## 2 Investigation of Five Parametric Models
### MRE with MWCNTs Models  
Carbon nanotubes are small sized materials that have low density and high aspect ratio. The bonding between elastomers and magnetic particles can be generally improved with reinforcing MWCNTs, thus the corresponding MR effect is higher. Experimental study was conducted to modify the previously discussed parametric models. With the dispersed MWCNTs in the liquid elastomer with sonication, which was later mixed with magnetic particles, the curing process could be completed under a steady magnetic field. Carbonyl iron powders with average diameter of 8 \(\upmu\)m were used as magnetic particles, and silicone rubber was as elastomer. Iron particles are chosen in MRE because of their high permeability, low remnant and saturated magnetization ability. MWCNTs with a diameter of 10 nm and length ranging from 10-15 \(\upmu\)m and 95 wt% purity were used as additives. About 30 wt% of iron particles and 1.5 wt % of MWCNTs were mixed in the solution and fabricated in the matrix. The mixture molds were placed between permanent NdFeB bar magnets to form stable polarized structured particle chains under the field strength of 1.35 T.  
For M1 to M4, a parallel spring element (\(\tau=G_{\text{MWCNT}}\gamma\)) is added to modify the aforementioned MRE models by considering the effect of MWCNTs, which functionally enhance the traditional magnetorheological elastomer. Model 5 can be improved by modifying relative permeability of matrix \(\upmu\)m. As shown in Figure 3, the shear stress-strain curves for magnetorheological elastomer nanocomposites are measured at the magnetic field 1 T, strain amplitude 11% and loading frequency 1 Hz. In Figure 3, for MRE, the elastic part of the curve represents initial modulus, i.e., 1.74 MPa, 1.81 MPa, 1.62 MPa, 2.13 MPa and 1.96 MPa for M1-5, respectively. Compared with initial stiffness at the strain amplitude  
Figure 2: Microscopic Model. of 0.11, the modulus for M4 and M5 are 1.95 MPa and 1.81 MPa, which show, respectively, 8.45% and 7.65% loss of initial modulus. For MRE with MWCNTs, the values of modulus for the elastic parts are 1.94 MPa, 2.03 MPa, 1.77 MPa, 2.38 MPa, 2.17 MPa, respectively, for modified M1-M5. Compared with initial stiffness at the strain amplitude of 0.11, the modulus for M4 and M5 are 2.15 MPa and 2.01 MPa, which show, respectively, 9.66% and 7.37% loss of initial modulus. These models are modified due to the fact that stronger interaction between fillers and matrix has larger positive effect on the storage modulus, and the loss factor or damping is more influenced by the friction between particles and matrix molecular chains that leads to more degrees of molecular mobility and energy dissipation. Payne effect can be obviously observed in macroscopic M4 and microscopic M5. Large magnetic field effect and strain effect will increase the energy of magnetic particles, which move away from each other and thus, reduce the interaction and cause breakage of physical bonding. In general, with strain amplitude increasing, both storage modulus and loss factor show a descending trend. This phenomenon is in accordance with the Payne effect. When the applied magnetic field increases, MRE and MR nanocomposites perform better storage modulus and loss factor, which indicates stronger MR effect. MR nanocomposites generally show higher MR effect than conventional MRE under the same condition. With applied current increasing, the increased magnetic field will form stronger magnetic flux density which results in enhanced storage modulus with related to stiffness and loss modulus related to damping property. Moreover, storage modulus decreases with increased magnetic intensity. On the contrary, loss modulus has consistent trend of change in regard to magnetic intensity. These magneto-viscoelastic properties and related models have been tested and verified and will be later applied to modeling viscoelastic behavior of MRE material as a core structure of the isolator [12; 32; 33; 51].