Compression shape factor

Most mechanical and civil engineering applications involving elastomers use the elastomer in compression and/or shear. In compression, a parameter known as shape factor (S—the ratio of one loaded area to the total force-free area) is required as well as the material modulus to predict the stress versus strain properties. In most cases, elastomer components are bonded to metal-constraining plates, so that the shape factor S remains essentially constant during and after compression. For example, the compression modulus E. for a squat block will be... 

The shape factor term is usually seen as (1 + 2S ) where S is the ratio of one bonded surface to the free surface. A useful review of the compression of rubber blocks is given by Gent2. 

When the shape factor is high, such that Ec/K (where K = bulk modulus) exceeds 0.1, the effective modulus will be below that expected, due to the bulk compression being appreciable. The effective modulus can then be estimated from99 ... 

The ISO standard clearly differentiates between bonded and unbonded test pieces and in an appendix gives the stress strain relationships, taking account of shape factor. In the scope it is pointed out that comparable results will only be obtained for bonded test pieces if they are of the same shape, and that lubricated and bonded test pieces do not give the same results. There is, however, a very curious little introduction that gives a very narrow view of when compression data is needed and makes a dubious claim about use on thin samples when hardness measurement would be difficult - so is an accurate compression test on thicknesses below 2 mm. 

The viscoelastic behaviour of rubbers is not linear stress is not proportional to strain, particularly at high strains. The non-linearity is more pronounced in tension or compression than in shear. The result in practice is that dynamic stiffness and moduli are strain dependent and the hysteresis loop will not be a perfect ellipse. If the strain in the test piece is not uniform, it is necessary to apply a shape factor in the same manner as for static tests. This is usually the case in compression and even in shear there may be bending in addition to pure shear. Relationships for shear, compression and tension taking these factors into account have been given by Payne3 and Davey and Payne4 but, because the relationships between dynamic stiffness and the basic moduli may be complex and only approximate, it may be preferable for many engineering applications to work in stiffness, particularly if products are tested. 

Only a constant strain method is specified with a standard strain of 25% for rubbers up to 80 IRHD, 15% for those 80 - 89, and 10% for those over 90. The compression is made between very smooth platens which are lubricated and, hence, the compression is made with some attempt at perfect slippage. Fairly obviously, the degree of slip and the test piece shape factor can affect the measured values of set. At one time it was standard to use glass-paper between the test piece and the platens to prevent slip but this produces greater concavity of the ends after release. 

The compressive strain properties of urethanes show that polyurethanes have very good load-bearing properties. Softer materials (below shore hardness of 75 A) all have very similar response curves. The shape of these curves is influenced to a large degree by the ratio of the constrained polyurethane to the free area. The ratio is commonly called the shape factor. In calculating the shape factor, only the area of one loaded surface is taken. The... 

The shape factor is an important consideration in the response to any applied load. Shape factor is defined as the ratio of the area of one loaded surface to the total of the unloaded surface that is free to bulge. The ability of the part to move when placed under load is important. If the surfaces are bonded to metal plates, the compressive stress to the compressive strain relationship is quite different. Figure 8.1 illustrates the load bearing of a series of polyurethanes compared to SBR and neoprene rubber compounds. 

The correct choice of the shape factor in polyurethane parts can help the bulging of parts under compression. This is shown in Figure 8.9. A cylindrical part with the same loaded area will deflect less than a rectangular part... 

Figure 3.24 Compressive stress-strain curves for natural rubber vulcanizate showing the effect of shape factor S. (From Ref. 24.)...

Back-up or backing A compressible material used in the bottom of sealant joints to reduce the depth of sealant and to improve its shape factor. Bacteria Microorganisms often composed of single cells, in the form of straight or curved rods (bacilli), spheres (cocci), or spiral structures. Barrier cream Preparatory protective creams that are applied to skin (paricularly hands) before working in a chemical environment. 

Once this function is determined, it could be applied to any substance, provided its critical constants Pc, T, and V are known. One way of applying this principle is to choose a reference substance for which accurate PVT data are available. The properties of other substances are then related to it, based on the assumption of comparable reduced properties. This straightforward application of the principle is valid for components having similar chemical structure. In order to broaden its applicability to disparate substances, additional characterizing parameters have been introduced, such as shape factors, the acentric factor, and the critical compressibility factor. Another difficulty that must be overcome before the principle of corresponding states can successfully be applied to real fluids is the handling of mixtures. The problem concerns the definitions of Pq P(> and Vc for a mixture. It is evident that mixing rules of some sort need to be formulated. One method that is commonly used follows the Kay s rules (Kay, 1936), which define mixture pseudocritical constants in terms of constituent component critical constants ... 

