# Magneto-Rheological Elastomer Composites. A Review
## 1 Introduction
### Physical Static Mechanism of MR Effect  
Coupling the MR's magnetization with the anisotropic interaction we obtain the \"particle magnetization model\", which attributes the MR effect to the magnetic permeability mismatch among the constituent phases, both the continuous and dispersed one. Considering the solid phase as the magnetic one, it could be possible to observe that the larger size particles behave as magnetic multi-domains. Considering the particle's magnetic moment is field-induced, their Brownian motion is neglectable; furthermore, the solvent thermal forces (\(\propto\) KT) are generally much smaller than magnetic and hydrodynamic ones. A simplified model could have been achieved by neglecting multibody and multipole magneto static interactions among filler molecules. The magnetic moment of a particle with single domain and magnetizable sphere of radius, in a linear magnetization regime, is  
\[m=4\pi\mu_{0}\mu_{cr}\beta a^{3}H_{0} \tag{2}\]  
where \(\mu_{0}\) is the permeability of the vacuum and is equal to \(\mu_{0}=4\pi\times 10^{-7}T_{m}A^{-1}\), \(\mu\)\({}_{cr}\) is the relative permeability of the continuous phase, and \(\beta\) is the contrast factor or coupling parameter \(\beta=\mu_{pr}-\mu_{cr}/\mu_{pr}+2\mu_{cr}\), \(\mu_{pr}\) is the relative permeability of the particles, and \(H_{0}\) is the magnetic field strength. Contrariwise, at higher magnitude, the particle magnetization saturates and the magnetic moment becomes independent of the field strength:  
\[m=\frac{4}{3}\pi\mu_{0}\mu_{cr}\alpha^{3}M_{s} \tag{3}\]  
In a linear regime, the interaction energy's magnitude between two magnetic moments is  
\[\lambda=\frac{\pi\mu_{0}\mu_{cr}\beta^{2}\alpha^{3}H_{0}^{2}}{2k_{B}T} \tag{4}\]  
where \(0<\beta<1\) for strong MR fields (conventional conformation), and \(-0.5<\beta<0\) for weak MR fields (not conventional conformation). The magnetostatic particles interaction dominates overall thermal motion in case of chain-like particle aggregates. The particles movement within the elastomer during fabrication of MRE composite results in kinetic aggregations into two different regimes. Firstly, the aggregates average length increases as a power law, then single width chain-like structure laterally aggregate forming columnar structures. Thermal fluctuations of particles in position correlate to their mechanism of lateral aggregation. Iron particles based MRE composite show coarsening defect of tip-to-tip stage of aggregation, leading to a local variation in the magnetic field surrounding to the chain-like structure. The lateral interactions between dipolar chains play an important role in the isotropic structure of MRE samples. In the MREs, the elastomer shows steady shear flow, as a result, the Mason number (\(M_{\rm n}\)), is basically dimensionless; thus, shear rate can be defined as the ratio between the hydrodynamic drag and the magnetostatic forces (estimated as the dipole force magnitude) acting on the particles [16]. \[M_{n}=\frac{8\eta_{\rm c}\dot{\gamma}}{\mu_{0}\mu_{cr}B^{2}H^{2}} \tag{5}\]  
\(\eta_{\rm c}\) is the continuous phase viscosity and \(\dot{\gamma}\) is the magnitude of the shear rate tensor. MR vulcanized elastomer behave as Newtonian fluids in the absence of a magnetic field, whilst if a magnetic field is applied transverse to the direction of flow, it is possible to observe a yielding, shear, and viscoelastic behavior. Yield stress is based on macroscopic and microscopic models. Macroscopic models employ minimum magnetic energy distribution that leads to homogenous structures of MRE composite reinforced with fillers such as spherical or layered particles in aggregates. This model contributes towards shear anisotropy of the strained aggregates under small deformation. Contrariwise, microscopic models treat inter-particle interactions, such as single particle chains sheared under an external magnetic field.