Experimental tests and analytical model of high damping rubber dissipating devices

Transcription

1 Engineering Structures 28 (2006) Experimental tests and analytical model of high damping rubber dissipating devices A. Dall Asta a,, L. Ragni b a Dipartimento di Progettazione e Costruzione dell Ambiente, Università di Camerino, Ascoli Piceno, Italy b Dipartimento di Architettura Costruzioni e Strutture, Università Politecnica delle Marche, Ancona, Italy Received 14 ovember 2005; received in revised form 16 March 2006; accepted 17 March 2006 Available 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 HDR as a dissipating device in structural systems is very promising in terms of controlling the response under live actions like wind or earthquake. The use of HDR does however entail some problems because its dynamic behaviour is not completely understood and the few HDR models that exist are not completely satisfactory for seismic analysis of structures equipped with HDR-base dissipation devices. Experimental tests were performed to obtain more accurate information about the behaviour of the material under cyclic shear paths with different strain rate and strain amplitude. A nonlinear 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 high damping rubber (HDR) due to its increasing use in industry, for example in vibration isolators, earthquake bearings, dissipating devices, but also because of its extensive use in vehicle tyres. HDR consists of natural rubber to which black carbon filler is added in order to improve a wide range of desirable material properties such as the strength and damping capacities. The addition of this filler however also has other effects, that are not always desirable, such as the Mullins effect described below. The use of HDR as a dissipating device in structural systems is very promising in terms of controlling the structural response under live actions like wind or earthquake. This type of dissipating device can in fact be used to realize dissipative steel bracings which may be placed in the interior of reinforced concrete or steel frames. The dampers may be connected directly to the bottom of the beams and to the rigid braces so as to endure shear strain under store drift. The result is an Corresponding author. Tel.: ; fax: address: andrea.dallasta@unicam.it (A. Dall Asta). increase of the frame stiffness and energy dissipation capacity so that both the control of lateral displacements in the case of small tremors and the reduction of damage in the case of strong motions are ensured [1,2]. With respect to other types of damper devices, based on elasto-plastic, viscous or shape memory materials, the HDRbased damper seems to be a promising energy dissipating device for a number of reasons. First, it is preferable with respect to dissipating devices based on elasto-plastic behaviour because the filled rubber is a fading memory material so that no permanent strains exist even after strong seismic events. In addition it permits dissipating energy even for the small lateral displacements produced by wind or minor earthquakes. Similar properties are also common to visco-elastic and viscous devices, but their energy dissipation capacity is very sensitive to the strain rate, contrary to HDR-based devices which show a lower strain-rate sensitivity. The difficulty in the use of HDR material is that its behaviour is quite complex because it is strain-rate, strain-amplitude and process dependent. The dependence on the process is known as the Mullins effect which consists of a rapid decrease of stiffness in the early load cycles (stress softening) due to a straininduced evolution of the microstructure of the material [3]. This /$ - see front matter c 2006 Elsevier Ltd. All rights reserved. doi: /j.engstruct

