Edited by Jessica Shapiro
Finite-element analysis (FEA) is an indispensable tool for predicting a product’s behavior in real-world situations and over time. However, an FEA model is only as good as the material parameters it’s based on. Getting good material data can be challenging enough for linear-design purposes, but elastomeric materials complicate matters with nonlinear and viscoelastic behavior.
Nonlinearity arises from rotations or translations large enough to be noticeable to the naked eye, from part-to-part contact, and from the material’s nonlinear response to loads or strains. For rubber products that undergo sealing, damping, or crimping, for example, these scenarios make analysis fully nonlinear.
Rubbers’ viscoelastic component deals with temperature and the material’s behavior over its lifetime. Add to this manufacturing tolerances characterized by minimum and maximum material conditions (LMC and MMC). An O-ring application, for example, involves sealing over the life of a device when it is minimally loaded at LMC, while MMC could deal with properties like failure stresses or strains after long-term exposure to shearing deformation modes.
Rubbers deform uniaxially, equibiaxially, in-plane, and volumetrically. Because each of these modes can also be tensile or compressive, anyone wishing to fully characterize rubberlike materials needs eight sets of test data, compared to the single tensile test required to characterize materials like rigid plastics and steels in their linear regimes.
To generate reliable FEA models for rubber parts, designers must have data from carefully conducted laboratory tests that can be reduced to critical parameters in constitutive equations. Each elastomeric compound and material condition — including temperature, aging history, and humidity — needs its own set of material parameters.
Unfortunately, ASTM test standards only cover uniaxial deformation with ASTM D412 for tension and ASTM D575 for compression. So, testing rubber under the remaining independent modes of deformation is still a research and development topic.
One advancement of note in rubber testing is remote extensometry. Strain measurements should be made independently of test-machine movement whenever possible. This keeps clamping effects from influencing strain results and makes measuring fixture and machine compliances less necessary. However, the strain-measuring device itself can also influence test results if it contacts the specimen. Instron Corp., Norwood, Mass., has a video-camera system that measures strain remotely.
Outfits like Windsor Industrial Development Laboratory (WIDL), Windsor, Ont., have successfully used such setups to monitor various deformation modes. Test machine companies MTS, Eden Prairie, Minn., and Zwick, Kennesaw, Ga., also have noncontact strain-measurement equipment based on lasers, video, or both.
Independent labs specializing in rubbers and rubber-product testing have developed methods that supplement the ASTM standards. Akron Rubber Development Laboratory (ARDL), Akron, Ohio, for example, proposed a micro- dumbbell tension setup. Its compact test chamber can reach 150°C or higher. A small test sample conserves material and lets engineers closely control test temperature.
Test conditions matter
WIDL conducted a series of tests to quantify the effects of precompression, strain rate, lubrication, platen material, surface finish, and sample shape factor.
Models derived from test data showed that ignoring friction returned pure uniaxial-compression results. However, introducing a small coeffcient of friction — 0.025 — yielded a triaxial stress state with a steeper force-defection curve. The triaxiality was particularly profound for buttons tested on platens that were not polished or lubricated.
However, the sample and chamber are too small to permit remote extensometry or direct strain measurement. Because of this and the clamping geometry, ARDL test results tend not to match those obtained using the ASTM 412 method.
For equibiaxial tension (ET), one test method uses an Iwawoto machine from Japan. U.S.-based engineers who want to use this test will have to get in line at either the Polymer Science Department of the University of Akron, Akron, Ohio, or at NASA in Houston, Tex.
The cost and lead time involved with equibiaxial testing has led those in the industry to develop other methods. These include a balloon technique initially developed by Joh Rubber, Ontario, in which a 1-mm-thick sheet of rubber is molded between a flat plate and a grid-marked plate. Technicians then clamp the gridded, conditioned sheet onto a cylindrical fixture and blow it into a spherical shape while monitoring the balloon’s dimensions and internal pressure.
