# Performance Analysis of Magnetorheological Porous Fabric Composite
## 3 Discussion
### Dynamic Viscoelasticity  
The storage modulus \(G^{\prime}\) and loss modulus \(G^{\prime\prime}\) are the key parameters to characterize the dynamic viscoelasticity of MRF materials. The storage modulus is mainly affected by the force between magnetic particles generated by the MR effect, which belongs to elastic energy. The loss modulus is affected by the energy dissipation of friction between different phases when the material is sheared [34]. In general, MRF can be approximated as a linear viscoelastic material under small shear strain. Its dynamic shear moduli can be equivalent to the complex modulus \(G^{\star}\) related to the excitation magnetic field \(H\) and the angular frequency \(\omega\). The shear stress \(\tau\) of the material can be expressed as follows:  
\[\tau(t)=G^{*}(H,\omega)\gamma(t) \tag{1}\]  
where the complex modulus \(G^{\star}\) is:  
\[G*=G\prime+iG^{\prime\prime} \tag{2}\]  
Therefore, the friction of fluid and the motion of magnetic particles can be neglected under the low shear strain and fixed oscillation frequency, and the energy dissipation of MRF material is mainly the friction between magnetic particles. According to Equations (1) and (2), the shear stress of materials can be regarded as magnetic stress \(\tau_{H}\) and friction stress \(\tau_{f}\), which can be expressed as follows:  
\[\tau=\tau_{H}+\tau_{f} \tag{3}\]  
Figure 8: Comparison of shear stress between different samples.  
Among them:  
\[\tau_{H}(t)=G^{\prime}(H,\omega)\gamma(t) \tag{4}\]  
\[\tau_{f}(t)=G^{\prime\prime}(H,\omega)\gamma(t) \tag{5}\]  
In order to further test the sample with 1 wt% lithium stearate, dynamic viscoelastic experiments were carried out. Figure 9 shows the variation curve of dynamic shear moduli with shear strain at a frequency of 10 rad\(\cdot\)s\({}^{-1}\) for the 1 wt% sample. As can be seen from the figure, the dynamic shear moduli of the sample decrease with the increase of strain amplitude under the zero field, and the storage modulus decays faster. When the shear strain exceeds a critical value (about 2.2%), the storage modulus is less than the loss modulus, so the composite is in a flow state.  
With the continuous increase of the excitation magnetic field, the storage modulus of the composite has been significantly improved, and it is maintained at a relatively stable value under a low strain amplitude. When the shear strain continues to increase to a certain threshold, the storage modulus begins to decline. As for the loss modulus, it has an upward trend and then a downward trend when the shear strain is low. At this time, the composite presents viscoelastic. This threshold divides the MRF material into linear viscoelastic (LVE) and nonlinear viscoelastic (NLVE). It can be seen from the figure that the dynamic shear moduli of the sample increase with the increase of the magnetic flux density, and the range of the linear viscoelastic also increases slightly. It is mainly due to the rheological effect inside the composite that makes the magnetic particles form a more stable chain-like structure with the increase of the magnetic flux density, resulting in viscoelasticity.  
The chain-like structure of the magnetic particles starts to be destroyed when the shear strain exceeds the threshold. At this time, this structure is in the dynamic equilibrium stage of \"break-recover\". When the shear strain continues to increase, the dynamic equilibrium of the structure is broken, and the composite is in a flowing state. It can also be seen from Figure 9 that the storage modulus of the 1 wt% sample is relatively large under a magnetic field. The higher storage modulus of 1 wt% samples is mainly attributed to its higher viscosity, which makes the chain-like structure formed by the rheological effect more stable.  
Figure 9: Dynamic shear moduli at the constant frequency.