2 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) Fig. 1. Dissipating device. phenomenon is not completely understood and few models of HDR exist. Additionally, the behaviour of rubber is affected by temperature, but only marginally in the temperature range of interest for seismic applications [4]. It should be noted that the use of rubber with enhanced dissipating properties is not new in the mitigation of seismic effects although, up to now, it has almost exclusively been adopted to produce bearings for seismic isolation of bridges or buildings. In the case of seismic isolation the main aim was to obtain a shift of the natural frequencies by means of very deformable supports. The dissipative properties of the material may be considered as a secondary effect. Very simplified models neglecting strain-rate dependence and the Mullins effect may be acceptable for the design [5 7]. These models do not furnish an adequate description of the dynamic behaviour of HDR devices analyzed in this paper that are usually used to increase dissipation and stiffness. It should also be noted that the rubber of these devices undergoes strain (homogeneous pure shear strain) which is different from the strain experienced by isolator rubber (simple shear and compression). A number of experimental works on carbon filled rubber have been published in the scientific literature and a complete overview may be found in [8] and [9]. These works show that the behaviour of HRD materials is mainly influenced by nonlinear elasticity coupled with a number of inelastic effects: nonlinear rate dependence, the Mullins effect and its dependence on strain amplitude. Several analytical papers do in fact propose models for these inelastic behaviours. In particular, in some works the quasi-static behaviour was studied and rate independent models of the Mullins effect based on the elasticity theory [3,10], the pseudo-elasticity theory [11] and the continuum damage theory were proposed. Only in work [12] is the damage theory applied to viscoelasticity in order to obtain a rate dependent damage model. The Mullins effect has however usually been analyzed as a phenomenon occurring on the virgin material only whereas further investigation is required to evaluate if the initial stiffness may be recovered after a sufficiently long period. This aspect is particularly important in devices used for reducing the effect of seismic events which rarely happen. In other works, like [9,13], the dynamic behaviour of rubbers under cyclic loads was studied by experimental tests and uniaxial rheological models, successively extended to the threedimensional case, were proposed on the bases of experimental data. In these models the nonlinear strain-rate dependence and the small rate independent hysteresis of the stable loops are included in order to match the energy dissipating property, but the Mullins effect related to early cycles was not considered. In general, these models and the experimental tests did not aim at analyzing rubber based dissipation devices where pure shear strain occurs, but their main aim was to characterize the tension compression behaviour under loading unloading paths. There is thus a lack of experimental information in this regard and the proposed relations between stress and strain tensors are not as accurate in describing the pure shear, as required in foreseeing the dynamic behaviour of structures. In order to define a model for the dynamic analysis of a structure equipped with HDR devices, the authors carried out a test program that aims at overcoming the previously cited limitations of existing tests and focuses on describing the device behaviour in the range of strain and strain rate of interest to mitigate seismic effects. Lastly, an analytical model is proposed. It is based on a rheological, thermodynamically compatible, approach and permits describing the main phenomena of relevance in the dynamic response of structures equipped with HDR-based dissipation 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 Research Center). They are based on the enhanced damping properties of a compound of natural rubber with addition of black carbon filler and they are designed to undergo a pure shear strain in one direction. A single device is made by the superposition of

3 1876 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) Fig. 2. Cyclic strain history (a) Cyclic tests (b) γ max = 0.5 and γ = 1.0 s 1, different time intervals. two rubber layers with area A = mm 2 and thickness h l = 5 mm separated by an intermediate 2 mm thick steel shim. They are usually arranged in vertical or horizontal position and they are disposed to connect rigid bracings to concrete or steel frames so that the relative displacements between the steel plates usually coincide with the inter-storey drift [1,2]. The dynamic behaviour of rubber dampers was characterized by performing a series of tests with a displacement feedback. A couple of dampers were tested in order to have a vertical plane of symmetry. A 100 k AMSLER servo-controlled hydraulic actuator was used to assign the displacement to the system. The actuator was mounted in such a way as not to give eccentric loads and it was positioned in series with the load cell. The actuator was connected to a reaction rigid frame, as shown by Fig. 1(a). During the entire duration of the test, the total displacements of the rubber devices were measured by means of inductive displacement transducers. The strain was measured as the ratio γ = δ/h between the relative displacement δ of the plates and the total rubber thickness h = 10 mm. The devices achieve the collapse at a strain value of about γ = 4.0 and are designed to sustain deformations up to γ = 2.0. The tests were performed by limiting the maximum value of the shear strain to γ = 2.0 and considering a strain rate range from 0.01 s 1 to 10 s 1. The first set of cyclic tests is intended to characterize the Mullins effect which is responsible for a softening in the material in the initial loading path. In particular, it was necessary to clarify whether the Mullins effect occurred in virgin material only or if it may be observed even on a device that has not worked for a long period of time. The first test was carried out on a device that had been subjected to very numerous tests in the past and that had not successively worked for three years so that data as to a possible aging effect of the material could also be obtained. Further tests were carried out after 1 week and after 1 day in order to obtain information regarding the recovery times of the Mullins effect. The tests consisted in applying a cyclic shear deformation where shear strain varies with a constant rate between an amplitude of γ = +0.5 and γ = 0.5 (Fig. 2(a)). The results reported refer to a strain rate γ = 1.0 s 1. The force strain diagrams are reported in Fig. 2(b) (here and hereafter, force F of the diagram refers to the total force of the couple of devices). Results show that the first loop is very similar to the stress strain diagram of the virgin material (furnished by the manufacturer), thus no damage, induced by age and previous activities, occurs in this type of rubber. Furthermore, the test clarified that the Mullins effect does not occur on virgin material only but that it may be observed even in non-virgin material that has not worked for a long period of time. As a matter of fact, successive tests showed that the softening due to the Mullins effect can be recovered in quite a short period. A remarkable Mullins effect can be observed even on material which has not worked for about 1 day. This effect strongly influences both the stiffness and the dissipating properties of the device. In this paper the stiffness and dissipating properties under cyclic paths are analyzed by introducing three parameters: K eff, R, and ξ. The first parameter furnishes a conventional measure of the stiffness and it can be obtained by the following ratio between the extreme values of force F and strain γ : K eff = F max F min γ max γ min. (1) The second parameter R furnishes information about the dissipation capacity for cycles with different amplitudes and it may be evaluated by the ratio R = W γ max (2) where W is the external work done for every cycle and γ max is the maximum strain attained. Finally, the third parameter ξ (equivalent viscous damping coefficient) furnishes approximate information about the ratio between the energy dissipated within a cycle and the maximum energy stored during the strain path. It may be defined by equating the external works done in a cycle for the considered material and the external work done in a linear viscous system with stiffness K eff at resonance condition [14]. The expression obtained for ξ is ξ = W 2π K eff (hγ max ) 2. (3)

