Engineering Structures 28 (2006) 1874–1884www.elsevier.com/locate/engstruct

Experimental tests and analytical model of high dampingrubber dissipating devices

A. Dall’Astaa,∗, L. Ragnib

a Dipartimento di Progettazione e Costruzione dell’Ambiente, Universita di Camerino, Ascoli Piceno, Italyb Dipartimento di Architettura Costruzioni e Strutture, Universita Politecnica delle Marche, Ancona, Italy

Received 14 November 2005; received in revised form 16 March 2006; accepted 17 March 2006Available online 22 May 2006

Abstract

High damping rubber (HDR) consists of natural rubber to which black carbon filler is added to increase its damping properties. The use of HDRas a dissipating device in structural systems is very promising in terms of controlling the response under live actions like wind or earthquake. Theuse of HDR does however entail some problems because its dynamic behaviour is not completely understood and the few HDR models that existare not completely satisfactory for seismic analysis of structures equipped with HDR-base dissipation devices. Experimental tests were performedto obtain more accurate information about the behaviour of the material under cyclic shear paths with different strain rate and strain amplitude. Anonlinear viscoelastic damage model was proposed to describe the behaviour of rubber under cyclic loads.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: High damping rubber; Mullins effect; Experimental tests; Dynamic behaviour of damper devices; Rheological model

1. Introduction

In the last few years great interest has been generated in highdamping rubber (HDR) due to its increasing use in industry, forexample in vibration isolators, earthquake bearings, dissipatingdevices, but also because of its extensive use in vehicle tyres.HDR consists of natural rubber to which black carbon filler isadded in order to improve a wide range of desirable materialproperties such as the strength and damping capacities. Theaddition of this filler however also has other effects, that arenot always desirable, such as the Mullins effect describedbelow.

The use of HDR as a dissipating device in structural systemsis very promising in terms of controlling the structural responseunder live actions like wind or earthquake. This type ofdissipating device can in fact be used to realize dissipativesteel bracings which may be placed in the interior of reinforcedconcrete or steel frames. The dampers may be connecteddirectly to the bottom of the beams and to the rigid bracesso as to endure shear strain under store drift. The result is an

∗ Corresponding author. Tel.: +39 0736 249620; fax: +39 071 2204576.E-mail address: andrea.dallasta@unicam.it (A. Dall’Asta).

0141-0296/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2006.03.025

increase of the frame stiffness and energy dissipation capacityso that both the control of lateral displacements in the case ofsmall tremors and the reduction of damage in the case of strongmotions are ensured [1,2].

With respect to other types of damper devices, based onelasto-plastic, viscous or shape memory materials, the HDR-based damper seems to be a promising energy dissipatingdevice for a number of reasons. First, it is preferable withrespect to dissipating devices based on elasto-plastic behaviourbecause the filled rubber is a fading memory material so thatno permanent strains exist even after strong seismic events.In addition it permits dissipating energy even for the smalllateral displacements produced by wind or minor earthquakes.Similar properties are also common to visco-elastic and viscousdevices, but their energy dissipation capacity is very sensitiveto the strain rate, contrary to HDR-based devices which show alower strain-rate sensitivity.

The difficulty in the use of HDR material is that its behaviouris quite complex because it is strain-rate, strain-amplitude andprocess dependent. The dependence on the process is known asthe Mullins effect which consists of a rapid decrease of stiffnessin the early load cycles (stress softening) due to a strain-induced evolution of the microstructure of the material [3]. This

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Fig. 1. Dissipating device.

phenomenon is not completely understood and few models ofHDR exist. Additionally, the behaviour of rubber is affected bytemperature, but only marginally in the temperature range ofinterest for seismic applications [4].

It should be noted that the use of rubber with enhanceddissipating properties is not new in the mitigation of seismiceffects although, up to now, it has almost exclusively beenadopted to produce bearings for seismic isolation of bridges orbuildings. In the case of seismic isolation the main aim wasto obtain a shift of the natural frequencies by means of verydeformable supports. The dissipative properties of the materialmay be considered as a secondary effect. Very simplifiedmodels neglecting strain-rate dependence and the Mullins effectmay be acceptable for the design [5–7]. These models do notfurnish an adequate description of the dynamic behaviour ofHDR devices analyzed in this paper that are usually used toincrease dissipation and stiffness. It should also be noted thatthe rubber of these devices undergoes strain (homogeneous pureshear strain) which is different from the strain experienced byisolator rubber (simple shear and compression).

