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Elastomers · biaxial testing · tearing energy · material modelling · thermo-mechanical analysis

For the investigation of elastomers un-der near-realistic loading, a new Biaxial Tester was designed and tested. The clamping system of the device was par-ticularly optimized for biaxial testing of rubber materials with high elongations. Measurements under different loading conditions enable a more reliable esti-mation of the parameters of the mate-rial models. Modified specimen geome-try is proposed to increase the size of the homogeneous stretched region of a planar specimen. Testing possibilities will be discussed for such applications like the parametrization of constitutive material models and the description of crack propagation with the J-integral approach. Finally, we will describe how to combine biaxial testing with ther-mography and separate energy storage and dissipation during biaxial mechani-cal tests.

Zweiachsiges Testsystem für die statische und dynamische Prüfung von Elastomeren und seine Anwendungen Elastomere · Biaxiale Prüfung · Weiter-reissenergie · Materialmodellierung · Thermomechanische Analyse

Für die Untersuchung von Elastomeren unter praktisch relevanten Belastungen wurde ein neuer Biaxial Tester aufge-baut. Das Klemmsystem wurde insbe-sondere für größere biaxiale Dehnungen optimiert. Eine Prüfung unter komple-xen Belastungsbedingungen ermöglicht eine zuverlässige Parametrisierung von Materialmodellen. Für eine Vergröße-rung des homogen gedehnten Bereichs wurde eine modifizierte Prüfkörpergeo-metrie vorgeschlagen. Die Möglichkei-ten für die Parametrisierung von konsti-tutiven Materialmodellen und die Ana-lyse des Risswachstums mittels des J-In-tegral werden diskutiert. Zum Schluss wird eine Möglichkeit, den Biaxial Tester mit Thermographie zu kombinieren, be-schrieben, um die Beiträge von Energie-speicherung und Dissipation bei den mechanischen Tests zu separieren.

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

IntroductionElastomers are a widely used group of soft materials with specific properties allowing a variety of technical applica-tions. These include for example tires, V-belts, gaskets, dampers and more. To use the elastomers in an appropriate way it is necessary to analyze their behavior under near realistic loading conditions; especially in the case of damaged mate-rial, this is very important.

The classical methods for the investi-gation of mechanical properties of rub-bers include tests with uniaxial loading, e.g. the uniaxial tension and the so called “pure shear” specimen tests. These measurements can be done relatively simply and quickly and nowadays they are used for several industrial applica-tions. However, the disadvantage of these methods is their restriction to a narrow window of loading profiles and the limitation in describing only a few of the loading states. So these classical methods are less appropriate for com-prehensive investigations of constitutive rubber material models and, as a conse-quence, are less suitable when modelling the complex loading states of new rub-ber parts.

Similarly, this is the case for a fatigue and fracture analysis of rubber materials. The classical methods are based on the uniaxial tests with different geometries of specimens. The SENT-geometry („sin-gle edge notched tension“), DENT-geom-etry („double edge notched tension“) or „pure shear“– geometry are the best known. Using SENT or DENT specimens, crack growth can be found under com-plex but not well-defined loading condi-tions. The results are of limited value if the complex loading is not described ap-propriately by the used material models. The so-called “pure-shear” geometry re-alizes nearly pure shear stress conditions [1], [2]. The most popular approach for the global analysis of crack growth is fol-lowing Rivlin and Thomas [3]. However, the obtained tearing energy does not represent a concise physical quantity; instead energy terms contributing to crack growth and dissipation in the crack

tip far field are lumped together. For a more local approach, Cherepanov and Rice [4], [5] introduced the J-integral. It corresponds to the tearing energy for the special case of straight crack propagation in an elastic material under static condi-tions. However, the mechanical behavior of rubbers is far from being perfectly elastic and the cracks often propagate in a direction which is not straight. This causes errors when using these simpli-fied approaches [6]. Therefore, the meth-od needs to be expanded for hyperelastic and viscoelastic materials.

This complex analysis of elastomer behavior needs enhanced methods and corresponding equipment. The Leibniz-Institut für Polymerforschung Dresden e.V. (IPF) together with Fa. Coesfeld Gm-bH & Co. KG, Dortmund, Germany, de-signed and built an innovative Biaxial Tester [7], [8], [9] for the non-destructive testing of rubber material under quasi-static and cyclic loading conditions, as well as carrying out investigations with respect to crack propagation. It opens new experimental possibilities, gener-ates new and advanced knowledge about the behavior of elastomer materi-als under different practice-oriented complex loading conditions and finally supports better design and optimization of rubber parts.

