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Equi-biaxial tension tests on magneto-rheological elastomers

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2016 Smart Mater. Struct. 25 015015

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Equi-biaxial tension tests on magneto-rheological elastomers

Gerlind Schubert and Philip Harrison

School of Engineering, University of Glasgow, University Avenue, Glasgow G12 8QQ, UK

E-mail: Philip.Harrison@glasgow.ac.uk

Received 9 August 2015, revised 30 September 2015Accepted for publication 19 October 2015Published 30 November 2015

AbstractA bespoke test rig has been designed to facilitate testing of magneto-rheological (MR)elastomers (MREs) under equi-biaxial tension using a standard universal test machine. Testswere performed up to 10% strain on both isotropic and anisotropic MREs with and without theapplication of an external magnetic field. Assumptions regarding the material’s response wereused to analyse stress–strain results in the two stretching directions. The assumptions have beenverified previously by uniaxial tension tests and by simulations of the magnetic flux distributionperformed using a commercial multi-physics finite element software. The MR effect, which isdefined as the increase in tangent modulus at a given strain, has been studied versus engineeringstrain. The latter was measured optically in the experiments using a digital image correlationsystem. Relative MR effects up to 74% were found when the particle alignment of anisotropicMREs was oriented parallel to an applied magnetic induction of just 67.5 mT.

Keywords: magneto-rheological elastomers, large strain, magneto-rheological effect, digitalimage correlation, equi-biaxial tension, magnetic field distribution

(Some figures may appear in colour only in the online journal)

1. Introduction

Magneto-rheological elastomers (MREs) are smart materialswhich can alter some of their properties reversibly and almostinstantaneously by the application of external magnetic fields.This behaviour is caused by the interaction of micron-sizedmagnetic particles dispersed in an elastomeric material. InMREs the magnetic particles are locked in position in the finalproduct. Isotropic and anisotropic materials can be prepared,the latter by exposing the MRE mixture to a magnetic fieldwhile curing; this forces the magnetic particles to align inchains, which results in a strong mechanical and magneticanisotropy [1]. The dynamic small strain behaviour of MREsis well explored [1–3]; different types of matrix materials andmagnetic particles have been studied. Recently, the magne-tostriction [4], and the magnetic properties of MREs [5] havealso been investigated. Nevertheless, there is still a distinct

lack of knowledge concerning the constitutive response ofthese smart materials when subject to large strains. Whileuniaxial compression and tension, and simple shear tests havebeen performed to large strains [6–9], the variety of materialsused in these experiments makes it difficult to compareresults. In order to develop constitutive models able toaccurately describe the complex behaviour of MREs com-prehensive datasets are required. This involves testing both inthe absence of and in the presence of magnetic fields, undervarious deformation modes, including multi-axial deforma-tions, all on the same type of material [10, 11]. Fitting theparameters of complex constitutive models can be a chal-lenging task and the challenge increases with the complexityof the constitutive model due to the increase of independentparameters within the model. For example, constitutiveequations for incompressible isotropic materials depend onjust two invariants [12]. Models for incompressible aniso-tropic materials depend on four invariants [13]. For incom-pressible magneto-sensitive anisotropic materials theconstitutive model can depend on as many as nine invariants[14]. The number of independent parameters increases withthe number of invariants; consequently, in order to uniquely

Smart Materials and Structures

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determine all the model parameters for such complex materialbehaviour, multiple independent experiments, each inducingdifferent deformation modes, must be conducted. Thisrequirement for multiple independent tests provides themotivation for the current investigation.

While Zhou [15] used the bubble-inflation method tostudy the fatigue behaviour of MREs in the absence ofmagnetic fields, so far the behaviour of MREs subject tomulti-axial deformations in the presence of applied magneticfields has yet to be studied. In this research, the response ofMREs has been studied under equi-biaxial tension up to 10%strain. A bespoke test rig was designed to facilitate testingusing a standard universal test machine. Both isotropic andanisotropic MREs with 10%, 20%, and 30% volume particleconcentration were studied in the absence and in the presenceof a 67.5 mT magnetic field applied parallel and perpendicularto the direction of particle alignment in anisotropic MREs.Full-field strains were measured optically across the samplesusing a digital image correlation (DIC) system, and themagnetic field distribution was simulated using the commer-cial finite element software Comsol. Analysis of the principalstresses in the experiments is based on assumptions verifiedpreviously by uniaxial tension tests on the same type ofmaterial [16] and by the calculated distribution of the mag-netic field strength [17]. These assumptions are improvedcompared to those used in an earlier published conferencepaper [18]. The stress–strain results and the magneto-rheo-logical (MR) effect, defined as the increase in tangent modulidue to an applied field, are both reported. Together with large-strain compression, tension and pure shear tests performed onthe same type of material [16], a set of experimental data ispresented that characterizes MREs under several deformationmodes up to considerably large strain. This is an essentialbasis for the development of accurate constitutive modelsfor MREs.

