# Innovative Orthopedic Solutions for AI-Optimized Piezoelectric Implants for Superior Patient Care
## 2 Materials and Methods
### Mathematical Modeling for Improved Implant Performance
#### 2.3.1 Piezoelectric Constitutive Equations  
The fundamental equations for piezoelectric materials that relate mechanical stress (\(\sigma\)) and the electric field (\(E\)) are given by the stress-strain relationship as follows:  
\[\sigma=c^{E}\,\epsilon-e^{T}\,E \tag{1}\]  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Property** & **Symbol** & **Typical Value** & **Units** \\ \hline Elastic Modulus (Young’s Modulus) & E\({}_{33}\) & 63–120 & GPa \\ \hline Poisson’s Ratio & \(\
u\) & 0.31–0.34 & - \\ \hline Density & \(\rho\) & 7.5–8.0 & g/cm\({}^{3}\) \\ \hline Piezoelectric Constant d\({}_{33}\) & d\({}_{33}\) & 300–600 & pC/N \\ \hline Piezolectric Constant d\({}_{31}\) & d\({}_{31}\) & \(-\)120 to \(-\)190 & pC/N \\ \hline Relative Permittivity (at constant stress) & \(\epsilon_{\tau}\) & 1700–2000 & - \\ \hline Dielectric Loss Factor & - & 0.002–0.02 & - \\ \hline Coupling Coefficient & k\({}_{33}\) & 0.65–0.75 & - \\ \hline Mechanical Quality Factor & Q\({}_{m}\) & 70–100 & - \\ \hline Curie Temperature & - & 250–350 & \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summarizes the material properties of lead zirconate titanate (PZT) used in finite element modeling (FEM) [53].  
\begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Displacement (\(\mu\)m)** & \begin{tabular}{l} **Radiated** \\ **Power (W)** \\ \end{tabular} & \begin{tabular}{l} **Phase** \\ **Response (\(\mu\)m)** \\ \end{tabular} & \begin{tabular}{l} **Electric Field** \\ **(V\(\mu\)m\({}^{-1}\))** \\ \end{tabular} & \
\begin{tabular}{l} **Potential** \\ **Energy (J)** \\ \end{tabular} &
\begin{tabular}{l} **Piezo** \\ **Impedance (f)** \\ \end{tabular} & **Voltage (V)** \\ \hline \(-\)1000 & 1.82 & 436.12 & 2907.34 & 2.10 & 7.36 & 1.18 \\ \(-\)1500 & 4.10 & 654.29 & 4361.13 & 4.73 & 0.019 & 1.77 \\ \(-\)500 & 0.46 & 217.95 & 1453.55 & 0.53 & 0.003 & 0.59 \\ \hline \end{tabular}
\end{table}
Table 2: Design points of experiments on the central composite structures of the different energy elements generated in the optimization process.  
where \(c^{E}\) is the stiffness matrix at a constant electric field, \(e\) is the strain tensor, \(e^{T}\) is the transpose of the piezoelectric stress matrix, and \(E\) is the electric field.  
Expanding for a specific case in a 3D matrix form electric displacement gives the following:  
\[\begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}c_{11}^{E}&c_{12}^{E}&c_{13}^{E}&0&0&0 \\ c_{21}^{E}&c_{22}^{E}&c_{23}^{E}&0&0&0\\ c_{31}^{E}&c_{32}^{E}&c_{33}^{E}&0&0&0\\ 0&0&0&c_{44}^{E}&0&0\\ 0&0&0&0&c_{55}^{E}&0\\ 0&0&0&0&0&c_{66}^{E}\end{bmatrix}\begin{bmatrix}\epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{41}^{E}&\epsilon_{42}^{E}&\epsilon_{43}^{E}\\ 2\epsilon_{31}\\ 2\epsilon_{12}\end{bmatrix}-\begin{bmatrix}\epsilon_{11}^{T}&\epsilon_{12}^{T} &\epsilon_{13}^{T}\\ \epsilon_{22}^{T}&\epsilon_{23}^{T}\\ \epsilon_{33}^{T}&\epsilon_{33}^{T}\\ \epsilon_{41}^{T}&\epsilon_{42}^{T}&\epsilon_{43}^{T}\\ 2\epsilon_{31}\\ 2\epsilon_{12}\end{bmatrix}\]  
The electric displacement is shown below.  
\[D=ce+e^{S}E \tag{2}\]  
where \(D\) is the electric displacement vector, \(e\) is the piezoelectric coupling coefficient matrix, and \(e^{S}\) is the permittivity matrix at a constant strain.  
Expanding similarly gives the following:  
\[\begin{bmatrix}D_{1}\\ D_{2}\\ D_{3}\end{bmatrix}=\begin{bmatrix}e_{11}&e_{12}&e_{13}&e_{14}&e_{15}&e_{16}\\ e_{21}&e_{22}&e_{23}&e_{24}&e_{25}&e_{26}\\ e_{31}&e_{23}&e_{33}&e_{34}&e_{35}&e_{36}\end{bmatrix}\begin{bmatrix}\epsilon _{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \epsilon_{22}\\ \epsilon_{31}\\ 2\epsilon_{12}\end{bmatrix}+\begin{bmatrix}\epsilon_{11}^{S}&0&0\\ 0&\epsilon_{22}^{S}&0\\ 0&0&\epsilon_{33}^{S}\end{bmatrix}\begin{bmatrix}E_{1}\\ E_{2}\\ E_{3}\end{bmatrix}\]