# Fiber Bragg Gratings Based Cyclic Strain Measuring of Weld Toes of Cruciform Joints
## 2 Theoretical Context  
### SCF of Welds  
The formation of welded joints is the way in which two or more pieces meet together by the welding process. They are generally made of base metal (BM), weld metal (WM), fusion zone (FZ) and heat affected zone (HAZ). Welded joints experience a series of complicated processes such as local heating, metal melting and solidification, etc., which result in a region with heterogeneous ingredients, organization and mechanical properties. According to the geometry of the weld, different welded joints have distinct SCF degrees. If the nonlinear plasticity is considered, stress and strain have different SCF, which are described as follows:  
\[SCF_{\sigma}=\frac{\sigma_{\text{max}}}{\sigma_{\text{n}}} \tag{1}\]  
\[SCF_{\varepsilon}=\frac{\varepsilon_{\text{max}}}{\varepsilon_{\text{n}}} \tag{2}\]  
where \(SCF_{\sigma}\) is the stress concentration factor, \(SCF_{\varepsilon}\) is the strain concentration factor, \(\sigma_{\text{max}}\) is the maximum stress, \(\sigma_{\text{n}}\) is the nominal stress, \(\varepsilon_{\text{max}}\) is the maximum strain and \(\varepsilon_{\text{n}}\) is the nominal strain. In the research of fatigue fields, stress range, or strain range is the much more important controlling parameter. As such, this paper defines corresponding SCF parameters as follows:  
\[SCF_{\Delta\sigma}=\frac{\Delta\sigma_{\text{max}}}{\Delta\sigma_{\text{n}}} \tag{3}\]  
\[SCF_{\Delta\varepsilon}=\frac{\Delta\varepsilon_{\text{max}}}{\Delta \varepsilon_{\text{n}}} \tag{4}\]  
where \(SCF_{\Delta\sigma}\) is the stress range concentration factor, \(SCF_{\Delta\varepsilon}\) is the strain range concentration factor, \(\Delta\sigma_{\text{max}}\) is the maximum stress range, \(\Delta\sigma_{\text{n}}\) is the nominal stress range, \(\Delta\varepsilon_{\text{max}}\) is the maximum strain range and \(\Delta\varepsilon_{\text{n}}\) is the nominal strain range. There are two kinds of typical welded joints commonly used in engineering, which are fillet joints (as shown in Figure 1) and butt joint (as shown in Figure 2). Their stress SCF formulas are shown as Equations (5) and (6) [7]. \[SCF_{\sigma}=1+0.35(\tan\theta)^{\frac{1}{5}}[1+1.1(c/l)^{\frac{3}{5}}]^{ \frac{1}{2}}(\frac{t}{\rho})^{1/2} \tag{5}\]  
\[SCF_{\sigma}=1+0.27(\tan\theta^{\prime})^{1/4}\Bigg{(}\frac{t}{\rho}\Bigg{)}^ {1/2} \tag{6}\]  
where \(\theta\) is the weld inclination angle, \(\theta^{\prime}\) is the reinforcement angle, \(t\) is the plate thickness, \(l\) is the fillet leg length, \(\rho\) is the weld toe radius and \(c\) is the weld un-penetration size. It can be seen from the above formulas that for fillet joints, \(SCF_{o}\) mainly depends on the weld inclination angle, the ratio of the weld un-penetration size to the fillet leg length and the ratio of the plate thickness to the weld toe radius. Smaller weld inclination angle, larger weld toe radius and larger fillet leg length will result in smaller SCF at the weld toe. For butt joints, \(SCF_{o}\) mainly depends on the reinforcement angle, \(\theta^{\prime}\), and the ratio of the plate thickness to the weld toe radius. With constant plate thickness and weld width, smaller reinforcement angle and larger weld toe radius will be helpful to decrease the SCF. Compared with butt joints, fillet joints usually have larger weld inclination angle, and the fillet leg length cannot be large without limitation due to the plate thickness. Therefore, the weld shape transition of fillet joints is much sharper, which is prone to produce severe SCF in the weld toe. On the contrary, the transition geometry of butt joints changes mildly and the SCF is small. Therefore, fillet joints are generally inferior to butt joints in their fatigue performances. In fact, the transitional area of the weld toe is not an ideal rounded shape, as shown in Figure 3. It can be seen that the local weld toe of the real weld is often composed by a series of small arcs with different radii, in which the smaller radius is close to the dimension of surface roughness.