Incompressibility rubber

If the particle is bonded firmly to the matrix (we will discuss this point later), the initial uniaxial tension stress is changed into a triaxial tension stress field, due to the low rubber incompressibility. The stress field around rubbery particles is not the same as that around a void. 

The low hardness has led to uses in printers rollers and stereos. It is, however, to be noted that when the material has been used to replace cellular rubbers or flexible polyurethane foams in sealing applications, problems have arisen where it has not been appreciated that although the rubber is very soft it is for practical purposes incompressible. 

For a rectangular rubber block, plane strain conditions were imposed in the width direction and the rubber was assumed to be an incompressible elastic solid obeying the simplest nonhnear constitutive relation (neo-Hookean). Hence, the elastic properties could be described by only one elastic constant, the shear modulus jx. The shear stress t 2 is then linearly related to the amount of shear y [1,2] ... 

We conclude that high internal stresses are generated by simple shear of a long incompressible rectangular rubber block, if the end surfaces are stress-free. These internal stresses are due to restraints at the bonded plates. One consequence is that a high hydrostatic tension may be set up in the interior of the sheared block. For example, at an imposed shear strain of 3, the negative pressure in the interior is predicted to be about three times the shear modulus p. This is sufficiently high to cause internal fracture in a soft rubbery solid [5]. 

So for an incompressible material v = 0.5 and Equation 2.4 is recovered. The value of Poisson s ratio for rubber is usually close to 0.5 but for many other solids the value is lower and we find 0.25 < v < 0.33. We may also describe a Bulk rigidity modulus, K, such as we would measure when we compress a material with hydrostatic pressure, in terms of Young s modulus ... 

This technique is used where charge density as well as uniformity are critical and are of paramount importance. The formulation is contained in a rubber bag which is evacuated before the application of pressure to it by means of an incompressible oil (Figure 3.2) Although wall friction is eliminated, some residual pressure gradients do occur because the bag rests on the surface at one end. 

The samples we used were vulcanizates of natural rubber (NR) and styrene-butadiene copolymer rubbers (SBR), carbon-filled and unfilled. Table 1 summarizes their preparative data. Incompressibility of these vulcanizates and some other vulcanizates were checked by dipping, stretching uniaxially, and weighing a specimen in water. 

Also, because the assumption is made that the rubber is incompressible ... 

Rather surprisingly, all these kinds of deformation can be described in terms of a single modulus. This is a result of the assumption that rubber is virtually incompressible (i.e. bulk modulus much greater than shear modulus). Young s modulus E = 3G (for fdled rubbers the numerical factor may be in fact as high as 4). Indeed, these relationships by no means fully describe the complete stress strain behaviour of real rubbers but may be taken as first approximations. The shear stress relationship is usually good up to strains of 0.4 and the tension relationship approximately true up to 50% extension. 

When a material is stretched there is also contraction in the direction perpendicular to the direction of stretching. The ratio of the lateral contraction to the longitudinal extension is Poisson s ratio. For incompressible materials, Poisson s ratio is 0.5 and as rubbers are very nearly incompressible they have values close to this. 

In rubber and viscoelastic fluids, these two quantities are sufficient since, when incompressibility is taken into account, I3 = 1. 

We now consider an extensional deformation of an incompressible rubber network (Fig. 3-7), where the stretch axes are oriented along the coordinate axes apd where the stretch ratios X, k2, and I3 are in directions 1, 2, and 3, respectively. For the example in Fig. 3-7, the deformation is a uniaxial extension that increases the length of the cylinder by a factor of X1 over its initial length. By volume conservation, the radius of the cylinder then shrinks to times the original radius. If the cross-link points are convected with... 

An example of the success of the temporary network model for a practical application is shown in Fig. 3-11. Here, the predictions of Eq. (3-24) are compared to experimental force-deflection data for impact tests in which a heavy flat-bottomed object is dropped onto a flat circular pad of dissipative Sorbothane rubber at various velocities and two different temperatures. Since the material is nearly incompressible under these conditions, the impact... 

The rubber in an inflated balloon is stretched biaxially, with initial diameter t/o and initial thickness of its walls to- This incompressible rubber contains u elastic strands per unit volume. 

For uniaxial deformations, two distinct principal extension ratios Ax and Ap, along and perpendicular to the director respectively, satisfy the relation Xz Xp = 1 because of the incompressibility of rubbers. 

