Energy damping estimation in automotive magneto-rheological elastomers through finite element method

: Magneto-rheological elastomers (MRE), which undergo upon the smart materials, are one of the very common products frequently used in modern vehicle production nowadays. For their visco-hyper properties, elastomers find a prevalent use as an energy absorber. Inducing a magnetic field would make their energy absorbing characteristics vary upon the desired ones by adjusting the applied potential field. A precise calculation of the amount of their absorption with respect to the potential field has a key role in the prediction of MREs’ responses. Therefore, in this study, a hyper-visco-magnetic constitutive model is utilized in COMSOL commercial software for the energy damping estimation in magneto-rheological elastomers. A representative volume element has been considered for the calculation of hysteresis loop areas as a characteristic of energy damping behaviour in MREs. Finally, the effects of magnetic flux intensity and mechanical load frequency in the energy damping behaviour of automotive magneto-rheological elastomers are evaluated.


Introduction
Magneto-rheological elastomers as smart composite materials are recently utilized in automotive suspensions.[1] [2] [3] [4].Elastomers due to their visco-hyper elastic behaviour have been comprehensively used as energy absorbers or energy dampers in automobiles.Since the automotive manufacturing companies have a serious competition in the ride and performance attributes of their automotive productions, the controllability of the systems plays a key role in this matter.The magnetization of elastomers enables one to control the energy damping and stiffness of the system.The more accurate the predictions of mechanical response in the magneto-rheological elastomers, the more precise the control on the energy absorption of the suspension system.
In order to predict magneto-hyper behaviour of the elastomer, some models were proposed in the literature [5].Since magneto-rheological elastomers are subjected to dynamic loadings, they need a visco-hyper-magnetic model so as to accurately represent their mechanical response.
In this study, a semi-coupled visco-hyper-magnetic model is utilized in COMSOL Multiphysics commercial finite element software to estimate energy damping in an automotive magnetorheological elastomer. 293

Finite element model
In this section, the details of the finite element simulation for the prediction of the hysteresis loops of automotive magneto-rheological elastomers are presented.The simulation could be subsidized in Geometry, Material model, Load and boundary condition and Mesh subsections as follows.

Geometry
Most of the elastomers utilized as energy dampers in the joining of the suspension links have been made in hollow cylindrical shapes that are subjected to the axial and torsional loadings.Since the loadings lead to the emergence of two planar components, namely normal and shear stress, the simplified model for the evaluation of energy damping behaviour of the elastomers was normally provided in 2D rectangular shape.Therefore, as can be seen in Figure 2, a 2D rectangular geometry with a width of a=12 mm, and a height of b=11.8 mm, is considered as a simplified automotive magneto-rheological elastomer.

Material model
Marvalová and Petríková [5] implemented a magneto-hyper coupling behaviour of a magnetorheological elastomers in COMSOL Multiphysics.The total stress tensor,  hyper-magnetic , and the magnetic field, H, are proposed as follows: in which, the strain energy density function, W, is defined as: where, C10 and C01 are the hyper-elastic material constant, α and β are the magnetic material constant, μ0 is the magnetic vacuum permittivity and the invariants, Ii , are calculated as follows: in which, B, c, F and J represent the magnetic flux, the right Cauchy-Green strain tensor, the deformation gradient tensor and the determinant of the deformation gradient tensor (Jacobian) respectively.
A visco-hyper-magnetic model could be extracted by substituting the above stress in a viscoelasticity model as: in which gi , t , τi and n are dimensionless material parameter, time, relaxation time constant and the required number of relaxation time constant respectively, and g∞ is [6]: The proposed formulation for a constant magnetic flux, B=constant, is implemented in the commercial finite element software COMSOL Multiphysics to simulate mechanical response of automotive magneto-rheological elastomers subjected to cyclic dynamic loads.

Load and boundary condition
In order to model axial and torsional loadings on the simplified automotive magneto-rheological elastomers, two separate loads are considered including a cycle of a uniaxial tensilecompressive load and a cycle of a fully-inversed shear load.The vertical displacement range, Δy=3.6 mm, and the horizontal displacement range, Δx=4 mm, are prescribed on the upper boundary of the domain to create cyclic normal and shear stretches respectively.The cyclic loads are applied in three frequencies, f=0.1, 1 and 10 Hz.As shown in figure 3, the lower boundary of the domain is fixed and the magneto-rheological elastomer is embedded in different magnetic fluxes, B=0, 0.