A number of experimental works on carbon filled rubberhave been published in the scientific literature and a completeoverview may be found in [8] and [9]. These works showthat the behaviour of HRD materials is mainly influencedby nonlinear elasticity coupled with a number of inelasticeffects: nonlinear rate dependence, the Mullins effect andits dependence on strain amplitude. Several analytical papersdo in fact propose models for these inelastic behaviours. Inparticular, in some works the quasi-static behaviour was studiedand rate independent models of the Mullins effect based onthe elasticity theory [3,10], the pseudo-elasticity theory [11]and the continuum damage theory were proposed. Only inwork [12] is the damage theory applied to viscoelasticityin order to obtain a rate dependent damage model. TheMullins effect has however usually been analyzed as aphenomenon occurring on the virgin material only whereasfurther investigation is required to evaluate if the initial stiffnessmay be recovered after a sufficiently long period. This aspect is

particularly important in devices used for reducing the effect ofseismic events which rarely happen.

In other works, like [9,13], the dynamic behaviour of rubbersunder cyclic loads was studied by experimental tests and uni-axial rheological models, successively extended to the three-dimensional case, were proposed on the bases of experimentaldata. In these models the nonlinear strain-rate dependence andthe small rate independent hysteresis of the stable loops areincluded in order to match the energy dissipating property, butthe Mullins effect related to early cycles was not considered.

In general, these models and the experimental tests did notaim at analyzing rubber based dissipation devices where pureshear strain occurs, but their main aim was to characterizethe tension–compression behaviour under loading–unloadingpaths. There is thus a lack of experimental information in thisregard and the proposed relations between stress and straintensors are not as accurate in describing the pure shear, asrequired in foreseeing the dynamic behaviour of structures.

In order to define a model for the dynamic analysis of astructure equipped with HDR devices, the authors carried outa test program that aims at overcoming the previously citedlimitations of existing tests and focuses on describing the devicebehaviour in the range of strain and strain rate of interest tomitigate seismic effects.

Lastly, an analytical model is proposed. It is based ona rheological, thermodynamically compatible, approach andpermits describing the main phenomena of relevance in thedynamic response of structures equipped with HDR-baseddissipation devices.

2. Experimental tests

The rubber dampers used in the experimental tests (Fig. 1)were manufactured by T.A.R.R.C. (Tun Abdul Razak ResearchCenter). They are based on the enhanced damping propertiesof a compound of natural rubber with addition of black carbonfiller and they are designed to undergo a pure shear strain inone direction. A single device is made by the superposition of

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Fig. 2. Cyclic strain history (a) Cyclic tests (b) γmax = 0.5 and γ = 1.0 s−1, different time intervals.

two rubber layers with area A = 170 × 230 mm2 and thicknesshl = 5 mm separated by an intermediate 2 mm thick steel shim.They are usually arranged in vertical or horizontal positionand they are disposed to connect rigid bracings to concrete orsteel frames so that the relative displacements between the steelplates usually coincide with the inter-storey drift [1,2].

The dynamic behaviour of rubber dampers was character-ized by performing a series of tests with a displacement feed-back. A couple of dampers were tested in order to have a ver-tical plane of symmetry. A 100 kN AMSLER servo-controlledhydraulic actuator was used to assign the displacement to thesystem. The actuator was mounted in such a way as not to giveeccentric loads and it was positioned in series with the load cell.The actuator was connected to a reaction rigid frame, as shownby Fig. 1(a). During the entire duration of the test, the total dis-placements of the rubber devices were measured by means ofinductive displacement transducers.

The strain was measured as the ratio γ = δ/h betweenthe relative displacement δ of the plates and the total rubberthickness h = 10 mm. The devices achieve the collapse ata strain value of about γ = 4.0 and are designed to sustaindeformations up to γ = 2.0. The tests were performed bylimiting the maximum value of the shear strain to γ = 2.0 andconsidering a strain rate range from 0.01 s−1 to 10 s−1.

