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Relaxation ∙ Recovery ∙ Steady-state dynamic Tests ∙ Material Models ∙ Filled Rubber

This paper presents new experimental approaches for investigating relaxation and recovery behaviour in steady-state dynamic tests on rubber filled with car-bon black, illustrated here using SBR double shear test pieces. Mathematical and experimental tools are presented and it is pointed out how to interpret the results in terms of relaxation and recovery. The overall behaviour in the designed experiments is found to be in qualitative agreement with Boltzmann’s superposition principle, suggesting that superposition of fading memories captures the essence of the time-dependency. The same characteri-zation techniques can be used for asses-sing more elaborate non-linear models.

Besonderheiten des Relaxa-tions- und Erholungsverhaltens gefüllter Elastomere Relaxation ∙ Erholung ∙ stationäre dyna-mische Experimente ∙ Materialmodelle ∙ gefüllte Elastomere

In diesem Paper werden neue experi-mentelle Ansätze zur Untersuchung des Relaxations- und Erholungsverhaltens rußgefüllter SBR-Proben in periodi-schen, dynamischen Versuchen vorge-stellt. Ein besonderes Augenmerk wird auf benötigte mathematische und ex-perimentelle Hilfsmittel gelegt. Im An-schluss wird erläutert, wie die Ergebnis-se in Hinblick auf Relaxation und Erho-lung interpretiert werden können. Das in den neu konzipierten Versuchen zu-tage tretende Materialverhalten stimmt qualitativ mit Boltzmann’s Su-perpositionsprinzip überein. Die darge-stellten Methoden und Versuchsdaten können genutzt werden um sowohl aufwendigere nichtlineare Modelle als auch andere Elastomere zu bewerten.

Figures and Tables:By a kind approval of the authors.

IntroductionSteady-state dynamic stiffness proper-ties are of great significance for assess-ing performance of rubbers for vibration isolation. Transient behaviour is equally important for assessing performance of shock isolators. In practice these two ap-plications generally involve elements of both aspects of rubber behaviour, mak-ing the decision whether to use a fre-quency-domain or time-domain model in analyses less than clear-cut. In order to test and enhance the existing material models or develop new ones, the proper-ties of rubber need to be studied experi-mentally. As it is difficult to cover all cas-es of application in one experiment, a great number of specific tests is neces-sary to entirely understand the material behaviour of rubber.

For investigating the recovery behav-iour of mean stress in a steady state dy-namic experiment, a suitable test se-quence has been developed by Ahmadi, Ihlemann and Muhr [1], see Figure 1. The purpose was to analyse the mean stress response during sinusoidal vibrations in nominal homogeneous simple shear. The approach was applied to filled SBR and NR double shear test pieces after they had been strained by a number of 100% strain cycles. The basic idea was to keep the strain rate continuous (except for the special case of changing to a vi-bration of zero amplitude), rather than keeping the frequency constant when changing the amplitude. This enables investigation of the effect of strain am-plitude without the influence of any ef-fect of strain rate.

This paper builds on the investigations from [1], which itself uses the experimen-tal scheme from [2], and concentrates on the experimental investigation of relaxa-tion and recovery phenomena. To provide a common basis for discussion, the terms “relaxation” and “recovery” are defined clearly. The test programme from [1] with carefully designed sequences of periodic blocks is modified, so that it covers both viscoelastic recovery and relaxation char-acteristics of the rubber and their interac-tion. It is found that recovery of the Mul-lins stress-softening is significant after one week, but that the complication of

the Mullins effect can be largely removed by experiment design, enabling the focus to be on viscoelastic behaviour. Experi-mental techniques using uniaxial simple shear excitations are presented together with mathematical tools for filtering and presenting the test data. The mean stress responses for vibrations of different am-plitudes are superposed and compared. The motivation is to test both expecta-tion and candidate stress-strain models against the carefully derived and pre-sented data.

Relaxation and RecoveryWhen straining viscoelastic material the resulting stress appears to be not in phase with the strain. The effects that arise due to this time-dependence are called “re-laxation”, “recovery” and “creep”. These terms are defined precisely in a paper from Alan Gent [3]. For defining the terms “creep” and “relaxation” Gent distin-guishes between load or displacement controlled experiments. The effect of in-creasing deformation over time under a constant load is called “creep” and the ef-fect of decreasing stress over time under constant strain is called “relaxation”. Fig-ure 2 illustrates both effects.