2. Materials

Silicone rubber MM 240TV mixed with 30 w% silicone oilACC 34, both purchased from the company ACC Silicones

were used as the elastomeric matrix material. Carbonyl ironparticles provided by the company BASF were used as themagnetic particles; the type SQ was chosen with an averageparticle size from 3.7 to 4.7 μm. Isotropic and anisotropicMREs with volume particle concentrations of 10%, 20% and30% were prepared, together with specimens made simply ofthe pure matrix material, i.e. 0% particle volume concentration.The manufacturing process of a commercially available sili-cone rubber is straightforward; all the components were mixedthoroughly for three minutes with a hand mixer, and themixture was degassed in a vacuum chamber for 10min, bothbefore and again after the mixture was poured in the moulds.The MREs were cured for 1.5 hours at 100 °C. To prepareanisotropic MREs the mixture inside the moulds was exposedto a 400 mT magnetic field during the curing process. Samplesheets with dimensions 50×50×2 mm were manufactured.The moulds used to prepare these samples were made of alu-minium and brass to avoid any unwanted magnetization. Themould and the manufacturing process are illustrated in figure 1.

3. Test setup and procedure

Tests were performed on isotropic and anisotropic MREs with10%, 20% and 30% iron content. A special test rig wasdesigned to enable equi-biaxial testing using an universal testmachine (Zwick Z250), which measured vertical force anddisplacement. Figure 2 shows an isotropic MRE specimenclamped in the rig. The rig was designed in accordance withthe British Standard [11] and consisted of upper and lowerframes, each attached to the test machine. The two frameswere not connected and did not contact one another during thetest. Test specimens were held in the rig using three slidingclamps on each side of the specimen; the clamps were free tomove along the side length of the frame as the test proceeded,ensuring an almost uniform biaxial stretch of the MRE sam-ples. To reduce friction and to avoid interactions between thetest rig and the magnetic field, the rig and the sliding clampswere made of polytetrafluoroethylene (Teflon). The clampsfor holding the rubber were made of aluminium with brassscrews. The test rig did not allow strains larger than 10% due

Figure 1. The moulds to manufacture samples for equi-biaxial tension tests. (a) Completely open, (b) ready for the pouring process, (c) withthe MRE mixture inside, and (d) between the heater plates and in between the poles of the electromagnet during the curing process.

2

Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

to the clamping system and the soft materials used. 175 mmlong aluminium connecting rods attaching the biaxial test rigto the Zwick test machine ensured sufficient distance betweenthe permanent magnets, the load-cell and the steel parts of thetest machine to ensure again any undesirable effects (such asfalse force readings). Four strong permanent neodymium N52magnets, two placed on either side of the test rig, were usedto create an average magnetic induction of 67.5 mT in themain direction and 7.1 mT perpendicular to this direction.The magnetic field was calculated using finite element ana-lysis in the absence of an MRE sample (the field strength isexpected to change once the sample is introduced in thesetup). The inter-magnet distance of 140 mm was kept con-stant during all experiments. The permanent magnets and thedirection of the particle alignment in an anisotropic MREspecimen are shown in figure 3. A naming convention is usedwhen describing the different types of experiment. Case 1 isillustrated in figure 2 and case 2 to case 5 are illustrated infigure 3.

Figure 2. The biaxial test rig attached to the uniaxial test machine(Zwick Z250). An isotropic MRE in the absence of amagnetic field is illustrated (case 1). The vertical recorded load, PV,the displacement, d, and the coordinate system are indicated.

Figure 3. Biaxial test setup with (a) an isotropic MRE with applied magnetic induction in the x-direction (case 2), (b) an anisotropic MREwith particle alignment in the y-direction without magnetic field (case 3), and with magnetic field (c) in the y-direction parallel to the particlealignment (case 4), and (d) in the x-direction perpendicular to the particle alignment (case 5).

3

Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

The tests were conducted at a test speed of 10 mmmin−1

with up to 10mm vertical displacements resulting in approxi-mately 10% strain in the stretching directions. AnisotropicMREs with 30% iron particles were only stretched up to 7%strain due to tearing of samples at higher strains. MREs werefirst tested without an applied magnetic field, followed by testswith 67.5 mT in the main magnetic field direction. All tests were4-cycle tests and the third loading cycle was used to analyse andinterpret the stress–strain results. A DIC system (Limess) wasused to measure full-field engineering strains within the MREsamples during the experiments. This measurement ensured thatany additional strains that might be present in the test rig areexcluded from the presented stress–strain results.

