## 2 Materials and Methods
### Squeeze Strengthening Effect of the MRF  
The material properties, as well as the magnetization curve and magnetic flux density-shear yield stress curve, for MRF-J01T provided by MRF for Chongqing Instrument Materials Research Institute, are shown in Figure 4.  
Based on the performance parameters of MRF in Figure 2, the relations between the magnetic induction strength, \(B\), and the shear yield stress before squeeze, \(\tau_{y}(B)\), can be obtained with the fitting equation, as follows:  
\[\tau_{y}(B)=965.5B^{4}-1651B^{3}+86.17B^{2}-23.99B+0.9684 \tag{2}\]  
According to the experiments of [PERSON] et al. [18], it is known that the shear yield stress of the magnetorheological fluid increases exponentially when it is subjected to axial squeezing pressure. Among them, when the magnetorheological fluid is shear-deformed, the internal deformation of the magnetic chain is as shown in Figure 5.  
Figure 4: MRF (MRF-J01T) material characteristic curve. Figure 3: Magnetization curves of materials. The shear yield stress of the MRF can be mainly divided into two parts: the magnetically induced yield stress of the modified dipole model, \(\tau_{m}\), and the friction yield stress by tribology, \(\tau_{f}\). The additional shear stress approximation caused by the applied magnetic field can be obtained from [19]:  
\[\tau_{m}=3\phi\mu_{f}\mu_{0}\beta^{2}H_{0}^{2}\bigg{(}\frac{R}{d_{0}}\bigg{)}^{3 }\zeta\bigg{(}\bigg{(}\frac{10}{A^{2}}+\frac{2}{B^{2}}\bigg{)}+\frac{48\beta \text{s}}{A^{3}}\bigg{(}\frac{R}{d_{0}}\bigg{)}^{3}\bigg{)}\gamma \tag{3}\]  
where \(A=1-4\beta\text{cos}^{3}\theta(R/d_{0})^{3}\zeta\), \(B=1+2\beta\text{cos}^{3}\theta(R/d_{0})^{3}\zeta\), \(\zeta=1.202\), \(\beta=(\mu_{p}-\mu_{f})/(\mu_{f}+2\mu_{p})\), \(H_{0}\) is the initial magnetic field strength, \(R\) is the radius of magnetic particles, and \(d_{0}\) is the distance between two adjacent magnetic particles. The remaining parameters are shown in Table 2.  
From Figure 5, the shear strain, \(\gamma\), of the magnetic chain can be expressed as:  
\[\gamma=\frac{b}{d_{0}}=\tan\theta \tag{4}\]  
Assume that the length of the magnetic chain is constant in the case of shear deformation, then the relationship between the shear strain, \(\gamma\), and the adjacent particles' distance, \(\delta\), within the chain can be expressed according to Equation (4), as [20]:  
\[\frac{\delta}{R}=2(\sqrt{1+\gamma^{2}}-1) \tag{5}\]  
From Equations (3)-(5), when \(H_{0}\) = 100 kA/m, the shear yield stress is influenced by the shear strain, and the change trend is shown in Figure 6. The magnetic yield stress rises rapidly as the shear strain increases when the shear strain is less than 0.5. However, when MRF produces large deformation, the magnetic chain will break and the shear yield stress will be reduced, and the MRF regains its fluidity. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Volume fraction (\(\phi\))/\% & 25 & Zero-field viscosity (\(\eta\))/Pa s & 0.8 \\ Density/g/cm\({}^{3}\) & 2.65 & Base liquid & Silicone oil \\ Relative permeability of the medium (\(\mu_{f}\)) & 400 & Relative permeability of particles (\(\mu_{p}\)) & 10 \\ Vacuum permeability (\(\mu_{0}\))/T-m/A & \(4\pi\times 10^{-7}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: MRF (MRF-J01T) material performance parameters. Figure 5: Particles in a magnetic chain. Now, the MRF is in the case of deformation under axial extrusion, the single chain becomes shorter, bends, and finally aggregates into a more compact and robust BCT structure, in which the chain behaves as low-yield strain and high-yield stress [21].