Physical properties and their meaning.
Breaking stresses and extensions in equibiaxial tension are often significantly greater than those in uniaxial tension (ref. 23). A probable source of these improved properties is the lack of a preferred direction for crack growth under biaxial tension. An inflated test specimen provides one means to obtain biaxial tension. Since this specimen does not have a cut edge, it doesn't have stress raisers at its edge like the cut edge of a tensile specimen. Therefore, both reduced crack growth and reduced stress raisers could improve properties in biaxially deformed specimens.
Squeezing a flat rubber specimen between lubricated metal plates also deforms a specimen biaxially. The lubricant permits slippage at the boundary between rubber and metal surface. The effect and importance of boundary conditions was described in an early paper by Kimmich (ref. 27) (figure 15). This figure shows that compressive stress at a given deflection is greatest for the adhered specimen (A); the response with sandpaper (S) is not too different. For a given deflection, compressive stress is considerably lower with graphite (G) and lowest with petrolatum (P), because of slippage. At 10% deflection, for instance, stress for A is 1.2 MPa (180 psi) while stress for P is only 0.2 MPa (30 psi). Thus, differences in boundary conditions cause a sixfold difference in compressive stress. Therefore, proper control of boundary conditions is necessary to obtain meaningful compression-deflection properties for rubber.
Compression-deflection properties are measured by ASTM D 575. This method specifies the use of sandpaper against the rubber test specimen, unless the rubber specimen is adhered. Figure 15 shows that the use of sandpaper or an adhered specimen gives similar compression-deflection results. By ASTM D 575, a specimen with a thickness of 12.6 mm (0.500 in.) and a diameter of 28.6 mm (1.129 in.) is compressed at either constant load or constant deflection. Two deflection cycles condition or stress soften a specimen before a measurement is made on the third cycle. Because rubber is stress softened by this laboratory test, laboratory properties should correlate better with those for an end-use product that is stressed in service.
In compression-deflection tests like ASTM D 575, specimen geometry is very important. The shape factor (SF) is frequently used to describe geometry. By definition SF is the ratio. area of one loaded face/total area free to bulge. This definition is limited, of course, to specimens where there is no (or virtually no) slippage at the rubber interface. Figure 16 shows examples of several shape factors (ref. 28). Each block of rubber has the same area over which the load is applied, but a different SF. For the SF range shown, 0.25-10.0, compression stiffness of the loaded rubber blocks varies by about two decades. Even higher shape factors increase stiffness to three decades as described later in the section on triaxial compression.
But first, let's consider specific SF effects for a 70 hardness (Shore A) composition over a limited SF range (ref. 28). Figure 17 shows that a SF range of 0.25-6.0 causes a large difference in compressive stress for this rubber composition. At a SF of 6.0, for example, a compressive stress of 7.5 MPa (1,088 psi) is reached at only about 2%, compressive strain. A strain of 42% is required to reach the same stress at a SF of 0.25. Even a limited range of SF values causes large differences in stress-strain behavior.
When SF is low, the compression-deflection behavior of rubber corresponds to rubber properties that are measured typically, like hardness and stress at 100% elongation (S-100). When SF is high, it is SF that dominates compression/deflection behavior, not these typically measured properties. Hence, rubber products with very high stiffness in compression can be made from low hardness rubber compounds, by incorporating high SF in their design. If the SF is sufficiently high, compressive stiffness approaches that obtained under triaxial compression.
Triaxial or bulk compression (figure 18) occurs when compressive force is applied uniformly to all surfaces of a rubber specimen. Under this condition, rubber does not change shape. Instead, rubber volume decreases. No case of fracture is known for rubber under uniform triaxial compression (ref. 23).
