Material property characterization for finite element analysis of tires.
In recent years, FEA has been extensively used in the tire design process to reduce the design cycle time by minimizing prototype making and testing. However, the use of FEA in tire analysis is a complex task for the designers because it is highly non-linear in terms of both material and geometric behavior. Large deformation of tires, often as large as 20% (which is far larger than other mechanical structures), is the best representation of geometrical non-linearity. Some advanced nonlinear FEA codes enable us to handle problems associated with tire analysis. It is very important to recognize that an accurate mathematical model selection to describe the mechanical behavior of various tire components is essential in FEA to relate strains in the tire model to stresses.
Extensive research (refs. 1-5) on constitutive modeling of rubber materials has resulted in the availability of a number of material models describing the mechanical behavior of rubber more accurately. In the past few years, the introduction of some popular material models into the commercially available finite element codes has made the design engineer's tasks easier. It is important to mention that the choice of material model is largely dictated by various factors like type of application, strain range, availability of required test data, etc. It is obvious that the accuracy of design analysis using these material models is directly related to the quality of the input as material constants. The accurate determination of material constants for rubber material depends on several aspects like loading rate, strain history, amount of strain, type of deformation, etc. It has been observed that deformation results in the softening of the rubber, which is a major obstacle to getting equilibrium stress-strain data. Most of the softening occurs in the first deformation, and after a few deformation cycles the rubber approaches a steady state with a constant stress-strain curve. Softening in this way occurs in vulcanizates with or without fillers. This phenomenon has been termed as the Mullins (ref. 6) effect. To obtain equilibrium stress-strain data, sufficient mechanical conditioning of the test sample is required. In the subsequent sections, the details about the material models used and a brief description of experimental methodologies for material constant determination will be discussed.
Hyperelastic material modeling
Generally, hyperelastic models used to describe the high deformation, mainly elastic and reversible loading, simultaneously assume highly non-linear and nearly incompressible material behavior. Many theoretical models have been developed to characterize the hyperelastic material properties of rubber. Some are based on statistical thermodynamics, while others take the phenomenological approach.
The statistical thermodynamics approach is based on observations that the rubber elastic forces arise almost entirely from the decrease in entropy with increase in applied extension, which follows from the structure of unstretched rubber being highly amorphous and hence of high entropy. This approach has generally dealt with assumed statistical distributions of the lengths, orientations and structure of rubber molecular networks, but appears to be inadequate for more than moderate strains. More information can be found (ref. 1).
The phenomenological approach treats the problem from the viewpoint of continuum mechanics. A mathematical framework is constructed to characterize the stress-strain behavior without reference to the microscopic structure. It assumes rubber to be an isotropic material in its unstrained state; that is, the long chain molecules of the rubber are assumed to be randomly oriented. Stretching of rubber causes orientation of rubber molecules, but as the orientation is in the direction of stretching, the assumption of isotropy can be said to remain valid. This assumption of isotropy is fundamental to the characterization of rubber by a quantity that is known as the strain energy density; the stored energy per unit volume. Some of the well known constitutive models of this type are: NeoHookean, Mooney-Rivlin, Ogden, Yeoh and the Van der Waals model. Reviews on hyperelastic material models can be found in published literature (refs. 7 and 8).
Many empirical rubber elasticity models have been developed based on the general polynomial form of the strain energy function. The most general strain energy function proposed by Rivlin is:
(1) W = ijk = n[summation over (ijk = 0)] [C.sub.ijk] [([I.sub.1] - 3).sup.i] [([I.sub.2] - 3).sup.j] [([I.sub.3] - 1).sup.k]
Where [I.sub.1], [I.sub.2] and [I.sub.3] are the principal strain invariants. In case of incompressible material, [I.sub.3] = 1, then the strain energy function reduces to
(2) W = ij = n[summation over (ij = 0)] [C.sub.ij] [([I.sub.1] - 3).sup.i][([I.sub.2] - 3).sup.j]
Taking only the first and second term of the equation 2
(3) W = [C.sub.10] ([I.sub.1] - 3) + [C.sub.01] ([I.sub.2] - 3)
This is known as the Mooney-Rivlin material model. This is the most popular material model used for analysis of rubber products. However, this model has an inherent limitation with respect to the following aspects. This model is not capable of predicting large strain behavior of rubber material. It has also been observed that Mooney-Rivlin constants determined from one deformation mode have limited value for predicting behavior in other deformation modes. Furthermore, this model predicts a linear shear stress-strain relationship with a constant shear modulus, which is not observed for carbon black filled rubber materials in practice. The Neo-Hookean model also has similar types of limitations. Although the Ogden model can predict large strain behavior (including upturn of stress strain curve) very well, it works best when multiaxial test data are available (ref. 9). It is worth mentioning here that the above-mentioned limitations are very much crucial for tire analysis, and therefore a suitable material model is required. With this background, Yeoh (ref. 10) proposed a novel strain energy function for characterization of rubber materials. The Yeoh model assumes that the strain energy density is a function of only the first principal strain invariant [I.sub.1]. The reason for not taking [I.sub.2] into consideration is that its contribution is much less compared to [I.sub.1], which is also supported by published experimental data (ref. 11). This is a simple cubic strain energy function in terms of ([I.sub.1-3):
(4) W = [C.sub.10] ([I.sub.1] - 3) + [C.sub.20] [([I.sub.1] - 3).sup.2] + [C.sub.30] [([I.sub.1] - 3).sup.3]
The stress-strain relation in uniaxial extension is given by:
(5) [sigma]/([lambda] - [[lambda].sup.2]) = 2[C.sub.10] + 4[C.sub.20] ([I.sub.1] - 3) + 6[C.sub.30] [([I.sub.1] - 3).sup.2]
Some unique features of the Yeoh model distinguish it from other models and include:
* Unlike other material models, the Yeoh model is applicable over a much wider range of deformation.
