On the shear test of a MR elastomer under magnetic field applied at various angles

In this paper the influence of the angle between the applied external magnetic field and the mechanical shear direction on the measured elasticity of a magnetorheological (MR) elastomer is addressed. The whole range of magnetic field angles from 0∘ to 90∘ is analyzed in steps of 5∘. Though this dependence is of highest importance for practical applications this issue is rather neglected in previous studies. The work uses MR elastomer specimens based on a silicone rubber matrix containing iron powder in weight fractions of 82 and 89 wt.%. It has been shown that the measured modulus of elasticity of such composites decreases as the angle between an external magnetic field and applied shear deformation increases. As the framework for the discussion of the findings a macroscopic explanation associated with the magneto-deformation effect as well as an influence of the demagnetizing factor are considered.


Introduction
Derived from the industry's need for novel solutions with actively adaptable material properties for the variation and individualization of products, materials with tuneable properties are in the scientific focus. In particular, magnetically active (MA) materials are investigated and characterized as they allow continuous and non-contact adjustment of material properties [1]. In addition to magnetorheological (MR) fluids and MR plastics, MR elastomers belong to the group of MA materials, intensively studied due to their extreme mechanical * Author to whom any correspondence should be addressed.
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compliance. An extensive review on the progress in the field of magnetorheology was presented very recently, emphasizing physical properties and formulations [2]. In another recent review a progress of MR elastomers is considered, including used materials, fabrication strategies, MR effects and potential applications [3].
Sensitivity of a MR elastomer to a magnetic field in terms of magneto-mechanics is usually estimated by obtaining an absolute and relative MR effect. The absolute MR effect is the difference between the measured modulus in an applied magnetic field and the modulus in the zero field. The relative MR effect is calculated as a ratio of the absolute effect to the zero field modulus. According to the review of magneto-mechanical characterizations of MR elastomers, the known experimental studies consider only situations where an external magnetic field and mechanical loading are applied parallel or perpendicular to each other [3]. Hence, the measured MR effects are also only evaluated for these mutual field and load orientations.
Nevertheless, the orientation of the magnetic field relative to the loading direction as well as orientation of prestructures relative to the magnetic field direction are of importance for MA materials [4]. While the influence of the mutual orientation of the structures in initially anisotropic MR elastomers and the direction of the field and load in a subsequent experiment is addressed from time to time in various studies [2,3,5,6], the variation of the mutual orientations of the external field and mechanical load is limited to the two orientations mentioned above.
In order to systematically characterize the influence of the mutual orientation of the externally applied magnetic field and the mechanical shear loading on the MR elastomer response, two selected MR elastomer compositions are prepared and investigated within our current study. One representing a typical carbonyl iron based MR elastomer and the other one being a highly filled extreme with a quasi bimodal microparticle mixture. The utilization of such a mixture of large and small microparticles is needed to allow highest filler contents and thus maximum possible magnetic interactions while maintaining the elastic material properties [7,8]. The experiments are carried out for both types of samples under different magnetic field angles for discrete field strengths in order to focus on angle-dependent effects.
The paper is structured as follows. First, a brief description of the material components and of the manufacturing techniques for the studied MR elastomer samples is given. This is followed by the results for the magnetic properties of these composites measured using a vibration sample magnetometer (VSM). The main experiments within this work are divided into two different experimental setups. One for the field angle dependent experiments and one for the experiments on magneto-deformation. Both are necessary for the asfollowing discussion.

Samples
This study examines MR composite samples based on polydimethylsiloxane matrix (SIEL tm produced by GNIIChT-EOS, Moscow, Russia). Two types of magnetic microparticle powders were used as fillers. The first type is CC grade carbonyl iron manufactured by BASF, Germany, containing spherical particles with a size of ∼3-5 µm ( figure 1, left). The second type of powder contains iron microparticles of irregular shape from 3 µm up to 60 µm in size (figure 1, right), obtained by grinding. Before the particles were added to the SIEL tm composition, they were treated with hydride containing silicone for an improved compatibility with the matrix and to prevent the formation of particle agglomerates. The material was vacuumed after mixing and polymerised at 100 • C. Two types of samples with different filler compositions were manufactured as given in table 1. Ethanol was added to the compound before mixing to reduce initial viscosity and obtain highly filled specimens (sample 2). This allows large quantities of powder to be dispersed without difficulty. The final high concentration of filler in the specimens resulted due to the evaporation of the added ethanol.
For each manufactured compound the magnetisation behaviour was analyzed by VSM measurements utilising Lake Shore 7407 magnetometer. As shown in figure 2 the magnetic saturation for sample 2 is higher than for sample 1, as expected due to the higher filler content in sample 2. The same is true for the initial magnetization behaviour. Small visible hysteresis loops are associated with particle structuring effects as reported in [9].
For the experiments performed the two different specimen shapes shown in figure 3 were used, as needed for the corresponding test setups.
For the investigation of field angle as well as field strength dependent behaviour a rectangular specimen 20 mm × 10 mm is cut from the manufactured compound plates, while for investigating magneto-deformation a circular 13 mm diameter disc is prepared.

