## Abstract

Magnetoactive elastomers (MAEs) are composite materials consisting of nearly rigid, magnetically susceptible particles embedded in a soft, magnetically insensitive elastomer matrix. These multi-functional materials exhibit field-dependent strains and changes in stiffness. However, the strains that have been achieved experimentally to date are still relatively small (of the order of 1%). The reason for these small strains can be traced back to the dipolar nature of the forces between particles. Large particle concentrations are required to generate strong forces, but large concentrations also lead to large overall stiffness for the composite material, which, in turn, tends to reduce the overall strain. In this paper, we propose a new class of MAEs with doubly layered, herringbone-type microstructures capable of generating much larger field-induced strains of up to 100%. This is accomplished by combining the strong action of magnetic torques on suitably oriented magnetic layers, which interact directly with the applied magnetic field, together with the excitation of soft modes of simple shear deformation in the elastomer layers. Theoretical predictions, based on an exact analytical solution for the macroscopic magnetoelastic response of the materials, allow for the optimization of the microstructure for enhanced magnetostriction.

## 1. Introduction

Magnetostriction—a spontaneous change in length induced by the application of a magnetic field—is a phenomenon typically associated with ferromagnetic materials. In most such materials, the field-induced strains are extremely small (from 10^{−6} to 10^{−5}). However, in rare-earth alloys (e.g. Terfenol-D) relatively large magnetostrictive strains of the order of 0.2% have been observed [1]. Larger still strains of almost 10% have been achieved in single crystals of Ni–Mn–Ga shape-memory alloys [2]. Such ‘giant’ strains are produced by the motion of twin boundaries, which can be controlled by reorientation of the applied magnetic field [3]. A severe shortcoming of the Ni–Mn–Ga alloys is that the strains in easier-to-manufacture polycrystalline specimens are much smaller (of the order of 1%). Similar phenomena and field-induced strains have been observed in ferroelectric ceramics [4,5].

Magnetostriction has also been found to occur [6–9] in magnetoactive elastomers (MAEs), also known as magnetorheological elastomers (MREs). These materials are composites consisting of particles of a stiff, magnetically susceptible material that are distributed randomly, or aligned in chains, in a soft elastomeric matrix [10–13]. The application of an external magnetic field leads to the production of magnetic forces and torques on the particles, which interact mechanically by generating deformations of the surrounding elastomer. Much of the interest in these materials has derived from their ability to change their overall stiffness by means of suitably applied magnetic fields [11–14], which can be used to suppress vibrations, dampen noise and control structural compliance. In addition, they can be easily manufactured and formed into a myriad of shapes. Although it has been recognized for some time that field-induced strains can also be generated in these materials, the strains that have been achieved thus far are relatively small (less than 1%) [6–9], and this has hampered their potential use as ‘artificial muscles’ and as sensors and actuators in robotics and in other applications [15].

The particles that are typically used in MAEs are magnetically soft, isotropic and ‘equiaxed’ (i.e. roughly spherical shape) [9,12,13], although the use of magnetically hard particles has been investigated recently [16]. As illustrated in figure 1*a*, the magnetostriction in MAEs arises from the average effect of internal forces that develop because of the dipoles induced on the particles by the applied magnetic field. As a consequence, these internal forces are proportional to the square of the particle concentration (i.e. *c*^{2}), and relatively large concentrations are required to generate significant forces [17,18]. On the other hand, the particles are very stiff, and large particle concentrations also lead to large overall stiffness for the composite, with the net result that only relatively small strains can be achieved [19].