Biaxial testing System for high static and dynamic Loading of Elastomers and its Applications

AuthorsSofya Dedova, Konrad Schneider, Gert Heinrich, Dresden, Germany

Corresponding Author:Sofya DedovaLeibniz-Institut für Polymer- forschung Dresden e.V.Hohe Straße 601069 Dresden, GermanyE-Mail: dedova@ipfdd.de

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1. Construction aspects and possibilities of Biaxial TesterThe Biaxial Tester consists of four sepa-rately controlled electro-mechanical line-ar motors. Each of them is equipped with sensors measuring the load and displace-

ment. The clamping system consists of opposed sets of individual clamps.

The Biaxial Tester is shown in Figure 1 and some technical data are summarized in Table 1. More technical details are ex-plained in [8] - [10].

Fig. 1: Biaxial Tester of COESFELD Materialtest.

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Fig. 2: Analysis of a crack contour with bottom illumination (left), strain field measure-ment with a top illumination (middle) and analysis of a strain field (major strain) near the crack tip with top illumination and using ARAMIS Software (right) for a SBR-sample.

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During experiments with complex con-figuration, the load-displacement data will be reordered and used for further analysis of stress-strain conditions. Paral-lel to the mechanical measurement a synchronized optical analysis can be done. For the investigation of the strain fields, patterned specimens are used and re-corded using top illumination. For such experiments the estimation of a strain field is done by means of ARAMIS Soft-ware from GOM, Germany. For the investi-gation of a crack contour, bottom lighting of the specimens is suggested. An exam-ple of a SBR sample (CB-filled Styrene-Butadiene-Rubber) stretched equibiaxial up to ex = ey = 10% is shown in Figure 2.

An infrared camera can be used for a thermo-mechanical analysis. In this way, time dependent temperature fields can be measured and investigated under complex loading conditions, which can be related to stored and dissipated energy.

1.1. Clamping systemThe clamping system of the biaxial tester was designed and built for a stretching range of ex = ey = 150 %.

The system has to work not only under static but also under dynamic conditions as well. For this reason the limits of the dynamic loading of the clamping system were tested. The inertial force of the sys-tem was measured for different frequen-cies, using a harmonic loading. The maxi-mum inertial force can be described as:

Fmax=4meff π2 f 2 A (1)

with Fmax maximum of inertial forcemeff effective mass, calculated from the experiment as: meff=m+mnm,mn relevant mass without and with clamping systemf frequencyA amplitude during harmonic loading

Increasing the deformation range re-quires a high displacement range for the individual clamps in the direction nor-mal to the individual loading direction. This means that the whole clamping structure will be relatively large, leading to high inertial forces. However, a high inertial force disturbs the experimental procedure. On the one hand, the limits of the load cells are a restriction on the whole system. On the other hand, a heavy system needs more time (more cycles) to optimize the controller and to reach the required experiment settings.

1 Technical specification of the Biaxial Tester

Control configuration Separate or by pairs (every axis in the opposite direction)

Load configuration Sine, Gauss-pulse, triangle, rectangle, user-specificloading profile

Max. displacement 120 mm for each axis (about 150% strain), max. displacement of each motor up to 60 mm

Loading velocity Max. 50 Hz, useful 10 Hz

Force range 2000 N

Data recording Total number of cycles, individual cycles or individual data of selected cycles

Image recording conditions Continuously or individual triggered

Specimen geometry Rectangular or quadratic up to 77 mm x 77 mm with beading bulge at the edge

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Therefore, the clamping system was designed using carbon fiber rein-forced light weight composites, see Fig-ure .Main advantages of the new clamping system include the following aspects:

■ the light weight construction enables high dynamic loading

■ long clamp rails on two levels enable 150 % of strain

■ the system is further extendable for higher displacements

■ all clamps have the same geometry, this makes it easier to produce or re-pair the clamps

■ top and bottom positions of the clamp rails provide a symmetrical loading with respect to the load cells

The present amplitude-frequency limit is shown in the Figure .

1.2. Specimen geometryBiaxial specimen forms and testing methods for rubber are known from the literature [11], [12]. These concepts allow a biaxial loading state only in the middle of the specimen. It can be successfully used for measuring stress-strain curves, but cannot be appropriately adapted for fracture or fatigue tests.

For stiff materials (for example met-als) other specimen forms will be used. The corresponding concepts are de-scribed in [13] - [19]. Some of these methods allow the determination of crack propagation, but the specimens are not appropriate for high elongations.

