## 5 Theoretical Model of Accelerated Factor
### Accelerated Factor of Tensile Strength Retention  
When the tensile specimens are soaked in the solution, the tensile strength decreases due to the gradual increase of the ageing area. The ageing condition of tensile specimens can be represented by the change of the cross section as shown in Figure 15, which is assumed that the specimen is uniformly aged along the thickness direction on the upper and lower sides. Thus, the actual tensile strength \(\sigma\) can be expressed by Equation (15).  
\[\sigma=\frac{\sigma_{u}\cdot A_{u}+\sigma_{a}\cdot A_{a}}{bh}=\frac{\sigma_{u }\cdot b\cdot(h-x)+\sigma_{a}\cdot b\cdot x}{bh}=\sigma_{u}-(\sigma_{u}- \sigma_{a})\frac{x}{h} \tag{15}\]  
where \(\sigma_{u}\) and \(\sigma_{a}\) are the initial (unaged) and residual (aged) tensile strength of the specimen, respectively. \(A_{u}\) and \(A_{a}\) are the unaged area and aged area of the specimen, respectively. \(b\) and \(h\) are the width and thickness of the specimen, respectively. \(x\) is the total ageing depth along the thickness of the laminates.  
Figure 14: Two-stage model with the abscissa changed to \(t/h^{2}\).  
Figure 15: Schematic diagram of ageing along cross section for BFRP laminates.  
It has been reported by study [49; 52; 53] that the total ageing depth \(x\) of the laminates is proportional to the square root of the immersion time \(t\), which is as follows regarded as a function of  
\[x{=}\ \alpha\sqrt{t}\ \ \ (x<h) \tag{16}\]  
where \(\alpha\) is a constant of the FRP composite that is irrelevant to specimen thickness \(h\).  
Then Equation (15) can be transformed as follows  
\[\sigma=\sigma_{u}-(\sigma_{u}-\sigma_{a})\frac{\alpha\sqrt{t}}{h} \tag{17}\]  
According to the literature [50; 54], for the given resin matrix, fiber, fabrication and immersion condition, the specimens with different thicknesses will decrease to the same strength when the ageing time is long enough. As a result, the equilibrium equation can be expressed as Equation (18) when the tensile strengths of the specimens when two thicknesses, \(h_{1}\) and \(h_{2}\), decrease to the same value at ageing time \(t_{1}\) and \(t_{2}\), respectively.  
\[\sigma_{1}=\sigma_{u}-(\sigma_{u}-\sigma_{a})\frac{\alpha\sqrt{t_{1}}}{h_{1}} =\sigma_{2}=\sigma_{u}-(\sigma_{u}-\sigma_{a})\frac{\alpha\sqrt{t_{2}}}{h_{2}} \tag{18}\]  
Thus, the \(AF\) of tensile strength retention of BFRP laminates can also be written as Equation (10).  
From the previous theoretical derivation, for BFRP laminates with two thicknesses, when the water absorption or tensile strength retention of the BFRP laminates with two different thicknesses reaches the same value, the ratio of the ageing time of the BFRP laminates with the two thicknesses is proportional to the square of the ratio of the corresponding two thicknesses. The square of the ratio of the two thicknesses is considered as accelerated factor (\(AF\)). As a result, it is feasible to accelerate ageing by reducing the specimen thickness. For example, if the water absorptions or tensile strength retentions of two BFRP laminates with different thicknesses reach a certain same value, the predicted ageing time of thicker specimen (\(t_{1}\)) can be calculated/accelerated by multiplying the real ageing time of thinner specimen (\(t_{2}\)) by the corresponding accelerated factor, \(AF=(h_{1}/h_{2})^{2}\), based on the two thicknesses of the thicker specimen (\(h_{1}\)) and thinner specimen (\(h_{2}\)).