Rubber-like elasticity strain

It is also possible to estimate the cross-link density from the stress-strain data, using the statistical theory of rubber-like elasticity [47,58]. For a swollen rubber the relationship is... 

These Monte Carlo distributions can be used in the standard three-chain model for rubber-like elasticity to generate, for example, stress-strain isotherms [5]. Non-Gaussian effects can cause large increases in modulus at high... 

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. 

Birefringence of Phantom Networks. This theory is the basis for all theories that deal with birefringence of elastomeric polymer networks. It is based on the phantom network model of rubber-like elasticity. This model considers the network to consist of phantom (ie, non-interacting) chains. Consider the instantaneous end-to-end distance r for the ith network chain at equilibrium and at fixed strain. For a perfect (ie, no-defects) phantom network the birefringence induced... 

The phenomenological approach to rubber-like elasticity is based on continuum mechanics and symmetry arguments rather than on molecular concepts [2, 17, 26, 27]. It attempts to fit stress-strain data with a minimum number of parameters, which are then used to predict other mechanical properties of the same material. Its best-known result is the Mooney-Rivlin equation, which states that the modulus of an elastomer should vary linearly with reciprocal elongation [2],... 

In summary, the anomalous upturn in modulus observed for crystallizable polymers such as natural rubber and cw-1,4-polybutadiene is largely, if not entirely, due to strain-induced crystallization. In the case of the noncrystaUizable PDMS model networks it is clearly due to the limited chain extensibility, and thus the results on this system will be extremely useM for reliable evaluation of the various non-Gaussian theories of rubber-like elasticity. 

The initial modulus is determined in the limit of small strain. The initial portion of the force-length curve is usually reversible. The deformation of the disordered interlamellar region is involved and the lamellar structure remains essentially intact. Interpreting the modulus, in terms of the basic structural and molecular parameters that define a semicrystalline polymer, is complex. In this region of very small strain, the primary effect is a rubber-like elastic deformation, whereby chain entanglements and other topological features act as effective cross-links. The total system is constrained by the bounding lamellae and their broad basal planes. 

The tensile stress-strain curve of a stretched NR sample is shown in Fig. 1. As expected, the mechanical behaviour appears to obey the concept of rubber-like elasticity, where the application of stress is considered to cause molecules to change from a coiled to an extended configuration... 

One of the fascinating properties of the elastomeric materials is their rubber-like elasticity— that is, they have the ability to be deformed to quite large deformations and then elastically spring back to their original form. This results from crosslinks in the polymer that provide a force to restore the chains to their undeformed conformations. Elastomeric behavior was probably first observed in natural rubber however, the past several decades have brought about the synthesis of a large number of elastomers with a wide variety of properties. Typical stress-strain characteristics of elastomeric materials are displayed in Figure 15.1, curve C. Their moduli of elasticity are quite small, and, they vary with strain because the stress-strain curve is nonlinear. 

These elastomers also exhibited stress-strain isotherms in elongation that were closer in form to those expected from the simplest molecular theories of rubber-like elasticity. Specifically, there were large decreases in the Mooney-Rivlin 2C2 correction constant. 

This linear relationship between stress and strain is a very handy one when calculating the response of a solid to stress, but it must be remembered that most solids are elastic only to very small strains up to about 0.001. Beyond that some break and some become plastic - and this we will discuss in later chapters. A few solids like rubber are elastic up to very much larger strains of order 4 or 5, but they cease to be linearly elastic (that is the stress is no longer proportional to the strain) after a strain of about 0.01. 

A rubber-like solid is unique in that its physical properties resemble those of solids, liquids, and gases in various respects. It is solidlike in that it maintains dimensional stability, and its elastic response at small strains (<5%) is essentially Hookean. It behaves like a liquid because its coefficient of thermal expansion and isothermal compressibility are of the same order of magnitude as those of liquids. The implication of this is that the intermolecular forces in an elastomer are similar to those in liquids. It resembles gases in the sense that the stress in a deformed elastomer increases with increasing temperature, much as the pressure in a compressed gas increases with increasing temperature. This gas-like behavior was, in fact, what first provided the hint that rubbery stresses are entropic in origin. 

The assumption that the contraction process is ideally adiabatic, while perhaps not entirely permissible practically, seems indicated by modern theory of the behavior of molecular chains, which pictures these as undergoing, when freed of restraints, a sort of segmental diffusion, much like the adiabatic expansion of an ideal gas into a vacuum (155). In the case of the molecular chain, it diffuses to the most probable, randomly coiled configuration, which is much less asymmetric, hence shorter, than an initially extended chain. Because rubber most nearly presents this ideal behavior, those fibers which develop increased tension (a measure of the tendency toward assumption of the contracted form) when held isometrically under conditions of increasing temperature (favoring the diffusion ) are said to be rubber-like. Most normal elastic solids upon stress are strained from some stable structure and relax as the temperature is raised. 

The elastic portion of the curves was investigated from the recorded data in order to determine precisely the initial volume strain variations. Since we showed that lims oidev/dEs) = (1 — 2vei), the initial Poisson s ratio Vei could be determined with a precision of a few percent. The values thus obtained are displayed in Table 19.2. It is seen, by comparison with neat PP, that the elastic Poisson s ratio of the blends increases slightly with the alloying content. This is probably due (i) to the contribution of PA6 that exhibits a Poisson s ratio significantly higher than that of PP and (ii) to the contribution of the elastomer (POE) for which Vei is near to 0.5 like most rubber-like materials. 

These are essentially independent effects a polymer may exhibit all or any of them and they will all be temperature-dependent. Section 6.2 is concerned with the small-strain elasticity of polymers on time-scales short enough for the viscoelastic behaviour to be neglected. Sections 6.3 and 6.4 are concerned with materials that exhibit large strains and nonlinearity but (to a good approximation) none of the other departures from the behaviour of the ideal elastic solid. These are rubber-like materials or elastomers. Chapter 7 deals with materials that exhibit time-dependent effects at small strains but none of the other departures from the behaviour of the ideal elastic sohd. These are linear viscoelastic materials. Chapter 8 deals with yield, i.e. non-recoverable deformation, but this book does not deal with materials that exhibit non-linear viscoelasticity. Chapters 10 and 11 consider anisotropic materials. 

A quantitative treatment of the rubber-elastic effect has recently been proposed by Choy ei al [48], which has led to an understanding of the drastic difference in the expansivities of LDPE and HDPE. According to them, an oriented polymer may be considered as a composite made up of four phases, i.e. the crystallites, the amorphous region, bridges and tie molecules. The internal strain due to rubber-like tie molecules... 

Finite strain elasticity the behaviour of polymers in the rubber-like state... 

Simo [51] proposed to penalize the classic elastic strain energy densities, Wo(F)> designed to fit the hyperelastic stress-strain responses of rubber-like materials submitted to the deformation gradient F, by a reducing parameter of the Kachanov form [99]. 

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