4. Distribution of the magnetic field strength

The magnetic flux distributions within the volume occupiedby the MRE sample were simulated with the multi-physicssoftware Comsol (AC/DC package) [19]. The model geometryis shown in figure 4; symmetry conditions meant that onlyone quarter of the setup was required in the model. The testrig itself was made of non-magnetic materials and had noinfluence on the magnetic flux calculations, and was thus notmodelled. Average dimensions measured during the experi-ments on isotropic and anisotropic MREs were used in themodel. Further information about the finite element model canbe found in [17]. The simulation results are in excellentagreement with magnetic field strength measurements taken atvarious positions in the setup using a Gaussmeter (F.W. BellModel 5180). Contrary to the actual experimental setup (seefigure 3) the main magnetic field direction in the Comsolmodel was always the y-direction and the particle alignmentdirection of anisotropic MREs was rotated. This was done tosimplify the modelling procedure.

The magnetic flux density in the x- and y-directions hasto be considered as the field strength in each direction influ-ences the stiffness of the MRE sample in each of thesedirections [6]. The ratio of the magnetic field strength in thetwo orthogonal directions is used in analysing the results ofthe test and so some care has to be taken here to understandthe field distribution. Since the biaxial test samples are square,a significant component of the magnetic flux exists perpen-dicular to the main magnetic flux direction. The distributionof the magnetic flux density, BY and BX, within the volumeusually occupied by an MRE sample (but calculated in theabsence of a sample, i.e. with μr = 1) is shown as a contourplot in figure 5. The average value of BY taken over thevolume of the MRE sample is 67.5 mT, and the flux density inthe y-direction differs across this volume by about 90.4%. Incontrast, the average value of BX is only 7.1 mT, but under-goes a much greater variation in strength of up to 357.2%.The changes in the horizontal flux density, BX, are thereforefar larger than those in the vertical direction.

The magnetic flux distribution across the volume occu-pied by the MRE sample was also calculated in the presenceof the samples ( 1rm > ) by changing the material properties ofthe MRE samples in the finite element model. The magneticpermeability was defined as isotropic or anisotropic andvalues determined earlier [17] were used; those values are

Figure 4. Geometry implemented in Comsol to calculate thedistribution of the magnetic flux density in the biaxial test setup.Two magnets are placed on either side of the test rig with an inter-magnet distance of 140 mm in the y-direction.

Figure 5.Magnetic flux density, BY and BX, within the volume of theMRE sample (calculated in the absence of a MRE sample withμr = 1) with BY ,mean = 67.5 mT and B 7.1 mT.X,mean =

Table 1. Relative permeabilities for isotropic MREs, ,isom and foranisotropic MREs in the particle alignment direction, ,m and

perpendicular to the alignment direction, .m̂

isom m m̂

Isotropic 10% MREs 1.60Isotropic 20% MREs 2.20Isotropic 30% MREs 3.70Anisotropic 10% MREs 1.60 1.50Anisotropic 20% MREs 2.70 2.30Anisotropic 30% MREs 4.45 3.55

4

Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

listed in table 1. Results of the finite element calculations foreach type of MRE are summarized in table 2. The averagemagnetic flux densities and the differences in the flux acrossthe volume occupied by the MRE sample are provided as wellas the ratio between B BY = in the main magnetic fielddirection and B BX = ^ perpendicular to it. The terms BP andB⊥ are introduced to avoid confusion in section 6. The factorB B̂ is largest at 14.98 (when the magnetic field is appliedparallel to the direction of particle alignment), and is smallestat 8.59 (when the magnetic field is applied perpendicular tothe alignment direction).

5. Optical strain measurement

A DIC system (Limess) was used to measure the strains. Thesystem consists of 4 M pixel cameras, bulb lights, and the VIC3D software to perform the analysis. Test samples weresprayed with a white random speckle pattern; images of thesamples are shown in figures 2 and 3. Grid lines were alsodrawn on the samples to enable manual calculation of thestrains by measuring pixel positions using the software Ima-geJ. A series of images was recorded during the cyclic tests.The DIC software performed correlation analysis by com-paring the defined area of interest (AoI) in each image. TheAoI was defined on the MRE sample as a square area situatedin the centre. The software divides the pattern into smallerareas and follows the same areas in each image. During thetest, the speckle pattern is stretched and displaced. Bytracking the speckle pattern, the DIC software is able to cal-culate displacements and strains across the test sample. Theoutput of the DIC software are matrices containing values ofdisplacement and engineering strain in both the x andy-direction at each point in the AoI at any given time. Con-fidence values describing the match at each point are alsoprovided. The engineering strains in the x-direction of a pure

