Effect of shape and other design factors on rubber behavior.

Effect of shape and other design factors on rubber behavior

Rubber is well known for its ability to return to nearly its original shape after repeated deformation. Less well known is the dramatic effect of design on the stiffness of rubber composites. For example, a rubber-steel composite bearing might exhibit a stiffness 500 times that of the rubber by itself. This high-stiffness capability is used to considerable advantage in a variety of applications.

The applications discussed in this article are bridge bearings, earthquake bearings and rocket nozzle bearings. Also discussed is the effect of selected compounding and processing factors on rubber behavior in these composites. Geometry and basic rubber properties, along with these factors, govern mainly the behavior of rubber composites.

Properties

Table 1 lists several properties for steel and rubber that are important for use in the design of rubber composites.

Following are definitions for the three moduli listed and for Poisson's ratio; also listed are the methods used to obtain these properties for rubber.

Young's modulus ([E.sub.o]) is the slope of the stress-strain curve over small tensile or compressive strains. It can be determined by stretching rubber in uniaxial tension, or by squeezing rubber between platens with lubricated faces[1].

Shear modulus (G) is the ratio of the shear stress to the resulting shear strain, with the shear strain expressed as a fraction of the original thickness of the rubber measured at right angles to the force. It can be determined using a quadruple lap shear specimen[2].

Bulk modulus (B) is the ratio of triaxial compressive stress to the change in volume per unit volume; it can be determined by several methods[3]. Poisson's ratio (V) measures the volume contraction upon deformation; it is defined as the ratio: relative lateral contraction/relative longitudinal strain under unilateral stress. V can be determined[4] by direct measurement or calculated from measured values of B and G.

Relationships among properties

Rubber may be considered as obeying the classical theory of elasticity[5] at very low strains, where the elastic constants of isotropic materials are related by equations 1-3.

[E.sub.o] = 2G(1+V)     (1)
   = 3B(1-2V)     (2)
   = 9BG/(3B+G)     (3)

V, a measure of volume contraction upon deformation of rubber, equals 0.5 when no contraction occurs. Values of V for most rubber compounds are slightly less than 0.5. Rearranged equation 2 (2R) shows that B is extremely sensitive to the value of V.

B = [E.sub.o]/3(1-2V) (2R)

Table 1 shows a value for V of 0.4991 for a typical rubber[7]. This value varies with factors such as rubber type and carbon black level[5]. For example, V decreases with increasing carbon black level and V is slightly greater for natural rubber than for several other rubbers. Also shown in table 1 are other typical properties for rubber, along with typical properties for steel ([E.sub.o], G, B and V).

In contrast to steel and other materials, G for rubber is very low, typically 1.5 MPa; B for rubber is typically 1100 MPa. The low value for G means that even small forces will alter the shape of rubber significantly. The high value for B means that even high hydrostatic compressive forces will reduce the volume of rubber only slightly. Restricting changes in rubber shape, for example by adhering it to steel, will markedly alter the compression stiffness of rubber. In addition to these factors, shape factor and boundary conditions are also important.

Boundary conditions

When a rubber block is compressed between parallel metal plates, the nature of the boundary between block and plate affects significantly the stiffness behavior of the rubber. Composites were formed from rubber blocks (27mm thick by 149mm wide by 200mm long) and metal plates. The rubber blocks were either adhered to the metal or they were lubricated at the boundary between surfaces of rubber and metal[8].

Figure 1 shows that stiffness in compression is maximum for the adhered composite, minimum for the composite lubricated with petrolatum. Stiffness with graphite is intermediate between the adhered and petrolatum-lubricated composites.

These results demonstrate the large effect of boundary conditions on stiffness behavior of rubber composites. For example in figure 1, compressive stress of the adhered composite at 10% deflection is about six times that of the petrolatum lubricated composite. Materials on the surface of rubber, for example bloom, wax and dust, can act as lubricants on a rubber surface and cause erratic test results. Sandpaper, used between platens and rubber, minimizes slippage and gives a result similar to that obtained with adhered rubber (figure 1). Shape is another important factor in stiffness behavior.