For incompressible materials, the volume of the specimen remains constant during deformation, and V is 0.5. This is generally not true, although it is approached by natural rubber with v = 0.49. For most polymeric materials, there is a change in volume AV, which is related to Poisson s ratio by... 

Diaz-Calleja R, Sanchis MJ, Riande E (2009) Effect of an electric field on the deformation of incompressible rubbers bifurcation phenomena. J Electrost 67 158... 

An extension of rubber elasticity (i.e. of the description of large, static and incompressible deformations) to nematic elastomers has been given in a large number of papers [52, 61-66]. Abrupt transitions between different orientations of the director under external mechanical stress have been predicted in a model without spatial nonuniformities in the strain field [52,63]. The effect of electric fields on rubber elasticity of nematics has been incorporated [65]. Finally the approach of rubber elasticity was also applied recently to smectic A [67] and to smectic C [68] elastomers. Comparisons with experiments on smectic elastomers do not appear to exist at this time. Recently a rather detailed review of the model of an-... 

Rubbers are incompressible in the sense discussed in section 6.2.1 and, in terms of the extension ratios, the incompressibility is expressed by... 

Thus the relationships (6.21) and (6.21a) are compatible with the isotropy and incompressibility of a rubber and reduce to Hooke s law at small strains. Materials that obey these relationships are sometimes called neo-Hookeian solids. Equation (6.21a) is compared with experimental data in fig. 6.6, which shows that, although equation (6.21a) is only a simple generalisation of small-strain elastic behaviour, it describes the behaviour of a real rubber to a first approximation. In particular, it describes qualitatively the initial fall in the ratio of to k that occurs once k rises above a rather low level. It fails, however, to describe either the extent of this fall or the subsequent increase in this ratio for high values of k. 

Under small deformations rubbers are linearly elastic solids. Because of high modulus of bulk compression (about 2000 MN/m ) compared with the shear modulus G (about 0.2-5 MN/m ), they may be regarded as relatively incompressible. The elastic behavior under small strains can thus be described by a single elastic constant G. Poisson s ratio is effectively 1/2, and Young s modulus E is given by 3G, to good approximation. 

For crosslinked incompressible rubbers, Poisson s ratio is nearly equal to 0.5, so that (20)... 

Whenever the material retains its volume during stretching (incompressibility may occur with ductile or rubber-like materials), one derives a simplified correlation ... 

Here v represents the Poisson ratio, defined as the ratio between the linear contraction and the elongation in the axis of stretching. In the case of constant volume (an incompressible body like rubber), i> = 0.5 and therefore E = 3G for rigid materials p < 0.3. There is also a direct test for tear strength, mainly in the case of thin films, similar to those used in the paper industry. 

Rubbers are very nearly incompressible Le. G < jT (where K is the bulk modulus). 

Considering now substances with rigid globular particles, such as phenol-formaldehyde resins, rubber latex and sulphur sols, it appears from Table 8 that Voc is independent of the concentration, being therefore equal to Vo, This can be understood because these particles do not immobilise any solvent, and are practically incompressible. Referring once more to Fig. 18, attention is drawn to the curves for... 

Combined with the fact, that the r-T lines meet at the origin, so that for T = 0 also r = 0. Rubber is practically incompressible. 

These relations enable one to relate the shear viscoelastic functions to their tensile counterparts. At high compliance levels, rubbers are highly incompressible, and the proportional relation between the tensile and shear moduli and compliances holds. However, at lower compliances approaching Jg, the Poison ratio fi (which in an elongational deformation is -(Mw/dM, where w is the specimen s width and / is its length) is less than Eqs. (28) and (29) are then no longer exact. For a glass ju T. When G(t) = K(t), E t) = 2.25 Gif). 

Most interesting is the large increase in reinforcement even for small bound rubber thicknesses. Let us briefly discuss the advantages of the model. They find that the results obtained are realistic for small as well as intermediate filler concentrations, i.e. they are in accordance with experiments at least qualitatively. For the core-shell systems they have provided exact calculations of intrinsic moduli for various special forms of core-shell elasticity, i.e. soft spheres, hard spheres with soft surfaces, etc. These results contain no fit parameters and in principle both compressible and incompressible media are accessible. 

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