Modeling rubber products--materials characterization for "quasi-static" FEA. (Tech Service).

One purpose of rubber testing is to determine parameters (constants and functions) in constitutive equations. Such equations are used to represent hyper- and viscoelastic behavior of rubber for arbitrary strains or loading histories. In the case of linear materials such as steel, an example or a material parameter is Young's modulus. However, rubber deforms non-linearly and is capable in instances of sustaining reversible strains in excess of several hundred percent. Hence, more parameters are needed to characterize rubber than are required for a linear material. These are to determine from experiments on laboratory specimens, to deform in well-defined strain fields. Moreover, predicting the large deformation viscoelastic response of elastomeric components relies on nonlinear viscoelastic FEA code capabilities. Such a code contains constitutive equations in which parameters need determination prior to any modeling. Material parameters in a given constitutive equation are generally unique to each rubber, and testing needs to be repeated for each compound and test condition (temperature and aging media/period).

Hyper-elastic response of rubber

Strain energy density functions U, representing rubber hyperelastic behavior, can be based on polynomials of strain invariants:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where:

[C.sub.ij] are Rivlin coefficients;

[D.sub.i] defines material compressibility;

R represents volume change with temperature;

[I.sub.1] = [[lambda].sub.1.sup.2] + [[lambda].sub.2.sup.2] + [[lambda].sub.3.sup.2];

[I.sub.2] = [([[lambda].sub.1][[lambda].sub.2]).sup.2] + [([[lambda].sub.2][[lambda].sub.3]).sup.2] + [([[lambda].sub.1][[lambda].sub.3]).sup.2];

J = [[lambda].sub.1][[lambda].sub.2][[lambda].sub.3] and [[lambda].sub.1] = [J.sup.-1/3][[lambda].sub.1].

They can also be based on stretch ratios:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Series in both forms represent the deviatoric and volumetric components of stored energy. Constants [C.sub.ij], [[alpha].sub.i], [[mu].sub.i] and [D.sub.i] derive from curve fits to test data.

Tests to produce rubber basic modes of deformation

Rubber can deform under four basic modes of deformation: uniaxial, equi-biaxial, planar and volumetric. Besides, each of these modes can be tensile or compressive, making the number of tests to characterize rubber at "time-zero" eight, compared to a single test for materials such as plastics and steel. Moreover, only the uniaxial mode of deformation has seen a way into ASTM Standards (refs. 1 and 2). Testing rubber under remaining modes of deformation is still a research and development topic.

Akron Rubber Development Laboratory (ARDL) proposed a micro-dumbbell sample to capture properties of rubber in tension. In particular, the compact chamber in figure 1 was designed to reach high temperatures (over 150[degrees]C). Results of ARDL's micro dumbbell did, unfortunately, not match those on Die C samples because of clamping; confinement does not allow the use of extensometers. Figure 2 shows a self-lubricating silicone rubber under uniaxial tension, as per ASTM E412. A video camera followed the sample's deformation remotely. Results of testing are shown in figure 3.

[FIGURES 1-3 OMITTED]

Figure 4 highlights rubber equi-biaxial stretching on an Iwawoto machine. Only two Iwawotos exist in the U.S.; one at the Polymer Science Department of the University of Akron, OH; the other at NASA in Houston, TX. The cost and lead-time to perform equi-biaxial tests on rubber using the Iwawoto led to the development of cheaper testers, including German Joh Rubber's "balloon," and a jig Ford Scientific Labs made at Ann Arbor Axel Products, in Michigan.

[FIGURE 4 OMITTED]

In Joh's technique, a millimeter thin sheet is molded between a flat plate and one with a grid. The sheet is then clamped onto a cylindrical fixture and blown to a semi-spherical shape. Stress is later calculated from internal pressure and height of the "balloon"; strain is differentiated from actual and original grid sizes. Unfortunately, Joh's technique is operator-dependent, in addition to being difficult to apply to conditions rather than the ambient. Axel Products' jig, on the other hand, stretches a round sheet of rubber with cables passing by pulleys, while a laser monitors deformation of a grid similar to Joh's. Fortunately, Trealor observed an equivalence between uniaxial tension and equi-biaxial compression (ref. 3). Since rubber is nearly incompressible, equi biaxial tension and compression tests are similar to uniaxial compression and tension, while these are independent.

Figure 5 replicates planar straining at WIDL. Stress normal to the sheet is zero, whereas in-plane stresses are equal for a six-inch wide sample. Beads retain the sample from slipping and define gauge length; contraction raised lateral strains to 4% (which should be zero in planar conditions) for an isobutyl compound.

[FIGURE 5 OMITTED]

Uniaxial, equi-biaxial and planar tests mimic rubber deviatoric behavior (constants [C.sub.ij] in the polynomial strain energy density form, and [[alpha].sub.i] and [[mu].sub.i] in the stretch ratios-based one). Figure 6 highlights fitting deviatoric test data on a 50 durometer A ethylene acrylic elastomer to a Yeoh polynomial model: a non-linear curve tit to an Ogden form of two terms makes for figure 7. Besides, compressibility constants [D.sub.i] require volumetric testing; a half inch diameter button, half an inch tall, a piston squaches in a cavity is used at ARDL and WIDL for such a purpose. Figure 8 interprets collected data for a 60 durometer A EPDM compound. Initial rubber reaction till filling the cavity is ignored.

[FIGURES 6-8 OMITTED]

While the uniaxial compression test is well documented in ASTM D575, roughness and lubrication were found to affect force-deflections quite significantly. Discs were made to study surface finish effect on compressing buttons and washers (3/4 and 3/8 inch in outer and inner diameters, die cut from 1/10 of an inch thick slabs for CSR testing). All discs were scanned. Figure 9 shows roughness and profile curves of an aluminum disc, labeled "very rough" or "3" in the test program.