In an effort to get around these limitations, an alternative mechanism for enhancing the field-induced strains was proposed [18]. As illustrated in figure 1*b*, magnetic torques can be generated on the particles when they have elongated shapes and when they are not aligned with the applied magnetic field (recall that the magnetic torque on a given particle is proportional to the cross product of the magnetization and the magnetic field in the particle [20,21]). Because these torques are induced through the direct interaction of the particles with the applied magnetic field, their magnitudes are of order *c* (as opposed to *c*^{2} for the dipolar forces), which means that significant field-induced effects may be achieved for relatively small particle concentrations, before the overall compliance of the composite is significantly compromised. The particle torques have the largest macroscopic effects when they can be made to work collectively so that their rotations are more effectively converted into macroscopic deformation [22]. Although composites with randomly oriented, or textured orientations would perhaps be easier to manufacture, cancellation effects would be expected due to particles tending to rotate in opposite directions. To get around this problem, it has been recently proposed [22] to make use of the layered structures depicted in figure 1*c*, where aligned long particles in each layer work together to maximize the effective deformation in the field direction, the alternating orientation of the layers being used to cancel out the undesired deformations in the transverse direction. Simple laminates of magnetoactive phases have also been considered recently [23] in connection with the possible development of magnetoelastic instabilities.

## 2. Theoretical analysis

Inspired by the above-discussed ideas, we consider here MAEs with simpler multi-layered structures, as depicted in figure 2, and investigate their potential use for actuation purposes. In particular, we focus on optimizing the microstructures in these systems to maximize the field-induced strains. At the microscale, the composite consists of two materials layered together: the soft, non-magnetic, incompressible elastomer phase and the rigid, magnetic inclusion phase. At this scale, the material can deform only by a simple shear along the layer direction. To focus the deformation in one direction, these MAE laminates can then be subsequently laminated at a larger length scale—the mesoscale—yielding a double-layered material with herringbone microstructure (figure 2*b*). For symmetry purposes, the two phases in the macroscale laminate are chosen to have equal volume fractions and such that the layer normals at the microscale level are arranged at angles +*θ*_{0} and −*θ*_{0} relative to the horizontal direction, defining the (+) and (−) phases of the macroscale laminate (with layer normal along the horizontal). It is assumed here that the rigid layers within a larger mesoscale layers are not ‘welded’ to the corresponding layers at a neighbouring mesolayer with opposite orientation angle. In practice, this could be achieved by adding a thin elastomer layer between the mesolayers, whose effect at the macroscopic level can be shown to be negligible (of the order of the volume fraction of the elastomer layer).

It is important to note that MAEs with these multi-layered structures should be possible to manufacture in a sequential manner. Thus, a simple laminate consisting of alternating layers of the magnetic and elastomeric phases could be fabricated by conventional means. Then, this laminated material could be sliced up into thicker layers (several times thicker than the layers in the original laminate) at the appropriate angle relative to the layer normals of the original laminate. Finally, these thicker mesolayers could be reassembled by alternating the orientations of their microlayers, and glued together with a thin layer of the elastomeric phase, to generate materials with the desired herringbone structure. While manufacturing these materials with herringbone microstructures is not completely trivial, it would seem to be a lot easier than generating the aligned-particle MAEs initially suggested by Galipeau & Ponte Castañeda [22]. We also anticipate that the properties of these herringbone-structured MAEs should be relatively insensitive to minor manufacturing imperfections, as long as the interfaces between the micro- and mesoscale layers can largely maintain their integrity.

As illustrated in the expanded view of the material in figure 2*b*, the herringbone microstructure evolves with the overall deformation. Extension or compression of the test sample, characterized by the uniaxial stretch , forces the microstructure to either straighten out or crumple up, because the magnetic layers are rigid and can rotate only to accommodate the macroscopic deformation. In particular, the normals to the magnetic layers, initially characterized by the angle *θ*_{0}, rotate to a new angle *θ* according to the relation
2.1Additionally, as already mentioned, the rotation of the rigid layers at the microscale must be accommodated by a simple shear of the (incompressible) elastomer matrix. Thus, simple kinematics shows that the simple shear (along the rigid layers) in the elastomer phase is given by
2.2where *c* is the volume fraction of the rigid layers. It is emphasized that equations (2.1) and (2.2) are purely kinematical and thus independent of the magnitude of the applied magnetic flux , which will allow for great simplification in the theoretical determination of the macroscopic response of these MAE systems. Note that combining these two expressions also allows the computation of the required amount of shear *γ* in the soft layers to produce a macroscopic stretch .

