# Comprehensive Numerical Analysis of a Porous Piezoelectric Ceramic for Axial Load Energy Harvesting
## 2 Modelling of Porous Piezoelectric Based Energy Harvester  
There are two types of composites that can give precise solutions: (i) serial circuit (\(\epsilon^{\rm e}\)) and, (ii) parallel circuit (\(\epsilon^{\rm p}\)). In the serial circuit, the phase boundary runs normally in the direction of the applied field and, in the parallel circuit, the boundaries between adjacent phases are parallel to the direction of the applied field [26; 27]:  
\[1/\epsilon^{\rm e}=1/\epsilon^{\rm s}=v^{\prime}/\epsilon^{\prime}+v^{\prime \prime}/\epsilon^{\prime\prime} \tag{1}\]  
\[\epsilon^{\rm e}=\epsilon^{\rm p}=\epsilon^{\rm p}\epsilon^{\prime}v^{\prime}+ \epsilon^{\prime\prime}v^{\prime\prime} \tag{2}\]  
The effective permittivity \(\epsilon^{\rm e}\) must be calculated from the permittivities \(\epsilon^{\prime}\) and \(\epsilon^{\prime\prime}\) of two components and their volume fractions \(v^{\prime}\) and \(v^{\prime\prime}=1-v^{\prime}\). The main purpose of this work is to study and analyse the output parameters of the piezoelectric energy generator. A piezoelectric energy harvesting device is a composite elastic and electroelastic body. The piezoelectric ceramics are made with various effective modules. The device is assumed to have small elastic oscillations in the moving coordinate system [28; 29; 30]. The mathematical model of piezoelectric energy generator devices can be presented, based on the initial boundary value problem of linear electric elasticity theory [31]:  
\[u(t)=\overline{u}\;e^{i\omega t} \tag{3}\]  
The mathematical formulation for the elastic medium:  
\[ \begin{split}&\rho\ddot{u}_{i}+\alpha\rho\dot{u}_{i}-\sigma_{ijj}=f _{i};\\ &\sigma_{ij}=c_{ijkl}(\epsilon_{kl}+\beta\dot{\epsilon}_{kl});\\ &\epsilon_{kl}=(u_{kl}+u_{lk})/2.\end{split}  \tag{4}\]  
The mathematical formulation for piezoelectric materials is written as  
\[ \begin{split}&\rho\ddot{u}_{i}+\alpha\rho\dot{u}_{i}-\sigma_{ijj}=f _{i};\\ & D_{i,i}=0;\\ &\sigma_{ij}=c_{ijkl}(\epsilon_{kl}+\beta\dot{\epsilon}_{kl})-e_{ ijk}E_{k};\\ & D_{i}+\zeta_{d}\dot{D}_{i}=e_{ikl}(\epsilon_{kl}+\zeta_{d} \dot{\epsilon}_{kl})-\epsilon_{ik}E_{k};\\ &\epsilon_{kl}=(u_{kl}+u_{lk})/2;\\ & E_{k}=-\varphi_{k};\end{split}  \tag{5}\]  
where \(\rho\) is the material density;  
\(u_{i}\) are the displacement vector components;  
\(\sigma_{ij}\) are the stress tensor components;  
\(f_{i}\) are the density vector components;  
\(D_{i}\) are the electric induction vector components;  
\(c_{ijkl}\) are the elastic moduli components of fourth rank tensor;  
\(e_{ikl}\) are the piezoelectric coefficients components of third rank tensor;  
\(\epsilon_{kl}\) are the strain tensor components;  
\(E_{k}\) are the electric field vector components;  
\(\varphi_{k}\) are the electric potential vector components;  
\(\epsilon_{ik}\) are the dielectric constants tensor components;  
\(\alpha\), \(\beta\), \(\zeta\) are the non-negative damping coefficients (the value of \(\zeta_{d}\) is used in ANSYS software). Since harmonic analysis is used in the calculations, the following operations are performed for the corresponding components in the equations:  
\[\dot{u} =i\omega u^{\prime} \tag{6}\] \[\dot{u} =i\omega u^{2}\]  
The axial piezoelectric generator model includes two piezoelectric cylinders, two piezoelectric bimorphs, duralumin substrates, proof mass, and L-shaped steel plates. The end of the piezo cylinder is fixed to the L-shaped steel plate, while the other end is fixed by a rigid pinch. The L-shaped steel plate and the duralumin substrate are glued together. The proof mass is fixed near the fixed end of the duralumin substrate. The two piezo-bimorphs are fixed near to L-shaped steel plate in the duralumin substrate as shown in Figure 1. The dimension characteristics of the elements of the piezoelectric energy generators are listed in Table 1. The material properties are summarized in Table 2. The porous piezoelectric ceramic materials are analysed by various researchers at different connectivity [32; 33].