rubber sample are shown in figure 6(a). DIC data were post-processed using Matlab. The AoI was cropped, and alsorotated for better handling. Unreliable values were eliminatedusing the confidence values provided by the DIC software.The post-processed strain field is shown in figure 6(b). Thestrains in the x-direction within the post-processed AoI arerelatively uniform, differing between 8.6% and 10.2%. Meanvalues and standard deviations of the post-processed strainfield are calculated and plotted versus time in figure 7, wherethe strain in both the local x- and y-directions are shown to bein excellent agreement. Mean values of the original AoI andof the cropped strain field were compared and do agree;however the standard deviations in strains resulting from theoriginal AoI (see figure 6(a)) are larger than those shown inthe processed AoI (see figure 6(b)). The strains in the x- andy-directions are nearly identical; this implies that the framestructure is rigid enough to impose equi-biaxial deformationkinematics. This was also found to be the case even for ani-sotropic MREs with higher iron content. The load–displace-ment data recorded by the test machine are subsequentlyrelated to the strain–time data of the DIC analysis via the timerecorded by the test machine.

6. Stress calculation

Only the vertical force, PV, was recorded by the uniaxial testmachine. In order to calculate stresses in the two stretchingdirections, the structural system of the biaxial test has to beanalysed and several assumptions have to be made. The biaxialrig is assumed to be a rigid body with the top frame moving10mm in the vertical direction; the experiment was displace-ment controlled. This movement of the rig caused stresseswithin the soft rubber. The structural system is illustrated infigure 8. The rigid body assumption of the biaxial frame issupported by the optical strain measurement which confirmed

Table 2.Mean values, B mT ,mean [ ] and relative difference B B B100 % ,max min mean( ) ( )´ - of the magnetic flux distribution, B B ,Y = in themain magnetic field direction and, B B ,X = ^ perpendicular to the main direction, within the volume occupied by isotropic and anisotropicMRE samples ( 1r m ). The ratio B B̂ [-] is calculated with Bmean. Values for B̂ are calculated with absolute values.

MRE Sample

Iron%[ ]

Flux in y-direction Flux in x-directionRatioB B̂

BmeanB B

Bmax min

mean

- BmeanB B

Bmax min

mean

-

Pure rubber 0 67.53 90.4 7.09 357.2 9.52

Isotropic MREs10 103.95 80.3 10.47 328.4 9.9320 138.45 82.1 13.27 324.8 10.4330 217.74 85.3 18.61 352.1 11.69

Anisotropic MREsaligned in y-direction

10 104.53 77.2 9.9 289.3 10.5820 167.20 80.6 13.35 296.6 12.5230 257.03 83.6 17.16 306.1 14.98

Anisotropic MREsaligned in x-direction

10 97.62 83.0 11.16 329.0 8.7520 143.10 85.7 16.66 338.6 8.5930 208.12 87.9 23.20 358.7 8.97

5

Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

equal strains in both stretching directions. Rotation of thebiaxial frame was prevented by its connection with the testmachine. The forces, PV and PH, and the moment, M, weregenerated at the fixed support but only the vertical force, PV,was recorded by the test machine. The stresses are assumed tobe equally distributed across the length of the three clamps wholding the side of the MRE samples. The distance a is thedistance between the fixed support and the clamps. The stressesare calculated in the reference configuration. To derive theequations for the stresses, σX and ,Ys five cases are dis-tinguished (case 1 to case 5 as illustrated in figures 2 and 3).The analysis of each case is presented in the following sections.

6.1. Case 1—isotropic MREs without magnetic field

Isotropic MREs are supposed to have equal properties in alldirections, which implies that the stresses Xs and Ys are

identical. Figure 2 shows the setup of case 1. From equili-brium of stresses in the x- and y-directions, the stresses can becalculated as,

P

t w21X Y

V

· ·( )s s= =

with the vertical force, PV, the thickness of the sample, t, andthe width of the clamps, w. As the horizontal force PH and themoment M at the fixed support are zero, no assumptions arerequired in analysing case 1.

6.2. Case 2—isotropic MREs with magnetic induction in thex-direction

The magnetic field is assumed to change the stiffness of theMRE in the x- and y-directions by the same ratio as thatbetween the average magnetic induction in the two directionsB B̂ (see table 2). The setup of case 2 is shown infigure 3(a). When a magnetic induction is applied on the

Figure 6. Maximum strain in the x-direction in the third loading part of a pure rubber sample calculated by the DIC software. (a) Originalstrain field as calculated by the DIC software and (b) rotated strain field, after unreliable values were eliminated and borders cropped.

Figure 7. Engineering strain of a pure rubber sample in bothstretching directions versus time. Mean values and standarddeviations of the post-processed DIC field of the four-cycle test areillustrated.

Figure 8. Structural system of the biaxial test setup used to calculatestresses within the MRE sample.

