## 5 Theoretical Model of Accelerated Factor
### Accelerated Factor of Water Absorption  
Considerable researches show that the water absorption of FRP composites can be described by two models as shown in Figure 13, i.e., the [PERSON]'s model and Two-stage model [39; 42; 51]. The [PERSON]'s model assumes that the initial phase of water absorption increases linearly with \(t^{1/2}\) in the initial phase, and then increases non-linearly until the water absorption reaches a dynamic equilibrium without obvious changes. For the Two-stage model, the water absorption is identical to [PERSON]'s model in the initial phase. But the water absorption cannot reach the equilibrium stage due to the water immersion constantly and the degradation of resin matrix. In the current study, the test results of water absorption conform to the Two-stage model. According to the ageing mechanism reported in the literature [49; 50], the following assumptions can be used to establish the water absorption model. First, the specimens with different thicknesses have the same diffusion rate for the same immersion solution because the diffusion rate is controlled by the concentration gradient of the immersion solution. Second, although the water absorption at the regular time varies with specimen thickness, the ageing mechanism of the composite remains unchanged. Third, the specimens with different thicknesses fabricated by the same material will eventually reach the same water absorption rate under the same immersion solution. The [PERSON]'s and Two-stage models can be used to describe the relationship between specimen thickness and ageing time in stage I (mass gain due to swelling of FRP composites) and stage II (mass loss due to relaxation of FRP composites), respectively. In the two models, the stage I of water absorption is identical. The [PERSON]'s model is shown in Equation (5), but it is difficult to find the relationship among different parameters in this expression, which is needed to be simplified as presented in Equation (6) [36; 40]. \[M(t)=M_{\infty}\Bigg{\{}1-\frac{8}{\pi^{2}}\sum_{j=0}^{\infty}\frac{\exp[-(2j+ 1)^{2}\pi^{2}\Big{(}\frac{Dt}{B^{2}}\Big{)}}{(2j+1)^{2}}\Bigg{\}} \tag{5}\]  
Figure 13: Typical model of water absorption of composites: (**a**) [PERSON] model; (**b**) Two-stage model. \[M(t)=M_{\infty}\left\{1-\exp\left[-7.3\left(\frac{Dt}{h^{2}}\right)^{0.75}\right]\right\} \tag{6}\]  
where \(M_{\infty}\) is the effective moisture equilibrium content, \(D\) is the diffusion coefficient. When two specimens with the thicknesses of \(h_{1}\) and \(h_{2}\) soaked in the same solution reach the same water absorption at ageing time \(t_{1}\) and \(t_{2}\), respectively. The equilibrium equation, i.e., \(M(t_{1})=M(t_{2})\) can be expressed as Equation (7) according to [PERSON]'s model, which can be simplified to Equation (8). \[M_{\infty}\left\{1-\exp\left[-7.3\left(\frac{Dt_{1}}{h_{1}^{2}}\right)^{0.75} \right]\right\}=M_{\infty}\left\{1-\exp\left[-7.3\left(\frac{Dt_{2}}{h_{2}^{2} }\right)^{0.75}\right]\right\} \tag{7}\]  
\[\exp\left[-7.3\left(\frac{Dt_{1}}{h_{1}^{2}}\right)^{0.75}\right]=\exp\left[-7.3\left(\frac{Dt_{2}}{h_{2}^{2}}\right)^{0.75}\right] \tag{8}\]  
According to the previous assumptions, the diffusion coefficient \(D\) is identical when the specimens are soaked in the same solution, yields  
\[\frac{t_{1}}{t_{2}}=\frac{h_{1}^{2}}{h_{2}^{2}} \tag{9}\]  
Therefore, the \(AF\) of water absorption in stage I of Two-stage model can be expressed in  
\[AF=\frac{t_{1}}{t_{2}}=\left(\frac{h_{1}}{h_{2}}\right)^{2} \tag{10}\]  
After the above derivation of \(AF\) in Stage I, it can be found that there is a certain relationship between ageing time \(t\) and specimen thickness \(h\) indeed.