In uniaxial tension, in contrast, shape changes occur with virtually no change in volume. Uniaxial tension is depicted in figure l@ by a tensile dumbbell. The arrows in figure 18 show the direction of the applied force. For triaxial compression or hydrostatic compression, force applied uniformly over the surface of the rubber causes only a very slight decrease in volume. Under hydrostatic compression, the bulk modulus (B) is defined as the ratio, hydrostatic pressure to volume strain (ref. 13): B= hydrostatic pressure/volume change per unit volume The value of B for rubber (ref. 29) is extremely high, about 2 GPa (3x[10.sup.5] psi). Because of the extreme value of B, a hydrostatic stress (pressure) of 1 MPa (145 psi) reduces (ref.3) the volume of a rubber block by only 0.05%. The same stress in uniaxial tension elongates a tensile specimen by about 100%. Hence, equal stress causes a 2,000-fold strain ratio for the specimens subjected to uniaxial tension and triaxial compression, respectively.
The different mechanisms involved in triaxial compression and uniaxial tension account for this large ratio (ref.3). In triaxial compression, high stress levels squeeze molecules closer together and reduce volume only slightly. In uniaxial tension, stress changes conformation of molecules rather than causing a significant change in specimen volume. Only low stress levels are required in uniaxial tension to change conformation. These features account for the large difference observed.
When a tensile specimen is stretched, its cross-sectional area decreases to accommodate the increase in length at virtually constant volume. These changes are related by Poisson's ratio ([upsilon]) as follows (ref. 13): [upsilon]= change in width per unit of width/change in length per unit of length The equation below (ref. 13) relates Young's modulus (E), bulk modulus (B) and [upsilon]. B= E/3(1-2[upsilon])
This equation is strictly valid only at small strains, where many materials can be treated as obeying the classical theory of elasticity. For rubber, which is virtually incompressible, [upsilon] approaches 0.5. Lindley (ref. 30) reports a value of [upsilon] of 0.49989 for a natural rubber gum, substituting this value of [upsilon] into the equation immediately above yields a value of 1,515 for the quotient B/E. Wood lists values of B = 2 x [10.sup.6] GPa and E =1.3 MPa for a gum vulcanizate (ref. 29), which gives a B/E quotient of 1,538. Hence, data from different sources produce comparable values for B/E.
For most materials, the value of [upsilon] is substantially less than that of rubber. For instance, [upsilon] for mild steel (ref. 30) is 0.291. Because of this low value, the compression modulus of steel changes little with shape factor. Hence, the effect of shape (SF) in mild steel can raise (ref. 30) its compression modulus to no more than about 30% above its Young's modulus (E).
Figure 19 shows that the SF effect with rubber is different and dramatic (ref. 31). Compression modulus ([E.sub.c]) is shown as a function of SF for three different hardness NR compounds, over a wide range of SF. Hardness is shown in International Rubber Hardness degrees (IRHD) and these are approximately equal to Shore A hardness units.
In figure 19, the effect of hardness on [E.sub.c] is most pronounced at low values of SF. For all three hardness compounds, [E.sub.c] increases sharply with increasing SF. [E.sub.c] of the 35 hardness compound increases by almost three decades over the SF range shown.
This unique behavior of rubber is used in the design of rubber products such as bearings to support buildings and bridges. A cut section from a bridge bearing (figure 20) shows steel plates that are bonded to rubber (black). By increasing the SF of this bearing, values of [E.sub.c] approach the value of the bulk modulus.
At sufficiently high SF, rubber does not bulge outward in a compressed bearing. This means that the rubber is changing volume without changing shape. At this high SF the ratio, compressive stress/compressive strain, is the bulk modulus (B). The value of B can be determined by squeezing a rubber specimen by a piston in a large steel cylinder under special conditions (ref. 32).
This stress mode is the opposite of triaxial compression. Instead of applying uniform inward pressure on all surfaces of rubber (triaxial compression in figure 18), all surfaces are pulled outward uniformly in triaxial tension. The effect of triaxial tension is approximated when thin rubber discs are bonded to metal and then subjected to a tensile loading. ASTM D 429 (Method A) describes the preparation of the bonded rubber-metal specimen. This specimen is intended for use in adhesive tests (figure 21), and either adhesive or cohesive failure can occur during test.
When cohesive failure occurs, it is a result of a small cavity in the central region of the specimen (figure 21) expanding under negative pressure (ref. 17). The negative pressure results from stresses caused by outwardly directed tension. These stresses cause an elastic instability in the specimens similar to those observed when an ordinary rubber balloon is inflated.