Gent [3] also gives a definition of time dependent recovery towards the original configuration after an imposed stress history returns to zero (Figure 3a). For the purpose of this paper Gent‘s

Characteristics of Relaxation and recovery Behavior of filled Rubber

AuthorsAnneke Lampe, Lars Kanzenbach, Jörn Ihlemann, Chemnitz, Deutschland , Alan Muhr, Hertford, United Kingdom Corresponding Authors:Anneke Lampe & Lars Kanzenbach Chemnitz University of TechnologyChair of Solid MechanicsReichenhainer Straße 7009126 Chemnitz, GermanyE-Mail: lars.kanzenbach@ mb.tu-chemnitz.de; anneke.lampe@online.de

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definition of the term “recovery” needs to be extended. Analogous to the dis-tinction of creep and stress relaxation, the term “recovery” is differentiated in-to “strain recovery” and “stress recov-ery” according to the responding pa-rameter, see Figure 3. Thus, “strain re-covery” describes the time dependent decrease of strain after a load has been removed (known from Gent’s definition of recovery). Now, a strain controlled test is considered in which the material is forced back to its original configura-tion (in which stress and strain were zero) after a period of non-zero strain. Since the stress “leads” the strain, in the response the stress is forced to pass through zero to a value of opposite sign, before recovering towards zero (Figure

3b). This effect is henceforth called “stress recovery”.

On the basis of these definitions a displacement-controlled test can be de-signed in which the phenomena of “re-laxation” and “stress recovery” interact. For this purpose a strained test piece is forced towards its original configuration but only part away. As described for the case of pure stress recovery, the de-crease in stress will be overstated in the moment of relief. Consequently, the stress response is influenced by both continued relaxation and stress recov-ery as illustrated in Figure 4. The possi-bility is foreseen that the evolution of stress, after a complicated strain history has settled on a fixed strain, may not be monotonic.

Test programmeThe influence of cyclic strain of small amplitude on stress recovery has been investigated by Ahmadi et al [1]. For the new experiments the same basic test command is used but modified in a way that covers the phenomena of relaxation and interaction of relaxation and stress recovery. Each test is driven by a displace-ment command signal that consists of three phases, as illustrated in Figure 5. During the first phase the amplitude is increased in a series of steps. In each step, or section, several cycles at constant amplitude are performed at a frequency such that the strain rate is always con-tinuous. The objective of this strain his-tory is to reduce the influence of the Mullins effect and anisotropic behaviour (for more details see [1] and [2]).

In the second phase the vibration sec-tions (parts with cyclic strain of small amplitude, see Figure 5) alternate with sections of cyclic strain of 100% ampli-tude. In contrast to the basic tests used by Ahmadi et al [1], the vibration sec-tions are not necessarily about a mean strain of zero, but may also be shifted to a defined offset.

The vibration sections are carried out symmetrically such that a positive offset is followed by a negative one of the same amount. Due to this symmetry of the displacement history the stress response

Fig. 1: Displacement command for a stress recovery test in [1] where Fxy is a coefficient of the deformation gradient.

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Fig. 2a: „Creep“ (load controlled): command: constant stress, response: increasing strain.

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Fig. 2b: „Relaxation“ (displacement controlled): command: constant strain, response: decreasing stress.

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Fig. 3a: „Strain recovery“ (load controlled): command: zero stress after previous loading, response: some strain remains (“set”) but slowly recovers towards zero strain.

Fig. 3b: „Stress recovery“ (displacement controlled): command: zero strain after previous straining, response: stress in the oppo-site direction which decreases with time.

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is expected to be symmetric as well, but it appears that in most cases the sym-metry axis of the stress is shifted. This observation is most naturally attributa-ble to a small zero error in the load cell or displacement imperfection in the con-figuration of nominal zero stress and ze-ro strain. This is taken to justify defini-tion of the symmetry axis as zero stress. Furthermore, the symmetric design of the displacement command is beneficial since the results for the positive and negative vibration sections can be com-pared in terms of reproducibility.

