Rubber elasticity theory, extension

It will be shown in Chapter 11 that the correlations developed in this monograph can be combined with other correlations that are found in the literature (preferably with the equations developed by Seitz in the case of thermoplastics, and with the equations of rubber elasticity theory with finite chain extensibility for elastomers), to predict many of the key mechanical properties of polymers. These properties include the elastic (bulk, shear and tensile) moduli as well as the shear yield stress and the brittle fracture stress. In addition, new correlations in terms of connectivity indices will be developed for the molar Rao function and the molar Hartmann function whose importance in our opinion is more of a historical nature. A large amount of the most reliable literature data on the mechanical properties of polymers will also be listed. The observed trends for the mechanical properties of thermosets will also be discussed. Finally, the important and challenging topic of the durability of polymers under mechanical deformation will be addressed, to review the state-of-the-art in this area where the existing modeling tools are of a correlative (rather than truly predictive) nature at this time. 

As rubbers deform without change of volume, the extension ratios in the two lateral directions are 1/VX. Rubber elasticity theory leads to the prediction that the true stress (the force divided by the deformed cross-sectional area) is given by... 

Historically, the question of mechanism of elasticity has been one of evaluating the relative contributions of three different proposed mechanisms (1) the random chain network (classic rubber elasticity) theory, - (2) the solvent entropy theory, and (3) the damping of internal chain dynamics on extension. Ttie first is due to the Flory school the second was initiated by Weis-Fogh and Andersen, and the third is due to the present author and coworkers of the last quarter century. 

If these new results stand the test of time, they may help answer one of the important challenges to the Mark-Flory random coil model (see Section 5.3). In this challenge the considerable discrepancy between rubber elasticity theory and experiment is blamed on the presumed nonrandom coiling of the polymer chain in space. These discrepancies may, however, lie rather in the realm of the mode of chain disentanglement on extension. 

A stress-strain isotherm for the uniaxial deformation of natural rubber, at ambient temperature, that was cross-linked in the liquid state is shown in Fig. 8.1.(5) Here f is the nominal stress defined as the tensile force,/, in the stretching direction divided by the initial cross-section, and a is the extension ratio. Using the most rudimentary form of molecular rubber elasticity theory f can be expressed as (6-9)... 

Rubber elasticity theory can also be used to derive a general class of singleintegral equations, as opposed to a particular equation such as that for the Lodge rabberlike liquid. To proceed, consider a cube of rabber, initially of unit edge, in an extensional deformation as shown in Figure 14.23. In the deformed state, the block of rabber has dimensions Aj, X2, and /I3, which also happen to be the extension ratios. 

Figure 1. Stress relaxation curves for three different extension ratios. Uncross-linked high-vinyl polybutadiene with a weight average molecular weight of 2 million and a reference temperature of 283 K. G is the apparent rubber elasticity modulus calculated from classical affine theory. (Solid line is data from Ref. 1).

Gumbrell.S.M., Mullins,L Rivlin.R.S. Departures of the elastic behavior of rubbers in simple extensions from the kinetic theory. Trans. Faraday Soc. 49,1495-1506 (1953). 

Stress/strain relationships are commonly studied in tension, compression, shear or indentation. Because in theory all stress/strain relationships except those at breaking point are a function of elastic modulus, it can be questioned as to why so many modes of test are required. The answer is partly because some tests have persisted by tradition, partly because certain tests are very convenient for particular geometry of specimens and partly because at high strains the physics of rubber elasticity is even now not fully understood so that exact relationships between the various moduli are not known. A practical extension of the third reason is that it is logical to test using the mode of deformation to be found in practice. 

Viscoelastic properties of molten polymers conditioning the major regularities of polymer extension are usually explained within the framework of the network concept according to which the interaction of polymer molecules is localized in individual, spaced rather far apart, engagement nodes. The early network theories were developed by Green and Tobolsky 49) and stemmed from successful network theories of rubber elasticity. These theories were elaborated more fully in works by Lodge50) and Yamamoto S1). The major elasticity. These theories is their simplicity. However, they have a serious drawback the absence of molecular weight in the theory. 

You should notice that the first term has the same form as that given by simple theories of rubber elasticity. The equation fits extension data for deformations up to about 300% very well, but cannot fit compression data using the same values of the constants C, and C2. Attempts to obtain the second term (in this form) using a molecular theory have not, as yet, been very successful, so we ll say no more about it. 

There exists as yet no rigorous extension of the statistical theory of rubber elasticity to a filled elastomer. Nevertheless, many attempts have been made to apply the theory to data on filled rubbers, usually with the objective of obtaining at least an approximate estimate of the number of filler-rubber attachments. These attempts have been discussed in earlier reviews (17, 126) and will not be considered here in full detail. We only restate briefly some of the experimental and theoretical difficulties inherent in this approach. 

Figure 11.15. Effects of input material parameters on stress-strain curves of elastomers under uniaxial tension, as calculated by the theory of rubber elasticity with finite chain extensibility. G denotes the shear modulus, while n denotes the average number of statistical chain segments (Kuhn segments) between elastically active network junctions, (a) Engineering stress a as a function of draw ratio X, as calculated by using Equation 11.41. (b) True stress (simply equal to aX for an elastomer) as a function of true strain [In (A,)].

According to the statistical theory of rubber elasticity, the elastic stress of an elastomer under uniaxial extension is directly proportional to the concentration... 

The simplest model is the statistical theory of rubber-like elasticity, also called the affine model or neo-Hookean in the solids mechanics community. It predicts the nonlinear behavior at high strains of a rubber in uniaxial extension with Fq. (1), where ctn is the nominal stress defined as F/Aq, with F the tensile force and Aq the initial cross-section of the adhesive layer, A is the extension ratio, and G is the shear modulus. 

The proposed basis for the nature of the ideal elasticity exhibited by the family of protein-based polymers using the generic sequence (GXGXP) , where X is a variable L-amino acid residue, became very controversial. The adherents to the classic (random chain network) theory of rubber elasticity took great exception to oiu proposal that the damping of internal chain dynamics on extension gave rise to entro-pic elasticity (for more extensive treatment of this controversy, see Urry and Parker. ). The physical basis for the different (even heretical, to some) mechanism of near-ideal elasticity provides insight into the remarkable biocompatibility of elastic protein-based polymers. 

---