The specimen for the Biaxial Tester is square-shaped with a dimension of 77 mm x 77 mm. Two specimen clamping concepts were chosen and tested for their suitability for high deformations. Both specimens have a bulge at the edge to re-alize a force-free fixing of the specimens by form fit. The first one has a simple bulge with a diameter of 4 mm, while the second one has an additional region be-tween the bulge and the middle of the specimen with 3 mm thickness for damp-ing of the clamping and free-space influ-ence between the clamps and a bulge di-ameter of 5 mm, see Figure , top.

To check the homogeneity of the strain field of the specimen, both geom-etries were analyzed by estimating the strain fields with the image correlation software ARAMIS from GOM, Germany (see Figure , bottom). A natural rubber with 20 phr carbon black, cross-linked with sulphur (NR20) was chosen for this experiment in which the nominal biaxial strain was 85 %. It is clearly visible that

the specimen with the first geometry displayed larger inhomogeneities. The positions “between clamps” with mini-mum values of strain and “clamp posi-tions” with maximum values of strain lie in an area of about 20 mm. The equiva-

lent strain ∆e spreads here between 80 - 100 %. In the middle area of the speci-men there are no significant deviations of e.

In the second case of specimen geom-etry the influence on the border area was

Fig. 4: Amplitude-frequency range of the biaxial-tester with the expanded clamping system.

Fig. 5: Edge of the NR20-sample with simple (top left) and improved geometry (top right) as well as strain field of samples with the conventional (left side) and the new sample geometry (right side).

Fig. 3: Clamping system in starting position (left) and with a biaxial stretched squared sample (right).

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damped by the thick region, which in-creases the region with a homogeneous strain in the middle of the specimen. The heavily disturbed external area is only about 9 % of the stretched length of the specimen.

Another positive aspect of such mac-roscale specimens is the eligibility for the fracture and fatigue experiments. For the analysis of crack growth rate, crack form and crack deviation the first geometry is sufficient. However, for the analysis of the energetic aspects of crack growth or crack tip, the highly homogeneous stretched area is needed. In this case, the second geometry is recommended.

2. Applications of the biaxial testing methodThe high biaxial deformation range with a relatively wide measuring field and homogeneous deformation and optical analysis provides a lot of new possibili-ties for the investigation of rubber mate-rial properties. It also opens new possi-bilities for fatigue and fracture tests for elastomers. Basically, all the tests could be done under nearly realistic loading conditions. High stretching (up to ex = ey = 150 %) especially gives the possibility to better parametrize material models and to evaluate their efficiency for fur-ther applications.

2.1. Parameterization of material modelsNowadays, the successful designing and production of rubber parts will be first modelled and tested using FEM. For this reason, correct and precisely measured data and correct modelled data are need-ed. The biaxial testing method gives the possibility to test the material under complex loading conditions and there-fore provides more data for the parame-terization of some material models.

From the literature, different constitu-tive models for rubber are known. The models have general differences: there are phenomenological models (e. g. Ogden Model) and physical based mod-els (e. g. Extended Tube Model (ETM)). For the analysis of hyperelasic behavior here the ETM was chosen [20], [21]. The mod-el takes into account the entanglements and mobility restrictions of molecular chains and fits the measured data better than other models [22], [23]. Further-more, due to physical mining of the pa-rameters, only one parameter set is needed to describe all the loading condi-tions. Using the stress-strain-values of 3 experiments (3 different loading condi-tions: equibiaxial (EB), “pure shear” (PS), and uniaxial (UA)) leads to precise fitting and parameterization of some chosen materials.

Fig. 6: Stress-strain-curve for the SBR-sample, case 1 with “pure shear” demullinisation up to 100% on the Zwick.

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The ETM describes the strain energy den-sity as:

𝑊𝑊(𝜆𝜆𝑖𝑖) =𝐺𝐺𝑐𝑐2 �

(1 − 𝛿𝛿2)(𝐼𝐼1 − 3)1 − 𝛿𝛿2(𝐼𝐼1 − 3)

+ 𝑙𝑙𝑙𝑙[1 − 𝛿𝛿2(𝐼𝐼1 − 3)]� +2𝐺𝐺𝑒𝑒𝛽𝛽2 (𝐼𝐼𝑔𝑔𝑒𝑒𝑙𝑙 − 3)

𝑊𝑊(𝜆𝜆𝑖𝑖) =𝐺𝐺𝑐𝑐2 �

(1 − 𝛿𝛿2)(𝐼𝐼1 − 3)1 − 𝛿𝛿2(𝐼𝐼1 − 3)

+ 𝑙𝑙𝑙𝑙[1 − 𝛿𝛿2(𝐼𝐼1 − 3)]� +2𝐺𝐺𝑒𝑒𝛽𝛽2 (𝐼𝐼𝑔𝑔𝑒𝑒𝑙𝑙 − 3)

(2)

here W is the strain energy density, λi are the principal stretches, I1 and Igen are the first and the generalized invariant, defined as I1 = λ1

2 + λ22 + λ3

2 and Igen = λ1-β + λ1

-β + λ-β3 .