[FIGURE 9 OMITTED]

Testing met to shade light on: pre-compressions; testing speed; lubrication; platen make (steel or aluminum) and surface finish; and shape factors of compressed samples. Figure 10 compiles results on a 50 durometer A poly-acrylic, for up to 50% nominal strains. Dry tests are reported, along with results using oil for lubrication and three surface finishes from smooth to very rough.

[FIGURE 10 OMITTED]

Figure 11 highlights deformations of an ASTM-D575 button made of Atlas (copolymer of tetrafluoroethylene and propylene) between rigid platens. Friction was ignored in the modeling, which led to a true uniaxial compression mode of deformation. Obviously, a frictionless compression is practically impossible, and further modeling included a 0.025 coefficient of friction in Coulomb's model. Figure 12 shows deformations and stresses through the Aflas button, at various compression levels. Figure 13 points out numerical differences in force-deflections with and without friction.

[FIGURES 11-13 OMITTED]

From figure 12, it is clear that bulging induces triaxiality to a test that is supposed to capture rubber uniaxial stresses and strains. This is the case when testing ASTM-D575 buttons, especially if mating components are not polished, and lubrication is omitted.

For ASTM-D575 button compression testing, smooth, industrial platens were used along with steel ones, with the roughest surface "3," as scanned on figure 9. Force-deflection characteristics of buttons made of the same PA material as the washers are presented in figure 14. Lubrication, this time around, shows quite an effect of reducing force-deflections, compared to the CSR-washer compression case. It is clear that the washer cross-section compressed is only 0.405 [inch.sup.2] compared to a unit square inch for an ASTM-D575 button. Besides, compressing buttons between platens "3" increased force-deflections from dry conditions, especially at nominal strain levels above 25%.

[FIGURE 14 OMITTED]

Thus, while challenging, it is recommended to lubricate ASTM-D575 buttons and platens for compression testing. It is also recommended to establish contact conditions for FEA separately.

Summary

This article continues with design parameters for LMC and MMC analyses. In particular, experiments to establish minimum pressure to seal and stress relaxation characteristics will be discussed, along with failure assessment under deviatoric modes of deformation.

In summary, it is easier to conduct uniaxial tension and compression, along with planar tension and volumetric compression tests to establish constants within functions representing rubber hyper-elastic behavior, it is suggested that uniaxial tension takes advantage of new non-contact extensometry by Instron and MTS Systems. It is also suggested that uniaxial compression uses lubrication and polished platens or a washer specimen rather than a button. Pre-conditioning once is advised, as most rubber products get tested at manufacturers levels prior to shipping to customers.

Planar tension testing ought to use short, yet very wide rubber strips. Beads are very effective in retaining samples in grips and defining gauge length. Besides, volumetric compression testing is extremely important for modeling confined rubber products (such as bushing, void-volume seals, crimped hoses, etc.). A piston-cylinder apparatus is a simple way to establish rubber compressibility. As to modeling rubber products, starting from "time-zero" or "quasi-static" analysis, other ingredients are needed, among which friction undeniably holds an important role.

References

(1.) "ASTM Designation D638-91 (Re-approved 1992)--Standard Test Method for Tensile Properties of Plastics." 1994 Annual book of ASTM Standards, Vol. 08.01.

(2.) "ASTM Designation D132-86 (Re-approved 1992)--Standard Test Method for Poisson's Ratio at Room Temperature," 1994 Annual book of ASTM Standards, Vol. 08.01.

(3.) Trealor, L.R.G., "The Physics of rubber elasticity," Second Edition, Oxford University Press, England (1958).

(4.) Trealor, L.R.G., "Stress-strain data for vulcanized rubber under various types of deformation," Trans. Faraday Soc., Vol. 40, pp. 59-70 (1940).

(5.) Gregory, M.T., "The stress-strain behavior of filled rubbers at moderate strains," Plastics and Rubber: Material and Applications (1979).

(6.) Ogden, R.W., "Recent advances in the phenomenological theory of rubber elasticity," Rubber Chemistry and Technology, Vol. 59, 361 (1986).

(7.) Rivlin, R.S., "Large elastic deformations of isotropic materials, IV. Further developments of the general theory," Phil. Trans. Royal Soc. London A 241 (1948).

(8.) "ASTM Designation E412-87--Test Methods for Rubber Property in Tension," 1994 Annual Book of ASTM Standards, Section 9 Rubber, Vol. 09.02.

(9.) "ASTM Designation D575-88--Test Methods for Rubber Property in Compression," 1994 Annual book of ASTM Standards, Section 9 Rubber, Vol. 09.01.

(10.) "MARC User's Manual," version 6.2, MARC Analysis Research, Palo Alto, CA (1996).

(11.) "ABAQUS/Standard User's Manual," version 5.9, Hibbitt, Karlsson & Sorensen, Pawtucket, RI (1998).

(12.) "Pro\ Engineer User Manuals," version 18, Parametric Technology Corp., 28 Technology Dr., Waterham, MA (1998).

(13.) Twizell, E.H. and Ogden, R.W., "Non-linear optimization of the material constants in Ogden's stress-deformation function for incompressible isotropic elastic materials," J. Austral. Math. Soc., Series B, Vol. 24, pp. 424-434 (1983).

Dr. Ben Chouchaoui is president of Windsor Industrial Development Laboratory which specializes in materials characterization and the computer simulation of product performance and manufacturing.