## 2 Investigation of Five Parametric Models
### MRE Models  
Magnetorheological material is a type of smart material that consists of micron-sized magnetic particles dispersed in an elastomer matrix with specific additives. Its rheological features such as stiffness and damping can change and be controlled reversibly and rapidly by an externally applied magnetic field. If the particles inside the elastomer are uniformly distributed, this material is called isotropic MRE, which is unstructured, while if they have a magnetic-induced, aligned, chain-like columnar structure, it is an anisotropic MRE. In the latter case, a field-dependent aligned particle structure forms and becomes locked in place in the polymer composite during crosslinking processes. To investigate important parametric models derived from Maxwell (a spring and dash-pot in series) and Kelvin (a spring and dash-pot in parallel) models, Laplace transform is employed to present five mathematical expressions of models. Some of the models can be appropriately presented in the time-domain form while others in the frequency domain form. Figure 1 shows these model schematics and the corresponding mathematical expressions are given in Table 1.  
Figure 1: Parametric Model. (**a**) M1 (**b**) M2 (**c**) M3 (**d**) M4 (**e**) M5. The basic assumptions and applicability of these models include:  
1. The viscoelastic property is modeled by Maxwell and Kelvin models, and Payne effect is modeled by modified Kraus model. 2. The steady state response of shear stress-strain curve is obtained from a harmonic strain excitation \(\gamma=\gamma_{0}\text{sin}(\omega t+\varphi)\), where \(\gamma_{0}\) is the strain amplitude, \(\varphi\) is the phase angle, which is the phase angle difference between the input and the output and \(\omega\) is the driving frequency. Model 1 (M1) is a simple linear viscoelastic model. The model is based on the magnetic fluid material three-parameter model. Different from the matrix of magnetic fluid material, the matrix of MRE material is an elastomer, and the effect of matrix deformation is considered. The four-parameter linear viscoelastic model is composed of a three-parameter standard solid model with additional spring element as shown in Figure 1a. In this model, the viscoelastic characteristics of modulus and damping capability is modeled by a three-parameter standard solid model, and the introduced spring element represents field-induced modulus [6]. Model 2 (M2) is a refined model considering the slip between particles and matrix. It consists of three parts: (a) the viscoelasticity of the polymer composite, (b) the magnetic-field-induced mechanical properties, and (c) interfacial slippage between the matrix and the particles. M2 includes a standard linear solid model, a stiffness variable spring, and a spring-Coulomb friction slider as shown in Figure 1b. In this model, the standard three-parameter solid model describes the viscoelasticity of the polymer composite. Magnetic field-dependent property is indicated by various stiffness spring elements by considering the dipole interaction between magnetic particles within the chain, and the interfacial slippage between particles and matrix is modeled by a slider with a threshold of interfacial bond strength [7]. Model 3 (M3) introduces fractional derivative. The stress of time domain is transformed into a complex modulus (storage modulus and dissipation modulus) by Fourier transform. This is different from the traditional Newton pot (Figure 1a,b), which can only express the behavior of viscosity. The fractional order introduces frequency into the storage modulus, so the viscoelastic behavior becomes the spring pot in Figure 1c. The spring pot can express the behavior between elasticity (called storage modulus) and viscosity (called dissipation modulus), which is more consistent with the actual physical experimental behavior.