4 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) Fig. 3. K eff (a), R (b) and ξ (c) of each loop for different tests. The values of K eff, R and ξ observed during the tests are reported in Fig. 3. In the first test (3 years) a remarkable reduction of K eff, ξ and R occurred between the first and last cycle; the variation with respect to the stable cycle is about 22% for K eff, 58% for R and 15% for ξ. As a consequence, the way the Mullins effect influences the structural response cannot be neglected when HDR is employed to control the behaviour under live actions, like earthquakes, that rarely occur. A second aspect that also requires investigations is the dependence of the response on the strain amplitude. This aspect has already been observed in the uni-axial stretch test [9,13]. In order to investigate the phenomenon, a number of cyclic tests at a constant strain rate γ = 1.0 s 1 were carried out by increasing amplitude, and allowing 1 day of rest between two subsequent tests. The results reported in Fig. 4 show that after 6 8 cycles, once that transient contribution due to the Mullins effect has vanished, the response always attains a stable loop which has a typical butterfly shape. Furthermore, stable loops attained for different maximum strains differ from each other both in terms of stiffness and shape. More specifically, loops corresponding to 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 K eff, R and ξ for each cycle amplitude. The diagrams show that K eff decreases remarkably when the amplitude increases, while the ratio between the energy dissipated and the maximum strain increases with rising values of amplitude. As a result, the devices are less rigid and more efficient in dissipating energy for larger strain values. The change of dissipation properties that occurs when the strain amplitude is varied, is between the behaviour of elastoplastic materials, where R tends to a constant value, and linear viscoelastic materials, for which R linearly increases. Despite the stiffness and the energy dissipated strongly varying by varying the cycle amplitude, the equivalent damping coefficient is approximately constant and the average value is about for the first cycles and 0.11 for stable cycles. The difference between the stable loops reveals that the change of material behaviour related to the Mullins effect is influenced by the maximum strain. This is also confirmed by a further test where a loop with amplitude of about γ = 0.5 is carried out after that larger strain has been applied by means of a test with amplitude γ = 2.0. In both the cases, before and after the test at γ = 2.0 the response attains a stable loop but cycle stiffness is different as shown by Fig. 7. In order to investigate the dependence of the response on the strain rate a number of cyclic tests with different rates were conducted. Results concerning a similar amplitude of around γ = 2.0 and different strain rate spanning from 0.01 s 1 to 10 s 1 are reported in Fig. 8. In general, a remarkable increase in stiffness and dissipating properties was observed when the rate exceeds the value of 1.0 s 1 that is usual in structural systems undergoing earthquake or wind tremors. It is interesting to observe that the strain rate affects both the transient response and the stable loops: the Mullins effect is very small for strain rates below 1.0 s 1 and becomes more and more remarkable when strain rate increases. The stable loops, compared in Fig. 9, are also different: the stiffness increases and the butterfly shape becomes more evident for high values of strain rate. As previously, the values of K eff, R and ξ are reported in Fig. 10. In the range considered both K eff and R increase from slow cycles to fast cycles. A limit