Making use of the classical theoretical formulations of magnetoelasticity [21,24–26], and building on earlier work for elastic composites [27], a variational framework has recently been developed for estimating the macroscopic response of magnetoelastic composites [18]. Thus, the macroscopic behaviour of the MAE laminates can be described in terms of the effective free-energy density function , which is computed from the volume average of the magnetoelastic energy that is stored in a representative volume element of the composite (i.e. under the separation of length scales assumption that the width of the layers is small compared with the overall size of the specimen), when appropriately chosen boundary conditions are applied leading to a macroscopically uniform uniaxial stretch and magnetic flux . Then, the macroscopic magnetization that develops in the sample and the mechanical traction that is required on the specimen's boundary to maintain equilibrium are determined by the relations [18,22] 2.3and 2.4where is the average density of the composite. It should be emphasized that equation (2.4), for the mechanical traction, properly accounts for the presence of the Maxwell stress in the vacuum surrounding the specimen (see [28]).

Because, as has been seen in connection with equations (2.1) and (2.2), the current state of the microstructure of the composite is completely determined by the applied stretch , the free-energy density function can be written as 2.5where , depending only on , is the energy density associated with the purely mechanical problem (when ), and corresponds to the magnetostatic energy that is stored in the MAE laminate in the current configuration, as determined by and .

Before detailing the expressions for and , we describe the constitutive behaviour of the elastomeric and magnetic phases. Thus, the mechanical behaviour of the elastomer phase will be taken to be of the generalized neo-Hookean type, such that *ϕ*^{(1)}_{me} is a function of the first invariant of the deformation gradient **F**, *I*_{1}=tr(**F**^{T}**F**). For convenience, we will make use of the invariant , which reduces exactly to *γ* for a state of simple shear. In particular, we will make use of the incompressible Gent model [29], where the material is described by the parameters *G* and *J*_{m}, which characterize respectively, the ground-state shear modulus and the limit of polymer chain extensibility, such that
2.6It is noted that there is no magnetic contribution to the free-energy density function of the elastomer, because the elastomer is magnetically insensitive.

The magnetic material will be taken to be mechanically rigid, so that it can only undergo rigid body rotations (and translations), whereas its magnetization behaviour will be taken to be isotropic and assumed to be described by a Langevin function with magnetic saturation *m*_{s} and initial susceptibility *χ*. Thus, the magnetic behaviour of the material will be determined by
2.7such that, according to equation (2.3), the magnetization vector **m** is related to **b** within the material via the relation **m**=*m*^{(2)}(*b*)**b**, together with
2.8

Under the assumed separation of length scales (micro ≪ meso ≪ macro), the mechanical and magnetic fields can be shown to be (essentially) uniform in the layers of the laminate and are then simply determined by the appropriate jump conditions between the lamination layers at each length scale [30,31]. By using these results, the magnetic and mechanical contributions to the energy can be easily determined. Thus, the mechanical contribution to the free energy can be shown to be given by
2.9where we have used the fact that no mechanical energy is stored in the rigid phase, and *γ* is determined in terms of by means of equations (2.1) and (2.2).

The corresponding magnetic contribution is given by (see [30], for the mathematically analogous problem for a dielectric laminate)
2.10where and are the magnitudes of the magnetic induction fields in the elastomer and magnetic phases of the (+) phase of the laminate. (The corresponding fields in the (−) phase are similarly given.) The horizontal and vertical components of , respectively labelled and , are determined as the solution of the two nonlinear algebraic equations:
2.11and
2.12where it is recalled that *m*^{(2)} is a nonlinear function of as determined by equation (2.8). Given and , the corresponding components of are easily determined from the equations:
2.13It should be emphasized that the magnetic contribution to the free-energy density of the composite has been computed in the current (deformed) configuration, and is therefore also a function of the macroscopic deformation (through the angle *θ*, which according to equation (2.1) is a function of ).