The first set of cyclic tests is intended to characterizethe Mullins effect which is responsible for a softening inthe material in the initial loading path. In particular, it wasnecessary to clarify whether the Mullins effect occurred invirgin material only or if it may be observed even on a devicethat has not worked for a long period of time. The first testwas carried out on a device that had been subjected to verynumerous tests in the past and that had not successively workedfor three years so that data as to a possible aging effect of thematerial could also be obtained. Further tests were carried outafter 1 week and after 1 day in order to obtain informationregarding the recovery times of the Mullins effect. The testsconsisted in applying a cyclic shear deformation where shearstrain varies with a constant rate between an amplitude ofγ = +0.5 and γ = −0.5 (Fig. 2(a)). The results reported referto a strain rate γ = 1.0 s−1.

The force–strain diagrams are reported in Fig. 2(b) (hereand hereafter, force F of the diagram refers to the total force

of the couple of devices). Results show that the first loop isvery similar to the stress–strain diagram of the virgin material(furnished by the manufacturer), thus no damage, inducedby age and previous activities, occurs in this type of rubber.Furthermore, the test clarified that the Mullins effect does notoccur on virgin material only but that it may be observed evenin non-virgin material that has not worked for a long periodof time. As a matter of fact, successive tests showed that thesoftening due to the Mullins effect can be recovered in quitea short period. A remarkable Mullins effect can be observedeven on material which has not worked for about 1 day. Thiseffect strongly influences both the stiffness and the dissipatingproperties of the device.

In this paper the stiffness and dissipating properties undercyclic paths are analyzed by introducing three parameters: Keff,R, and ξ .

The first parameter furnishes a conventional measure of thestiffness and it can be obtained by the following ratio betweenthe extreme values of force F and strain γ :

Keff =Fmax − Fmin

γmax − γmin. (1)

The second parameter R furnishes information about thedissipation capacity for cycles with different amplitudes and itmay be evaluated by the ratio

R =W

γmax(2)

where W is the external work done for every cycle and γmax isthe maximum strain attained.

Finally, the third parameter ξ (equivalent viscous dampingcoefficient) furnishes approximate information about the ratiobetween the energy dissipated within a cycle and the maximumenergy stored during the strain path. It may be defined byequating the external works done in a cycle for the consideredmaterial and the external work done in a linear viscous systemwith stiffness Keff at resonance condition [14]. The expressionobtained for ξ is

ξ =W

2π Keff (hγmax)2 . (3)

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Fig. 3. Keff (a), R (b) and ξ (c) of each loop for different tests.

The values of Keff, R and ξ observed during the tests arereported in Fig. 3. In the first test (3 years) a remarkablereduction of Keff, ξ and R occurred between the first and lastcycle; the variation with respect to the stable cycle is about22% for Keff, 58% for R and 15% for ξ . As a consequence, theway the Mullins effect influences the structural response cannotbe neglected when HDR is employed to control the behaviourunder live actions, like earthquakes, that rarely occur.

A second aspect that also requires investigations is thedependence of the response on the strain amplitude. This aspecthas already been observed in the uni-axial stretch test [9,13].In order to investigate the phenomenon, a number of cyclictests at a constant strain rate γ = 1.0 s−1 were carried out byincreasing amplitude, and allowing 1 day of rest between twosubsequent tests.

The results reported in Fig. 4 show that after 6–8 cycles,once that transient contribution due to the Mullins effect hasvanished, the response always attains a stable loop which has atypical “butterfly” shape. Furthermore, stable loops attained fordifferent maximum strains differ from each other both in termsof stiffness and shape. More specifically, loops correspondingto lower strain exhibit a lower dissipation and a higher stiffness.A comparison between these and loops is reported in Fig. 5.

