Mathematical modeling of magneto-sensitive elastomers

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Abstract

Magneto-sensitive (MS) elastomers are a class of smart materials whose mechanical properties change instantly by the application of a magnetic field. These materials typically consist of micron-sized ferrous particles dispersed in an elastomer. The full system of equations for deformable MS solids in an electro-magnetic field is first considered. Then, the strain-energy functions for isotropic MS elastomers are presented and a simple phenomenological model is suggested. Finally, to illustrate some of the features of the derived model, a MS elastomer confined by parallel top and bottom plates is subjected to shear deformation under the influence of a magnetic field normal to the plates. An acceptable agreement is illustrated between numerical simulation and experimental observation.

Introduction

Magneto-sensitive (MS) (in literature magnetorheological––MR) elastomers are materials that respond to an applied magnetic field with an instantaneous change in the mechanical behavior. An improved understanding of MS elastomers is demanded by the prospect to provide simple, reliable and rapid-response interfaces between controls laws and mechanical systems. It is now well recognized that MS elastomers have the potential to improve the design of electromechanical devices and their operation. For example, an elastomer with field dependent properties may be used as a device with a variable stiffness. Therefore, this wide range of potential applications and associated economic benefits are the reason for the intense research on these materials in recent years, see for example Borcea and Bruno (2001), Carlson and Jolly (2000) and Jolly et al. (1996a).

The magnetic efficiency to change field dependent mechanical properties is optimized by choosing a particle material with a high magnetic saturation. Cobalt has the largest saturation of 2.4 T of all known elements, however it is not used commercially. In general an alloy of iron with a magnetic saturation of 2.1 T is used as an additive in the mixing process of MS elastomeric compounds.

MS solid elastomers are less well publicized and recognized than their electro-sensitive (ES) counterparts. Both elastomers are composed of polarizable particles, dispersed in a polymer medium, having a size on the order of a few microns (typically from 10−7 to 10−5 m). Carrier fillers are selected based upon their electro-magnetic and thermo-mechanical properties: silicone and/or other rubber-like materials with very small electric conductivity. Typical particle volume fractions are between 0.1 and 0.5. During the manufacturing process of MS elastomers, the isotropy condition inherent of the filler material is maintained in the final composite. Therefore, these materials are considered isotropic and non-conductive. We assume, for simplicity, that they remain isotropic even during the application of a magnetic field. However, MS elastomers become non-homogeneous due to the presence and distribution of particles in the carrier filler.

Structurally, field responsive elastomers can be thought of as the solid analogs of field responsive non-colloidal suspensions or fluids, see for example Ginder and Davis (1994). There are however some distinct differences in the way in which these two classes of materials are intended to be used. The most important one is that the field sensitive particles within the elastomer composite are intended to always operate in the pre-yield regime, see Carlson and Jolly (2000). Therefore, field responsive elastomers can best be described by a field dependent modulus, see for example Borcea and Bruno (2001), Jolly et al. (1996a), Rigbi and Jilken (1983).

Jacob Rabinow at the US National Bureau of Standards introduced MS materials in the late 1940s, see Rabinow (1948). At about the same time, initial experimental results on electrorheological fluids were published by Winslow (1949). Following the discovery of these materials, the major interest was dedicated towards ES materials in the late 1940s and early 1950s, which is evident by the larger number of patents and publications. Except for the interest immediately following Rabinow’s work, there has been little new information about and publications on MS media. Recently however, MS elastomers have been recognized as a commercially viable product and thus, an increase in research and publication is recognized (see for example Kordonsky, 1993; Carlson and Jolly, 2000; Jolly et al., 1996a). A number of MS elastomers and various MS elastomer-based systems have successfully been brought into the market, including adaptive tuned vibration absorbers, stiffness tunable mounts, suspensions and automotive bushing.

In the following analysis, we consider the MS elastomer as a moving non-polar isotropic continuum. Section 2 starts with the classical work by Pao (1978). We summarize the full system of equations for the moving isotropic non-polar continuum medium in an electro-magnetic field such as the Maxwell equations, the mechanical and thermodynamical balance laws. In Section 3 we derive the basic system of constitutive equations for MS elastomers using a phenomenological approach based on experimental data by Carlson and Jolly (2000). The reduced system of constitutive equations is complemented by the system of boundary and initial conditions. In Section 4 we present and analyze a simple strain-energy function for MS elastomers. For illustration of the presented phenomenological model, in Section 5 we examine the basic operational features of controllable incompressible elastomeric devices. The final section is devoted to concluding remarks.

Section snippets

Physical laws for moving continuum media in an electro-magnetic field

Let a continuum deformable solid in the reference configuration occupy a domain Ω⊂R3. In the deformed configuration each point XΩ moves into the position x=X+u=χ(t,X)∈R3 where u and χ are the displacement and mapping, respectively, and t is the parameter describing the motion of the medium (usually, this is the physical time). We consider a one-to-one, i.e. locally invertible and orientation-preserving mapping in Ω for every t>0.

In this paper standard mathematical notations are used; for more

Reduction of the basic system for hyperelastic MS solids

It can easily be verified that the basic system is indeterminant, i.e. there are more unknown variables than equations. The system is rendered determinate by providing a sufficient number of constitutive material laws. Here we consider only hyperelastic solids, thus no viscosity dependence needs to be accounted for. For these solids the stress tensor and the work done by the stresses do not depend on the path of deformation, which takes the solid from the reference to the current configuration.

Constitutive relations for hyperelastic MS materials

Constitutive equations for hyperelastic materials, frequently used to describe the non-linear elastic behavior of filled and unfilled elastomers, are obtained as the derivative of a strain-energy or stored-energy function Φ (see for example Green and Zerna, 1975; Holzapfel, 2001; Lurie, 1990; Ogden, 1997; Treloar, 1975). This scalar energy function is defined per unit reference volume rather than per unit mass. From there, the Cauchy stress tensor has the well known formσ=J−1Φ(F)F·FT,where

Example: shear deformation between parallel plates

Let us consider the problem of a MS elastomer confined by two infinite parallel plates in the xy-plane and subjected to an unidirectional quasi-static shear deformation along the x-direction, see Fig. 1. Suppose that the mapping associated with the elastomer and that the magnetic field intensity perpendicular to the xy-plane, are given byx=X+u(z)i,H=Hk,z∈[0,h],where i, j, k are the Cartesian basis vectors, u is the displacement along the x-direction, and h is the distance between the plates.

We

Conclusions

In this paper, we have summarized the complete system of constitutive equations for an isotropic MS elastomer within the framework of the electro-dynamical and thermo-mechanical theories. For hyperelastic, isotropic MS elastomers a simple strain-energy function has been presented and verified by experimental data. An acceptable agreement was illustrated between results of the numerical simulation and experimental observation. It was shown that the effect of the magnetic field is to stiffen the

Acknowledgements

The research was partially supported by the Research Directorates General of the European Commission (through project GRD1-1999-11095). The authors gratefully acknowledge this support.

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