Two different ways of transition are investigated. In the first case the vibra-tion section starts after the previous sec-tion, of 100% strain amplitude, has passed zero displacement. This corre-sponds to a displacement command sec-tion for investigating relaxation phe-nomena. In the second case the vibration section (the mirrored sections in the negative area are not counted) starts af-ter the previous section has lately passed its peak, which corresponds to a dis-placement command section for investi-gating the interaction of recovery and relaxation. In the special case of “offset + amplitude = 100%”, a third transition point can be found at the top of the peak. However, in this paper only the first two types of transition are investigated.

Switching from cycles of 100% strain amplitude and zero mean to a smaller amplitude with an offset mean raises a new feature, since the strain rate of the large amplitude command now depends on the offset, and thus so would the fre-quency of the vibration if the switching were done precisely at the offset. In-stead, it was decided to keep the strain and strain rate continuous at the transi-tion point and the strain rate amplitude constant; in consequence, the transition is not at the mean level of the vibration. In order to keep the strain rate continu-ous at the transition point, both the

function value and the gradient at the end of section i have to be equal to the function value and the gradient at the beginning of the next section i+1:

𝑞𝑞i �𝑡𝑡i = 𝑡𝑡i,end � = 𝑞𝑞i+1 (𝑡𝑡i+1 = 0) (1)

𝑞𝑞′i �𝑡𝑡i = 𝑡𝑡i,end � = 𝑞𝑞′i+1 (𝑡𝑡i+1 = 0) (2)

We shall consider that in the ith section, ti is a variable running from the value ti = 0 to ti,end, so that ti+1 = t - ∑ ti,end. Then the following equations arise for a sinusoidal

function q, which can be used to describe the displacement command:

(3)

(4)

It follows that 𝑓𝑓i 𝑞𝑞�i , 𝑓𝑓i+1 𝑞𝑞�i+1, 𝑞𝑞i0 and 𝑞𝑞i+1

0 are known. Then this nonlinear system of two equations (3) and (4) is solved with an iterative approach and delivers

Fig. 4a: Com-mand for inves-tigating interac-tion of stress re-covery and rela-xation.

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Fig. 4b: Expec-ted response in-fluenced by su-perposition of stress recovery and continued relaxation.

Fig. 5: Basic design of the displacement command signal for an individual test, here with vibration sections of offset 50% and amplitude 10%.

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Fig. 6: Different ways of transition: after approach from zero (left) or after approach from peak (right). The dashed lines illustrate how to in-terpret the different transitions as relaxation (left) or interaction of relaxation and stress recovery (right).

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the required pair of values for ti,end and the phase shift ϕi+1 when appropriate start values are chosen. The second phase in Figure 5 is complete once vibra-tions for both ways of transition, and both signs of offset, have been imposed. The design of the third phase (Figure 5) is related to the second phase with the difference that the vibration sections are replaced by constant strain at the chosen offset. Thus, the displacement command in this phase is identical for different tests that have identical off-sets but differ in the amplitude. As a consequence, the stress response should be equal as well which means that the third phase additionally serves as an indicator for comparability of test sequences with identical offsets.

Evaluation of test dataDuring the vibration sections, the stress evolution is a superposed function con-sisting of the sinusoidal stress response and a decreasing or increasing mean stress response due to relaxation and

stress recovery phenomena. In order to filter out the cyclic component and gain a pure relaxation / stress recovery curve (henceforth called mean stress curve) the following equation is applied to cal-culate a marching mean stress [1]:

𝜎𝜎0 =1𝑇𝑇

� 𝜎𝜎(𝑡𝑡) 𝑑𝑑𝑡𝑡𝑡𝑡p + 𝑇𝑇/2

𝑡𝑡p − 𝑇𝑇/2

where

𝑡𝑡p ∈ �𝑇𝑇2

, … , 𝑡𝑡i,end −𝑇𝑇2�

(7)

Figure 7 compares a stress evolution (sol-id line) with the filtered mean stress (dashed line) versus τ, the time scale of the vibration section.

To improve comparability of different vibration sections it is recommended not to choose the transition point as the time to set τ = 0 in the evaluation. In-stead, the first time after transition that the strain function passes through the mean value (equivalent to offset) of the vibration section is taken as τ = 0, see Figure 8. Thus, all analyses of mean stress

evolution for the sections start with nearly the same phase.

