# Fiber Bragg Gratings Based Cyclic Strain Measuring of Weld Toes of Cruciform Joints
## 2 Theoretical Context
### Cyclic Stress-Strain Response Behavior  
Since the weld toe of welded joints has strong stress concentration effect, even if the nominal stress is in linear elastic phase, the actual stress and strain at the weld toe tend to be in plastic phase. Furthermore, compared with the elastic-plastic behavior under monotonic loading, the stress-strain response under cyclic loading is far more complicated.  
Lots of fatigue tests concerning different metals have shown that the strain amplitude decreases gradually as the number of cycles increases, even if under the symmetrical loading with constant amplitude, which is called cyclic hardening. Conversely, some metals have the property of cyclic softening, which means strain amplitude increases with cycles increasing. Cyclic hardening and cyclic softening phenomena occur in not only strain-controlled tests but also stress-controlled tests. It is generally believed that the cyclic loading causes the metal to repeatedly undergo plastic deformation, resulting in some changes to the plastic flow characteristics of the material. Then the ability to resist deformation is enhanced or reduced, which reflects cyclic hardening or softening behavior. Commonly, cyclic hardening or softening happen in the early stage of cyclic loading, then it will disappear, and the stress amplitude or strain amplitude will stabilize gradually. Such behaviors can generally be described by cyclic stress-strain curve (\(\sigma_{a}-\varepsilon_{a}\) curve) and hysteresis loop curve (\(\Delta\sigma-\Delta\sigma\) curve) [36].  
The cyclic stress-strain curve can be expressed by Equation (7), which does not reflect the loading path. The hysteresis loop curve can cover this shortcoming and can be expressed by Equation (8).  
\[\varepsilon_{a}=\varepsilon_{e}+\varepsilon_{p}=\frac{\sigma_{a}}{E}+\left( \frac{\sigma_{a}}{K^{\prime}}\right)^{1/n^{\prime}} \tag{7}\]  
\[\Delta\varepsilon=\Delta\varepsilon_{e}+\Delta\varepsilon_{p}=\frac{\Delta \sigma}{E}+2\left(\frac{\Delta\sigma}{2K^{\prime}}\right)^{1/n^{\prime}} \tag{8}\]  
where \(E\) is elastic modulus, \(K^{\prime}\) is cyclic strength coefficient, \(n^{\prime}\) is cyclic strain hardening exponent, \(\varepsilon_{a}\) is strain amplitude, \(\Delta\varepsilon\) is strain range, \(\varepsilon_{e}\) is elastic strain, \(\varepsilon_{p}\) is plastic strain, \(\sigma_{a}\) is stress amplitude and \(\Delta\sigma\) is stress range.  
For welded joints, it is difficult to determine the actual stress at the weld toe by the force signals recorded by the testing machine. Therefore, strain signals at the weld toe are often obtained and used to calculate the corresponding stresses by combining Equations (7) and (8). Furthermore, the stress-strain response curve can be drawn. The computing procedure is as follows:  
1. For the first loading cycle, \(\sigma_{a}\) is calculated according to \(\varepsilon_{a}\) by cyclic stress-strain curve (Equation (7)), in which \(\varepsilon_{a}\) is obtained from the strain testing data.
2. For other cycles, the hysteresis loop curve (Equation (8)) is used to calculate \(\Delta\sigma\) according to \(\Delta\varepsilon\), in which \(\Delta\varepsilon\) is obtained by rain flow counting for the strain testing data.  
It is worth noting that when the strain reaches the same reversal point for the second time, a closed-loop will be formed, and the material memory effect should be introduced. Since the closed-loop does not affect the subsequent response process, stress and strain here should be calculated according to the previous loading path.