Fig. 6 reports the values of Keff, R and ξ for each cycleamplitude. The diagrams show that Keff decreases remarkablywhen the amplitude increases, while the ratio between theenergy dissipated and the maximum strain increases with risingvalues of amplitude. As a result, the devices are less rigid andmore efficient in dissipating energy for larger strain values.The change of dissipation properties that occurs when thestrain amplitude is varied, is between the behaviour of elasto-

plastic materials, where R tends to a constant value, and linearviscoelastic materials, for which R linearly increases. Despitethe stiffness and the energy dissipated strongly varying byvarying the cycle amplitude, the equivalent damping coefficientis approximately constant and the average value is about 0.135for the first cycles and 0.11 for stable cycles.

The difference between the stable loops reveals that thechange of material behaviour related to the Mullins effect isinfluenced by the maximum strain. This is also confirmed by afurther test where a loop with amplitude of about γ = 0.5 iscarried out after that larger strain has been applied by means ofa test with amplitude γ = 2.0. In both the cases, before andafter the test at γ = 2.0 the response attains a stable loop butcycle stiffness is different as shown by Fig. 7.

In order to investigate the dependence of the response onthe strain rate a number of cyclic tests with different rates wereconducted. Results concerning a similar amplitude of aroundγ = 2.0 and different strain rate spanning from 0.01 s−1

to 10 s−1 are reported in Fig. 8. In general, a remarkableincrease in stiffness and dissipating properties was observedwhen the rate exceeds the value of 1.0 s−1 that is usual instructural systems undergoing earthquake or wind tremors. Itis interesting to observe that the strain rate affects both thetransient response and the stable loops: the Mullins effect isvery small for strain rates below 1.0 s−1 and becomes moreand more remarkable when strain rate increases. The stableloops, compared in Fig. 9, are also different: the stiffnessincreases and the “butterfly” shape becomes more evident forhigh values of strain rate. As previously, the values of Keff,R and ξ are reported in Fig. 10. In the range considered bothKeff and R increase from slow cycles to fast cycles. A limit

1878 A. Dall’Asta, L. Ragni / Engineering Structures 28 (2006) 1874–1884

Fig. 4. Cyclic tests at different amplitudes.

Fig. 5. Stable loop at different strain amplitudes.

value of the energy dissipated seems to be achieved however,for γ = 5.0 s−1. The cycle with the maximum strain rateshows a remarkable increase of Keff and R, of about 37% and40% respectively, with respect to the cycle with the minimumstrain rate. Regarding the equivalent damping coefficient ξ ,from Fig. 10(c) it can be observed that it remains about constantaround the value 0.14 for strain rate up to 5.0 s−1 while itstrongly increases up to 0.19 for larger strain rate values.

Finally in order to verify that no permanent strain occurs andin order to separate the elastic response from the time relaxingoverstress, relaxation tests were carried out with different strainvalues through to γ = 2.0. The force–strain diagrams arereported in Fig. 11(a). The overstress contribution vanishes in a

very long period and, in order to obtain an upper and lowerbound of the elastic contribution, the stress evolution at thesame constant strain was measured by applying two differenttime histories (type “a” and “b” of Fig. 11(b)) in which theequilibrium stress is obtained from lower and higher values ofstress.

There is a certain gap between the termination points ofrelaxation and this seems to indicate a very small equilibriumhysteresis. As shown in the force–time diagram however, therelaxation process may not be completed within the timeinterval observed. These observations lead to the conclusionthat there are relaxation processes leading to equilibriumwith different (very short and very long) relaxation times.Consequently, after a sufficiently long period of time fromthe application of the strain history, the device returns to itsnatural state (zero stress and zero strain) and the material canbe classified as a fading memory material.

In conclusion, experimental tests showed that the materialbehaviour is characterized by a transient contribution, usuallycalled the Mullins effect, which vanishes for a repeated cyclicstrain path and depends both on the strain rate and themaximum strain experienced. Furthermore, once the transientresponse disappears, the material exhibits stable loops whichare strain-rate dependent and have a typical “butterfly” shapethat becomes more evident when the strain amplitude increases.After the application of a strain history, the material relaxes toits initial “natural” state in just a few hours and recovers itsinitial characteristics, i.e. it shows a similar transient responsefor similar strain histories, in just a few days. The variation

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Fig. 6. Keff (a), R (b) and ξ (c) for different strain amplitudes.

of the properties in terms of effective stiffness and lost energyrelated to the Mullins effect and the strain-rate and amplitudedependence, can be quite remarkable even if the equivalentviscous coefficient exhibits less significant variations.