Mesh
In order to discretise the geometry into finite element model, 40 Quadrilateral elements are utilized.These 4-node elements have degree of freedoms in the displacement and the magnetic fields.

Results and discussion
In this section, numerical results obtained from finite element model for cyclic behaviour of the simplified automotive magneto-rheological elastomers are presented.The area in the hysteresis loops due to the viscous response of the material subjected to cyclic loading would result in the energy damping.
In Figure 4, hysteresis loops for a magneto-rheological elastomer embedded in different magnetic fluxes, B=0, 0.5 and 1 T and subjected to the cyclic normal stretch with amplitude (b+Δy)/b0.15 and frequency f=0.1 Hz are presented.The stretches smaller and larger than 1 indicate compressive and tensile loading modes respectively.As can be seen in Figure 3, mechanical behaviour of the magneto-rheological elastomer is different in compressive and tensile modes.Increasing the magnetic flux, the enlargement in the hardening behaviour of the elastomer in the tensile mode and, in contrast, the weakening of the elastomer in the compressive mode could be observed.Moreover, analyses indicate that there is a higher reduction in the energy damping of the magneto-rheological elastomer when subjected to a larger magnetic flux.
The magneto-rheological effect has been defined as a parameter in percentage of variation in the mechanical response of the material embedded in a magnetic flux with respect to those mechanical responses in the absence of any magnetic flux.The results reveal that the magnetorheological effect in the tensile mode is greater than those in the compressive mode.Also, the enlargement of the magnetic flux would lead to an increase in the magneto-rheological effect.
In figures 5 and 6, the hysteresis loops for a magneto-rheological elastomer subjected to cyclic normal stretches with frequencies f=1 and 10 Hz are presented.The results reveal the same pattern in the variation of mechanical behaviour in compressive and tensile loading modes, as mentioned for cyclic normal stretch with frequency f=0.1 Hz.Although, a comparison between results shows that the increase of the load frequency would stiffen up the material, there is no clear trend in the effects of load frequency on the energy damping in the material.This is due to the significant impact of natural frequency of the magneto-rheological elastomer on the magnitude of energy damping in each cycle, on the other hand, as aforementioned, the variation of the magnetic flux would lead to the variation of the stiffness and as a result the variation of the natural frequency.
In Figures 7-9, hysteresis loops for a magneto-rheological elastomer embedded in different magnetic fluxes, B=0, 0.5 and 1 T and subjected to cyclic shear stretches with amplitude of (a+Δx)/a=0.16 and different frequencies (i.e.f=0.1, 1 and 10 Hz) are presented.Similar to normal stretch, increasing the magnetic flux would enhance the shear modulus and the hardening of the material.Unlike the compressive-tensile loading, in fully-inversed shear load there is a symmetric response in hysteresis loops.
Finally, for all cases the estimated energy damping and magneto-rheological effect are summarized in Table 1.

Conclusion
In this paper, a visco-hyper-magnetic model was implemented in COMSOL Multiphysics commercial finite element software.The hysteresis loops due to cyclic loading of a magnetorheological elastomer embedded in a magnetic flux were estimated.The results reveal that the magnetization of the elastomer would lead to a significant variation in the energy damping of the material.Moreover, the increase of the load frequency has a major role in stiffening the elastomer.

Figure 1 .
Figure 1.Schematics of a magneto-rheological elastomer in an automotive McPherson suspension.

Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .Figure 9 .
Figure 4. Hysteresis loops (Normal stress-Normal stretch) obtained with finite element method for a magneto-rheological elastomer embedded in different magnetic flux and subjected to a cyclic normal stretch with frequency f=0.1 Hz.

Table 1
Damping energy (mJ) in each hysteresis loop and Magneto-rheological effect (%) for a magneto-rheological elastomer embedded in different magnetic flux and subjected to cyclic normal and shear stretches with different frequencies