6

Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

isotropic MREs, the force measured in the vertical direction,PV, is larger than the force measured on isotropic MREswithout an applied induction P .V,CASE1 The increase,P P P ,inc V V,CASE1= - is attributed to increases in the forcegenerated in the two stretching directions. The relationbetween the stresses x,incs and y,incs due to Pinc is determinedby the factor B B̂ with BP acting in the x-direction:

B

B. 2x

y

,inc

,inc( )s

s=

^

Equation (2) is a significant assumption, and other morecomplex non-linear descriptions, such as x y,inc ,incs s =B B( )a^ could be more appropriate. Here α is a parameterthat should be determined through more experiments. Never-theless, in order to simplify the subsequent analysis and alsobecause of the time constraints governing this investigation,the assumption of equation (2) is currently a necessaryconcession. Attention is drawn to this point as a possiblesource of error in the method and work to clarify the accuracyof this assumption is deferred to the future. For example,uniaxial tension tests with the particle alignment in theloading direction but with a magnetic field perpendicular tothe loading direction could be conducted to examine theaccuracy of equation (2). With the relation of equation (2) andthe equilibrium of forces in the vertical direction the stressdefinitions can be derived as,

P

t w

P P

B

Bt w

2

2

1

, 3

X x x,CASE1 ,incV,CASE1

V V,CASE1( )· ·

· ·( )⎛

⎝⎜⎞⎠⎟

s s s= + =

+-

+^

P

t w

P P

B

Bt w

2

2

1

. 4

Y y y,CASE1 ,incV,CASE1

V V,CASE1( )· ·

· ·( )⎛

⎝⎜⎞⎠⎟

s s s= + =

+-

+^

The value PV,CASE1 is the mean value of forces measuredon isotropic MREs of the same iron particle volume fractiontested without an applied field. Note, the horizontal force PH

and the moment M are not zero in this and in all followingcases.

6.3. Case 3—anisotropic MREs without magnetic field

The particle alignment direction of the anisotropic MREs wasoriented in the y-direction for all tests. The setup of case 3 isillustrated in figure 3(b). These MREs are much stiffer in theparticle alignment direction, and the stresses in the y-directionare therefore higher than in the x-direction, i.e. .Y Xs s> Asonly the vertical force, PV, was measured by the test machine,an assumption is required in order to attribute the verticalforce to the two stretching directions. In uniaxial tension tests[16], measurements on anisotropic MREs with the particle

alignment in the loading direction (A) and perpendicular to it(AP) were performed. The relative stress factor betweenstresses of anisotropic samples with alignment in the loadingdirection, ,As and perpendicular to the loading direction, ,APsis used to evaluate the relationship between Xs and Ys in theequi-biaxial tension tests. In the case of biaxial tension testboth directions are the loading directions, thus in they-direction the particle alignment is parallel and in thex-direction the alignment is perpendicular to the load:

f . 5Y

X

A

AP( ) ( )

( )( )e

s es e

ss

= =

Mean values from at least three repeated uniaxial tension testswere taken to calculate the relative stress factors. These areplotted versus uniaxial strain in figure 9. With the stress factordefined in equation (5), combined with the equilibriumcondition, the stresses in both stretching directions can becalculated as,

P

f t w

2

1, 6X

V·( ( ) ) · ·

( )se

=+

P f

f t w

2

1. 7Y

V· · ( )( ( ) ) · ·

( )se

e=

+

This method of using the relative stress factor obtained fromuniaxial tension tests provides a first approximate analysis ofthe biaxial test data. It might be possible in future to improvethe analysis method by using results of constitutive model-ling. For now, however, this is the only information availableto describe the behaviour of anisotropic MREs.

Figure 9. Relative stress factor f ( )e between anisotropic MREs withalignment in (A) and perpendicular to (AP) the loading directiontested in uniaxial tension are shown. The stress factor is plottedversus the engineering strain in the uniaxial stretching direction.Note that the stress factor tends to infinity for small strains, so valuesbelow 2% strain are unreliable.

7

Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

6.4. Case 4—anisotropic MREs with magnetic induction in they-direction

Anisotropic MREs with particle alignment in the y-directionwere tested with a magnetic induction also applied in they-direction, thus the direction of the particle alignment wasparallel to the main magnetic field direction; the setup isshown in figure 3(c). Two assumptions are required to analysethe test results of case 4.

(i) The force PV increases due to the applied magneticinduction by P ;V,inc thus P P P .V V,CASE3 V,inc= + Theincreased force, P ,V,inc can be attributed to the twostretching directions in the same way as in case 2. Theonly difference here is that the magnetic field is nowapplied in the y-direction rather than in the x-direction.The relation between the magnetic factor B B̂ and thestresses Xs and Ys of the biaxial tests are thus defined asfollows:

B

B. 8

y

x

,inc

,inc( )

ss

=^

(ii) The stresses resulting from PV,CASE3 (equations (6) and(7)) were calculated with a stress factor f ( )e obtainedfrom uniaxial tension tests.