Setup for investigating the magnetic field dependent shear properties
To investigate field angle and field strength dependent behaviour the magnetic field testing device (MFATD) introduced in [10] was utilized in combination with a DynaMess TP 5kN HF testing machine. The assembled setup is illustrated in figure 4. The MFATD consists of four separate but mechanically coupled shear cells. For the shear cells variable magnetic fields from 0 to 120 kA m −1 can be realized at any field angle by means of two continuously rotatable, rotationally coupled electromagnets. The shear cells are arranged in pairs in between the pole shoes of each electromagnet. The field angle coupling of the electromagnets is achieved via a coupling plate. To investigate the field angle dependence of a material, each shear cell is equipped with a rectangular specimen (figure 3 left). The double-symmetrical quadruple arrangement of the shear cells averages out thickness variations, orientation variations, normal force differences and inhomogeneities of the specimens while also minimizing tilting moments and field inhomogeneities. To prepare experiments the retaining screws of each bearing block are loosened slightly to enable positioning of specimens and accordingly the adjustment of the shear gaps. Each specimen is then positioned between a fixed bearing block and the central, movable shear plate, which is connected to the lifting cylinder of the testing machine. While slightly pressing a pair of opposing bearing blocks towards each other the retaining screws are tightened, clamping each of the corresponding two specimens in between a bearing block and the central shear plate. This procedure is repeated for the other pair of bearing blocks. To prevent rapid warming of the electromagnets and due to limitations in the power supply the maximum driving current for this device is 14.8 A resulting in a maximum field strength of H max ≈ 120 kA m −1 for air. To ensure that the specimen temperature does not exceed 45 • C during a complete test run, the experimental durations per current step are additionally  limited to 15 s. Coil temperatures are monitored independently by four thermocouples. For details on this setup also refer to [11]. For clarity, in figure 5 an illustration of the test principle is shown including all relevant experimental variables.

Setup for evaluation of the magneto-deformational effect
To measure the mechanical stress produced by a composite as a result of the magneto-deformational effect in the direction of the applied magnetic field, a MARS IV rheometer was used. The rheometer is equipped with a normal force sensor integrated into the air bearing. The force sensor has a force resolution of 0.01 N. A homogeneous magnetic field was provided by an electromagnetic cell designed in our laboratory for MR characterisations of various magnetic composites [12][13][14]. The cell and measurement configuration is shown schematically in figure 6. The sample was placed between the plates of the rheometer measuring geometry. The preliminary axial load on the sample did not exceed 0.1 N. In order to ensure a stress relaxation samples were left in the rheometer for at least 1 h without any load and without a magnetic field applied. The external magnetic field strength was then changed step by step, while the change in normal force due to the magneto-deformational effect inside the MR elastomer was measured. The duration of one step for each applied magnetic field strength was set to 30 s. During these measurements, test specimens were not subjected to any additional loading (e.g. shear). Measurements were repeated at least three times for at least three different specimens of each sample type. For each measurement, samples were anew placed between the plates of the rheometer's measuring geometry.

