# Study on Vibration Reduction Performance of Gear Pairs Made by a High-Strength Fe-Mn Damping Alloy
## 4 Constitutive Model of a New Type of Fe-Mn Damping Alloy Material
### The Constitutive Equation for Unloading  
To determine the constitutive equation of the damping alloy material at different unloading points, the unloading curve was approximately located in the nonlinear elastic response part, according to the experimental stress-strain curve before the stress of material reached the yield limit. The unloading and loading curves were approximately symmetrical about the straight line connecting any unloading point and the origin. Assume that unloading begins in \((\varepsilon_{0},\sigma_{0})\); then, the straight-line equation of its origin was \(\sigma=\sigma_{0}/\varepsilon_{0}\cdot\varepsilon\). After making the loading curve \(\sigma=E\varepsilon+\alpha\varepsilon^{2}+\beta\varepsilon^{3}\) symmetric about this line, the unloading curve \(\sigma=a\varepsilon+be^{2}+c\varepsilon^{3}\) could be obtained, as shown in Figure 10.  
We dimmed the unloading curve to \(\sigma=a\varepsilon+be^{2}+c\varepsilon^{3}\) and the loading curve to \(\sigma=E\varepsilon+\alpha\varepsilon^{2}+\beta\varepsilon^{3}\). The tangential slope of the loading curve at a certain point was \(k_{1}\), the tangential slope of the corresponding point on the unloading curve was \(k_{2}\), and the slope of the symmetrical curve was \(k_{3}\). According to the angle relationship of the symmetrical curve, the following could be obtained:  
\[\frac{k_{3}-k_{1}}{1+k_{3}k_{1}}=\frac{k_{2}-k_{3}}{1+k_{2}k_{3}} \tag{17}\]  
\[k_{2}=\frac{\varepsilon_{0}{}^{2}k_{1}-2\sigma_{0}\varepsilon_{0}-\sigma_{0} {}^{2}k_{1}}{\sigma_{0}{}^{2}-2\sigma_{0}\varepsilon_{0}k_{1}-\varepsilon_{0 }{}^{2}} \tag{18}\]  
At the origin, the derivative of the loading curve was \(k_{1}=\sigma\prime|_{\varepsilon=0}=E\). The derivative of the unloading curve was \(k_{2}=\sigma\prime|_{\varepsilon=0}=a\), and when \(k_{1}=E\), \(k_{2}=\frac{\varepsilon_{0}{}^{2}E-2\sigma_{0}\varepsilon_{0}-\sigma_{0}{}^{2 }E}{\sigma_{0}{}^{2}-2\sigma_{0}\varepsilon_{0}E-\varepsilon_{0}{}^{2}}\). Then, we obtained  
\[a=\frac{\varepsilon_{0}{}^{2}E-2\sigma_{0}\varepsilon_{0}-\sigma_{0}{}^{2}E} {\sigma_{0}{}^{2}-2\sigma_{0}\varepsilon_{0}E-\varepsilon_{0}{}^{2}} \tag{19}\]  
At the unloading point \((\varepsilon_{0},\sigma_{0})\), the derivative of the loading curve was \(k_{1}=\sigma\prime|_{\varepsilon=\varepsilon_{0}}=E+2\alpha\varepsilon_{0}+3 \beta\varepsilon_{0}{}^{2}\), and the derivative of the unloading curve was \(k_{2}=\sigma\prime|_{\varepsilon=\varepsilon_{0}}=a+2b\varepsilon_{0}+3c \varepsilon_{0}{}^{2}\). After substituting them into Equation (17), it was equal to:  
\[a+2b\varepsilon_{0}+3c\varepsilon_{0}{}^{2}=\frac{\varepsilon_{0}{}^{2}k_{1} -2\sigma_{0}\varepsilon_{0}-\sigma_{0}{}^{2}k_{1}}{\sigma_{0}{}^{2}-2\sigma _{0}\varepsilon_{0}k_{1}-\varepsilon_{0}{}^{2}} \tag{20}\]  
In addition, the unloading curve passes the unloading point \((\varepsilon_{0},\sigma_{0})\). Then  
\[\sigma_{0}=a\varepsilon_{0}+b\varepsilon_{0}{}^{2}+c\varepsilon_{0}{}^{3} \tag{21}\]  
From Equations (18) and (19), we obtained \(a=\frac{\varepsilon_{0}{}^{2}E-2\sigma_{0}\varepsilon_{0}-\sigma_{0}{}^{2}E} {\sigma_{0}{}^{2}-2\sigma_{0}\varepsilon_{0}E-\varepsilon_{0}{}^{2}}\), \(b=\frac{-k_{2}\varepsilon_{0}-2\alpha\varepsilon_{0}+3\sigma_{0}}{\varepsilon _{0}{}^{2}}\), and \(c=\frac{k_{2}\varepsilon_{0}+4\alpha_{0}-2\sigma_{0}}{\varepsilon_{0}{}^{3}}\), where \(k_{1}=E+2\alpha\varepsilon_{0}+3\beta\varepsilon_{0}{}^{2}\) and \(k_{2}=\frac{\varepsilon_{0}{}^{2}k_{1}-2\sigma_{0}\varepsilon_{0}-\sigma_{0} {}^{2}k_{1}}{\sigma_{0}{}^{2}-2\sigma_{0}\varepsilon_{0}k_{1}-\sigma_{0}{}^{2}}\). Thus, the unloading curve at the unloading point \((\varepsilon_{0},\sigma_{0})\) was \(\sigma=a\varepsilon+be^{2}+c\varepsilon^{3}\). Figure 10: Loading curve and assumed unloading point.