3. Constitutive model

The aim of this section is to formulate a constitutive modelfor the devices tested, to describe the transient and stableresponses, in the range of strain rate and strain amplitudeof interest for practical applications. The proposed modelfurnishes a relation between the strain γ (previously defined)and the shear force in the dissipating devices which is expressedby the ratio τ between the force and the area of the rubber.1

The response of the material has been decomposed as thesum of two contributions: the former exists for every strainhistory and the latter describes a transient response (Mullinseffect) which vanishes as the strain history progresses.

The former component of the stress τ0 has been describedby assuming a rheological model consisting of a nonlinearelastic spring acting in parallel with two generalized Maxwellelements with linear springs. This approach has the advantageof furnishing models whose thermodynamic compatibility maybe easily checked [15]. The total stress can consequently beexpressed in the form:

τ0 = τe + τv1 + τv2 (4)

1 The pure shear strain of the rubber also produces a reactive traction on thesteel plates. It may be of interest to design the device–structure connection. Thisforce is usually small and was not considered in this paper.

Fig. 7. Cyclic tests before (a) and after (b) test at γ = 2.0.

where

τe = fe(γ ) (5a)

τv1 = Ev1 [γ − γv1] (5b)

τv2 = Ev2 [γ − γv2] . (5c)

The first term represents an elastic contribution and the othertwo terms are overstresses relaxing in time. At least two termsare required to describe different material behaviours related tolong-time (Eq. (5b)) and short-time relaxation (Eq. (5c)).

1880 A. Dall’Asta, L. Ragni / Engineering Structures 28 (2006) 1874–1884

Fig. 8. Cyclic tests at different strain rates.

The internal variables γv1 and γv2 describe inelastic strainsand their evolution is controlled by the two different laws

γv1 =

[|γ |

η1 (γ )+ ν1

]τv1 (6a)

γv2 =

[H (−γ γ )

η2|γ | + ν2

]τv2 (6b)

where η1 (γ ) = ξ0 + ξ1|γ | and H is the Heaviside function(H (x) = 1 if x > 0 and H (x) = 0 if x ≤ 0). The constantparameters ν1 and ν2 control the rate of relaxation in time whilethe other terms control the shape of the τ–γ diagram.

This partial model can describe the limit loops experimen-tally observed for strain cycles ranging from γ = −2.0 toγ = 2.0 and involving different strain rates. The comparisonbetween model and experimental data is reported in Fig. 12 forγ = ±2.0 and strain rate ranging from 0.1 s−1 to 10 s−1.

The approach proposed with internal variables is similar tothat presented by [13] but the evolution laws proposed in thispaper are different and permit describing the particular shapeof the stable loops and the change observed for different strainrates.

The other contribution due to the Mullins effect is modelledhereafter. The experimental tests show that the material tendsto different stable loops when strain cycles involve differentmaximum strains and this phenomenon may be described bymeans of a damage parameter qe which tends to a limit valuedepending on the maximum strain experienced. Its evolutionlaw may be posed in the following form:

qe = ζe|γ | (0.5|γ | − qe) if qe < 0.5|γ | (7a)

Fig. 9. Stable loop at different strain rates.

qe = 0 if 0.5|γ | ≤ qe ≤ 1. (7b)

The experimental tests also showed that a strain-rate dependentcontribution to stress exists in the transient response. Thiscompletely vanishes as the strain history progresses and canbe described by a second damage parameter qv with a simplerevolution law:

qv = ζv|γ | (1 − qv) . (8)

The total contribution to stress τm from the Mullins effect,controlled by the two damage parameters, can now be describedas the sum of the two contributions

τm = τme + τmv (9)

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Fig. 10. Keff (a), R (b) and ξ (c) at different strain rates.