Testing methodsIn order to transmit the desired displace-ment history, the traverse of the testing machine is controlled by an external sig-nal. Here, the data points of the desired displacement command are read by a LabVIEW programme which controls a Servotest machine via a National Instru-ments NI USB-6002 Multifunctional Da-ta Acquisition unit (DAQ). The „NI box“ converts the control file to a voltage function of time with which the machine can be driven. The machine control for the presented tests was developed and established by Orlando [4]. A similar method is described in a paper from Kanzenbach et al [5].

The tests were carried out on filled SBR test pieces (material formulation is given in [2]). To avoid the uncertainty that two rubber test pieces may not have identical material behaviour, a sin-gle test piece is best used for all tests when sections of different test sequenc-es are to be compared. However, the in-fluence of the material “memory” needs to be considered. In order to investigate the influence of the rest period prior to a test, a virgin filled SBR test piece was subjected 5 times to the same basic test (offset: 90%, amplitude: 10%) with rest periods in between as specified in Fig-ure 9.

Figure 10 shows the resulting mean stress response during one vibration section. It can be observed that the

Fig. 7: Analysis of the stress response during a vibration section: Original superposed stress evolution (solid line) and filtered relaxation / stress recovery curve (dashed line).

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Fig. 8: Start point τ = 0 for analysis of vibrati-on sections should not be the transition point (x) for reasons of com-parability, but rather the first coincidence of strain with the mean, or „offset“ value (*).

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Fig. 9: Course of test execution with defined rest periods in between.

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curves of the first and second test are almost coincident. Due to the Mullins effect the stress level of the third test is significantly lower compared to the sec-ond test. A slight influence of the rest period between the third and fourth test is visible, but in general the curves of tests no. 3, 4, 5 approximately coin-cide as well. We may conclude that each test causes softening that influences the results of subsequent tests done within 2 hours, but the softening al-most completely recovers after a rest period of one week. These results sug-gest that either the test piece should be left to rest for at least a week between two consecutive tests or that different test sequences should be carried out in quick succession. For reasons of time the second option is chosen. Further studies were done, which revealed that the test piece should be strained by at least 3 preliminary strain sequences prior to the real tests to minimize fur-ther effects of stress softening.

Experimental resultsEach test is of the generic form indicated in Figure 5 and comprises four vibration sections with magnitudes of offset and amplitude characteristic of the test, to-gether with four sections at fixed strain. The chosen vibration amplitudes were always 2% or 10% (in addition to 0%) and the chosen offsets were 10%, 50% or 90%. Each combination of amplitude and offset was tested in a separate test se-quence. These examples of combinations of amplitude and offset serve to point out how to interpret experimental re-sults in terms of relaxation and recovery.

Figure 11 gives an overview of the re-sulting mean stress curves. For reasons of clarity the fixed strain curves are not de-picted in these figures. The dashed curves result from the sections that are equiva-lent to a relaxation experiment (hence-forth “relaxation curves”), the solid curves from the sections that represent an inter-action of relaxation and stress recovery (henceforth “interaction curves”).

All curves approach a quasi steady state after some seconds. As expected from simple relaxation experiments, the curvature of the relaxation curves increas-es with increasing offset. The curvature of the interaction curves decreases with in-creasing offset. In the case of amplitude 10% and offset 90% it even reverses. This observation leads to the assumption that the stake of relaxation in the interaction grows with increasing offset.

Superposition An interesting aspect of evaluation is to superpose, using vertical shifts, the me-an stress curves of different vibration amplitudes (10%, 2%, 0% = fixed strain; offset constant) as was done for the reco-very curves in the paper of Ahmadi et al [1]. As illustrated in Figure 12 the super-position works well for the offsets 10% and 50%. In the case of offset 90% the relaxation curves still coincide but not the interaction curves; this may be due to the fact that it is a special case. In Tab-le 1 the magnitudes of all the vertical shifts to fixed strain curves for the diffe-rent offsets and amplitudes can be seen.Looking back to the stress recovery tests in the paper from Ahmadi et al [1], the investigated vibration amplitudes were at most 14%. In a thought experiment these amplitudes are to be increased up

to 100%. In this extreme case the result-ing mean stress curve would be a straight line at zero stress, which could not be superposed with the other curves.

Both the observation of the interac-tion curves and the thought experiment regarding the recovery curves lead to the notion that the superposition of recovery effects is limited by a boundary, which would be interesting to investigate quantitatively.