Gc and Ge, are crosslink and topological constraint moduli; β is the constrained release fitting parameter and δ is the large strain parameter describing the limited network chain extensibility. All those parameters are physical parame-ters, describing certain aspects of the polymer network.

For the fitting of curves, derivatives of equation 2 were used. The detailed de-scription of the analytical solution for the ETM is given in [20].

For the experiments, a commercial SBR-rubber (SBR) was used. The equibi-axial experiments were performed by the Biaxial Tester. The uniaxial and “pure shear” experiments were done by a Zwick/Roell Z010 for reaching high elon-gations using conventional “pure shear” geometry (80 x 10 x 1,5 mm3).

The ETM describes hyperelastic equi-librium conditions of elastomers. There-fore, first the equilibrium curves were estimated. For approaching hyperelastic conditions (equilibrium curve) a quasi-static step experiment was done. For equibiaxial loading the strain of the specimen was gradually increased in steps of 10 % up to 80 %, and gradually decreased in steps of 10 % with the strain rate of 20 %/sec, the holding time

2 Parameters for the Extended Tube Model with variations of loading history of the “pure shear” test for the SBR-sample.Parameter: Gc Ge δ β

Case 1 0.2120 0.3405 0.2417 1

Case 2 0.2701 0.2720 0.2345 1

Fig. 7: Stress-strain-curve for the SBR-sample, case 2 with “pure shear” demullinisation up to 300% on the Zwick.

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for each step was 3 min. The same was done for a uniaxial loading; a gradual increase of 10 % each time up to 280 % and a gradual decrease of 10 % each time, with a strain rate of 2,16 %/sec, the holding time for each step was also 3 min. The “pure shear” conditions were tested up to different maximal loadings. Case 1 contained gradual increases of 10 % each up to 80 % and gradual de-creases of 10 % each. Case 2 contained gradual increases up to 220 % with 6 holding steps and gradual decreases with a strain rate of 16,7 %/sec. Using the relaxation rate for every holding point (lambda step) the equilibrium stress was extrapolated.

Before testing, the specimens were dy-namically loaded for the removal of Mull-ins effect: for equibiaxial conditions: 100%; “pure shear”: 100 % (case 1) and 300 % (case 2); uniaxial: 380 %. The variation was done for the “pure shear” conditions by possibility analyzing these conditions on both pieces of testing equipment. The measured and modelled data are shown in Figure 6 and Figure 7. The estimated coef-ficients for the ETM are shown in Table 2.

The measured values for materials with different loading history differ from each other, and so do the parame-ters of the ETM. These differences are higher in the region of high stretching. A set of ETM-parameters, which is esti-mated by the simultaneous use of measurements with different loading profile fits all measurements in a satis-factory manner and is relatively stable with respect to the small scattering of measured values.

2.2. Viscoelastic material behavior and dissipative heatingElastomer parts are often used under dynamic loading conditions. Under these loading conditions the material shows mainly viscoelastic behavior. Precise ana-lyzation of the materials under these conditions is necessary to describe the behavior of loaded parts in a correct way. For this reason, the dynamic stress-strain experiments were combined with a measurement of internal heating by a thermo-camera VarioCam from InfraTec, Dresden, Germany, with the analyzing software IRBIS.

Experiments were done using the NR20 sample (natural rubber with 20 phr of carbon black, cross-linked with Sulphur) once more. The tests were per-formed with equibiaxial dynamical si-nusoidal loading (1 Hz) with ex = ey =

30 %, 60 % and 90 % respectively. A typi-cal temperature distribution of the specimen is shown in Figure 8 (left and middle).

In these experiments the relationship between mechanical energy, dissipative sample heating and heat loss to the sur-rounding was investigated. It has been found that most of the energy, which is provided by mechanical work, goes into the internal energy of the specimen, but only a small part is dissipative heating. The mean temperature of the specimen at the point of maximal load increases exponentially (Figure 8 right). Finally, the energy loss to the surrounding equals the dissipated energy of a whole loading cycle. The different shares of energy are shown in Figure 9 and investigated in detail in [24].

