Ideal-rubber approximation

Show that, for an ideal rubber obeying Gaussian statistics (and hence eqn 3.38), the nominal tensile stress-strain curve is approximated to a high accuracy at small strains by... 

Equation (3.31) is called the equation of state of the rubber. This equation was firstly derived by Guth and James in 1941 (Guth and James 1941). We conventirai-ally make an ideal-chain approximation with C = 1. 

Ideal rubbers and liquids deform at constant volume, for which Vp is equal to 0.5. Poisson s ratio for real elastomers may be experimentally determined by applying extensometers in the transverse and axial directions of a sample. Approximate values of Vp thus determined are 0.33 for glassy polymers, 0.4 for semicrystalline pol5miers, and 0.49 for elastomers. Young s modulus E is the ratio of normal stress f/A to corresponding strain e ... 

A sample of a certain ideal rubber has a number average RMM between crosslinks of 5 000 and a density of 900 kg/m. A block of this rubber, a cube of side 100 mm, is tested at a temperature of 300 K. What is its tensile modulus What is its shear modulus Axes X, Y, and Z are chosen parallel to the edges of the cube. A compressive force Fx is applied in the -direction, to reduce the X-dimension from 100 mm to 75 mm. Calculate F. What do the Y and Z dimensions become A further compressive force Fy is now applied in the Y-direction to reduce the Y-dimension to 75 mm, the X-dimension remaining at 75 mm. What now are the magnitudes of the forces f and F,. What has the Z-dimension now become How much strain energy is now stored in the block Assume the rubber obeys Gaussian statistics and use the approximation... 

As we shall see, ideal rubbers possess properties which reproduce at least qualitatively the main features in the behavior of real rubbers and thus can provide an approximate first description. Equation (7.6) implies that for an ideal rubber, force and temperature are linearly related... 

It is found experimentally that the stretching of a mbber object approximately obeys three properties (1) the volume remains constant (2) the tension force is proportional to the absolute temperature and (3) the energy is independent of the length at constant temperature. An ideal rubber exactly conforms to these three properties. Since the volume is constant, the first term on the right-hand side of Eq. (28.9-1) vanishes for an ideal rubber. For reversible processes in a closed system made of ideal mbber, the first and second laws of thermodynamics give the relation ... 

However, no real material shows either ideal elastic behavior or pure viscous flow. Some materials, for example, steel, obey Hooke s law over a wide range of stress and strain, but no material responds without inertial effects. Similarly, the behavior of some fluids, like water, approximate Newtonian response. Typical deviations from linear elastic response are shown by rubber elasticity and viscoelasticity. 

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 spring, with one fixed end and another free end, gives an approximate idea of a pole in the domain of elasticity of solids. Ideally, only one end of the spring should be considered. Obviously, this is not practically feasible and this is a pnrely virtual concept. The only real system has two ends, making a dipole, or no ends at all in the case of a tore (e.g., rubber band). 

The major reason that butyl rubber is used in the tire industry is its superior resistance to air permeability, as weii as other gases. Butyi has approximately 13 times greater resistance to air permeability than natural rubber. This makes it ideal for use in making tire innerliners and inner tubes. While some polar specialty elastomers can also provide good air permeability resistance, they are far more expensive and generally do not possess the right combination of other needed properties. 

Rubber and plastic melts can be considered, to a first approximation, as extremely high-viscosity fluids. This is only an approximation and it must be remembered that polymers generally show viscoelastic properties—a combination of viscous flow and elastic recovery. Viscosity, in turn, is the quantitative measure of resistance to flow under a given set of circumstances. The Greek letter that usually designates viscosity is Tj. For an ideal, Newtonian fluid, viscosity is simply the ratio between Shear Stress (t), the pressure placed on the fluid to create flow, and the Shear Rate (y), the rate of flow over time as seen in Equation 16C.1 ... 

It is widely accepted that polymers in melts and rubbers are in random flight conformations, i.e. the molecule has a large choice of conformations and these differ by energies much less than kT, so that they can be considered as isoenergetical to a first approximation. The theoretician s picture is idealized as in Figure 1, where the crosslinks are indicated simply by crosses connecting the chains, indicated by lines. 

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