* This model is able to predict the stress-strain behavior in different deformation modes from data obtained in one simple deformation mode like uniaxial extension. This circumvents the need for a rather cumbersome experiment, like biaxial extension or a shear test, which directly relates the reduction in testing time, as well as cost.
* This model can also predict the variation of shear modulus with increasing deformation.
The Yeoh material model has been chosen to describe the hyperelastic properties of rubber compounds for this article.
Viscoelastic material modeling
Viscoelasticity, or delayed response to stress change, of elastomers is essentially important for tire performance prediction (ref. 12). Equilibrium or reversible stress-strain relations from the kinetic theory become submerged in viscoelastic effects of such materials. It is well known that elastomers exhibit time-dependent phenomena such as stress relaxation, creep and frequency dependence of the dynamic properties. These viscoelastic properties have a significant influence towards the traction (wet and dry), steering response and rolling resistance (fuel efficiency) properties of a tire. Therefore, the consideration of viscoelasticity in tire analysis is also required to predict tire performance properties more accurately.
The mathematical theory of linear viscoelasticity (ref. 13) can be presented by an analysis of ideal spring-dashpot models to represent, respectively, the elastic and viscous components of the response of the material to stress. The first is the Maxwell model, which consists of a spring and a viscous dashpot (damper) in series. The sudden application of a load induces an immediate deflection of the elastic spring, which is followed by creep of the dashpot. On the other hand, a sudden deformation produces an immediate reaction by the spring, which is followed by stress relaxation according to an exponential law. The second is the Kelvin (also called Voigt or Kelvin-Voigt) model, which consists of a spring and dashpot in parallel. A sudden application of force produces no immediate deflection, because the dashpot (arranged in parallel with the spring) will not move instantaneously. Instead, a deformation builds up gradually, while the spring assumes an increasing share of the load. The dashpot displacement relaxes exponentially. A third model is the standard linear solid, which is a combination of two springs and a dashpot. Its behavior is a combination of the Maxwell and Kelvin models.
In most of the commercially available finite analysis programs, time dependent material behaviors are modeled with viscoelastic constitutive laws that have hyperelastic strain energy functions. The viscoelastic response can be defined either in time domain or frequency domain, depending upon the type of analysis. Time domain viscoelastic material response is defined by a Prony series expansion of the dimensionless relaxation modulus (ref. 14):
(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where, [g.sub.R] (t) is the normalized relaxation modulus, N = no. of Prony terms, [g.sup.-p.sub.i] and [[tau].sup.G.sub.i] are the material constants.
All the rubber compounds used for different components (like tread, sidewall, belt, etc.) in a passenger radial tire are studied here. The uniaxial extension method was used to characterize the mechanical properties (stress/strain) of rubber vulcanizates using a tensile testing machine. The tensile test samples were cut out of a molded sheet according to ASTM D-412. Samples were pre-stretched at the speed of 50 mm/min, up to 100% strain for ten cycles before the actual measurement was taken. This pre-conditioning is required to eliminate the stress-softening effect. During the pre-conditioning, five minutes relaxation time was provided between each unloading and reloading of the sample. At the end of the pre-conditioning cycles, samples were removed from the grip and allowed to relax for a period of 30 minutes before performing the final test. Then, samples were tested and stress values were recorded at different strain intervals. All the measurements, including pre-conditioning tests, were carried out at room temperature only.
Stress relaxation experiments were carried out in tension mode using the tensile testing machine to capture the viscoelastic behavior of the rubber compounds. Pre-conditioning of the test samples was performed by stretching the samples at a speed of 500 mm/min, for a few cycles and, at the end, the samples were allowed to relax for 30 minutes. Then, stress relaxation tests were carried out by stretching the samples to 100% strain level and kept for 900 seconds in this condition. Stress values at different time intervals were recorded.
Hyperelastic, viscoelastic material constant determination
The hyperelastic material constants were determined for given sets of experimental stress-strain data through a least squares curve fitting procedure in Abaqus (ref. 14), which minimizes the relative error in stress. The relative error:
(7) E = [sigma] [(1 - [T.sup.th.sub.i] / [T.sup.test.sub.i]).sup.2]
Where [T.sup.test.sub.i] is a stress value from the test data and Tith comes from the nominal stress expression obtained from the polynomial strain energy function. Drucker stability criterions for all the materials have also been checked. The values of material constants of the Yeoh material model ([C.sub.10], [C.sub.20], [C.sub.30]) for all the rubber compounds are given in table 1. It has been observed that all the [C.sub.20] values reported here are negative and smaller in magnitude than [C.sub.10], while the [C.sub.30] values are positive, but smaller in magnitude compared to [C.sub.20] values. These magnitudes will create the typical s-shape of the stress-strain behavior of rubber: At low strains [C.sub.10] represents the initial shear modulus, which softens at moderate strains due to the effect of the negative second coefficient ([C.sub.20]) and is followed by an upturn at large strain due to the positive third coefficient ([C.sub.30]). Figure 1 shows the comparison between the Yeoh and Mooney-Rivlin models, and it is observed that the Yeoh model agrees well with the test data. Figure 2 shows the suitability of the Yeoh model at large strain (300%).
The viscoelastic material parameters were calculated using the stress vs. time data obtained from the stress relaxation test. These stress relaxation data were converted into shear relaxation data, assuming near incompressibility of the material. Figure 3 shows the comparison of the Abaqus model (N = 2) fitted to test data for belt material. Prony series parameters are then determined through a non-linear least squares fit procedure. As a general guideline, the number of terms in the Prony series should be typically not more than the number of logarithmic decades spanned by the test data. As the present test data covers up to 900 sec ([approximately equal to] 3 decades), therefore 2-3 Prony series terms are required for a good fit. For the present case, 2 term Prony series constants are used to represent viscoelasticity for all the rubber compounds, and the values are given in table 2.
Case study: Analysis of 155/70 R 13 tire
A radial tire of 155/70 R 13 size with one polyester body ply, two steel belts and one nylon cap strip (covering the belt edges) construction has been chosen for this study. The full tire cross section was modeled using a four noded axisymmettic element, CGAX4H (continuum, generalized, four noded axisymmetric element with a hybrid element formulation to handle incompressibility and twist). MSC/Patran software was used for generating the FEA model. The model consists of 615 nodes and 530 elements. Figure 4 shows the axisymmetric tire layout. Since the reinforcements in body ply and belts are the major load carrying members of the tire, therefore it is necessary to model reinforcement very accurately. The carcass and belt reinforcements of the tire were modeled by the unidirectional rebar elements superimposed with the rubber matrix element. Another important point regarding reinforcement in a tire is that the cord direction and end counts of the plies and belts change due to the shape change between the cylindrical uncured tire on the building drum and the cured tire. The variable cord direction (angle with respect to the tire center line) and end counts (ends per inch) in a tire cross-section have been assigned using an in-house developed Fortran program.
The 3D tire model is generated by using Symmetric model generation techniques (ref. 14) from the 2D axisymmetric model. The model consists of 36 segments with 44,280 nodes and 38,160 elements (C3D8H, continuum three dimensional, eight noded with hybrid formulation), which include eight-noded continuum elements. The road is modeled using rigid solid elements. Contact with friction is considered between the tire and the road. Due to its complexity, the tread pattern geometry is not considered in this analysis. The 3D tire model is shown in figure 5.
All the rubber components were modeled as incompressible continuum elements. Hyperelastic properties were defined using the Yeoh material model. Prony series parameters were used to assign viscoelastic properties of rubber compounds. The polyester cord and steel belt materials were modeled as linear elastic material. The Young's modulus and Poisson's ratio were used to define material properties of polyester and steel cord. The bead bundle was modeled with linear elastic continuum elements.
Two sets of analysis were carried out using Abaqus/Standard solver, one with only hyperelastic material constants (Analysis-I) and the second one in combination with viscoelastic material properties (Analysis-II) to see the effect of viscoelastic properties on the tire performance prediction. It is important to mention here that the proper analysis option has been chosen for the Analysis-II to capture viscoelastic effects. Axisymmetric inflation analysis was performed by applying a distributed surface pressure of 240 KPa. In the next step, the road (rigid surface) was displaced vertically to the estimated static deflection while the tire was in the inflated condition. A subsequent step was used to adjust the load on the rigid surface to the actual desired load of 3,800 N.
Results for various output parameters obtained from the prediction have been provided in table 3. Inflation and load deflection tests for the same tire size were also carried out to verify the predicted results (table 3). All the measurements were taken at the same inflation pressure and vertical load used for prediction purposes. It is observed that inflated dimensions agree well with the measured dimensions. Figure 6 shows the predicted and experimental load deflection curves. The experimental curve is slightly stiffer than that of the predicted one. However, the deviation is very small, and it can be taken as a reasonably good prediction. Measured contact area and contact pressure values match more closely when the analysis is carried out using hyperelastic and viscoelastic material constants together as compared with using the hyperelastic material properties alone. The explanation will be given in the next section.
Some more parameters such as Von Mises stress and strain energy values of belt edges for both types of analysis have been reported in table 4, keeping in mind that this region is highly prone to failure during service. Although measured values are not available for these parameters, it gives good insight about the effect of material properties (viscoelastic) on critical durability parameters like stress and strain energy density. It is evident from the results reported in table 4 that there are significant differences in output parameters obtained from the two types of analysis. It is obvious that the inclusion of viscoelasticity is causing these differences to occur, which can be explained nicely by describing the mechanism of viscoelastic material behavior. When a load is applied on a viscoelastic material, there is an instantaneous elastic response, followed by a slow time-dependent viscoelastic response. This viscoelastic response tends exponentially to a solution. The results of Analysis -II give the complete response (instantaneous, as well as time dependent part). However, this time-dependent phenomenon cannot be captured when one uses only hyperelastic material properties. It is also observed that the Von Mises stress and strain energy values obtained from Analysis-II are lower than that of Analysis-I, as expected. Findings of this study reveal that the inclusion of viscoelasticity is important for tire performance prediction.
Simple experimental methodologies for characterizing hyperelastic and viscoelastic material constants for rubber material have been developed. A uniaxial tension test and stress relaxation tests (in tension mode) were performed to determine the hyperelastic and viscoelastic material properties, respectively. Subsequently, a case study of tire analysis has been presented using these material constants, and it showed good correlation between predicted and experimental results. It has been demonstrated that inclusion of the viscoelastic material description improves prediction accuracy. Later, we plan to develop more advanced viscoelastic test methodology to predict tire rolling resistance and temperature distribution.
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Table 1 - hyperelastic material constants Material constants Compound [C.sub.10] [C.sub.20] [C.sub.30] (N/[mm.sup.2]) (N/[mm.sup.2]) (N/[mm.sup.2]) Tread 0.7769 -0.2759 0.0953 Sidewall 0.4876 -0.1413 0.0386 Body ply 0.4795 -0.1356 0.0436 Belt 1.0239 -0.4272 0.1732 Filler 1.0414 -0.3908 0.1343 Rim strip 0.6659 -0.2085 0.0651 Inner liner 0.4861 -0.1152 0.0253 Chafer 0.5998 -0.1218 0.0267 Table 2 - viscoelastic material constants Material constants Compound Relaxation Relaxation Modulus (g) Time ([tau]), sec. [g.sub.1] [g.sub.2] [[tau].sub.1] [[tau].sub.2] Tread 0.1296 0.1209 1.08 51.42 Sidewall 0.0619 0.0689 1.70 109.3 Body ply 0.0567 0.0681 2.44 99.51 Belt 0.0853 0.0834 0.92 54.63 Filler 0.1132 0.1113 0.97 51.70 Rim strip 0.0868 0.0889 1.18 60.31 Inner liner 0.0910 0.0944 1.01 60.15 Chafer 0.0844 0.0959 1.30 109.1 Table 3 - predicted and measured results Parameters Prediction Experiment Analysis -I Analysis -II (without (with viscoelasticity) viscoelasticity) Overall diameter 547.10 547.10 546.90 (mm) Section width 160.83 160.83 161.50 (mm) Contact area 14,819.3 15,625.7 16,354.8 ([mm.sup.2]) Average contact 258.1 244.8 233.8 pressure (KPa) Table 4 - predicted Von Mises stress and strain energy density values at belt edges Parameters Analysis -I Analysis -II Difference (without (with (%) visco- visco- elasticity) elasticity) Von Mises stress @ 0.5722 0.5171 10 belt 1 edge (N/[mm.sup.2]) Von Mises Stress @ 1.282 1.151 11 belt 2 edge (N/[mm.sup.2]) Strain energy density 2.93E-05 2.30E-05 27 @ belt 1 edge (J/[mm.sup.3]) Strain energy density @ 1.54E-04 1.20E-04 28 belt 2 edge (J/[mm.sup.3])
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|Date:||Jan 1, 2006|
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