Fig. 11. Relaxation tests at different strain values (a). Time history “a” and “b” (b).

where

τme = αm (1 − qe) fe(γ ) (10a)

τmv = Ev3 (1 − qv) [γ − γv3] . (10b)

The former is elastic and describes a stiffness increment whichtends to a strain dependent value and the latter is a strain-rate dependent contribution similar to τv2 which requires thedefinition of the related evolution law

γv3 =

[H (−γ γ )

(1 − qv)

|γ |

η3+ ν3

]τmv. (11)

It should be observed that qe also affects the parameterη1, previously introduced, whose complete expression isη1 (γ, qe) = ξ01 + ξ02 (1 − qe)

2+ ξ1|γ |. The total stress τ is

consequently the sum of τ0 (Eq. (4)) and τm (Eq. (9)). Fig. 13gives a comparison between the experimental data and theanalytical model with different maximum strains to test theability of qe to describe the asymptotic behaviour. In Fig. 14experimental data and the analytical model for different strainrates are compared to test the other damage parameter qv .

In describing the Mullins effect, the authors followedthe basic idea of introducing a growing damage, already

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Fig. 12. Stable loops at different strain rates: comparison between experimental data and analytical model.

Fig. 13. Transient response at different strain amplitudes: comparison between experimental data and analytical model.

proposed by [12] but, in the model proposed in this paper,two different damage parameters, with different evolutionlaws, were introduced to describe both the dependence on themaximum strain and the strain rate.

The constitutive behaviour is completely defined once thestrain γ and five internal variables, which may be collected ina vector y = [γv1, γv2γv3, qeqv], are known. The specific freeenergy per unit volume has the following form

ϕ (γ, y) = [1 + αm (1 − qe)] ϕe (γ ) +Ev1

2(γ − γv1)

2

+Ev2

2(γ − γv2)

2+ (1 − qv)

Ev3

2(γ − γv3)

2 (12)

where ϕe (γ ) is the strain energy of the elastic component,such that fe = dϕe/dγ (see Eqs. (5a) and (10a)), and theother terms describe the strain-rate dependent contributions.Thermodynamic compatibility requires that (repeated indexdenotes summation).

τ γ − ϕ = −∂ϕ

∂ykyk ≥ 0 (13)

and is ensured once Evi ≥ 0, ηi > 0, νi ≥ 0, ζe ≥ 0,ζv ≥ 0, (i = 1, 2, 3). In this work no attempt to describe therecovery of the initial stiffness of the material in time was madebecause it is not of particular interest in studying the response

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Fig. 14. Transient response at different strain rates: comparison between experimental data and analytical model.

Table 1Constant values

fe(γ ) Ev1 ν1 η1 (γ, qe) Ev2 ν2 η2(N

mm2

) (N

mm2

) (mm2

N s

) (N

mm2

) (N

mm2

) (mm2

N s

) (N

mm2

)0.029γ 5

− 0.082γ 3+ 0.29γ 2.56 0.078 0.179 − 0.127(1 − qe)

2+ 0.047|γ | 0.447 26 0.025

αm E3v ν3 η3 ζe ζv(N

mm2

) (N

mm2

) (mm2

N s

) (N

mm2

)1.5 0.256 2.23 0.025 1.0 0.2

under seismic actions which act rarely and for a short period oftime so that a complete recovery of the initial stiffness usuallyoccurs between two subsequent events. The results reported inthe diagrams were obtained by adopting the parameter valuesreported in Table 1.

4. Conclusions

An experimental test program was performed in order tocharacterize the cyclic behaviour of high damping rubber underpure shear strain and investigate some aspects not previouslycompletely understood more thoroughly.

Experimental tests demonstrated that material behaviour ischaracterized by a transient contribution. Once the transientresponse has disappeared, the material exhibits stable loopswhich are strain-rate dependent and have a typical “butterfly”shape. After applying a strain history, the material relaxes to itsinitial “natural” state and recovers its initial characteristics.

On the basis of experimental results, an analytical model,able to describe the material behaviour in the range of interestfor seismic applications, was developed. The constitutivebehaviour was described by means of a thermodynamicallycompatible rheological model, in which internal variables wereintroduced to describe inelastic phenomena. The results arequite accurate and permit overcoming the limitations of modelspreviously adopted in seismic analysis, that were generallybased on hysteretic laws neglecting the dependence on thestrain rate and the Mullins effect.

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