Shape factor

Many engineered rubber composites consist of rubber-metal laminates. These composites can assume a variety of shapes, but a laminate with rubber adhered to the parallel faces of metal plates finds frequent application. Shape factor (SF) is defined as the ratio: area of one loaded face/total area free to bulge. Figure 2 illustrates shape factors of 0.25, 1.0 and 10.0.

Figure 3 shows the pronounced effect of compressive strain on compressive stress over an SF range of 0.25-6. For example, a strain of only about 2% causes a 6 MPa stress at an SF of 6; at an SF of 0.25, a strain of about 38% is required to produce the same stress. Hence, even these relatively low shape factors change stiffness substantially.

Figure 4 shows the effect of a wide SF range on compression modulus for three different hardness compounds[9]. International hardness degrees (IRHD) are comparable to Shore A hardness. The hardness values shown, 35-75, represent the range of hardness for compounds typically used in engineered rubber composites. This hardness range in figure 4 results in less than an order of magnitude change in compression modulus.

In contrast SF, over the range shown in figure 4, can increase compression modulus by almost three orders of magnitude. Hence, SF can affect stiffness one hundred fold more than typical compounding changes. SF effects have been established for rubber with different shapes, for example hollow squares and hollow cylinders[10].

The use of SF to alter stiffness in laminated rubber-metal composites is a powerful design tool. Composites can be fabricated that have extremely high stiffness in a direction normal to the plates in a composite, accompanied by quite low stiffness in the lateral direction. These features are included in a variety of applications, including laminated composites used as bridge bearings, earthquake bearings and nozzle bearings for solid rockets.

Applications

Rubber at the bonded edges of these composite bearings is free to bulge. Tensile stresses tend to be maximum at the edge under most conditions[11]. Incorporation of a radius in the rubber[12], along with other design modifications[13], at the edges reduces stress and extends the life of composites. As an alternative to radii, bridge bearings can be encapsulated in rubber.

Bridge bearings Figure 5 shows a section cut from a bridge bearing, where the uncut rear and right faces are encapsulated with rubber[14]. Encapsulation reduces stress concentrations at the edge of the plates and also protects the metal plates from corrosion. Rubber is adhered in these metal plates to yield a composite that is extremely stiff in the vertical or load bearing direction. Stiffness in the horizontal direction is very low because of the low shear modulus of the rubber. The presence of the metal plates does not alter shear stiffness.

Bearings with these characteristics support the considerable weight of a bridge with allowable changes in bridge (roadway) level. The bearings do this while accommodating the thermal contraction and expansion of bridges. Because shear stiffness of these bearings is low, a contracting and expanding bridge transfers only acceptably low lateral forces to bridge support structures, for example concrete columns. Lower lateral forces mean that concrete columns with reduced diameter might be used.

Pressure measurements on bridge bearings established that pressure concentrations did not exceed the design strength of concrete[10], which is about 21 MPa. Figure 6 shows the cross section of a bridge bearing subjected to an average compressive stress of 5.5 MPa. Measured pressure over the central area of this bearing was 7.4 MPa, with the pressure tapering off to zero at the edges.

Figure 7 shows the same size bearing with a centrally located hole[10]. The hole alters both the magnitude and distribution of pressure. Maximum pressure increases to about 10.3 MPa and again tapers off to zero at the edges. Bridge bearings require no maintenance and have a long history of successful service. Another application for similar type bearings is in earthquake protection for buildings.

Earthquake bearings Laminated rubber bearings, similar in construction to the bridge bearing in figure 5, are used as base isolators between earth and structure to provide protection during earthquakes[15]. Earthquake bearings must be able to support the considerable vertical load of a structure. such as a building, with a large safety factor. In an earthquake during sidewise movement of the building, bearings must continue to support this vertical load. The bearing or isolator must be sufficiently stiff vertically to prevent significant amplification of any vertical components induced by the earthquake.

The main purpose of the isolator is to protect structures from the horizontal component of ground motion during an earthquake. This protection results from a spring mounting system that prevents most of the earth-induced forces from transferring into the structure. A structure mounted on rubber bearings will have a horizontal natural frequency that depends upon bearing stiffness[15]. This stiffness depends upon both rubber modulus and bearing geometry.