Shear test
The cyclic shear test performed provide force-displacement curves which are converted to stress-strain curves taking into account the geometric parameters of the test specimen. For a viscoelastic material, such as typical MR elastomers, these curves are hysteresis loops. In figure 7 selected hysteresis curves extracted from the experiments performed are given for two field orientations (α = 0 • and 90 • ) for both samples investigated. The corresponding complex shear moduli are calculated by minima and maxima. From these and from the area enclosed by the curves the corresponding shear storage and shear loss moduli are calculated (G ′ and G ′′ respectively). The obtained moduli for these examples are listed together with the resulting MR effects in table 2.
The magneto-mechanical behaviour of the MR elastomer samples is as expected. The moduli of the material have increased significantly under the influence of the external magnetic field. The obtained values are generally in accordance with the literature data for the type of MR composites in question [3]. It is noteworthy that for sample 2, which has a higher filler concentration and a fraction of large microparticles, the resulting MR effect is slightly higher compared to the effect for the standard sample 1 when measuring for α = 0 • . At the same time, this is not the case for α = 90 • . The fact that no significantly improved magnetic field response is observed for sample 2, despite its composition, is due to the relatively high range of applied mechanical strain (γ = 5%).
The opposed stress-strain hysteresis curves considered for figure 7 were obtained for selected values of α from more comprehensive results presented below.
The angular dependencies for different magnetic field strengths were investigated in angular steps of ∆α = 5 • for both samples in more than 1000 individual measurements   The moduli of both MR composite samples under examination increase with increasing magnetic field strength and decrease with increasing angle of deviation of the magnetic field direction from the shear direction (figure 8). In the field range investigated the storage moduli G ′ (H) either correspond linearly to H or by the power of two, which would be H 2 . For illustration, both possible fit functions and the experimental results obtained at several α are given in figure 9. For a field angle of α = 90 • a linear fit describes the material behaviour sufficiently, while for decreasing field angles up to α = 0 • a quadratic fit correlates in a better way for the field range investigated. It has to be noted that for higher fields above approximately 80 kA m −1 a higher deviation resulted for both fitting functions.
The loss modulus G ′′ of sample 2, which has a higher concentration of magnetic filler, is initially higher (at H = 0) and increases with increasing field to a lesser extent compared to the loss modulus of the conventional sample 1 (compare to figure 8(b)).
The field angle dependence of G ′ and G ′′ for H 120 = 120 kA m −1 are illustrated in figure 10. The decrease in storage modulus G ′ with increasing angle α is almost the same for both samples. At the same time, the loss modulus of sample 2 is less sensitive to variation in angle α, particularly in the range of 0 • -60 • . A slight plateau can be identified for the upper field angles and a slight overshoot for low field angles, which both is understood to be an issue of misalignment of the two separate electromagnets used for the four independent shear gaps of the MFATD. Due to the misalignment the effective field angle for two of the shear gaps is lower (e.g. 83 • ), while for the two other shear gaps the field angle is precisely oriented (e.g. 85 • ). This slight misalignment has a higher influence for higher gradients of property changes and therefore is particularly noticeable for angles close to 0 • and 90 • . Results obtained for mean loss factor presented in figure 8(c) certainly correspond to the dependencies G ′ (H, α) and G ′ ′ (H, α) described above and are in accordance with the known relation tan(δ) = G ′ ′ /G ′ .
Before turning to a discussion about the obtained field angle dependencies of the investigated samples, results for the magneto-deformational effect should be considered.

Magneto-deformation
It is well known that one of the characteristic features of MR elastomers is the magneto-deformational effect [15]. The effect can be quantified both by measuring changes in the geometric parameters of the sample or by estimating the macroscopic mechanical stresses generated by the sample in an external magnetic field. We used the second approach, i.e. estimating the mechanical stress σ generated by the MR elastomer specimens in the direction of the applied magnetic field.
The measurement results for the stresses σ induced by both types of samples in an external field H are shown in figure 11.
The measured stresses correspond to the elongation of the samples in the direction of the applied field. The dependence is as expected for the filled composites due to short ranged order effects by spatial dispositioning of filler particles. This has been predicted theoretically [16] and was observed in many experimental studies before, e.g. [15,[17][18][19][20][21]. For the sample containing a fraction of larger microparticles (sample 2), the stresses are higher than for the sample with the standard filler (sample 1). This is to be explained with different magnetic interactions in the specimens with a different compositions. In general, for a fixed sample, the macroscopic stress induced in a material by an applied magnetic field is proportional to the change of the magnetic free energy and consequently is proportional to the product of magnetic susceptibility and the square of the field strength, i.e. σ ∼ χH 2 [22,23]. Thus, stress σ is greater in a material with a greater susceptibility χ. This is true for sample 2 as can been seen comparing results given in figures 2 and 11).
The presented results show that significant stresses are generated in a fixed sample in the direction of the applied magnetic field. The absolute values of σ are of comparable magnitude to shear stress τ (see figures 7 and 11), though a direct comparison is not valid from a mechanical point of view. Thus, obviously, the stress σ should have an effect on the shear test results. It should be mentioned as well, that the shape of the specimens used for shear test (rectangular plates) differs from the one of the specimens for magneto-deformation experiments (cylindrical discs) (see figure 3). On the other hand, the samples are approximately equal in thickness. Thus, the geometric demagnetizing factor and therefore the effective magnetic field influencing the sample in these experiments are approximately the same for α = 90 • . For other values of α the effective magnetic field may be different in these experiments and can have a quantitative influence on magneto-deformation.

