## 5 A Hysteresis Model with Damage Accumulation of Welding Materials
### Comparison between the Test Results and Results Calculated by the Proposal Model  
The stress-strain hysteresis curves of the welded specimens are calculated by using the hysteresis model and considering the damage accumulation, constructed by this paper (Equation (13)). The results are compared with the experimental results, as shown in Figure 16, which contain the comparison results of any six semi-cycles referred to as twenty, forty, and eighty percent of the approximate cyclic life of each specimen. It is shown that the calculated results of the model agree well with the experimental curves. The comparison indicates that the model has a good applicability. It can be seen that the amplitude of cyclic stress, elastic modulus, and hysteretic energy decrease with the increase of cyclic cycles under the same loading condition, which is consistent with the experimental results. It shows that the damage accumulation evolution law of the welding materials can be reflected effectively by the hysteretic model considering the damage accumulation.  
Figure 16: _Cont._  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specimens** & \(\mathbf{x_{B}}\)**/\%** & \(\mathbf{K^{\prime}}\) & \(\mathbf{n^{\prime}}\) \\ \hline  & WA1(1) & 0.25 & \(2.26\times 10^{7}\) & 0.546 \\ WA1 & WA1(2) & 0.35 & \(2.65\times 10^{7}\) & 0.499 \\ WA1 & WA1(3) & 0.45 & \(3.05\times 10^{7}\) & 0.459 \\  & Model Parameters & \(\mathbf{k^{\prime}}=3.95\times 10^{6}\)\(\mathbf{s}_{1}+1.27\times 10^{7}\) & \(n^{\prime}=-43.5\)\(\mathbf{s}_{B}+0.654\) \\ \hline  & WA2(1) & 0.25 & \(2.23\times 10^{7}\) & 0.547 \\ WA2 & WA2(2) & 0.35 & \(2.64\times 10^{7}\) & 0.504 \\ WA2 & WA2(3) & 0.45 & \(3.05\times 10^{7}\) & 0.461 \\  & Model Parameters & \(\mathbf{k^{\prime}}=4.10\times 10^{6}\)\(\mathbf{s}_{B}+1.21\times 10^{7}\) & \(n^{\prime}=-43.0\)\(\mathbf{s}_{B}+0.655\) \\ \hline  & WA3(1) & 0.25 & \(2.24\times 10^{7}\) & 0.546 \\ WA3(2) & WA3(2) & 0.35 & \(2.64\times 10^{7}\) & 0.505 \\ WA3(3) & 0.45 & \(3.04\times 10^{7}\) & 0.464 \\  & Model Parameters & \(\mathbf{k^{\prime}}=4.00\times 10^{6}\)\(\mathbf{s}_{B}+1.24\times 10^{7}\) & \(n^{\prime}=-41.0\)\(\mathbf{s}_{B}+0.649\) \\ \hline  & WB1(1) & 0.25 & \(1.40\times 10^{7}\) & 0.567 \\ WB1 & WB1(2) & 0.35 & \(1.63\times 10^{7}\) & 0.521 \\ WB1 & Model Parameters & \(\mathbf{k^{\prime}}=2.30\times 10^{6}\)\(\mathbf{s}_{B}+8.25\times 10^{6}\) & \(n^{\prime}=-46.0\)\(\mathbf{s}_{B}+0.628\) \\ \hline  & WB2(1) & 0.25 & \(1.41\times 10^{7}\) & 0.566 \\ WB2 & WB2(2) & 0.35 & \(1.63\times 10^{7}\) & 0.521 \\ WB2 & WB2(3) & 0.45 & \(1.85\times 10^{7}\) & 0.476 \\ Model Parameters & \(\mathbf{k^{\prime}}=2.25\times 10^{6}\)\(\mathbf{s}_{B}+8.46\times 10^{6}\) & \(n^{\prime}=-45.0\)\(\mathbf{s}_{B}+0.679\) \\ \hline  & WB3(1) & 0.25 & \(1.40\times 10^{7}\) & 0.566 \\ WB3(2) & 0.35 & \(1.64\times 10^{7}\) & 0.520 \\ WB3(3) & 0.45 & \(1.86\times 10^{7}\) & 0.475 \\  & Model Parameters & \(\mathbf{k^{\prime}}=2.30\times 10^{6}\)\(\mathbf{s}_{B}+8.28\times 10^{6}\) & \(n^{\prime}=-45.5\)\(\mathbf{s}_{B}+0.680\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Values of parameters \(\mathbf{K^{\prime}}\) and \(\mathbf{n^{\prime}}\).