## 3 Metal Tubing Mechanical Properties Experiment
### Establishment of Constitutive Model  
Due to its simple form, few measured parameters and clear meaning, the Johnson-Cook model can well reflect the influence of strain, strain rate and temperature on the mechanical properties of ductile metal materials during the deformation process, which is widely used in the study of mechanical properties of materials [18]. The Johnson-Cook model expression is as follows:  
\[\sigma=\big{(}A+B\epsilon^{n}\big{)}\Big{(}1+C\ln\dot{\epsilon}^{*}\Big{)}(1-T _{*}^{m}) \tag{1}\]  
where \(A\) is the yield strength of the material at the reference temperature and reference strain rate; \(B\) and \(n\) are the strain hardening coefficients; \(C\) is the strain rate sensitivity  
Figure 4: Tensile results of section sample and section.  
Figure 5: 304 stainless steel tubing true stress–true strain curve.  
coefficient; \(m\) is the temperature softening index; \(\dot{\varepsilon}\) is the equivalent plastic strain; \(\dot{\varepsilon}^{*}\) is the dimensionless equivalent plastic strain rate. In \(\dot{\varepsilon}^{*}=\dot{\varepsilon}/\dot{\varepsilon}_{0}\), \(\dot{\varepsilon}\) is the actual strain rate, and \(\dot{\varepsilon}_{0}\) is the reference strain rate. In \(T_{*}=(T-T_{r})(T_{m}-T_{r})\), \(T_{r}\) is the reference temperature, \(T_{m}\) is the melting point of the material, and \(T\) is the experimental temperature. Due to the small effect of temperature during the tubing fatigue precision fracture separation process, this tensile experiment is carried out at room temperature. Only four parameters, \(A\), \(B\), \(n\) and \(C\), need to be obtained.  
(1) Determine the parameters \(A\), \(B\) and \(n\).  
Since the experiment is carried out at room temperature, both the experimental temperature and the reference temperature are 25 \({}^{\circ}\)C. \(\dot{\varepsilon}=0.001\) s\({}^{-1}\) Therefore, Equation (1) can be simplified as follows:  
\[\sigma=(A+B\varepsilon^{n})\bigg{(}1+C\ln\frac{\dot{\varepsilon}}{0.001} \bigg{)} \tag{2}\]  
When the strain rate \(\dot{\varepsilon}=0.001\) s\({}^{-1}\), Equation (2) can be simplified as follows:  
\[\sigma=A+B\varepsilon^{n} \tag{3}\]  
At this point, the yield strength of the material under stretch \(\dot{\varepsilon}=0.001\) s\({}^{-1}\) is the value of \(A\). According to Figure 5, \(A\) = 245 MPa can be obtained. By taking the logarithm of both sides, Equation (3) can be transformed into the following:  
\[\ln(\sigma-A)=\ln B+n\ln\varepsilon \tag{4}\]  
By substituting the obtained true stress-true strain data into the above equation, \(n\) = 0.35 and \(B\) = 460 MPa.  
(2) Determine the parameter \(C\).  
At room temperature, when the plastic strain \(\varepsilon\) = 0, Equation (2) can be simplified as follows:  
\[\sigma=A\Big{(}1+C\ln\dot{\varepsilon}^{*}\Big{)} \tag{5}\]  
According to the tensile experiment data, the parameter \(C\) under different strain rates can be obtained. Through nonlinear fitting, the parameter \(C\) = 0.0038 can be obtained.  
Therefore, the relationship between true stress and true strain of 304 stainless steel tubing at room temperature can be expressed as follows:  
\[\sigma=\Big{(}245+460\varepsilon^{0.35}\Big{)}\bigg{(}1+0.0038\ln\frac{\dot{ \varepsilon}}{0.001}\bigg{)} \tag{6}\]  
Through tensile experiment, the Johnson-Cook constitutive model parameters of 304 stainless steel tubing obtained are shown in Table 2.