The resulting stresses Xs and Ys are defined inequations (9) and (10). As both the particle alignment and theapplied magnetic field are in the y-direction, the stresses inthis direction are expected to be much higher than the stressesin the x-direction, i.e. ,Y Xs s

P

f t w

P P

t w

2

1

2

1, 9

X x x

B

B

,CASE3 ,incV,CASE3

V V,CASE3

( )( )

·( ( ) ) · ·

· ·( )

s s se

= + =+

+-

+^

P f

f t w

P P

t w

2

1

2

1. 10

Y y y

B

B

,CASE3 ,incV,CASE3

V V,CASE3

( )( )

· · ( )( ( ) ) · ·

· ·( )

s s se

e= + =

+

+-

+^

6.5. Case 5—anisotropic MREs with magnetic induction in thex-direction

Case 5 is very similar to case 4, with the only difference thatthe magnetic induction is applied in the x-direction rather thanin the y-direction. The setup of case 5 is shown in figure 3(d).The assumptions and the derived equations are analogous to

those explained in case 4:

P

f t w

P P

t w

2

1

2

1, 11

X x x

B

B

,CASE3 ,incV,CASE3

V V,CASE3

( )( )

·( ( ) ) · ·

· ·( )

∣∣

s s se

= + =+

+-

+^

P f

f t w

P P

t w

2

1

2

1. 12

Y y y

B

B

,CASE3 ,incV,CASE3

V V,CASE3

( )( )

· · ( )( ( ) ) · ·

· ·( )

∣∣

s s se

e= + =

+

+-

+^

Again, two assumptions are used to calculate the stresses inthe two stretching directions: the uniaxial stress factor(equation (5)) and the magnetic factor (equation (2)).

The assumptions used here are improved compared to anearlier published conference paper [18]. In particular, thestress factor obtained from uniaxial tension tests performed in[16] and used here to interpret the experimental data, is afunction of strain rather than of displacement. Further, themagnetic field assumption is based on a factor between theaverage levels of magnetic induction in the two stretchingdirections instead of assuming changes in material behaviouronly in the direction of the magnetic field. These modifica-tions provide improved accuracy and reliability when inter-preting the results.

7. Stress–strain results and MR effect

The experimental results from tests on isotropic and aniso-tropic MREs conducted in the absence of a magnetic field(case 1 and case 3), and those conducted in the presence ofmagnetic field (case 2, case 4 and case 5) are reported in thissection.

The load–displacement data recorded by the uniaxial testmachine are converted to stress–strain data in the twostretching directions (X and Y) using the optically measuredstrains described section 5 and the stress equations derived insection 6. First, the MRE behaviour in the absence of amagnetic field is discussed. The effect of changing the ironparticle volume concentration is considered and the differ-ences between the isotropic and anisotropic MREs isascertained.

Second, the MR effect is studied by comparing stress–strain results of tests conducted in the absence of magneticfields with those conducted in the presence of magnetic fields.The increase in the tangent moduli versus large-strain isreported for various applied magnetic field strengths. Abso-lute and relative MR effects are defined as

E EMR 13Mabs 0 ( )= -

E EMR 14Mrel 0 ( )=

or expressed as the MR increase with E E 1 100M 0( )- ´defined as a percentage value. In these equations E0 and EM

are the tangent moduli calculated as the linear slope of the

8

Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

stress–strain curves (using 1% strain increments), from testsin the absence and in the presence of a magnetic field,respectively. Use of this small strain increment produces areasonable approximation of the first derivative of the stress–strain curves.

7.1. Results of tests in the absence of a magnetic field (cases 1and 3)

The vertical load versus displacement data for all types ofMREs tested in the absence of a magnetic field are shown infigure 10(a). The measured force increases with increasingiron content. Also, forces are larger in anisotropic MREscompared to the equivalent isotropic MREs, and this differ-ence increases with increasing iron particle volume

concentration. The associated stress–strain curves are shownin figure 10(b). In the case of anisotropic MREs theassumption discussed in section 6.3 has been applied. Thetangent moduli, E0, are plotted versus engineering strain infigure 11(a), and the maximum values between 1% and 2%strain are plotted versus the iron volume fraction infigure 11(b). The moduli increase in a non-linear manner withincreasing iron volume content, where the MREs containing30% iron particles exceed the linear level.

7.2. Characterization of the MR effect (cases 2, 4 and 5)

A magnetic field strength of 67.5 mT was applied in the mainmagnetic field direction to study the increase in stiffness ofthe MRE material in the equi-biaxial tension test series. The

Figure 10.Vertical load–displacement curve and the stress–strain curves comparing the different types of MRE samples tested in the absenceof a magnetic field. Stresses of the isotropic MREs are compared with the stresses parallel (P) and perpendicular (⊥) to the particle alignmentof anisotropic MREs.