Assessing material modelsAnother interesting aspect of evaluation is to compare the experimental results to those delivered by material models. As an example an integral approach based on Boltzmann’s superposition principle [6] is evaluated. It is given by:

𝜎𝜎(𝑡𝑡) = � 𝐺𝐺 (𝑡𝑡 − 𝑠𝑠)𝛾𝛾 ̇ 𝑑𝑑𝑠𝑠𝑡𝑡

0 (8)

Fig. 10: Influence of rest period between two tests: Mean stress curves with rest time of 1 week between (1) and (2), 75 minutes between (3) and (4) and 30 minutes between (4) and (5), (offset 90%, amplitude 10%).

Fig. 11: Overview of the resulting mean stress response for all combinations of offset and amplitude. Left: amplitude 2%, right: amplitude 10%. Offsets: +/-10%, +/-50%, +/-90%.

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1 Magnitudes of all the vertical shifts to fixed strain curves.Amplitude 2% Amplitude 10%

Offset +/- 10% +0.0059 / -0.0091 +0.0135 / -0.0183Offset +/- 50% +0.0081 / -0.0080 +0.0141 / -0.0285Offset +/- 90% +0.0086 / -0.0042 +0.0091 / -0.0227

where G(t-s) is the shear relaxation mod-ulus and γ⋅ is the strain rate. In principle the Boltzmann theory captures all the effects – relaxation, recovery, hysteresis, the effect of frequency on loss and stor-age moduli. The relaxation modulus G(t) is evaluated separately for each offset by dividing the stress response in a simple relaxation test by the strain [7; 8], see Figure 13. Note that G(t) depends on strain, in conflict with the assumption of linearity underlying Boltzmann’s princi-ple. For the calculations the function G(t) was taken to be that of the offset strain. The calculated mean stress curves for the same offsets and amplitudes as in the experiments are illustrated in Figure 14.

The graphs agree qualitatively with the experimental results of Figure 11. However, the theoretical relaxation and interaction curves approach one another and coincide in quasi steady state. Addi-tionally, the increase/decrease of curva-ture with increasing offset is different to the experimental results.

DiscussionFigure 13 shows that filled SBR does not have linear viscoelastic behaviour, since the shear relaxation modulus depends on

the strain. The modulus falls as the strain is increased, as previously reported for the same material both for the quasi static stress-strain curve and for the am-plitude dependence of the dynamic mod-ulus [2]. The latter effect has little de-pendence on the static strain, so it is not simply an effect of nonlinear elasticity, although a unifying inelastic model has been proposed by Ahmadi et al, [9, 10, 11] who investigated the same material. However, if the relaxation modulus is normalised by the stress at a reference time, this dimensionless relaxation rate is only weakly dependent on strain, being a little greater at lower strains and varying by 33% over the strain range 10% to 90%. This compares to a 65% variation in the magnitude of the relaxation modulus.

This suggests that whatever the origin of the nonlinearity of the stress-strain behaviour, it does not strongly affect the evolution of relative stresses with time

over the time scale covered in Figure 12. The semiquantative correspondence be-tween Figures 11 and 14 support this view. The most striking difference be-tween the experimental and theoretical plots is the absence of convergence of the experimental „relaxation“ and „interac-tion“ plots. Thus the nonlinearity of the material seems to manifest itself as sub-stantially time-independent stress off-sets. It is anticipated that similar time-in-dependent deformation or “set” would result from a similar stress-history com-mand. Such time-independence is sug-gestive of elasto-plastic effects in materi-als. The amplitude dependence of the dy-namic modulus of filled rubber has been modelled with some success based on such an approach [9, 10, 11].

Conclusion Clear definitions of the terms relaxation and recovery extending those given by

Fig. 12: Super-position of the mean stress curves (with the negative quad-rant mirrored into the positi-ve) during vibra-tion sections of amplitudes 10%, 2%, 0% (fixed strain): All curves are shifted verti-cally towards the fixed strain.

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Fig. 12a: Offset 10%: redefined zero and mirroredFig. 12b: Offset 10%: vertical shifted

Fig. 12c: Offset 50%: all curves coincideFig. 12d: Offset 90%: Relaxation curves (upper curves) coincide, superposition not

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Fig. 14: Overview of the resulting mean stress response for all combinations of offset and amplitude, calculated by the hereditary integral model of Boltzmann. Left: amplitude 2%, right: amplitude 10%. Offsets: +/-10%, +/-50%, +/-90%.

