## 2 Modeling of Pieoresistive Behavior
### Electrical Circuit Model for Plain-Woven CFRP  
SHM on the woven CFRP is usually implemented by installing electrodes forming a network on the top and bottom surfaces. The electrodes flow electric current to monitor the electric resistance of the CFRP, which directly corresponds to deformation or damages. In this work, plain-woven CFRP is used as a specimen. This CFRP has the simplest fiber pattern compared to other woven composite types in which it has symmetric properties in wrap and weft directions. Thus, the characteristic of plain-woven CFRP can be modeled as an electrical circuit with electrical resistances along with the wrap and thickness directions.  
The model that represents the electrical resistances in the CFRP is shown in Figure 2a. Two electrodes are installed on the top surface of the plain-woven CFRP with a certain distance (\(L_{c}\)) to investigate electric resistance alteration due to deformation. There is contact resistance (\(R_{c}\)) at the interface between these electrodes and composite surfaces. The \(R_{c}\) depends on surface treatment prior to the attachment of electrodes. Since the plain-woven CFRP has identic geometry for each lamina, the resistance of each lamina for thickness direction (\(R_{t}\)) has identic value. Furthermore, considering each lamina in plain-woven CFRP has identic geometry in wrap and weft directions, the electric resistance along wrap direction (\(R_{w}\)) can also be considered having similar values with resistance along weft direction.  
Total resistance (\(R_{Total}\)) among two electrodes installed on the top surface of the woven CFRP can be calculated as follows:  
\[R_{Total}\ =R_{CFRP}+2R_{c}, \tag{1}\]  
where \(R_{CFRP}\) is the total resistance of the CFRP. It can be calculated as follows:  
\[R_{CFRP}\ =\ \frac{\Big{(}2R_{t(n-1)}+R_{(n-1)}\Big{)}R_{t(n)}}{R_{w(n)}+2R_{ t(n-1)}+R_{(n-1)}}, \tag{2}\]  
where \(n\) denotes the number of CFRP lamina.  
Figure 1: Illustration of the fiber arrangement in plain-woven carbon fiber reinforced polymer (CFRP) under tensile test.  
In case that the electrodes are attached on the top and bottom surfaces of the woven composite, a series \(R_{t}\) model can be implemented as shown in Figure 2b. In this model, the thickness of the specimen (\(t_{(n)}\)) is the addition of lamina thicknesses that have identic value. The \(R_{CFRP}\) can be then calculated as follows:  
\[R_{CFRP}=nR_{t}, \tag{3}\]  
Electrical resistivity of lamina n in wrap direction (\(\rho_{w(n)}\)) and thickness direction (\(\rho_{t(n)}\)) can be then calculated as follows:  
\[\rho_{w(n)}\ =\ \frac{R_{w(n)}t_{(n)}b}{L_{c}}, \tag{4}\]  
\[\rho_{t(n)}\ =\ \frac{R_{t(n)}L_{c}b}{2t_{(n)}}, \tag{5}\]  
where \(b\) is width of specimen.  
Another important parameter required for implementing SHM in the woven CFRP is the gauge factor (\(G_{CFRP}\)). It can be calculated as follows:  
\[G_{CFRP}\ =\ \frac{\varepsilon_{e}}{\varepsilon_{m}}, \tag{6}\]  
where \(\varepsilon_{e}\) and \(\varepsilon_{m}\) are electric and mechanical strains, respectively. Those can be calculated as follows:  
\[\varepsilon_{e}=\frac{\Delta R}{(R_{CFRP}-\Delta R)}, \tag{7}\]  
\[\varepsilon_{m}\ =\ \frac{\Delta L}{(L_{c}-\Delta L)}, \tag{8}\]  
Figure 2: (**a**) Electrical circuit to laminate the plain-woven composite with both electrodes installed on the top surface and (**b**) electrodes installed on the top and bottom surface.  
where \(\Delta R\) is an electrical resistance alteration and \(\Delta L\) is a deformation of the specimen. Positive value of \(G_{CFRP}\) indicates positive piezoresistive behavior and vice versa. In case that nonlinear piezoresistive behavior appears, the \(G_{CFRP}\) under loading and unloading conditions would have different values. The above difference must be investigated experimentally for developing SHM. The selection of \(G_{CFRP}\) value applied to the electrical resistance model of the plain-woven composite must consider the nonlinearity and the existence of residual resistance.