Elasticity behaviour

A polymer chain can be approximated by a set of balls connected by springs. The springs account for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

Dislocations are known to be responsible for die short-term plastic (nonelastic) properties of substances, which represents departure from die elastic behaviour described by Hooke s law. Their concentration determines, in part, not only dris immediate transport of planes of atoms drrough die solid at moderate temperatures, but also plays a decisive role in die behaviour of metals under long-term stress. In processes which occur slowly over a long period of time such as secondaiy creep, die dislocation distribution cannot be considered geometrically fixed widrin a solid because of die applied suess. 

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic behaviour. This is the behaviour characterised by Hooke s Law (Chapter 3). All solids are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1%. The slope of the stress-strain line, which is the same in compression as in tension, is of... 

Finally, Fig. 8.3 shows a third form of elastic behaviour found in certain materials. This is called anelasfic behaviour. All solids are anelastic to a small extent even in the regime where they are nominally elastic, the loading curve does not exactly follow the unloading curve, and energy is dissipated (equal to the shaded area) when the solid is cycled. Sometimes this is useful - if you wish to damp out vibrations or noise, for example you... 

In recent years impact testing of plastics has been rationalised to a certain extent by the use of fracture mechanics. The most successful results have been achieved by assuming that LEFM assumptions (bulk linear elastic behaviour and presence of sharp notch) apply during the Izod and Charpy testing of a plastic. 

Zvi Hashin and S. Shtrikman, A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials, Journal of the Mechanics and Physics of Solids, March-April 1963, pp. 127-140. 

D.J. Gunton and G.A. Saunders, The Elastic Behaviour of In-Tl alloys in the vicinity of the martensitic transition, Solid State Commun. 14 865 (1974). 

Many microbial polysaccharides show pseudoplastic flow, also known as shear thinning. When solutions of these polysaccharides are sheared, the molecules align in the shear field and the effective viscosity is reduced. This reduction of viscosity is not a consequence of degradation (unless the shear rate exceeds 105 s 1) since the viscosity recovers immediately when die shear rate is decreased. This combination of viscous and elastic behaviour, known as viscoelasticity, distinguishes microbial viscosifiers from solutions of other thickeners. Examples of microbial viscosifiers are ... 

A further important property which may be shown by a non-Newtonian fluid is elasticity-which causes the fluid to try to regain its former condition as soon as the stress is removed. Again, the material is showing some of the characteristics of both a solid and a liquid. An ideal (Newtonian) liquid is one in which the stress is proportional to the rate of shear (or rate of strain). On the other hand, for an ideal solid (obeying Hooke s Law) the stress is proportional to the strain. A fluid showing elastic behaviour is termed viscoelastic or elastoviseous. 

In reality the ideal elastic rubber does not exist. Real rubbery materials do have a small element of viscosity about their mechanical behaviour, even though their behaviour is dominated by the elastic element. Even so, real rubbers only demonstrate essentially elastic behaviour, i.e. instantaneous strain proportional to the applied stress, at small strains. 

Mechanical properties per se concerns with the qualities which determine the behaviour of a material towards applied forces. The ability to support weight without rupture or permanent deformation, to withstand impact without breaking, to be mechanically formed into different shapes - all these depend upon a combination of mechanical properties characteristic of metals. Four types of behaviour of a material under stress are very important linear or elastic behaviour, plastic behaviour, creep behaviour and fatigue behaviour. 

In a first-order fluid (Newtonian) only significant dimensionless groups can be derived which include elastic behaviour [88]. 

The reduced stress is defined as the force per cross-sectional area of the undeformed sample, divided by the term X-X- with X being the relative elongation L/L0. With undiluted rubber, this is not found experimentally. In most cases, however, the elastic behaviour in a moderate elongation range is satisfactorily described fcy the empirical Mooney-Rivlin equation, which predicts a linear dependence of on reciprocal elongation X- (32-34)... 

As for the derivation of Eqs. 122,123 and 124 only the transitions 1—>2 have been counted, these equations do not describe recovery processes, where the transitions 2 —>1 are important as well. These approximations have been made for convenience s sake, but neither imply a limitation for the model, nor are they essential to the results of the calculations. Equation 124 is the well-known formula for the relaxation time of an Eyring process. In Fig. 65 the relaxation time for this plastic shear transition has been plotted versus the stress for two temperature values. It can be observed from this figure that in the limit of low temperatures, the relaxation time changes very abruptly at the shear yield stress Ty = U0/Q.. Below this stress the relaxation time is very long, which corresponds with an approximation of elastic behaviour. 