5 1878 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) 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 rate shows a remarkable increase of K eff and R, of about 37% and 40% respectively, with respect to the cycle with the minimum strain rate. Regarding the equivalent damping coefficient ξ, from Fig. 10(c) it can be observed that it remains about constant around the value 0.14 for strain rate up to 5.0 s 1 while it strongly increases up to 0.19 for larger strain rate values. Finally in order to verify that no permanent strain occurs and in order to separate the elastic response from the time relaxing overstress, relaxation tests were carried out with different strain values through to γ = 2.0. The force strain diagrams are reported in Fig. 11(a). The overstress contribution vanishes in a very long period and, in order to obtain an upper and lower bound of the elastic contribution, the stress evolution at the same constant strain was measured by applying two different time histories (type a and b of Fig. 11(b)) in which the equilibrium stress is obtained from lower and higher values of stress. There is a certain gap between the termination points of relaxation and this seems to indicate a very small equilibrium hysteresis. As shown in the force time diagram however, the relaxation process may not be completed within the time interval observed. These observations lead to the conclusion that there are relaxation processes leading to equilibrium with different (very short and very long) relaxation times. Consequently, after a sufficiently long period of time from the application of the strain history, the device returns to its natural state (zero stress and zero strain) and the material can be classified as a fading memory material. In conclusion, experimental tests showed that the material behaviour is characterized by a transient contribution, usually called the Mullins effect, which vanishes for a repeated cyclic strain path and depends both on the strain rate and the maximum strain experienced. Furthermore, once the transient response disappears, the material exhibits stable loops which are strain-rate dependent and have a typical butterfly shape that becomes more evident when the strain amplitude increases. After the application of a strain history, the material relaxes to its initial natural state in just a few hours and recovers its initial characteristics, i.e. it shows a similar transient response for similar strain histories, in just a few days. The variation

6 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) of the properties in terms of effective stiffness and lost energy related to the Mullins effect and the strain-rate and amplitude dependence, can be quite remarkable even if the equivalent viscous coefficient exhibits less significant variations. 3. Constitutive model Fig. 6. K eff (a), R (b) and ξ (c) for different strain amplitudes. The aim of this section is to formulate a constitutive model for the devices tested, to describe the transient and stable responses, in the range of strain rate and strain amplitude of interest for practical applications. The proposed model furnishes a relation between the strain γ (previously defined) and the shear force in the dissipating devices which is expressed by the ratio τ between the force and the area of the rubber. 1 The response of the material has been decomposed as the sum of two contributions: the former exists for every strain history and the latter describes a transient response (Mullins effect) which vanishes as the strain history progresses. The former component of the stress τ 0 has been described by assuming a rheological model consisting of a nonlinear elastic spring acting in parallel with two generalized Maxwell elements with linear springs. This approach has the advantage of furnishing models whose thermodynamic compatibility may be easily checked [15]. The total stress can consequently be expressed in the form: τ 0 = τ e + τ v1 + τ v2 (4) 1 The pure shear strain of the rubber also produces a reactive traction on the steel plates. It may be of interest to design the device structure connection. This force is usually small and was not considered in this paper. where τ e = f e (γ ) Fig. 7. Cyclic tests before (a) and after (b) test at γ = 2.0. τ v1 = E v1 [γ γ v1 ] τ v2 = E v2 [γ γ v2 ]. (5a) (5b) (5c) The first term represents an elastic contribution and the other two terms are overstresses relaxing in time. At least two terms are required to describe different material behaviours related to long-time (Eq. (5b)) and short-time relaxation (Eq. (5c)).

7 1880 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) Fig. 8. Cyclic tests at different strain rates. The internal variables γ v1 and γ v2 describe inelastic strains and their evolution is controlled by the two different laws [ ] γ γ v1 = η 1 (γ ) + ν 1 τ v1 (6a) [ ] H ( γ γ ) γ v2 = γ + ν 2 τ v2 (6b) η 2 where η 1 (γ ) = ξ 0 + ξ 1 γ and H is the Heaviside function (H (x) = 1 if x > 0 and H (x) = 0 if x 0). The constant parameters ν 1 and ν 2 control the rate of relaxation in time while the other terms control the shape of the τ γ diagram. This partial model can describe the limit loops experimentally observed for strain cycles ranging from γ = 2.0 to γ = 2.0 and involving different strain rates. The comparison between model and experimental data is reported in Fig. 12 for γ = ±2.0 and strain rate ranging from 0.1 s 1 to 10 s 1. The approach proposed with internal variables is similar to that presented by [13] but the evolution laws proposed in this paper are different and permit describing the particular shape of the stable loops and the change observed for different strain rates. The other contribution due to the Mullins effect is modelled hereafter. The experimental tests show that the material tends to different stable loops when strain cycles involve different maximum strains and this phenomenon may be described by means of a damage parameter q e which tends to a limit value depending on the maximum strain experienced. Its evolution law may be posed in the following form: q e = ζ e γ (0.5 γ q e ) if q e < 0.5 γ (7a) Fig. 9. Stable loop at different strain rates. q e = 0 if 0.5 γ q e 1. (7b) The experimental tests also showed that a strain-rate dependent contribution to stress exists in the transient response. This completely vanishes as the strain history progresses and can be described by a second damage parameter q v with a simpler evolution law: q v = ζ v γ (1 q v ). (8) The total contribution to stress τ m from the Mullins effect, controlled by the two damage parameters, can now be described as the sum of the two contributions τ m = τ me + τ mv (9)