In summary, the constitutive behaviour of the laminated MAEs depicted in figure 2 is determined by equations (2.3) and (2.4), where is given by equation (2.5) in terms of and , as given by equation (2.9) and (2.10), respectively. It can be deduced from these expressions that the magnetoelastic coupling in the MAE laminates depends on the interplay between the mechanical stiffness of the rubber layers and the magnetic stress generated by the metallic layers. The overall magnetoelastic coupling is thus characterized by the relative stiffness of the elastomer phase to the magnetic saturation of the metallic layers, which may be quantified by the non-dimensional parameter 2.14such that larger magnetoelastic coupling corresponds to larger saturation magnetization or to lower elastomer stiffness [18].

For a given value of *κ*, it is then of great practical interest to determine the structure that optimizes the magnetoelastic effects. Next, we explore the effects of the initial orientation angle of the microlayers *θ*_{0}, and the concentration of the rigid phase *c*. As will be seen, a wide range of behaviours may be achieved by varying these two parameters.

## 3. Results

Figure 3 shows plots of the overall magnetization of the MAE composite, as determined by equation (2.3). The results are presented as functions of the applied magnetic field for volume fraction *c*=0.4 and several values of the initial angle *θ*_{0}. The specific values of the relevant material parameters are displayed directly in figure 3. Note that the overall magnetization is rather insensitive to *θ*_{0} and is consistent with the Langevin-type behaviour of the magnetic layers, tending to *cm*_{s} for large values of the applied field. The magnetization behaviour for the pure magnetic layers is also shown for comparison purposes.

Figure 4 shows the effect of the magnetic field on plots of the applied (mechanical) traction as a function of the logarithmic strain for the MAE laminate, as determined by equation (2.4). The values of the microstructural parameters *c* and *θ*_{0}, and material properties of the elastomer and magnetic phases are specified on the plot. It can be seen that the magnetic field has the effect of progressively shifting down and to the left the traction–strain curve until a state of saturation is reached. Note that the shift is non-uniform (i.e. larger shifts for tensile strains than for compressive strains). As can be seen from figure 4, under the application of the magnetic field, a traction force is required to keep the material in its reference (undeformed) configuration (the actuation traction), whereas a strain will result if no mechanical traction is applied on the boundary of the specimen (the magnetostriction). It can also be observed that the slope of the traction–strain curves changes significantly with the application of the magnetic field, with the obvious implications for the magnetoelastic Young's modulus of the composite.

Figures 5 show the magnetostriction as a function of the applied magnetic flux for different values of the microstructural and material parameters. Figures 5*a* provides results for *c*=27% of carbonyl iron embedded in a silicon rubber with a modest value of *κ*=0.067. These properties were selected to be able to ‘calibrate’ the predictions of the theory to the experimental results (filled circles) of Guan et al. [8]. In this connection, it should be remarked that the MAEs considered by Guan et al. contained equiaxed particles instead of layers. For this reason, the theoretical predictions of reference [19] for MAEs with spherical inclusions are also included in the plot and seen to be in very good agreement with the experimental results for the above choice of material properties. Making use of the same material properties, figure 5*a* also shows the theoretical predictions for the MAEs with the layered, herringbone microstructures for several values of the initial layer orientation *θ*_{0}. Thus, even though experimental results are not available for the layered MAEs, it is possible to conclude from the results shown in figure 5*a* that the layered microstructures can lead to significant enhancements in the magnetostriction compared with the equiaxed particle microstructures. Furthermore, the level of improvement can be seen to be strongly dependent on the angle *θ*_{0}. Figure 5*b* shows the corresponding results for MAEs with a softer matrix, and a higher concentration *c*=40% of particles with improved magnetic properties (a higher saturation magnetization) leading to a larger value of *κ*=16. It can be seen that the strains produced are much larger than for the previous case (up to 50% instead of 1%), and again that the strain generated in the MAEs with herringbone microstructure are much larger than for the MAEs with spherical particles [19]. In addition, it can be seen from these plots that the magnetostriction initially grows quadratically with the applied field, which is consistent with the quadratic nature of the Maxwell stresses, but then turns around and tends to saturate as the magnetic field becomes large and the magnetization in the particles reaches saturation. (The saturation levels are shown in the plot in dashed lines and labelled ‘sat.’)

