{1,1}+J^{p(0)}_{3,3}+J^{p(0)}=-2 hq\Delta\dot{p}^{(0)}. \end{split}  \tag{12}\]  
Accompanying Equation (12), the zeroth-order constitutive equations in terms of displacements, electrical potential, and weak induced perturbations of hole and electron concentrations are as follows:  
\[ \begin{split} T^{(0)}_{11}&=2h\left(\overline{c}_{ 11}u^{(0)}_{1,1}+\overline{c}_{13}u^{(0)}_{3,3}+\overline{c}_{31}\phi^{(0)}_{ 3}-\overline{\lambda}_{1}\theta^{(0)}\right),\ T^{(0)}_{13}&=2h \left(c_{55}\left(u^{(0)}_{1,3}+u^{(0)}_{3,1}\right)+e_{15}\phi^{(0)}_{1} \right),\\ T^{(0)}_{33}&=2h\left(\overline{c}_{31}u^{(0)}_{1,1}+ \overline{c}_{33}u^{(0)}_{3,3}+\overline{c}_{33}\phi^{(0)}_{3}-\overline{ \lambda}_{3}\theta^{(0)}\right),\ D^{(0)}_{3}&=2h\left(\overline{ c}_{31}u^{(0)}_{1,1}+\overline{c}_{33}u^{(0)}_{3,3}-\overline{c}_{33}\phi^{(0)}_{3} +\overline{p}_{3}\theta^{(0)}\right),\\ D^{(0)}_{1}&=2h\left(e_{15}\left(u^{(0)}_{1,3}+u^{( 0)}_{3,1}\right)-e_{11}\phi^{(0)}_{1,1}\right),\ J^{n(0)}_{1}&=2h \left(-qn_{0}u^{n}_{11}\phi^{(0)}_{1,1}+qD^{n}_{11}\Delta n^{(0)}_{1,1}\right),\\ f^{n(0)}_{3}&=2h\left(-qn_{0}u^{n}_{33}\phi^{(0)}_{ 3}+qD^{n}_{33}\Delta n^{(0)}_{3}\right),\ J^{n(0)}_{1}&=2h\left(- qp_{0}u^{n}_{11}\phi^{(0)}_{1,1}-qD^{n}_{11}\Delta p^{(0)}_{1,1}\right),\\ f^{p(0)}_{3}&=2h\left(-qp_{0}u^{p}_{33}\phi^{(0)}_{ 3}-qD^{p}_{33}\Delta p^{(0)}_{3}\right).\end{split}  \tag{13}\]  
in which, the effective material constants related to the so-called stress relaxation of PS thin-film structure are defined as follows:  
\[ \begin{split}\overline{c}_{11}=c_{11}-\frac{c_{12}c_{21}}{c_{22} },\ \overline{c}_{13}=c_{13}-\frac{c_{12}c_{23}}{c_{22}},\ \overline{c}_{33}=c_{33}-\frac{c_{32}c_{22}}{c_{22}},\ \overline{c}_{31}=e_{31}-\frac{c_{12}c_{23}}{c_{22}},\ \overline{c}_{33}=e_{33}-\frac{c_{32}c_{22}}{c_{22}},\\ \overline{\lambda}_{1}=\lambda_{1}-\frac{c_{12}\lambda_{2}}{c_{22} },\ \overline{\lambda}_{3}=\lambda_{3}-\frac{c_{32}\lambda_{2}}{c_{22} },\ \overline{c}_{33}=\varepsilon_{33}+\frac{c_{32}c_{32}}{c_{22}},\ \overline{p}_{3}=p_{3}+\frac{c_{32}\lambda_{2}}{c_{22}}.\end{split}  \tag{14}\]  
Here, it should be stressed that thermo-PS can exhibit unique interactions between extensional deformations, polarization, and motion of charge carriers along the in-plane directions of a thin film. As shown in Equation (13), thermal fields produce effective polarization charges, i.e., zeroth-order electric displacements \(D^{(0)}_{1}\) and \(D^{(0)}_{3}\), the electric fields \(E^{(0)}_{1}=-\phi^{(0)}_{,1}\) and \(E^{(0)}_{3}=-\phi^{(0)}_{,3}\) associated with electric potential \(\phi^{(0)}\) or polarization then act on mobile charges, i.e., \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\), and further affect their motion and transport behavior, as well as the distribution of current density. For the static extensional deformation problem considered and investigated in this paper, according to Ref. [45], the mechanical and electric boundary conditions (including extensional deformations \(u^{(0)}_{1}\) and \(u^{(0)}_{3}\), extensional resultant forces \(T^{(0)}_{11}\) and \(T^{(0)}_{33}\), electric potential \(\phi^{(0)}\), electron and hole concentration perturbations \(\Delta n^{(0)}\) and \(\Delta p^{(0)}\), electric displacement \(D^{(0)}_{3}\), and electron and hole current density \(J^{n(0)}_{3}\) and \(J^{p(0)}_{3}\)) at four edges of the PS thin film are taken to be the following:  
\[