To reduce the force intensity of the earthquake, the horizontal natural frequency ([f.sub.hn]) must be well below the dominant earthquake frequencies. Since these dominant frequencies are usually in the 1.5 to 8 Hz range for firm soil sites, a mounting frequency of about 0.5 Hz is usually chosen for the structure[15]. Equation 4 relates ([f.sub.hn]), bearing stiffness (K), and the supported mass (m). (4) [f.sub.hn] = 1/2 [Pi] [(K/m).sup.0.5]

For a typical bearing designed to support 200 tons, and with a [f.sub.hn] value of 0.5 Hz, a horizontal stiffness of 200 ton/meter is obtained. Equation 5 can be used[15] to determine the compression modulus ([E.sub.c]) of a bonded rubber layer. (5) [E.sub.c] = 5.7 G[(SF).sup2]

In Equation 5, SF is shape factor and G is shear modulus; the coefficient, 5.7, has been found to give good agreement with practice over the range of rubber hardness typically used in structural bearings[15]. Another factor that determines bearing stiffness is the number of layers in the laminated bearing. Variation in interplate placement in laminated bearings is claimed to affect bearing stiffness less than variations in compounding or in non-uniform cures.

Compounding techniques can be used to alter the wind sway characteristics of a structure[15]. A high wind load of 0.003g would produce a movement of about 30mm amplitude for a building mounted at an [f.sub.hn] of 0.5 Hz. While this movement does not endanger the building, it can disturb building occupants. Disturbance of occupants can be minimized by appropriately incorporating carbon black in the bearing compound so as to increase non linearity at low strains.

By this compounding technique, modulus is about five times higher in the low strain region (2% and lower) where wind control is desirable. At higher shear strains (50-125%) that might be experienced in an earthquake, a compound can demonstrate the lower shear modulus that is needed for building protection. Hence, wind sway control and earthquake protection can both be obtained by properly combining design and compound.

Other important considerations for earthquake bearings are fire protection, creep and aging behavior. Increasing the thickness of the outer layer of rubber on a bearing greatly increases its fire resistance[16] and this technique is used for fire protection. Creep measurements were made over seven years on a 1,400 ton building supported by natural rubber bearings[17]. Based on these measurements, the rate of creep is acceptable; settling of the building onto its foundation is expected to take more than a century. A century of service has already been demonstrated for bridge support pads installed in a rail viaduct in 1889 in Australia[15]. Both bridge and earthquake bearings use flat metal plates in their construction. In contrast, nozzle bearings for solid rockets use annular-shaped metal plates that have curved surfaces.

Rocket nozzle bearings Figure 8 shows a cross-sectional view of a laminated steel-rubber bearing that permits the nozzle to move with respect to the rocket case and thus steer the rocket[18]. During service, burning fuel produces high pressure gas that places the bearing under a considerable compressive load. Only limited compression of the bearing occurs because high shape factors of 15-20 are used in bearing design[18]. Because the rubber layers (compound A in table 2) have a shear modulus of only 0.12 MPa, lateral movement of the nozzle requires little force. Low force is desired so as to limit the weight of the force-actuating mechanism.

The rupture stress in shear for compound A is 4.8 MPa, even though its shear modulus is more than ten fold lower than a typical value shown in table 1 for G (1.5 MPa). The increased plasticizer level in compound B reduced G even further. Accompanying the increased plasticizer in compound B are lower viscosity and associated processing problems. Processing improved substantially through the use of the second and independent cure system shown for compound C in table 2.

This second system introduced a small number of crosslinks into compound C during its processing at 132 [degrees] C. This limited amount of crosslinking raised viscosity slightly and improved significantly the processing behavior of compound C relative to B. The two different cure systems resulted in two stages of crosslinking[19]. The two stages (two plateaus) were clearly evident on a Monsanto Rheograph for compound C run at 132 [degrees] C. Certain efficient vulcanization systems also exhibit two plateaus[ref. 20]. Thus it is possible to improve processing of low viscosity compounds that have an extremely low modulus.