Figure 6 shows the saturation magnetostriction *e*_{s} as a function of the concentration *c*, and the initial orientation angle *θ*_{0}. These parameters have a strong influence on the magnetostriction and reveal that the optimal magnetostriction depends sensitively on the microstructure. As can be seen in figure 6*a*, the magnetostriction vanishes when *c*→0, because the elastomer matrix is magnetically insensitive, as well as when *c*→1, because the composite becomes mechanically rigid. (Results are also shown for MAEs with spherical particles [19] for comparison purposes.) The optimal concentration for the layered MAEs takes place for concentrations near 40%. However, the optimal value depends on the precise value of the initial orientation angle *θ*_{0}. As can be seen from figure 6*b*, when *θ*_{0}→0^{°} or 90^{°} the torques on the fibres vanish, and the composite becomes rigid in the stretch direction. The optimal effect is obtained for values near 30^{°}, because these configurations allow the largest sustained torques on the magnetic layers, and the softest mechanical state as the composite is deformed to the magnetostricted state. In figure 6*b*, the ‘geometric limit’ of the magnetostriction is also shown for reference purposes. This geometric limit, which is obtained by letting in equations (2.1) and (2.2), shows the largest possible strain that is achievable for given initial layer orientation *θ*_{0} for sufficiently large values of *κ*.

Following up on this last point, figure 7 shows the maximal theoretically possible saturation strain for various choices of the microstructural variables and material constants, as shown in table 1. Clearly, by making the elastomer matrix soft enough and choosing the magnetic layers to have sufficiently high saturation magnetization, magnetostrictions nearing 100% can be achieved. In this connection, it should be emphasized that the layered MAEs should be much more robust at large fields than the spherical-particle MAEs, which tend to fail by particle debonding at large applied fields.

Finally, it should be noted that the results of this work are consistent with the earlier theoretical findings of reference [22] for MAEs with aligned, cylindrical fibres of elliptical cross section, in the limit as the aspect ratio of these fibres tends to infinity and the fibres become layers.

## 4. Conclusion

We have shown that much larger strains (up to 100%) are theoretically possible in MAEs with especially designed herringbone microstructures than for standard MAEs with equiaxed particles, for which it has been difficult to generate strains larger than 1%. The basic idea is to exploit the collective effect of magnetic torques on the stiff, magnetic phase acting through simple shears in the soft, elastomeric phase to generate large uniaxial strains along the ‘average’ direction of the layers—a mechanism that should be fairly robust and insensitive to small manufacturing imperfections. The exact analytical estimates for the layered MAEs have been obtained by making use of an energetic formulation together with certain simplifications allowed by the use of layered structures with piecewise constant fields, as well as incompressibility of the elastic phase and rigidity of the magnetic phase. The largest value of the magnetostriction for given material properties, as characterized by the parameter *κ*, is obtained by optimizing the volume fraction *c* and the initial layer orientation *θ*_{0}. Systems with higher magnetic saturation and lower stiffness, corresponding to higher values of *κ*, exhibit larger overall magnetostriction. At a more fundamental level, the results of this paper demonstrate the importance of the microstructure and its evolution on the coupled, nonlinear macroscopic response of magnetoactive composite systems. Similar effects would be expected in electroactive composites. Indeed, although for different microstructures and in the context of a small-strain theory, the recent work of Tian *et al*. [34] has shown that sequentially laminated composites can also be used to enhance the electrostriction in dielectric elastomer composites. The magnetoactive systems, however, have some distinct advantages, including the possibility of untethered actuation, relatively high impedances, and, most importantly, freedom from the constraints imposed by dielectric breakdown.

## Funding statements

This material is based on work supported by the National Science Foundation (grant no. CMMI-1068769).

- Received June 7, 2013.
- Accepted July 24, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.