Elastic behavior, metals

In this chapter we examine the elastic behavior of polymers. We shall see that this behavior is quite different from the elasticity displayed by metals and substances composed of small molecules. This is a direct consequence of the chain structure of the polymer molecules. In many polymers elasticity does not occur alone, but coupled with viscous phenomena. The combination of these effects is called viscoelasticity. We shall examine this behavior as well. 

Elastic Behavior. Elastic deformation is defined as the reversible deformation that occurs when a load is appHed. Most ceramics deform in a linear elastic fashion, ie, the amount of reversible deformation is a linear function of the appHed stress up to a certain stress level. If the appHed stress is increased any further the ceramic fractures catastrophically. This is in contrast to most metals which initially deform elastically and then begin to deform plastically. Plastic deformation allows stresses to be dissipated rather than building to the point where bonds break irreversibly. 

Displacement Strains The concepts of strain imposed by restraint of thermal expansion or contraction and by external movement described for metallic piping apply in principle to nonmetals. Nevertheless, the assumption that stresses throughout the piping system can be predic ted from these strains because of fully elastic behavior of the piping materials is not generally valid for nonmetals. 

Early in the history of crystal dislocations, the lack of resistance to motion in pure metal-like crystals was provided by the Bragg bubble model, although it was not taken seriously. By adjusting the size of the bubbles in a raft, it was found that the elastic behavior of the raft could be made comparable with that of a selected metal such as copper (Bragg and Lomer, 1949). In such a raft, it was further found that, as expected, the force needed to form a dislocation is large. However, the force needed to move a bubble is too small to measure. 

An analysis of the transfer function of this system can be made using the matrix method described by Okano et al. (1987). However, the stiffness of the rubber pieces is highly nonlinear. Okano et al. (1987) found that the measured transfer function does not fit theoretical predictions based on a constant stiffness. A nonlinear elastic behavior must be taken into account. Another problem with the metal-stack system is that the resonance frequency is around... 

The mechanical behavior of Zr02-Ni system strongly depends on constitutional variation. The Ni-rich materials exhibit typical behavior of elasto-plastic deformation and ductile fracture similar to metallic material. The materials containing PSZ from 40-80 vol% mainly presents typically linear elastic behavior and macroscopic brittle fracture. However, the material with 60 vol% PSZ behaves as non-linear elastic behavior after the linear stage. 

The elastic behavior upon applied shear stress is primarily typical in the case of solids. The nature of elasticity is in the reversibility of small deformations of interatomic (or intermolecular) bonds. In the limit of small deformations the potential energy curve is approximated by a quadratic parabola, which corresponds to a linear t(y) dependence. Elasticity modulus of solids depends on the type of interactions. For molecular crystals it is 109 N m 2, while for metals and covalent crystals it is 1011 N m"2 or higher. The value of elasticity modulus is only weakly dependent (or nearly independent) on temperature. 

Polymers of all types, glassy or semi-crystalline, have a much more protracted transition from small-strain elastic behavior to fully developed plastic flow than do metals, which can stretch the transition over a quite large strain of the order of 0.05. This is a consequence of the much lower level of crystallinity in polymers than in metals and because the thermally assisted unit inelastic transformation events, occurring primarily in the amorphous component, are in the form of isolated sessile shear transformations in relatively equi-axed small-volume... 

Moreover, rubbers exhibit unique thermo-elastic effects unknown in metals, as noted first by Gough as early as 1805. Gough (1805) reported two distinctive responses, namely that (a) a rubber when held stretched under a constant force contracts reversibly on heating, and (b) it gives off heat reversibly when stretched at constant temperature (Treloar 1975). These important characteristics that were confirmed later by Joule (1859) are now referred to as the Gough-Joule effect and are key in the mechanistic understanding of the elastic behavior of rubbers. 

High elasticity behavior is different to that one what is usually observed in reversible (or elastic) deformations of soUds (e.g. metals). Stress subjected to a specimen causes deformation. If the stress is removed and the specimen returns to its original shape, the deformation is called elastic or reversible. Hooke s law is the base law of deformation of a perfect elastic body (Chapter 7, Equation 15) deformation (strain) stress... 

More details concerning the elastic behavior of the rare earth metals can be found in the extensive review by Scott (1978). 

Since whiskers have high tensile strengths they are also capable of withstanding exceptionally large elastic strains. Metallic and even some oxide whiskers support strains of 2 to 5% before fracture or yield occurs. Towards the higher strains the stress-strain behavior is often nonlinear and substantial deviations from Hooke s law are observed. The stress-strain curves are similar to the one shown in Fig. 45 for the amorphous iron alloy fiber. At the highest strain some stress relaxation may also occur, giving rise to an irreversible residual deformation. 

Viscoeiasticity. As already noted, the time-dependent properties of polymer-based materials are due to the phenomenon of viscoelasticity (qv), a combination of solid-like elastic behavior with liquid-like flow behavior. During deformation, equations 3 and 6 above applied to an isotropic, perfectly elastic solid. The work done on such a solid is stored as the energy of deformation that energy is released completely when the stresses are removed and the original shape is restored. A metal spring approximates this behavior. 

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