# A Novel Structure of Rubber Ring for Hydraulic Buffer Seal Based on Numerical Simulation
## 2 Materials and Structure  
### Material  
The material used in this buffer seal is Nitrile rubber(NBR), which is widely used in landing gear buffer, considered the pressure resistance of NBR. Based on the assumption of hyper-elasticity theory, the deformation and stress of rubber-like material which is employed by the strain-energy density function, is written as a function of the strain invariant [19].  
\[W=f(I_{1},I_{2},I_{3}) \tag{1}\]  
where \(W\) is the strain-energy function, \(\lambda\) represents for the invariant of the Cauchy-Green strain tensor [20]. Furthermore, the invariants of the Cauchy-Green strain tensor could be expressed by the principal stretches ratios \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\).  
\[\left\{\begin{array}{ll}I_{1}&=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^ {2}\\ I_{2}&=(\lambda_{1}\lambda_{2})^{2}+(\lambda_{2}\lambda_{3})^{2}+(\lambda_{3} \lambda_{1})^{2}\\ I_{3}&=(\lambda_{1}\lambda_{2}\lambda_{3})^{2}\end{array}\right. \tag{2}\]  
In addition, the formula of the Cauchy stress tensor \(\sigma^{ij}\) is expressed in terms of the principal stretches,  
\[\left\{\begin{array}{ll}\sigma_{1}=2\lambda_{1}\Big{[}\frac{\partial W}{ \partial I_{1}}+\left(\lambda_{2}^{2}+\lambda_{3}^{2}\right)\frac{\partial W} {\partial I_{2}}+\lambda_{2}^{2}\lambda_{3}^{2}\frac{\partial W}{\partial I_ {3}}\Big{]}\\ \sigma_{2}=2\lambda_{2}\Big{[}\frac{\partial W}{\partial I_{1}}+\left(\lambda_ {1}^{2}+\lambda_{3}^{2}\right)\frac{\partial W}{\partial I_{2}}+\lambda_{1}^{2 }\lambda_{3}^{2}\frac{\partial W}{\partial I_{3}}\Big{]}\\ \sigma_{3}=2\lambda_{3}\Big{[}\frac{\partial W}{\partial I_{1}}+\left(\lambda_ {1}^{2}+\lambda_{2}^{2}\right)\frac{\partial W}{\partial I_{2}}+\lambda_{1}^{2 }\lambda_{2}^{2}\frac{\partial W}{\partial I_{3}}\Big{]}\end{array}\right. \tag{3}\]  
The two parameter M-R model for approximately incompressible material is adopted,  
\[W=C_{10}(I_{1}-3)+C_{01}(I_{2}-3)+\left(\frac{C_{10}}{2}+C_{01}\right)\left( \frac{1}{I_{3}^{2}}-1\right)+\frac{C_{10}(5v-2)+C_{01}(11v-5)}{2(1-2v)}(I_{3} -1)^{2} \tag{4}\]  
where \(C_{10}\) and \(C_{01}\) are the material parameters. As the rubber material is assumed to be approximately incompressible [21], i.e., the Poisson ratio \(v=0.499\approx 0.5\), \(I_{3}=1\). Therefore, the two parameter M-R model is written as,  
\[W=C_{10}(I_{1}-3)+C_{01}(I_{2}-3) \tag{5}\]  
In addition, the uniaxial compression tests,  
\[\left\{\begin{array}{ll}\lambda_{1}=\frac{1}{\lambda_{2}^{2}}=\frac{1}{ \lambda_{3}^{2}}=\lambda\\ \sigma_{1}=2\lambda_{1}\Big{[}\frac{\partial W}{\partial I_{1}}+\left(\lambda _{2}^{2}+\lambda_{3}^{2}\right)\frac{\partial W}{\partial I_{2}}\Big{]}\\ \sigma_{2}=\sigma_{3}=0\end{array}\right. \tag{6}\]  
where \(\lambda\) is the strain value. By Equations (5) and (6), the Cauchy stress is developed as:  
\[\sigma=\frac{2\lambda^{4}}{\lambda^{3}-1}C_{10}+\frac{2\lambda^{3}}{\lambda^{ 3}-1}C_{01} \tag{7}\]  
According to the data of rubber compression experiment quoted from Reference [8], the material parameters of M-R are fitted as \(C_{10}=1.84\), \(C_{01}=0.47\) with Equation (7).