8 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) Fig. 10. K eff (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 τ me = α m (1 q e ) f e (γ ) τ mv = E v3 (1 q v ) [γ γ v3 ]. (10a) (10b) The former is elastic and describes a stiffness increment which tends to a strain dependent value and the latter is a strainrate dependent contribution similar to τ v2 which requires the definition of the related evolution law [ ] H ( γ γ ) γ γ v3 = + ν 3 τ mv. (11) (1 q v ) η 3 It should be observed that q e also affects the parameter η 1, previously introduced, whose complete expression is η 1 (γ, q e ) = ξ 01 + ξ 02 (1 q e ) 2 + ξ 1 γ. The total stress τ is consequently the sum of τ 0 (Eq. (4)) and τ m (Eq. (9)). Fig. 13 gives a comparison between the experimental data and the analytical model with different maximum strains to test the ability of q e to describe the asymptotic behaviour. In Fig. 14 experimental data and the analytical model for different strain rates are compared to test the other damage parameter q v. In describing the Mullins effect, the authors followed the basic idea of introducing a growing damage, already

9 1882 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) 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 evolution laws, were introduced to describe both the dependence on the maximum strain and the strain rate. The constitutive behaviour is completely defined once the strain γ and five internal variables, which may be collected in a vector y = [γ v1, γ v2 γ v3, q e q v ], are known. The specific free energy per unit volume has the following form ϕ (γ, y) = [1 + α m (1 q e )] ϕ e (γ ) + E v1 2 (γ γ v1) 2 + E v2 2 (γ γ v2) 2 + (1 q v ) E v3 2 (γ γ v3) 2 (12) where ϕ e (γ ) is the strain energy of the elastic component, such that f e = dϕ e /dγ (see Eqs. (5a) and (10a)), and the other terms describe the strain-rate dependent contributions. Thermodynamic compatibility requires that (repeated index denotes summation). τ γ ϕ = ϕ y k ẏ k 0 (13) and is ensured once E vi 0, η i > 0, ν i 0, ζ e 0, ζ v 0, (i = 1, 2, 3). In this work no attempt to describe the recovery of the initial stiffness of the material in time was made because it is not of particular interest in studying the response

10 A. Dall Asta, L. Ragni / Engineering Structures 28 (2006) Fig. 14. Transient response at different strain rates: comparison between experimental data and analytical model. Table 1 Constant values ( f e (γ ) ( E v1 ) ( ν 1 ) η ( 1 (γ, q ) e ) ( E v2 ) ( ν 2 ) ( η 2 ) mm 2 mm 2 mm 2 mm 2 s mm 2 mm 2 s mm γ γ γ (1 q e ) γ ( α m ) ( E 3v ) ( ν 3 ) ( η 3 ) ζ e ζ v mm 2 mm 2 mm 2 s mm under seismic actions which act rarely and for a short period of time so that a complete recovery of the initial stiffness usually occurs between two subsequent events. The results reported in the diagrams were obtained by adopting the parameter values reported in Table Conclusions An experimental test program was performed in order to characterize the cyclic behaviour of high damping rubber under pure shear strain and investigate some aspects not previously completely understood more thoroughly. Experimental tests demonstrated that material behaviour is characterized by a transient contribution. Once the transient response has disappeared, the material exhibits stable loops which are strain-rate dependent and have a typical butterfly shape. After applying a strain history, the material relaxes to its initial 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 interest for seismic applications, was developed. The constitutive behaviour was described by means of a thermodynamically compatible rheological model, in which internal variables were introduced to describe inelastic phenomena. The results are quite accurate and permit overcoming the limitations of models previously adopted in seismic analysis, that were generally based on hysteretic laws neglecting the dependence on the strain rate and the Mullins effect. References [1] Dall Asta A, Dezi L, Giacchetti R, Leoni G, Ragni L. Dynamic response of composite frames with rubber-based dissipating devices: experimental tests. In: Proceedings of ICASS, fourth international conference on advances in steel structures [2] Bartera F, Giacchetti R. Steel dissipating braces for upgrading existing building frames. J Construct Steel Res 2004;60(3):

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