Nozzle bearings were fabricated using low-modulus compound A. Shear tests as well as tensile tests were run on these bearings, even though bearings do not normally experience tension in service. Subsequent to these tensile tests, pock marks or voids were observed in the failed rubber regions of the bearings tested in tension. Molding defects were thought to be the cause of these voids.

To establish whether the voids were caused by molding or by testing in tension, a number of specimens (figure 9) with an SF = 3.1 were prepared using compound A[18]. The rubber in half of these specimens was cut through with a sharp knife; the other half of the specimens was tested to failure in uniaxial tension. Voids were evident on the failed surfaces of all specimens that were tested in tension, but no voids were evident on the cut surfaces. Hence tensile testing caused the voids, not molding.

These voids result from triaxial tension that occurs in bonded rubber discs at relatively small tensile loads[21]. Voids form when the local triaxial stress[22] attains a value of 5/6 [E.sub.0]. Assuming [E.sub.0] equals approximately 3G (see equation 1), Young's modulus for compound A would be only about 0.4 MPa, which is an extremely low value. Hence, triaxial stress must be avoided or restricted to very low levels if failure is to be avoided in low modulus rubber.

The rocket nozzle example illustrates the effect not only of design, but also the effect of compounding, processing and testing. High SF and associated high compressive stiffness of the nozzle limit vertical compressive strain in the nozzle. Only low force is needed to move the nozzle horizontally because the rubber possesses an extremely low shear modulus. Associated with this rubber is a very low viscosity that caused processing difficulties. Processing was improved by rising viscosity through special compounding techniques. Finally, tests were chosen to ensure that they were relevant to the end use application.

Hence, greatest success will be realized when design is considered along with a number of other important factors. [Table 1 and 2 Omitted] [Figure 1 to 9 Omitted]

References [1]O.H. Yeoh, Plastics and Rubber Processing and Applications, 4 (2)141 (1984). [2]Determination of rubber modulus in shear, International Standard Organization Standard 1827. [3]R.P. Brown "Physical testing of rubbers," Applied Science Publishers, London, 1979, p. 143. [4]M. Stanojevic, Polymer Testing, 3 (3) 183 1983. [5]P.K. Freakley and A.R. Payne "Theory and practice of engineering with rubber," Applied Science Publishers Ltd., London, 1978, p. 24 [6]T. Baumeister, "Standard handbook for mechanical engineers," McGraw-Hill Book Company, New York, 1967, p. 5-6. [7]A.N. Gent and Y.-C. Hwang, Rubb. Chem. Technol. 61 630 (1988). [8]E.G. Kimmich, India Rubber World 103 (3) 45 (1940). [9]P.W. Allen, P.B. Lindley and A.R. Payne, "Use of rubber in engineering," Maclaren and Sons Ltd., London, 1967, p. 10. [10]D.A. Meyer and J.A. Welch, Rubb. Chem. Technol. 50 145 (1977). [11]A. Stevenson, Rubb. Chem. Technol. 59 208 (1986). [12]Ref. 9, ibid., p. 70. [13]J.G. Sommer, "Rubber parts," Chapter 6.10 in "Handbook of product design for manufacturing," J.G. Bralla, Ed., McGraw-Hill, New York (1986). [14]J.G. Sommer, Rubb. Chem. Technol. 58 662 (1985). [15]C.J. Derham, Rubber World, 195 (6) 32 (1987). [16]L. Mullins, Phys. Technol., 15 177 (1984). [17]C.J. Derham, NAPRA 3rd Rubber in Engineering Conference, London, 1973, p. F/1. [18]J.G. Sommer, "Physical properties and their meaning," in "Textbook for intermediate correspondence course," Part I, H.L. Stephens, Ed., Rubber Division ACS, 1985. [19]J.G. Sommer, unpublished work, GenCorp. [20]C.T. Loo, M. Porter, and B.K. Tidd, Rubb. Chem. Technol. 61 173 (1988). [21]A.N. Gent, Proceedings of the Royal Society of London, Series A, 249 195 (1959). [22]K. Cho and A.N. Gent, Journal of Materials Science 23 (1) 141 (1988).