They then calculate a stress from these measurements using the strength-of-materials equations for linear shells. Strains are determined by measuring the change in grid size. This technique is somewhat operator-dependent, and it is difficult to perform the test at nonambient conditions. Ford Motor Company’s Scientific Laboratory, Dearborn, Mich., had Axel Products, Ann Arbor, Mich., make a jig that uses a cable-and-pulley system to stretch a round sheet of rubber equibiaxially. A laser beam remotely monitors deformation of a grid drawn on the sample with a marker.
Fortunately some researchers, including L. R. G. Treloar, observed that when a sample stretches axially, the two transverse axes undergo compression. Each transverse axis’ compressive stress is equal to the absolute value of half the axial tensile stress, making uniaxial tension (UT) and equibiaxial compression (EC) equivalent. This is due to rubber’s near incompressibility.
Likewise, engineers trying to determine material constants can perform the relatively easier uniaxial compression (UC) test and skip the more problematic ET test.
Planar tension (PT) tests use wide samples to minimize edge effects and integrally molded beads to define gage length and prevent slipping. The sample’s design originated in a European division of Ford. Tests at WIDL have indicated that stress normal to a 6-in.-wide sample is zero while inplane stresses are equal across sample width. Poisson contraction of the sample raised lateral strains — which should be zero in planar conditions — to 4% for an isobutyl compound at around 50% axial strain.
In many applications, a rubber’s volumetric behavior is one of its most important attributes. Parts like void-volume gaskets, seals, hydrostatic pads, and confined bushings are often volumetrically constrained during longterm loading.
One way to get volumetric data is to compress a ¼-in.- diameter, ¼-in.-tall button. ASTM D575 test setups use a sample compressed between two parallel platens to determine UC behavior. By swapping the platens for a cavity and piston and cutting sample size, WIDL has been able to determine the volumetric behavior of various rubbers.
WIDL’s method ignores initial test data in favor of the force-deflection slope after the sample has completely filled the cavity. Researchers also found that the fixture’s surface roughness and lubrication significantly affect force-deflection data. (See Test Conditions Matter.) Test engineers should also be aware of these factors when testing rubbers, especially when determining UC properties as a substitute to ET testing.
Fitting the model
Once tests of a rubber’s hyperelastic response are complete, designers can derive strain-energy-density functions, U, which are the constitutive equations underlying FEA models. Although complete characterization of the rubber may seem arduous, the resulting strain-energy-density functions are valid for any product made of the characterized material.
These functions use polynomials of strain invariants to represent elastomers’ hyperelastic behavior, their ability to reversibly strain to several times their original dimensions.
Strain-energy-density functions can be written as:
U = Σi+j=1 Cij (I1 – 3)i (I2 – 3)j + Σi=1 (1/Di) (J – 1 – R)2i
where Cij = material coefficients, Di = material compressibility, and R = volume change with temperature. The elastic volume ratio, J, and the invariant functions, I1 and I2, are related to the extension coefficients λi = 1 – εi where ε1, ε2, and ε3 = the principal strains:
I1 = λ12 + λ22 + λ32
I2 = (λ1λ2)2 + (λ2λ3)2 + (λ1λ3)2
J = λ1λ2λ3
Strain-energy-density functions can also be based on the extension ratios, λi, themselves. This approach, known as the Ogden model, gives U as:
U = Σi=1 (2μi/αi2) (λ1α + λ2α + λ3α – 3)
+ Σ (1/Di) (J – 1 – R)2i
Series in both forms of U represent the material’s deviatoric response, or its tendency to deform without changing size, as well as the volumetric or hydrostatic component of the stored energy — contraction or expansion of the rubber without a shape change.
In both cases, designers can determine Cij , αi , and μi by fitting curves to stress-strain curves derived from uniaxial and planar test data.
The Ogden model given above and the Yeoh model, a special case of the first U equation in which j = 0, are examples of equations analysts can choose to fit to test data. Here, an analyst’s experience and judgment will guide the selection of a fit model that matches the anticipated deformation of the product at the relevant region of the stress-strain curve.
Volumetric compression testing, such as that discussed in Test Conditions Matter, provides data analysts can use to get the Di compressibility constants. The slope of the stress-strain curve corresponds to 1/D1. Because the material’s volumetric response is almost linear, analysts can neglect higher values of i without a significant loss of fidelity.