Discussion
To discuss the field angle dependencies, figure 12 shows results for G ′ for both types of the MR composites at three selected fields strengths. Values of shear storage modulus G ′ can obviously be considered constant for H = 0, wherefore MR effect has the same field angle dependence as the G ′ .
Effective field strengths due to angular dependent demagnetisation factors are not considered within the presented experimental work, meaning that all experiments were performed for the same steps of driving currents for the electromagnets and therefore for the same steps of external magnetic field strengths applied. Though, the influence of demagnetization factors will be discussed at the end of this section. Figure 13 illustrates the challenge for determining demagnetization factors D for field angles 0 • < α < 90 • using the method introduced by Osborn in [25] by assuming an ellipsoidal geometry with comparable aspect ratios, as by this method the demagnetization factors can only be calculated for the main axes of the ellipsoid. Though, it has to be pointed out that there are other methods for different shapes which could be used, e.g. for rectangular shapes by   Aharoni [26]. Therefore the minor errors for e.g. α = 90 • increase accordingly with geometric deviations from an ideal ellipsoid. Shear storage modulus decreases with increasing deviations of the direction of the magnetic field from the mechanical loading direction. The qualitative course of the curves at H ̸ = 0 is comparable for both samples. The underlying principle of function suggested by the authors is illustrated in figure 14, presuming as main reason for the angular dependence of the experimentally determined moduli an angular influence of magneto-deformation on the overall stress condition and therefore on the forces measured. Accordingly, it is assumed that the force F is a combination of a reaction force due to shear deformation of the sample F γ and the y component of a force due to magnetodeformation F md_y . Therefore the force measured is F = F γ + F md_y , where the angular dependent y component of the magnetodeformational force F md_y (α) can be calculated by F md_y = F md cos α. Following our suggestion no magnetodeformational influences are measured for 90 • , as magnetodeformation occurs perpendicular to the loading direction. Though it has to be kept in mind that the superimposed normal stress caused by magnetodeformation could effect shear results as previously reported for different normal force preloadings [24]. The field angle dependence for both MR elastomer samples tested follows in good agreement a cosine function, though for values of α close to 0 • and 90 • the deviation is higher. This could be interpreted as an issue of misalignment of both electromagnets of the MFATD to each other. Accordingly results for one angle would be an average of two slightly deviating angles. For higher gradients of change in moduli (e.g. close to α = 90 • ) misalignments would have a higher influence than elsewhere, as can be seen for G ′ and G ′′ for angles α ⩾ 60 • . This misalignment could be as well an explanation for angles α ⩽ 20 • and the shift of the expected maximum at 0 • to approximately   To check whether the previous argumentation interpreting magnetodeformational forces as the main reason for the field angle dependent behaviour is tenable, we estimate the resulting maximum shear stresses caused by magnetodeformation in our shear experiments. Therefore, assuming magnetodeformational forces as volume force f md in the direction of the magnetic field and taking into account the thickness of the cylindrical disc for magnetodeformational experiments t md as well as volume of the rectangular shear specimens V shear and shear plate area A shear , the discussed effects of magnetodeformation are calculated from the experimentally determined magnetodeformational stress σ and plotted in figure 15 for the normal force direction (z direction) as well as the resulting shear stresses for the shear direction (y direction).
The values given for magnetodeformational shear stresses are to be understood as rough estimates as mechanical constraints were simplified with resulting forces in y direction being halved on both shear plates (assuming both as fixed to estimate resulting reaction forces), which are calculated as follows.
Therefore maximal magnetodeformational stress occurs at α = 0 • (cos 0 • = 1). For sample 2 magnetodeformational stress ∆τ md is approximately 1.3 kPa at H = 120 kA m −1 .  Demagnetization factor D illustrated for three different magnetic field angle α (with the magnetic field direction ⃗ H fixed to vertical direction for ease of understanding). For the extrema at α = 0 • and α = 90 • values of D can be calculated by aspect ratios with minor errors assuming an ideal ellipsoidal geometry [25]-using this procedure for all other field angles the errors would increase according to the increasing geometric deviations from an ideal ellipsoid.
As forces induced by magnetodeformation in y direction are not affected regarding of the mechanical loading in the cyclic experiments being tension or compression forces by magnetodeformation would just cause a shift of load level and asymmetrical hysteresis curves, however not a change in storage modulus.
Thus, as discussed magnetodeformation may affect results due to superpositioning of a compressive load additionally to the shear load, though these influences are not a decisive factor for the field angle dependent changes in storage modulus experimentally measured. The authors are aware that a realistic consideration of residual stresses and pre-stresses would be much more complex, however an experimental separation of the individual effects is highly complex and an independent field of research, which is why the strongly simplified approach presented here is used. A holistic consideration of the internal stresses should be addressed in depth elsewhere.
The angular dependencies measured are qualitatively diametrically different compared to those reported for MR fluids for which the MR effect in contrast increases with increasing field angles from 0 • to 90 • [10,11].
It is well known that the effective field inside a sample is affected by the magnetic poles distribution resulting from the externally applied field. It is widely accepted that this distribution generates a so-called demagnetizing field having opposite sense with respect to the overall magnetization, which for isotropic samples is directed along the polarizing field [25]. As an analytical determination of effective field strengths or demagnetization factors is not trivial. Therefore finite element analysis (FEA) were performed with ANSYS magnetostatic without user subroutines, taking an inside view on the rectangular shear specimens. The mesh was generated with tetrahedron and brick elements by adaptive meshing with a minimal element edge length of 0.25 mm. Demagnetization factor D was derived from the FEA results. The internal effective field was read from the result table for all elements of the specimen volume and averaged. The external field strength was taken in the same manor for the elements in between the shear gaps corresponding to the sensor position in the experiments. The results plotted in figure 16 show a comparable angle dependence as found within our experimental work. These simulations were performed for constant coil currents of the electromagnets-with the currents being proportional to the resulting external field strengths. To ensure comparability of the shear results between different field angles all experiments within this work were performed at identical current levels for all angles.
To identify influence of a constant effective magnetic field strength within test samples on the field dependent storage modulus an iterative optimization study was performed utilizing ANSYS magnetostatic. Therefore the currents were varied iteratively for all angles heading for a constant average effective field strength for all elements of the sample and not exceeding the maximum realizable current I max = 14.8 A. With this limitation in maximum current a constant effective field strength of 27.1 ± 0.2 kA m −1 can be realized for all angles by adapting current for each angle. With the currents determined iteratively via simulation I var_sim corresponding experimental data was identified-choosing results for experimental currents closest to the theoretical values determined. These are plotted in figure 17 all-together with the corresponding storage moduli of both samples.
As can be seen, for the unstructured MR elastomers investigated within this study, nearly no field angle influence occurs when taking into account effective field strength. Values for storage modulus at 90 • still deviate and are approximately 50% higher. This could be due to imprecision in alignment of electromagnets or inaccuracies in simulation of effective field strength.

