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By using a representative volume element (RVE) approach, this paper investigates the effective mechanical properties of anisotropic magnetorheological elastomers (MREs) in which particles are aligned and form chain-like structure under magnetic field during curing. Firstly, a three-dimensional RVE in zero magnetic field is presented in ABAQUS/Standard to calculate the macroscopic mechanical properties of MREs. It is shown that the initial shear modulus of MREs increases by 56% with a 20% volume fraction of particles compared to that of pure rubber. Then by introducing the Maxwell stress tensor, a two-dimensional plane stress RVE for the MRE is developed in COMSOL Multiphysics to study its response under a magnetic field. The influences of magnetic field intensity, radius of particles, and distance between two adjacent particles on the macroscopic mechanical properties of MRE are also investigated. The results show that the shear modulus increases with the increase of the applied magnetic field intensity and the radius of particles and the decrease of the distance between two adjacent particles in a chain. The predicted numerical results are consistent with theoretical results from Mori-Tanaka model, double inclusion model, and dipole model.

Increasing interest in magnetorheological elastomers (MREs), in which micron sized ferrous particles are dispersed in soft matrix such as rubber, is driven by their unique property of magnetic-mechanical coupling [

In order to study the unique response of anisotropic MREs, the property of a field under localized and microscopic condition has to be homogenized to get macroequivalent properties. Therefore, both macro- and microscales should be considered. This paper aims to study the effective mechanical properties of anisotropic MREs by using FEM. A three-dimensional representative volume element (RVE) with microscale for the MRE is developed in ABAQUS/Standard and solved by assigning periodic boundary conditions to obtain its macroscopic mechanical properties with no field applied. Secondly, by introducing the Maxwell stress tensor, a two-dimensional plane stress RVE for the MRE is developed in COMSOL Multiphysics to study its response under a magnetic field. Finally, influences of magnetic field intensity, radius of particles, and distance between two adjacent particles in a chain on the macroscopic mechanical properties of MRE are also investigated.

MREs consist of micron sized silicon rubber matrix and magnetizable carbonyl sphere particles (about 3 to 5

Sketches of MREs.

Isotropic MREs

Anisotropic MREs

Several methods can be used to compute the magnetic force, for example, the Lorentz, the Maxwell stress tensor, and the virtual work method [

In the theory of composite materials, representative volume element (RVE) is the smallest volume over which a measurement can be made that will yield a value representing the whole. Periodic boundary conditions (PBC) are often used to simulate a large system by modeling a small part that is far from its edge. Many researchers [

There are two ways to get the macromechanical properties of composites. One is the direct FEM; the other is mean field homogenization. FEM is based on a RVE and gives accurate and detailed microfield. The second approach is based on the theory of Eshelby inclusion of various approximate models [

Particulate filler herein is carbonyl iron particle, which is a typical high permeability material having low remanence and high magnetic saturation rate. The material parameters are given as follows: Young’s modulus is 210 GPa, Poisson’s ratio is 0.33 [

The silicon rubber used as matrix in MREs can be modeled by a form of free energy function of Mooney-Rivlin [

As shown in Figure

Several particles of orthotropic MREs.

RVE of MREs

Tension deformation of MREs

Figure

Stress-strain curve without a magnetic field.

Then a RVE with zero magnetic field which has one particle with a volume fraction of 0.2 is established. As can be seen from Figure

Orthotropic MREs.

RVE of the orthotropic MRE

Deformation of the orthotropic MRE

Shear stress-strain curve without a magnetic field in the orthotropic MRE.

Dipole interaction in a straight chain.

In the dipole model, the free energy function in MREs can be decomposed into three parts nominally representing the energy contributions from the matrix, particle, and the coupled interaction of magnetic filed and particles. The magnetic dipolar interaction between adjacent particles in a chain is illustrated in Figure

The dipole model assumes that the interaction energy of the two dipoles is [

In a current free region, where

As shown in Figure

Magnet flux density in MRE.

Figure

Shear deformation in MRE.

As shown in Figures

Different magnetic field strength.

Different radius of particles.

Different distance between two adjacent particles.

In this paper, by using ABAQUS and COMSOL Multiphysics, magnetic-mechanical behaviors of MREs are investigated via FE simulations on RVEs with periodic boundary conditions being applied. An analysis of 3D RVE for the MRE without a magnetic field shows that the initial shear modulus of the MRE increases to nearly 1.56 times compared to that of the pure rubber acting as the matrix. A simple shear deformation of plane stress RVE using the Maxwell stress tensor is further implemented to characterize the properties of anisotropic MRE in the presence of a magnetic field. The numerical results show that an increase in initial shear modulus can be achieved when the intensity of magnetic field and radius of particles increase and the distance between two adjacent particles decreases. All the modeling results are in good agreement with theoretical results. Future works will be concentrated on the macro-/micromechanical properties of anisotropic MREs by using three-dimensional FEM.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The supports from the National Natural Science Foundation of China (11172171 and 11272362) and Ph.D. Programs Foundation of Ministry of Education of China (20130073110054) are gratefully acknowledged.

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