## 5 A Hysteresis Model with Damage Accumulation of Welding Materials
### A Model of Hysteretic Curve with Damage Accumulation  
The Ramberg-Osgood model has proven to be a good description of the cyclic stress-strain relationship of the metallic materials by many studies, and the Masing criterion shows that the stress-strain path under cyclic loading can be described by amplifying 1 time the cyclic stress-strain curve in a steady state [62; 63; 64; 65], such as explained in Figure 13. Based on the theory, and considering the effect of the damage, the evolution equation of damage variable \(D\) is introduced to construct a model of the hysteretic curve for the welding material. As shown in Figure 14, a hysteresis model with damage accumulation of welding materials is presented, in which the No. \(n\) semi-cycle is shown as an example to describe the hysteresis criterion. Figure 12: Schematic diagram of cyclic stress-strain curve. Figure 13: Masing behavior. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**With Q235 Base Metal**} & \multicolumn{4}{c}{**With Q345 Base Metal**} \\ \hline
**Specimens** & \(E_{0}\)**/GPa** & \(K_{0}\)**/MPa** & \(n_{0}\) & **Specimens** & \(E_{0}\)**/GPa** & \(K_{0}\)**/MPa** & \(n_{0}\) \\ \hline WA1 & 239.700 & 1266 & 4.784 & WB1 & 247.710 & 953 & 8.099 \\ WA2 & 223.000 & 1331 & 5.531 & WB2 & 241.070 & 939 & 8.610 \\ WA3 & 223.000 & 1466 & 6.203 & WB3 & 241.070 & 885 & 9.272 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters of the cyclic stress-strain curves for welding materials. Curve \(A_{n}\)-\(C_{n}\)-\(B_{n}\) is the loading curve of No. \(n\) semi-cycle, which is constructed by considering nonlinear loading. Point \(A_{n}\) is the starting point of the \(n\) semi-cycle loading curve, and \(B_{n}\) is the final point of the loading curve, but is the starting point of the \(n\) semi-cycle unloading curve. \(\sigma_{m}^{D(n)}\) is the stress amplitude of the No. \(n\) semi-cyclic stress-strain curve and the ordinate of point \(B_{n}\). The stress amplitude of each semi-cycle will be degraded with damage accumulating, so \(\sigma_{m}^{D(n)}\) is thought of as the damage stress amplitude, which can be calculated by Equation (8) based on damage mechanics theory, \(\sigma_{m}\) is the initial stress amplitude, \(D_{n}\) is the cumulative damage variable after \(n\)th semi-cycle, which can be calculated by Equation (3)\(\eta\) and \(\xi\) are the damage parameters that can be fitted by the low-cycle fatigue test results of the welding material. Curve \(B_{n}\)-\(A_{n+1}\) is the unloading curve of No. \(n\) semi-cycle, and the unloading process is approximately elastic. \(E_{n}^{D}\) is the elastic modulus with damage accumulation, which in respect of \(D_{n}\) can be calculated by Equation (9) based on damage mechanics theory and Shen [30; 31]. \(E_{0}\) is the original elastic modulus, \(g\) and \(h\) are the damage parameters like \(\eta\) and \(\xi\). \[\sigma_{m}^{D(n)}=(\eta-\xi D_{n})\sigma_{m} \tag{8}\]  
\[E_{n}^{D}=(g-hD_{n})E_{0} \tag{9}\]  
Overall, the hysteresis model considering damage accumulation of welding material is depicted in Figure 14, the expressions of the model are as follows:  
(1) unloading curve (Linear)  
\[\sigma_{n}=E_{n}^{D}\varepsilon_{n}+t_{n} \tag{10}\]  
\[t_{n}=(\sigma_{m}^{D(n)}-E_{n}^{D}\varepsilon_{B}) \tag{11}\]  
In Equations (10) and (11), \(\sigma_{n}\) and \(\varepsilon_{n}\) are stress and strain values of point at \(n\)th semi-cycle curves, \(t_{n}\) is the parameter for determining the shape of the curve, \(\varepsilon_{B}\) is the strain amplitude. (2) loading curve (nonlinear)  
Figure 14: Hysteretic model of welding material considering cumulative damage under cyclic loading. The expression of the model is established based on the damage degenerating law, Ramberg-Osgood model and Masing behavior. \[\varepsilon_{n}=\left\{