## 2 Methodology  
### Establishment of Characteristic Matrix of IEPCM  
Due to the absence of piezoelectric capacity for the cement-based material, the piezoelectric property of IEPCM is up to that of the PZT powders [22]. That is, the higher the doping of PZT powder content acquires the better piezoelectric properties. Allowing for the requirements of the mechanical toughness and compatibility to concrete for the composite materials, we launched the simulation of the 0-3 type PCM mainly consisting of PZT-4 type powder and525-type sulphur aluminate cement by volume mixing ratio as 0.75:0.25. The polarization direction in the \(x\)-axis is set as positive, and conductive silver paste is coated on the top surface of PCM rectangular solid wafer as IEs pattern in the x-axis, as illustrated in Figure 1. The elastic stiffness coefficient matrix (**c**), dielectric constant matrix (\(\varepsilon\)) and piezoelectric stress constant matrix (**e**) of the IEPCM obtained by using ANSYS can be correspondingly given as follows [21]. The piezoelectric constant \(d_{33}\) of the IEPCM is calculated by the parallel model formula, as expressed in Equation (1),  
\[\textbf{d}_{33}=\frac{V^{1}\textbf{d}_{33}^{1}s_{33}^{2}+V^{2}\textbf{d}_{33}^ {2}s_{33}^{1}}{V^{1}s_{33}^{2}+V^{2}s_{33}^{2}} \tag{1}\]  
where \(V^{1}\) and \(V^{2}\) denotes the volume fraction of the component 1 (piezoelectric phase) and component 2 (hydrated cement phase); \(s_{33}^{1}\) and \(s_{33}^{2}\) is the elastic compliance constant of component 1 (piezoelectric phase) and 2 (hydrated cement phase), respectively. Allowing that the cement base phase has no any piezoelectric properties, the piezoelectric constant \(\textbf{d}_{33}\) of IEPCM means the piezoelectric constant \(\textbf{d}_{33}^{1}\) of piezoelectric ceramic. According to the series model formula of composite materials, we can get the dielectric constant of IEPCM, as shown in Equation (2). \[\varepsilon=V^{1}\varepsilon^{1}+V^{2}\varepsilon^{2} \tag{2}\]  
where \(\varepsilon^{1}\) and \(\varepsilon^{2}\) denotes the dielectric constant of piezoelectric phase and hydrated cement paste, respectively. The relative dielectric constant of common cement phase is around five, which is sufficiently small when compared with the piezoelectric ceramic. Therefore, the contribution of \(\varepsilon^{2}\) can be neglected and thus the dielectric constant of IEPCM becomes \(\varepsilon=V^{1}\varepsilon^{1}\). Similarly, the elastic stiffness coefficient of IEPCM is proportional to the elastic modulus of the two phases, and can also be acquired by the series model formula of composite materials, as given by  
\[\textbf{c}=\frac{\textbf{c}^{1}V^{1}E^{1}+\textbf{c}^{2}V^{2}E^{2}}{V^{1}E^{1 }+V^{2}E^{2}} \tag{3}\]  
where \(\textbf{c}^{1}\), \(\textbf{c}^{2}\) is the elastic stiffness coefficients of the piezoelectric and cement phase, respectively; \(E^{1}\) and \(E^{2}\) represents the corresponding elastic modulus, respectively. The \(E^{1}\) and \(E^{2}\) of piezoelectric and cement phase is generally equal to 76.5 and 25 GPa, respectively. The \(E^{2}\) value is comparable to \(E^{1}\), which is not to be neglected, but the coefficient of stiffness matrix of cement phase can be omitted when compared with that of the piezoelectric phase. The elastic stiffness coefficients of IEPCM can be represented by  
\[\textbf{c}=\frac{\textbf{c}^{1}V^{1}E^{1}}{V^{1}E^{1}+V^{2}E^{2}} \tag{4}\]  
The density of IEPCM is set as 7600 kg/m\({}^{3}\), and three coefficients in the **c**, \(\varepsilon\), and **e** matrices of PZT-4-type piezoelectric ceramics are selected in accordance to Reference [22].The corresponding **c**,  
Figure 1: The outline configuration of piezoelectric cement-based material with interdigitated electrodes (IEPCM). \(\varepsilon\), and **e** matrix of IEPCM in ANSYS simulation system are calculated by Equations (1)-(4), and the results are shown in Tables 1-3, respectively. It is worth to pointing out, the encoding sequence of **c**, **e** is consistent to the order of the input parameters in ANSYS.