Figure 11. Tangent moduli, E0, versus the iron volume fraction of isotropic MREs and anisotropic MREs parallel (P) and perpendicular (⊥)to the particle alignment direction. Tangent moduli between 1% and 2% strain are provided.

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Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

magnetic flux lines were not unidirectional; when interpretingexperimental data, this issue is taken into account using theanalysis described in section 6.

Pure rubber and isotropic MREs (case 2). The verticalforce–displacement and the stress–strain data of isotropicMREs both with and without an applied magnetic flux areshown in figure 12. The stress–strain data in the direction ofthe applied magnetic field (x-direction) were calculated usingequations (3) and (4). An increase in force is apparent whensamples are subjected to a magnetic flux density, but thisincrease is most significant for isotropic MREs with 20% ironcontent.

The relative MR effects (equation (14)) of pure rubberand isotropic MREs are plotted versus engineering strain infigure 13. The highest relative MR increase of about 25%(1.09MPa absolute MR effect) was achieved with an iso-tropic 20% MRE in the small-strain region. The isotropic30% MREs exhibit lower relative increases of about 21%; thisis a somewhat unexpected result but is probably due to thelarge no-field modulus of the isotropic 30% MRE specimens(see figure 11). As expected, the MR effects in the directionperpendicular to the applied induction are all close to 1.Results from pure rubber samples indicate an experimentaland analytical error of about 3%.

Anisotropic MREs (case 4 and case 5). The load–dis-placement and the stress–strain curves of anisotropic MREswith magnetic field applied parallel to the direction of particlealignment are compared with those of the NoField tests infigure 14. All of the samples show an increase in force.Stresses are calculated using equations (9) and (10). Largeincreases in stress are visible for anisotropic MREs with 20%and 30% iron content. The relative MR effects are evaluatedtogether with anisotropic MREs where the magnetic field isapplied perpendicular to the alignment direction in figure 16.

The vertical load–displacement and stress–strain data ofanisotropic MREs tested with a magnetic field applied per-pendicular to the direction of particle alignment are shown in

figure 15. Increases in the forces are small for MREs with 10%and 30% iron content and are relatively large in the case of theanisotropic 20% MREs. The stresses in the two principalstretching directions are found using equations (11) and (12),and show the same tendency as the load–displacement data.

The relative MR effects of anisotropic MREs with themagnetic field applied parallel and perpendicular to the par-ticle alignment direction are plotted versus strain in figure 16.The highest MR increase of about 74% (4.89MPa absoluteMR effect) was measured for the anisotropic MREs with 30%iron content when the magnetic field was applied parallel tothe particle alignment direction. This MR increase occurred atabout 4% strain, whereas in the small-strain region theincrease measured on the same MRE was just 27.9%. Thesteep increase in MR effect of the anisotropic 30% MREs

Figure 12. Vertical load–displacement and stress–strain curves in the x-direction of pure rubber and isotropic MRE samples with 10%–30%iron content comparing the NoField with the Magnet tests. The magnetic field, BP, was applied in the x-direction.

Figure 13. Relative MR effects of pure rubber and isotropic MREswith 10%–30% iron content are plotted versus strain. The magneticfield, BP, is applied in the x-direction.

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Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

with increasing strain is unusual and contrasts with resultsfound in uniaxial compression and tension experiments [16].Nevertheless an increase in MR effect with increasing strain isobserved in all MREs containing 30% particle content in thisinvestigation (see figures 13 and 16(b)). It is not yet clear ifthis is a real effect or if due to either experimental error orerrors due to the assumptions used in the analysis of theresults. Further testing, or perhaps numerical modelling at themicro-scale e.g. [20] would be useful in clarifying this point.The MR effects perpendicular to the main magnetic fielddirection are all close to 1.

When the magnetic field was applied perpendicular to theparticle alignment direction, the highest MR increase of 46%(2.24MPa absolute MR effect) was achieved with MREscontaining 20% iron content. The MR effects of anisotropicMREs with 10% and 20% iron content exceed the MR effects

found when the magnetic flux was applied parallel to thedirection of particle alignment. This contrasts with the resultsfound in uniaxial tension tests [16] and might be due to non-uniformity of the applied magnetic field. The results achievedin equi-biaxial tension are also influenced by the assumptionswhich were required to analyse the data, and could beimproved with the use of constitutive modelling in futureinvestigations.

