# Failure Mechanism and Structural Analysis of Chain Slings with Non-Standard Connections
## 4 Structural Analysis
### Analysis Model  
The exact material specifications of the chain used in this study were not clearly documented in the test certificates provided by the supplier. Therefore, AISI 4340 steel was selected as the representative material for the analysis, as it exhibits mechanical properties similar to those of Grade 100 alloy steel, which is commonly used for chain slings. It is noteworthy that AISI 4340 is known for its sensitivity to heat treatment conditions, which can significantly affect its mechanical properties such as yield strength and ultimate tensile strength. Given this variability, the material properties used in the present analysis were based on the mechanical characteristics of AISI 4340 under a normalizing heat treatment condition. The fundamental material properties adopted in the analysis, including Young's modulus, Poisson's ratio, and density, are summarized in Table 8.  
Meanwhile, to more accurately simulate the ductile deformation and eventual fracture behavior of the material under the loading conditions applied to the chain, the Johnson-Cook strength model was adopted in this analysis. This model is widely used in the analysis of various industrial materials, as it effectively represents the nonlinear stress-strain relationship and fracture characteristics exhibited by ductile metallic materials under practical conditions. The Johnson-Cook strength model is defined as shown in Equation (1), where the flow stress (\(\sigma\)) is determined by parameters such as the initial yield strength (A), strain hardening coefficient (B), strain rate sensitivity coefficient (C), strain hardening exponent (n), equivalent plastic strain (\(\epsilon\)) and thermal softening exponent (m). In this study, the plastic strain rate (\(\dot{\epsilon}_{0}\)) was set to 1 (/sec), and the influence of the reference plastic strain rate (\(\dot{\epsilon}_{0}\)), reference temperature (T\({}_{0}\)), and melting temperature (T\({}_{m}\)) was neglected. As a result, the equation can be simplified as shown in Equation (2). \[\sigma=(\mathrm{A}+\mathrm{B}\epsilon^{n})\bigg{(}1+\mathrm{C}\mathrm{I} \mathrm{n}\,\frac{\dot{\epsilon}}{\dot{\epsilon}_{0}}\bigg{)}\bigg{(}1-\bigg{(} \frac{\mathrm{T}-\mathrm{T}_{0}}{\mathrm{T}_{m}-\mathrm{T}_{0}}\bigg{)}^{m} \bigg{)} \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Structural Properties**} \\ \hline Young’s modulus & 205 GPa \\ Poisson’s ratio & 0.29 \\ Mass density & 7850 kg/m\({}^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Material properties of AISI 4340. \[\sigma=(\mathrm{A}+\mathrm{B}\epsilon^{n})(1+\mathrm{Cln}\,\dot{\epsilon}) \tag{2}\]  
Fracture damage modeling was performed based on the cumulative damage law of the Johnson-Cook fracture model, as expressed in Equation (3). This model is designed to predict the onset of fracture by quantitatively evaluating the accumulation of plastic strain in the material under applied external loads. Fracture is assumed to occur when the equivalent plastic strain accumulates beyond a certain threshold, and this is expressed by the relationship between the equivalent plastic strain increment (\(\Delta\overline{\epsilon}\)) and the plastic strain at failure (\(\overline{\epsilon}_{\ell}\)), where cumulative damage (D) reaches or exceeds a value of 1, as shown in Equation (4). The plastic strain at failure (\(\overline{\epsilon}_{\ell}\)) is determined by variables such as the strain rate ratio (\(\dot{\epsilon}/\dot{\epsilon}_{0}\)), the ratio of hydrostatic pressure to equivalent stress (\(\mathrm{P}/\overline{\sigma}\)), and a set of damage constants (D\({}_{1}\) through D\({}_{5}\)), as well as the reference temperature (T\({}_{0}\)) and melting temperature (T\({}_{\mathrm{m}}\)). If the effect of temperature is neglected, the model can be simplified as shown in Equation (5). The material constants used in the Johnson-Cook model and the parameters related to fracture are summarized in Tables 9 and 10.