The final main category of non-Newtonian behaviour is viscoelasticity. As the name implies, viscoelastic fluids exhibit a combination of ordinary liquid-like (viscous) and solid-like (elastic) behaviour. The most important viscoelastic fluids are molten polymers but other materials containing macromolecules or long flexible particles, such as fibre suspensions, are viscoelastic. An everyday example of purely viscous and viscoelastic behaviour can be seen with different types of soup. When a thin , watery soup is stirred in a bowl and the stirring then stopped, the soup continues to flow round the bowl and gradually comes to rest. This is an example of purely viscous behaviour. In contrast, with certain thick soups, on cessation of stirring the soup rapidly slows down and then recoils slightly. 

As noted in Chapter 1, viscoelastic fluids exhibit a combination of solid-like and liquid-like behaviour. Even a simple analysis of viscoelastic effects in process plant is beyond the scope of this book. This section is restricted to an outline of practical implications of elastic effects and a demonstration of the fact that viscoelastic liquids exhibit stronger elastic behaviour as the deformation rate is increased. 

As noted in Chapter 1, purely viscous behaviour corresponds to A = 0, while purely elastic behaviour is approached as A — It will now be... 

Thus purely viscous behaviour is approached as Aw 0 and purely elastic behaviour as Ao> — o°. 

The fluid s relaxation time A is the characteristic time of the fluid and, for oscillatory shearing, cu 1 can be taken as a measure of the characteristic time of the flow process, so De = A to. Thus, viscous behaviour occurs when the Deborah number is low, reflecting the fact that the fluid is able to relax. When the Deborah number is high, elastic behaviour is observed because the fluid is unable to relax sufficiently quickly. 

TPOs are at the boundary between thermoplastics, to which they are linked by some sources, and TPEs, with a high thermoplastic character and a weak elastic behaviour. Sold by the same producer, the cost of a TPO is approximately 60-65% of that for an equivalent TPV. 

Note 3 The names of Oseen, Zocher, and Frank are associated with the development of the theory for the elastic behaviour of nematics and so the elastic constants may also be described as the Oseen-Zocher-Frank constants, although the term Frank constants is frequently used. 

Blends of ethylene-vinyl acetate (EVA) copolymer with metallocene-catalysed elastomeric ethylene-alpha-olefin copolymer were investigated and were found to be immiscible in the melt and solid state but mechanically compatible. The morphology (SEM), thermal (DSC), rheological (viscosity), mechanical (including tensile, shear thinning and elastic behaviour) and optical properties of EVA-rich and ethylene-alpha-olefin copolymer-rich blends were studied and the results are discussed in terms of processibility in film applications. 24 refs. 

Atomic force microscopy and attenuated total reflection infrared spectroscopy were used to study the changes occurring in the micromorphology of a single strut of flexible polyurethane foam. A mathematical model of the deformation and orientation in the rubbery phase, but which takes account of the harder domains, is presented which may be successfully used to predict the shapes of the stress-strain curves for solid polyurethane elastomers with different hard phase contents. It may also be used for low density polyethylene at different temperatures. Yield and rubber crosslink density are given as explanations of departure from ideal elastic behaviour. 17 refs. 

Calculation of the critical pressure required to cause a phase transformation at 0 K can also be obtained from first-principle calculations. Assiuning the various phases exhibit normal elastic behaviour the tangency rule can be applied to energy vs volume plots to yield values for the critical pressure that would generate a phase transformation ... 

For an ideal network T is in the numerator of the formula for E, so that E is proportional to the absolute temperature. The log E - T curve thus shows a positive slope (not a straight line because of the log-scale but slightly curved upward). In reality this simple picture is often disturbed by deviations from ideal rubber-elastic behaviour. 

Secondly a broader MMD leads to a higher value of M, the average which is mainly responsible for the elastic behaviour of the melt, and thus for the recoil after deformation and solidification. These frozen-in rubber-elastic deformations cause the behaviour of shrink-films. 

Frozen-in stresses originate from the rabber-elastic behaviour of the melt the rubber- elastic deformations (chain orientations) are frozen-in upon cooling and remain present as latent stresses. 

An exception is negative elasticity, whereby the price for certain usually very expensive luxury products markedly decreases as they become more available. Examples include gem stones, scarce fine fragrance materials such as animal musks, and other luxury items. Elastic behaviour is promoted by alternative competing sources of supply of the product, the number of uses possible for the material and the utility and value of the product to the customer. 

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