The theory of rubber like elasticity predicts that the degree of expansion of a cavity will become indefinitely large (ref. 17. The cavity does this when the tensile stress reaches a critical value ([[sigma].sub.c]) [[sigma].c] is reached when the stress in triaxial tension is about 5E/6. Hence, rubber with a low value of E is prone to failure by internal rupture if it is placed in triaxial tension. To avoid internal rupture in triaxial tension, limit (ref. 23) the mean tensile stress applied to thin bonded blocks to less than about E/3.
The need to limit triaxial stress was demonstrated by a laminated steel/rubber seal for a rocket (figure 22) (ref. 33). One end of the seal attaches to the lower portion of a rocket case and the other end attaches to the nozzle. The seal consists of rubber (NR) bonded to annular shaped steel elements. The seal permits the nozzle to move with respect to the rockets case and thus steer the rocket.
When the rocket fires, hot gases from burning propellant place the seal under considerable compressive stress. Compressive strain in the seal is low because of its high shape factor (SF = 15-20). Although the seal compresses only slightly under high force, only low force is necessary to move the nozzle in the direction indicated by the arrows in figure 22. The low shear modulus rubber (0.12 MPa or 18 psi) used in the seal accounts for the low force required.
Since Young's modulus (E) is approximately equal to three times the shear modulus (G), E for this rubber is also low. Because E is low, a tensile force applied to the seal along its axis could cause seal failure. This force places the bonded low@modulus rubber in triaxial tension. Low modulus rubber is especially prone to failure in the triaxial strain mode.
When the seal was tested in this mode, cohesive failure did in fact occur. A large number of pock marks were observed on the surface of the jailed rubber. The source of these marks or voids might be gases trapped during molding, or they might be caused by cavitation during testing.
To establish the cause, the same rubber used in the seal was bonded into specimens like those in figure 21. The rubber in half of these specimens was sliced by a sharp knife. The other half of the specimens were tensile tested, and rupture occurred cohesively in the rubber. The ruptured surfaces of the tensile tested specimens were covered with pock marks or voids, while the cut surfaces were free of voids. Hence, the voids were caused by cavitation during testing since they were not present in the untested specimens.
This example demonstrates the importance of proper selection of a test method and subsequent interpretation of test results. The weakness of the seal in triaxial tension in this application is of less importance because the seal is not normally in triaxial tension in service. The soft rubber performed satisfactorily in compression because its shape factor was sufficiently high. Only low forces were required to move the nozzle because of the low shear modulus rubber used in the seal.
[3.] L.R.G. Treloar, Introduction to polymer science, Wykeham Publications, London, 1970. [13.] F.S. Conant, "Physical testing of vulcanizates," chapter 5 in Rubber Technology, M. Morton, Ed., Van Nostrand Reinhold Company, Second Edition, 1973, p. 114. [17.] A.N. Gent, "Rubber elasticity. Basic concepts and behavior," chapter 1 in Science and Technology of Rubber, F.R. Eirich, Ed., Academic Press, New York, 1978, p. 1. [23.] A.N. Gent, "Strength of elastomers," chapter 10 in ref. 17, p. 419. [27.] E.G. Kimmich, India Rubber World 103, 45 (1940). [28.] D.A. Meyer and J.A. Welch, Rubber Chem. Technol. 50, 145 (1977). [29.] L.A. Wood, Rubber Chem. Technol. 49, 189 (1976). [30.] P.B. Lindley, Journal of Strain Analysis 3, 142 (1968). [31.] P.B. Lindley, NR Technical Bulletin, "Engineering design with natural rubber," 1974 Edition. [32.] B.P. Holownia, chapter 8 in Elastomers: Criteria for engineering design, C. Hepburn and R.J.W. Reynolds, editors, Applied Science Publishers Ltd., London, 1979, p. 111. [33.] W.A. Hartz and J.G. Sommer, unpublished data, The General Tire & Rubber Company.
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|Title Annotation:||part 4|
|Author:||Sommer, John G.|
|Date:||Oct 1, 1996|
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