Summary and outlook
Within this study field angle dependent shear measurements were performed, where the field angle α is defined as the angle between the shear direction and the external magnetic field direction. Finely graduated field angle dependent measurements are presented by us for various field strengths for the first time and give a comprehensive experimental description of the magneto-mechanical behaviour of unstructured MR elastomers as presented for G ′ , G ′′ and tan δ in figure 8(c). It has to be highlighted, that the field angle dependent MR effects measured in the shear tests performed by us for MR elastomers drastically differ from the effects known for MR suspensions. As it is known from previous studies, for MR suspensions, the greatest MR effect occurs when the external magnetic field is applied perpendicular to the mechanical load and is minimal when the directions of shear and field are parallel. In this study we experimentally showed that for MR elastomers the situation is the opposite for constant external field strengths. The maximum effect is observed when the shear direction and the field are nearly the same. When the field direction deviates from the shear direction, the effect decreases and is minimal in the situation where the field is nearly perpendicular to the mechanical loading. This is confirmed for samples with different filler. It could be shown that angular dependent influences are if only subordinately related to magneto-deformational effects as these forces are too low and additionally would cause asymmetric hysteresis curves. Though, the field angle dependent behaviour is in good accordance with effects by field angle dependent change of the demagnetization factor and thus a change in the internal effective magnetic field of the shear specimens (see figure 16). Demagnetization factors were determined in magnetostatic simulations for the shear sample geometry. To investigate the effect of a constant effective field strength inside the shear samples for different field angles further magnetostatic simulations were carried out by iteratively adapting coil currents until a constant effective field strength for all field angles was achieved. For the corresponding current steps closest to the simulation results for a constant effective field strength the experimental data was then compared and showed no significant field angle effect for the unstructured MR elastomers used within this study. Solely the storage moduli measured at 90 • are significantly deviating when taking into account the effective magnetic field. As the maximum currents are limited for the existing device further investigations are needed at higher effective field strengths to fully clarify this and for more precisely adapted currents. To do so a much more powerful MFATD device would be needed.
This study uncovers the need of in-depth theoretical approaches on the influence of arbitrary field angles on the behaviour of MR elastomers. From a scientific point of view further research focused on MR elastomers with microstructural anisotropy can contribute a lot to an understanding of the underlying effects. From a practical perspective, with the dependencies found in this study for different field angles the basis for the development of a more precise material modelling is layed. Results obtained and discussed enable a better estimation, preliminary design and efficient optimization of magneto-active assemblies.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors.