Predicting properties of carbon black filled rubber compounds for FEA of tires.

Tire development has always been a costly and time consuming process because of the many requirements made on tires, including treadwear resistance, low initial cost, fuel efficiency, safety and traction on all kinds of road surfaces and conditions. But this cost is increasing because of increased performance demands by modern vehicles and consumers and also because of marketing's desire for more new tire products each year. Improved techniques for tire development are desired by all manufacturers to reduce development lead times and cost. Currently, finite element analysis (FEA) holds the most promise to eventually reduce tire development cost.

FEA is a technique that involves the mathematical modeling of a tire, by reducing it down into a large number of interacting elements. Appropriate property characteristics are then assigned to each element to allow the proper response to applied loads and/or deformations. Once a suitable FEA model is established, it is capable of generating an approximate numerical response to any load or deformation that might be applied to the tire. Of course, computers are essential in accomplishing the vast number of calculations that have to be completed.

FEA has been used for many years in the analysis of metal parts. However, adoption of FEA techniques in the rubber industry has not happened as quickly for very good reasons. Rubber or elastomeric materials are unique and bring a much higher degree of complicity to FEA applications. Rubber's uniqueness includes an ability to withstand large reversible deformations, being nearly incompressible, having a non-linear stress strain response curve, and exhibiting a behavior that is time-temperature dependent. In particular, the high deformations and near incompressibility lead to difficulties with FEA calculations.

Elasticity theory

In its unstrained state, rubber is assumed to be perfectly elastic and isotropic (long chain elastomeric molecules with completely random orientation). This assumption of the isotropic state is fundamental to the characterization of rubber by a quantity known as the strain energy function (W), which represents the stored strain energy per unit volume.

The stress strain relationship for rubber materials has to be determined experimentally in simple deformation modes, but converted to mathematical equations via a strain energy function to be used in FEA programs. Many strain energy functions (also called rubber constitutive models) have been proposed to express the non-linear stress strain relationship for rubber materials.

Rivlin (ref. 1) proposed that the strain energy function (W) be expressed as a polynomial of strain invariants, W = W([I.sub.1],[I.sub.2],[I.sub.3]).

Strain invariants of the Green deformation tensor are defined by Rivlin (ref. 2) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Lambda] is the extension ratio ([Lambda] = final length/original length = 1 + [Delta]L/[L.sub.0])

[L.sub.0] is the original length, and

1, 2 and 3 are the three orthogonal directions. Strain invariants are mathematical constants dependent on strain and therefore unitless (ref. 3).

The most general strain energy function (excluding finite compressibility) proposed by Rivlin (ref. 1) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

For an ideally incompressible material which maintains constant volume, [I.sub.3] = 1. Then the strain energy function proposed by Rivlin reduces to:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the term [coo equals zero, indicating by definition a zero energy state at the starting (unstretched) point. This equation generates an infinite series of terms that is normally truncated at some point for practical use.

Using only the first term, yields W = [c.sub.10]([I.sub.1] - 3), which indicates a linear relationship described as neo-Hookean by Rivlin (ref 1).

Confirming Mooney's original work (ref. 4), the first order equation (taking two terms) is called the Mooney-Rivlin model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This model provides a single curvature for the stress strain relationship. It is a widely used model, but it was developed on data from non-filled rubber materials. The Mooney-Rivlin model requires that the simple shear stress strain slope be linear, but that is typically not the case for carbon black filled rubber compounds as pointed out by Yeoh (ref 5).

The second order (taking five terms) Rivlin equation is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This higher order equation may yield an improved fit up through moderate strains as it provides a model that exhibits one inflection point on the stress strain curve.

Continuing, the third order (with nine terms) Rivlin equation is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This model will have two inflection points and may be needed especially if the product operates at high strains.

When the second or third order Rivlin equations are used along with stress measurements through high strains, the fit may be adversely affected at the more meaningful lower strain values. So, in effect, higher order Rivlin equations may provide a better fit at high strains, but at the expense of the fit at low and moderate strains. The choice of which equation to use may depend on how much strain the product is expected to experience.

Certainly, there is a wide range of strain levels within radial tires operating at normal conditions. De Eskinazi et al. (ref. 6) have studied shear strains at the belt edges of small radial passenger tires, experimentally measuring shear strains of 10 to 30% away from the footprint (dependent upon tire construction), but 60 to 80% shear strain within the footprint. Therefore, it would seem that a radial passenger tire normally operates at less than 80% strain (excepting the tread-road interface).

Mechanical analysis

Through mechanical analysis, Rivlin (ref. 1) has demonstrated that the following relationships exist between stress, strain, and W for a pure homogeneous strain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Where:

[Delta]W/[Delta][I.sub.1] and [Delta]W/[Delta][I.sub.2] are partial derivatives of W with respect to [I.sub.1] and [I.sub.2],

[Lambda] = principal extension ratios,

t = principal true stresses (referred to the deformed dimensions),

[Sigma] = engineering stresses (referred to original dimensions; therefore t = [Sigma][Lambda]),

and 1, 2 and 3 are the three orthogonal directions.

The left sides of equations 9 through 11 are called the "reduced stress" terms. Essentially, the above equations describe the stress differentials between the three orthogonal directions. From these three equations, the stress strain relationships for simple deformations, including uniaxial tension, uniaxial compression and planar tension, can be determined (refs. 5,7 and 8).

Cubic equation 

Yeoh (ref. 5) has proposed a cubic equation in the invariant ([I.sub.1] - 3) as the strain energy function to better characterize the elastic properties of carbon black filled rubber compounds. Yeoh's proposed cubic equation, which can be viewed as a simplified form of the third order Rivlin equation, has the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This proposed material model features a modulus in simple shear that varies with deformation, in contrast with the Mooney-Rivlin model which requires a linearity. Yeoh asserts that the cubic equation can also be used to predict stress strain behavior in different deformation modes. Gregory (ref. 9) first reported that plotting reduced stress versus the invariant ([I.sub.1] - 3) for different deformation tests gave a single curve, based on a review of carbon black filled natural rubber compounds.

With equation 12, the partial derivatives of W with respect to [I.sub.1] and [I.sub.2] are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Yeoh (ref. 5) states that published experimental data for carbon black filled rubber compounds indicate that [Delta]W/[Delta][I.sub.2] is much smaller than [Delta]W/[Delta][I.sub.1] and that [Delta]W/[Delta][I.sub.2] is numerically close to zero. This supports the suitability of the cubic equation to describe the stress strain relationship of non-linear materials.

But, [Delta]W/[Delta][I.sub.2] is not exactly zero. This means that when applying the cubic equation to carbon black filled rubber compounds at small strains, the absolute difference between the model and experimental may be small, but the relative difference may be large (refs. 5 and 10).

Application of the cubic equation

Deformation tests are physically set up to be geometrically simple to allow easy mechanical analysis per equations 9, 10 and 11. For uniaxial tension, uniaxial compression and planar tension (pure shear) testing, the force always acts in a single direction. Testing then measures stress and strain in this direction.

As an example, the planar tension (pure shear) test can be analyzed as follows (ref. 11). Assuming incompressibility, [I.sub.3] = ([Lambda.sub.1][Lambda.sub.2][Lambda.sub.3])[2] = 1. For moderate extension ratios, [Lambda.sub.1] = [Lambda] (measured), [Lambda.sub.2] = 1 (held to original dimension), and therefore [Lambda.sub.3] must equal [Lambda.sup.-1]. For the engineering stresses, [Sigma.sub.1] = [Sigma] (measured) and [Sigma.sub.3] = 0. The stress [Sigma.sub.2] is not equal to zero, it is supplied by the rigid clamps. Since true stress t = [Sigma][Lambda], then [t.sub.1] = [Sigma.sub.1][Lambda.sub.1] or [Sigma][Lambda] and [t.sub.3] = 0.

Substituting the above values into equation 10 allows the calculation of the stress differentials between the specimen length and thickness:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Since we are using the cubic equation as our strain energy model, equations 13 and 14 are then substituted into equation 15 yielding:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The uniaxial tension and compression tests can be similarly analyzed yielding the following expression for both tests:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The reduced stress term is expressed in the form of a quadratic equation in ([I.sub.1] - 3) for all three deformations tests with the general form being:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A summary of reduced stress term and ([I.sub.1] - 3) term for each deformation test is shown in table 1.

[TABULAR DATA NOT REPRODUCIBLE IN ASCII]

Plotting the reduced stress term versus the ([I.sub.1] - 3) term for each deformation test separately will yield equivalent curves (ref. 10). If the cubic equation is suitable, this allows the use of one simple test, like uniaxial tension, to predict stress strain data for other deformation tests that may be more difficult to conduct and control. Using uniaxial tension is particularly useful as a base test because almost every rubber laboratory has the capability to perform it.

Finite element analysis process

Although application of FEA may appear to be a straightforward process, in actual practice it is more complicated. Usually, product modeling and material (compound) selection are occurring at the same time. Since the FEA model and materials interact, adjustment of both at the same time is necessary. Complete determination of compound properties is a much slower process than regular laboratory testing, especially when Mullins effect, aging and product operating temperatures are considered. To allow the modeling and model verification process to continue unimpeded, a quick determination of approximate compound properties is desirable.

The objective of this article is to determine the suitability of the cubic equation as a strain energy function and to determine its capability to predict uniaxial compression and planar tension (pure shear) properties from uniaxial tension data for typical carbon black filled compounds used in tires.

If these property predictions are suitable, time and material testing expenses would be saved. When the final compounds are chosen, complete testing can be done to assure accuracy of the FEA work.

Experimental

Four typical carbon black filled tire compounds representing a range of physical properties were selected for verification of the cubic equation's suitability (table 2).

[TABULAR DATA NOT REPRODUCIBLE IN ASCII]

Physical measurements were done in uniaxial tension, uniaxial compression and planar tension (pure shear) after samples were prepared according to ASTM D 3182 procedures. Laboratory specimens were cured at 160[degrees]C for appropriate times for each compound to reflect its state of cure in the tire.

For FEA purposes, the equilibrium stress strain measurement is desired. However, to complete testing within reasonable time limits, compromises are normally made. Yeoh (ref. 5) has indicated that strain rates at 10%/minute and at 50%/minute produced stress values that were substantially equivalent. In this study, strain rates of less than 50%/minute were used for all testing.

For the uniaxial tension testing, standard the C dumbbells were prepared according to ASTM D 412 procedure. Duplicate specimens were tested at a strain rate of 5 mm/minute at room temperature on a T-2000 testing machine.

Uniaxial compression testing was done on compression set samples (ASTM D 395). Specimen size is 12.5 mm thickness and 29.0 mm diameter. Testing was completed in duplicate at a strain rate of 5 mm per minute at room temperature on a MTS 831 tester. Silicone oil was used as a lubricant between the specimen and tester platen to minimize side bulging of the specimen (ref. 12).

Planar tension (pure shear) testing was done on specimens with overall dimensions of 76.2 mm width, 80 mm length and 2 mm thick. Secure clamping keeps the specimen from significantly drawing in at the ends to provide for pure shear up to moderate strains (100%) (ref. 8). Duplicate testing was done at a strain rate of 5 mm/minute at room temperature on a T-2000 testing machine with a laser extensometer based on marks 12.7 mm apart.

Results and comparison

Uniaxial tension stress strain results for the four compounds are shown in figure 1. The stress data on this chart and all subsequent charts has been indexed using the stress value of compound C at 230% strain as a base of 100. All data are typical for carbon black filled rubber compounds.

The cubic strain energy function as given in equation 18 was applied to the uniaxial tension data by least squares parabola regression generating the curves in figure 2. The fit is a good approximation of the measured data points (also shown).

Using the terms for uniaxial tension as shown in table 1, reduced stress and ([I.sub.1] - 3) values were calculated from the uniaxial tension data and plotted as shown on figure 3.

From figure 3 data and using the terms in table 1, predictions of uniaxial compression and planar tension (pure shear) were made through interpolation for the four compounds as shown on figures 4 and 5. Note that the shapes of the curves appear to be normal.

Figures 6 to 9 show the predicted uniaxial compression curves compared to measured data points for each compound. Normally, uniaxial compression data above 40% strain are not used in FEA programs because the test generates unrealistic results above 40% strain. Compounds A, B and D show good agreement, while the highest modulus compound C shows only fair agreement.

Figures 10 to 13 show the predicted planar tension (pure shear) curves compared to measured data points for each compound. There is excellent agreement at the lower strains, particularly less than 80% strain where radial tires normally operate. The predicted curve does deviate more from the measured points above 100% strain as the pure shear assumption becomes less valid as the specimen ends start to draw inward.

Conclusions

Yeoh's (ref 5) proposed cubic equation in the invariant ([I.sub.1] - 3) as the strain energy function is a good approximation of the characteristics of carbon black filled rubber compounds, especially at strains less than 80% where radial tires normally operate.

When uniaxial tension data for four carbon black filled rubber compounds were used to predict uniaxial compression and planar tension (pure shear) data, the correlations with actual test measurements were reasonably accurate. The correlations were especially good at strains less than 80% where radial tires normally operate.

It is suggested that applying Yeoh's proposed cubic equation to uniaxial tension data to predict uniaxial compression and planar tension (pure shear) data is a quick and inexpensive method to generate a good base line of material properties.

References

[1.] R.S. Rivlin, "Rheology: Theory and applications, vol 1," F.R. Eirich, ed, Academic Press, New York, N.Y. 1956, ch. 10, p. 351.

[2.] R.S. Rivlin, Philos. Trans. R. Soc. London, A, 241, 379 (1948).

[3.] D.J. Charlton, J. Yang and K.K. Teh, Rubber Chem. Technol. 67, 481 (1994).

[4.] M. Mooney, J. Appl. Phys. 11, 582 (1940).

[5.] O.H. Yeoh, Rubber Chem. Technol. 63, 792, (1990).

[6.] J. De Eskinazi, K. Ishihara, H. Volk, and T.C. Warholic, "Towards predicting relative belt edge endurance with the finite element method," Tire Science and Technol., TSTCA, Vol. 18, No. 4, October-December 1990, pp. 216-235.

[7.] R.S. Rivlin, Rubber Chem. Technol. 65, pp. G51-G66, (1992).

[8.] A. Pannikottu, J. Seiler and J.J. Leyden, "Non-linear finite element analysis support testing for elastomer parts," paper no. 73 presented at a meeting of the Rubber Division, American Chemical Society, Cleveland, Ohio, Oct. 17-20, 1995; abstract in Rubber Chem. Technol. 69, 156, (1996).

[9.] M.J. Gregory, Plast. Rubber Mater. Appl. 4, 184 (1979).

[10.] O.H. Yeoh, Rubber Chem. Technol. 66, 754 (1993).

[11.] A.N. Gent, "Engineering with rubber, chapter 3," A.N. Gent, ed., (1992).

[12.] P.H. Mott and C.M. Roland, Rubber Chem. Technol. 68, 739 (1995).