2.3. Analysis of crack propagationA popular approach for the description of crack propagation in rubber is based on the global analysis of crack growth, fol-lowing Rivlin and Thomas [3]. The local approach according to Cherepanov and Rice [4], [5] by the J-integral is more suit-able for arbitrary crack paths. For the spe-cial case of straight cracks in an elastic

material under static conditions, the J-in-tegral corresponds to the tearing energy. However, due to the viscoelastic behav-iour of rubber, bulk dissipation must be taken into account. There are some re-search groups working on the adoption of the J-Integral concept for elastomer mate-rials (for example [25]. In the following experiments the calculation of tearing energy by means of J-Integral will be shown for biaxial load conditions.

In principle, the J-Integral is a closed-line integral, which measures the change in energy due to a change in stress distri-bution within the integration area:

J = ∫Γ(Wn- - σ=n-∇u- )ds (3)

with Γ path of integration; W elastic strain energy density; n- path normal; σ= stress tensor; u- displacement gradient; ds path element.

This J-integral method has the following advantages: It is independent of the ge-ometry of the specimen and the loading conditions; it can be used in the case of crack deviation and it is path independ-

Fig. 8: Temperature distribution of the NR20-sample during biaxial harmonic loading with a strain amplitude of 60 % at 1 Hz at minimal and maximal strain of the 250th loading cycle (left and middle) and curves of dissipative heating over time at different strains (right).

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Fig. 9: Temperature distribution of the NR20-sample during biaxial harmonic loading with a strain amplitude of 60 % at 1 Hz at minimal and maximal strain of the 250th loading cycle (left and middle) and curves of dissipative heating over time at different strains (right).

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ent. The last hypothesis is only correct for the elastic materials, but it leads to some deviation of results for the materials with dissipative effects (see the results below).

Distinct quasi-static and dynamic ex-periments were carried out with the Bi-axial Tester. The parameters for the con-stitutive ETM [20], [21] were estimated. For the estimation of the tearing energy with the J-Integral method, dynamic ex-periments were performed. The notched samples were tested with the frequency of 1 Hz and loaded up to 2000 cycles. Dif-ferent kinds of rubber were used for the validation of the method. In Figure 10 the example of filled SBR (Styrene-Buta-diene-Rubber) with the squared samples, tested under uniaxial, equibiaxial and “pure shear” loading is shown. The J-In-tegral was calculated for 4 different strains: 5, 8, 10 and 13 % at cycles be-tween 500 and 2000 during continuous crack growth.

Figure 10 shows the relation between the crack growth rate and the tearing energy for different loading conditions on a logarithmic scale. The values for the different loading regimes lie on a com-mon curve. The onset of accelerated crack growth (upward deviation from the linear regime in Figure 10) starts under uniaxial loading at lower tearing ener-gies compared to equibiaxial loading. These differences will be investigated in more detail in the future.

ConclusionsIn this work a method of biaxial analysis of rubber specimens is presented. The new possibilities of the Biaxial Tester are:• static and dynamic biaxial stretching

up to 150%• 2d-strain estimation by image correla-

tion• fracture and fatigue analysis for com-

plex loading conditions

Fig. 10: Estimation of the J-integral for the SBR-sample: Integration path around the crack tip (left), dependence of the crack growth rate during cyclic loading on the J-integral (right).

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• analysis of the 2-d-temperature distri-bution of the specimen for the estima-tion of energy storage and dissipation

The device opens new possibilities to in-vestigate material properties in a more precise manner. Some applications like investigation of complex material proper-ties, characterization of viscoelastic and thermal behavior, and fatigue analysis are shown. This many-sided experimental support leads to a more complex under-standing of rubber behavior and gives the possibility to use it for designing new elastomer parts in an optimal way.AcknowledgementsThe authors would like to thank the Company Goodyear for financial sup-port. Thank goes to Dr. Axel Spickenheu-er and Nicole Schneider for their support in designing and building the clamping system, Kristina Eichhorn for measure-ments with the Biaxial Tester and Zwick tensile machine, and to Lutz Zybell for help using thermography. The acknowl-edgements also go to Dr. Roberto Lom-bardi, Dr. Jan Domurath, Dr. Karsten Brüning, and Dr. Arnaud Vieyres for valu-able discussions and to Peter Dowidat (Fa. Coesfeld) for support with the Biaxial Tester software. References[1] R. Stoček, PhD-Thesis, Dynamische Rissaus-

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