# 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 Loading  
There are few currently available works of literature on the constitutive model of damping alloys. Damping alloys have high damping characteristics in general applications, and energy dissipation occurs. As shown in Figure 8, the engineering stress-strain curves of two kinds of damping alloy materials are obtained by conducting standard tensile tests on the Fe-22 Mn-12 Cr-4 Co-0.2 Nb alloy material and the Fe-22 Mn-12 Cr-4 Co alloy material. The stress exceeded the yield limit, and the plastic deformation of the two damping alloys was significantly larger. However, the stress increase was much smaller. The tensile limit of the two damping alloys is very close. After the tensile limit was reached, necking occurred with continuous stretching, the stress decreased sharply, and the final fracture occurred. After the yield stage of the alloy, the stress and strain showed a nonlinear variation that did not conform to [PERSON]'s law. The stress-strain relationship of the damping alloy was very similar to that of high polymers (rubber, plastics, etc.) before the tensile limit was reached. The Fe-Mn damping alloy was characterized by high strength and low cost, and its damping property increases with increased strain amplitude. The viscoelastic theory was therefore used to construct its constitutive model. The constitutive relations of linear, isothermal, homogeneous and isotropic viscoelastic materials in the time domain can be expressed as follows:  
\[\sigma(t)=E_{0}\varepsilon+\alpha\varepsilon^{2}+\beta\varepsilon^{3}+\int_{0} ^{t}c(t-\tau_{t})\frac{\partial\varepsilon(\tau_{t})}{\partial\tau_{t}}d\tau_ {t} \tag{1}\]  
The relaxation modulus function was a continuous monotone, non-increasing function in time, represented by the stress relaxation part and related to the material itself. When \(t<0\), \(c(t)=0\), \(\varepsilon(t)=0\). The constitutive Equation (1) was extended to a three-dimensional state by using symmetric Piola-Kirchhoff stress tensor \(S_{ij}\) and Green strain tensor \(E_{ij}\):  
\[S_{ij}=c_{0}[A]E_{kl}+\alpha E_{ij}{}^{2}+\beta E_{ij}{}^{3}+\int_{0}^{t}c(t- \tau_{t})[A]\frac{\partial E_{kl}}{\partial\tau_{t}}d\tau_{t} \tag{2}\]  
Figure 8: Engineering stress-strain curves of two damping alloy materials. When the Cauchy stress tensor \(S_{ij}\) was transformed to the Cauchy stress tensor \(\sigma_{ij}\) relative to the current configuration, the relationship was as follows:  
\[\sigma_{ij}=\frac{1}{J}F_{ik}S_{kl}F_{lj}^{T} \tag{3}\]  
where \(J\) was the determinant of deformation gradient stress tensor \(F_{ij}\), \(F_{ij}J=\left\|F_{ij}\right\|\). The expression of \(E_{ij}\) in Formula (2) was:  
\[E_{ij}=\frac{1}{2}\Big{[}{F_{ik}}^{T}F_{kj}-\delta_{ij}\Big{]}=\frac{1}{2} \Big{[}{F_{ki}F_{kj}-\delta_{ij}}\Big{]} \tag{4}\]  
\([A]\) was the Jacobian matrix of the constitutive model, which was:  
\[A=\left[ \begin{array}{ccccc}\frac{1-\
u}{(1+\
u)(1-2\
u)}&\frac{\
u}{(1+\
u )(1-2\
u)}&\frac{\
u}{(1+\
u)(1-2\
u)}&0&0&0\\ \frac{\
u}{1-\
u}&\frac{\
u}{(1+\
u)(1-2\
u)}&\frac{\
u}{(1+\
u)(1-2\
u)}&0&0 &0\\ \frac{\
u}{1+\
u)(1-2\
u)}&\frac{\
u}{(1+\
u)(1-2\
u)}&\frac{1-\
u}{(1+\
u)(1 -2\
u)}&0&0&0\\ 0&0&0&\frac{1}{2(1+\
u)}&0&0\\ 0&0&0&0&\frac{1}{2(1+\
u)}&0\\ 0&0&0&0&0&\frac{1}{2(1+\
u)}\end{array} \right] \tag{5}\]  
\(\
u\) was Poisson's ratio for the material. To solve the integrals in formula (2), the material parameters \(c(t)\) were expressed as Prony series. \[c(t)=\sum_{i=1}^{n}c_{i}e^{-\frac{t}{\delta_{i}}} \tag{6}\]  
By substituting Formula (5) with Formula (2), we could obtain:  
\[S_{ij}=E_{0}[A]E_{kl}+\alpha{E_{kl}}^{2}+\beta{E_{kl}}^{3}+\sum_{i=1}^{n}\int_ {0}^{t}c_{i}e^{-\frac{t-\eta_{i}}{\eta_{i}}}[A]\frac{\partial E_{kl}}{ \partial\tau_{l}}d\tau_{l} \tag{7}\]  
Considering the gear reducer as a research object, the input speed was 500 rpm-900 rpm, the pinion tooth number was 19, and the gear tooth number was 26.