Gent [3] are pointed out. Based on this, a suggestion for a steady state dynamic test sequence is given, which covers the phenomena of relaxation and recovery. Evaluation and testing methods are pre-sented. Results of carefully devised tests carried out on filled SBR test pieces and corresponding mean stress curves are presented and interpreted with respect to recovery and relaxation. It is shown how the results can be used for assessing the applicability of material models, e.g. Boltzmann’s superposition principle. The superposition, by vertical shifts, of mean stress responses of vibrations with differ-ent amplitudes delivers coinciding curves, provided the sums of offset strain and vibration section amplitudes are sig-nificantly less than the high amplitude (100% in this paper) conditioning cycles. The magnitudes of these vertical shifts are readily quantified, and it is suggested that their prediction would provide a

sensitive test for candidate nonlinear viscoplastic material models.

References[1] Ahmadi, H.R., Ihlemann, J., Muhr, A.H., Time-

dependent effects in dynamic tests of filled rubber (2008). Proceedings of the 5th Euro-pean Conference on Constitutive Models for Rubber, ECCMR 2007, 305.

[2] Besdo, D., Ihlemann, J., Kingston, J.G.R., Muhr, A.H., Modelling inelastic stress-strain phe-nomena and a scheme for efficient experi-mental characterization (2003). Constitutive Models for Rubber, 3, 309.

[3] Gent, A.N., Relaxation processes in vulcan-ized rubber. I. Relation among stress relaxa-tion, creep, recovery, and hysteresis (1962). Journal of Applied Polymer Science, 6 (22), 433.

[4] Orlando, V., Development of a Cost-Effective Versatile Servo-hydraulic Controller Using NI CompactRIO (2017). National Instruments Innovations Library.

[5] Kanzenbach, L., Stockmann, M., Ihlemann, J., Ex-

tension of the Testing Machine Control for High-precision Uniaxial Tension Measurements (2016). Materials Today: Proceedings, 3 (4) 993.

[6] Boltzmann, L., Zur Theorie der elastischen Nach-wirkung (1878). Annalen der Physik, 241 (11), 430.

[7] Ferry, J.D., Viscoelastic properties of polymers (1980). J. Wiley & Sons, New York, 3rd Edition.

[8] Tschoegl, N.W., The phenomenological theory of linear viscoelastic behaviour (1989). Springer-Verlag, Berlin.

[9] Ahmadi, H.R., Kingston, J.G.R., Muhr, A.H., Dy-namic properties of filled rubber (2008). Part I-experimental data and FE simulated re-sults. Rubber Chem & Tech, 81, 1-18.

[10] Ahmadi, H.R., Muhr, A.H., Kingston, J.G.R., Gra-cia, L.A., Gómez, B., Interpretation of the high low-strain modulus of filled rubbers as an in-elastic effect (2003). Constitutive Models for Rubber III - Proceedings of 3rd European Con-ference on Constitutive Models for Rubber, ed by J. Busfield and A. Muhr, Balkema.

[11] Ahmadi, H.R., Kingston, J.G.R., Muhr, A.H., Dynamic properties of filled rubber Part II Physical basis of contributions to the mod-el. Rubber Chem & Tech, 2011, 84, 24.

AcknowledgementsThanks are due to TARRC and its staff for their support, and notably to Vincenzo Orlando for his control interface and help which enabled the tests to be done on the servohydraulic machine.

The authorsDr. Alan Muhr is the Unit Head of Engineer-ing & Design at TARRC (Tun Abdul Razak Research Centre). Prof. Dr.-Ing. habil. Jörn Ihlemann is head and Lars Kanzenbach is a scientific assistant of the Professorship of Solid Mechanics at TU Chemnitz. Anneke Lampe is studying Computational Engi-neering/Mechanical Engineering (Master) at TU Darmstadt. This work was fulfilled during her time as TU Chemnitz student.

Fig. 13: The shear rela-xation modulus G(t) is gained from a simple relaxation test, in which the test piece is strained by 10%, 50% and 90%. The respon-ding stress (left) is divi-ded by the applied strain to calculate the relaxation modulus (right).

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