MR effect versus volume particle concentration. To studythe influence of the iron content, the maximum absolute andrelative MR effects are plotted versus iron particle volumeconcentration in figure 17, and are also listed together withthe no-field moduli E0 and the field moduli EM in table 3.Only the MR effects in the main magnetic field direction arepresented. MR effects increase with increasing iron content inthe MRE samples, but the results of isotropic and anisotropic

Figure 14.Vertical load–displacement and stress–strain curves in the y-direction of anisotropic MREs with 10%–30% iron content comparingthe NoField with the Magnet tests. Both the particle alignment and the applied magnetic induction, BP, were in the y-direction.

Figure 15.Vertical load–displacement and stress–strain curves in the x-direction of anisotropic MREs with 10%–30% iron content comparingthe NoField with theMagnet tests. The particles in the MREs were aligned in the y-direction while the magnetic induction, BP, was applied inthe x-direction.

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Smart Mater. Struct. 25 (2016) 015015 G Schubert and P Harrison

MREs containing 30% iron content are somewhat unex-pected, as discussed earlier.

8. Summary and conclusions

Equi-biaxial tension tests were performed on MREs. A spe-cial test rig was designed to enable testing on a universal testmachine. Two permanent magnets on each side of the test rigcreated a magnetic flux density of 67.5 mT in the mainmagnetic field direction. Tests on isotropic and anisotropicMREs with 10%–30% iron particle content were performedwith the magnetic field aligned both parallel and perpendi-cular to the particle alignment direction. A DIC system wasused to measure full-field strains. To calculate stresses in the

two stretching directions, several assumptions were employedin the data analysis:

(i) Frictionless clamps; forces along the frame axesare zero.

(ii) The magnetic field changes the properties of thematerial parallel and perpendicular to the magneticfield direction by the same ratio as the ratio of meanmagnetic field strengths in the two directions, B B .^

(iii) For anisotropic MRE samples the relation betweenstresses parallel and perpendicular to the particlealignment is the same as a relative stress factor f ( )etaken from uniaxial tension tests [16].

High MR increases of about 74% were found in aniso-tropic MREs containing 30% iron content when the particle

Figure 17. Maximum absolute and relative MR effects in the direction of the applied magnetic induction of all types of MREs versus thevolume particle concentration.

Figure 16. Relative MR effects of anisotropic MREs with 10%–30% iron content are plotted versus strain. The magnetic field, BP, is applied(a) in the y-direction (parallel to the direction of particle alignment), and (b) in the x-direction (perpendicular to the direction of particlealignment).

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alignment was oriented along the same direction as themagnetic field of 67.5 mT. To put these results into per-spective, previous tests reported by the authors on the samematerial under uniaxial compression revealed a 111% MRincrease under the influence of a magnetic field of 450 mT,and even higher increases of 284% in uniaxial tensile testswere measured when the MRE was under the influence of amagnetic field of 289 mT. In both tests samples were strainedup to 50% [16]. The equi-biaxial tension tests are an impor-tant part of a comprehensive dataset [16] and are required inorder to fully characterize the complex behaviour of MREs tofacilitate the future development and evaluation of accurateconstitutive models [10]. However, several improvements tothe test procedure reported in this paper can be suggested forfuture implementation:

(i) Perhaps the most important point is to improvemeasurement of either the horizontal force or themoment at the test machine (in addition to the measuredvertical force). This would make all the assumptionsdescribed in section 6 redundant and any errorsassociated with these assumptions would then beeliminated. Redesigning the rig to include biaxial ortorque load-cells would improve the quality of theresults obtained from the equi-biaxial tension tests.

(ii) Improvement of the clamping system, and a largerstiffness of the sliding clamps (see section 3) to enabletesting of the MRE samples up to larger strains.

Acknowledgments

The authors gratefully acknowledge support for this workprovided through a scholarship from the Glasgow Research

Partnership in Engineering, and EPSCR grant (reference EP/H016619/3) and the EPSRC loan pool for use of the DICequipment.

References

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Table 3. No-field moduli E0, the field moduli EM, together with themaximum absolute and relative MR effects measured with magneticinduction parallel (P) and perpendicular (⊥) to the particle alignmentdirection are listed. Only the MR effects in the main magnetic fielddirection are presented.

MRE sample Iron %( ) E0 EM MRabs MRrel

Pure rubber 0 1.5677 1.6039 0.0362 1.0235

IsotropicMREs

10 3.6120 4.2267 0.6147 1.161420 4.1569 5.2435 1.0866 1.251130 5.4031 6.5846 1.1815 1.2133

AnisotropicMREs (P)

10 2.5605 3.3381 0.7776 1.303720 4.1864 5.8756 1.6892 1.403530 6.6222 11.5133 4.8911 1.7386

AnisotropicMREs (⊥)

10 2.4732 3.3324 0.8592 1.352620 4.2857 6.5274 2.2390 1.462030 6.6266 8.2615 1.6349 1.2689

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