Although only compressibility factor calculations are used as an example in the explanation of the method, other properties can be predicted equally well. Because of the temperature and density dependence of the diameters and shape factors needed to relate them to critical constants it is best to determine separate values of them for each component. Three basic dimensionless properties should be determined. These are the ones best suited to the use of the HSE method with an equation of state in terms of temperature and density. These are the compressibility factor, z the internal energy deviation (U — V)/RT and a dimensionless fugacity ratio, ln(f/pRT). All other desired properties can be obtained from them. The ln(f/pRT) and z are calculated similarly. The computation scheme is outlined as shown in Table III. 

Table III. Shape Factors and Diameters for Nonconformal Fluids with Unknown Potentials Example for Compressibility Factor...

Table I. Component Liquid Molar Volumes (Vs) and Isothermal Compressibilities (ks) at 100 K and Vapor Pressure (Ps) Which Were Used along with Shape Factors (c or a) to Determine Equation-of-State Parameters a and b...

If there is complete absence of slippage, stress and strain are not uniform throughout the test piece, and barrelling takes place on compression. The relation between stress and strain is then dependent on the shape factor of the test piece. The stress-strain relationship can then be expressed as ... 

When the shape factor becomes very high such that approaches the bulk modulus K, the effective modulus is less than expected due to appreciable bulk compression and can be estimated from... 

The laminated spring of the last section are often compressed in a direction perpendicular to the layers. The top and bottom surfaces of the rubber layers cannot expand sideways because they are bonded to steel plates. The effect of this restraint on the compressive response depends on the layer shape factor S (Fig. 4.3) defined as... 

The bulk modulus of rubber, which depends on the strength of the van der Waals forces between the molecules, is 2 GPa. Therefore, the compressive modulus of a rubber layer increases by a factor of a thousand as the shape factor increases from 0.2 (Fig. 4.3). The responses are not shown for S < 0.2 such tall, thin rubber blocks would buckle elastically (Appendix C, Section C. 1.4), rather than deforming uniformly. When laminated rubber springs are designed, Eqs (4.5) and (4.7) allow the independent manipulation of the shear and compressive stiffness. The physical size of the bearing will be determined by factors such as the load bearing ability of the abutting concrete material, or a limit on the allowable rubber shear strain to 7 < 0.5 and the compressive strain e < -0.1. 

Compressive Young s modulus of a rubber spring vs. shape factor curves labelled with the rubber shear modulus in MPa (from Lindley, R B., Engineering Design with Natural Rubber, 4th Ed, Malayan Natural Rubber Producers Association, 1974). 

S is called the shape factor, and for simple applications can be taken as the ratio of stressed to unstressed area. The negative sign in the equation indicates compression, since A is less than 1 and the quantity inside the bracket is negative. 

X component of force, N (Ibj, dyn) compressibility constant, dimensionless mean surface renewal factor, s" conduction shape factor, m (ft)... 

In selecting this pure substance, an additional parameter accounting for the shape of the molecules should be used. The average critical compressibility factor Zg as defined by (33) is often used as an empirical shape factor parameter. However, when e molecular weights of the gases become small or at low temperatures, the pseudo-mass as calculated in (40) becomes important in addition to shape factors. If a reference substance with known properties is selected in this manner, it is convenient to define an equivalent temperature and pressure, 7 and P , as ... 

Limiting instantaneous live load deflections is important to ensure that deck expansion joints are not damaged. Steel reinforced elastomeric bearings exhibit nonlinear compressive load-deflection behavior. Compressive stiffness of an elastomeric layer substantially increases with increasing shape factor. The total compressive deformation of an elastomeric bearing is equal to the sum of the compressive deformation of all its constituent elastomeric layers. 

Note that the smallest acceptable value of shear modulus has been used. This will result in the largest compressive deformation. Because the bearing is composed of four interior layers and two exterior layers all having essentially the same shape factor, the total initial dead load deflection can be estimated as... 

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