Elastic extension, maximum

It can be demonstrated [6.27] that Elnl(N, m) has its maximum at m N/2, when the elastic compression fields generated by partial interstitials essentially compensate for the elastic extension fields surrounding partial vacancies. At m as N/2, the configurational entropy of the complex is also maximum. Therefore, m w N/2 is the most probable number of atoms disposed in the complex of N noncoincident sites, so that nearly one half of the noncoincident sites is occupied by atoms, and another half is vacant. The existence of two-level systems, high diffusion mobility of atoms along non-coincidence sections, low-energy structural fluctuations in polyclusters is connected with this circumstance (Sect. 6.6). 

The elasticity of rubbers is very different from that of materials such as metals or even glassy or semicrystalline polymers. Young s moduli for metals are typically of the order of 10 MPa (see table 6.1) and the maximum elastic extension is usually of order 1% for higher extensions fracture or permanent deformation occurs. The elastic restoring force in the metal is due to interatomic forces, which fall off extremely rapidly with distance, so that even moderate extension results in fracture or in the slipping of layers of atoms past each other, leading to non-elastic, i.e. non-recoverable, deformation. 

A clue to the reason for this behavior can be found from the load-diameter-extension data. At temperatures above which load serrations do not occur, the load increases uniformly as the diameter decreases until maximum load occurs, and further deformation is restricted to one neck. At the lower temperatures a different behavior occurs and the load extension curve is serrated. Each load drop corresponds to a diameter decrease at one or several necks, which have formed before "maximum load." Further, it must be realized that the rising load portion of each serration corresponds to an essentially elastic extension, whereas plastic deformation occurs during the rapidly decreasing load portions. During this decreasing load, at about constant strain, elastic strains are relieved and "exchanged" for an equivalent amount of plastic strain. Finally, consideration must be given to the fact that in these steels austenite is metastable and transforms under stress to yield martensite. 

Let us consider also the regularities common for different types of extension. Dependencies of extension a upon elastic strain a are given in Fig. 5. Continuous lines in Fig. 5 indicate dependencies a(a) in extension at different constant strain velocities x. The higher x, the higher passes the dependency a(cx). The points with maximum a... 

A typical stress-strain diagram for a metal is shown in Fig. 11. This metal follows Hook s law up to a proportional limit ox yield strength) of 2 x 109 Pa. The elastic limit, above which the metal undergoes plastic deformation, which is not recoverable when the stress is removed, is close to the proportional limit. The maximum stress that the metal can support is the ultimate strength (or tensile strength) of the metal, which occurs at the maximum extension of the material. 

Eqs. (7.8) - (7.11) indicate that a film containing a surfactant possesses indeed an elasticity ( /> 0). It follows also that the higher the surfactant surface activity and the thinner the film, the higher the modulus of elasticity is. However, since the bulk surfactant concentration in the film diminishes upon extension, the initial increase in the modulus can be followed by a decrease. The possibility of appearance of a maximum in elasticity at certain film thicknesses has been shown in [27]. 

The stress-strain curves in Figure 6.1 show the stiffness/elasticity of wood (initial slope), its extensibility (strain to the point of failure), work to failure (the area under the curves), and the failure stress or strength (the maximum stress). The crucial feature is the enormous increase in irrecoverable longitudinal extensibility beyond the elastic limit once the MFA exceeds 20°. 

Elastic strain gages have a linearity within 1% for 10% of the maximal extension. For extensions of up to 30% of the maximum, the non-linearity increases to 4%. Another problem is the dead band or initial non-linearity due... 

At the peak force (position A in Fig. 8.5a), there are two possibilities for the next strain state Elastic unloading along path AU, and further plastic straining along the path AN. A non-uniform strain state develops, as parts of the specimen elastically unload, and the plastic strain in one region increases to form a neck. The plastic deformation of the neck is partially driven by elastic energy release from the rest of the specimen. The condition that A is at the maximum in the force-extension or force-strain curve can be written... 

It is usual to define the tensile yield stress as the engineering stress F JAq calculated from the maximum E ,ax in the force-extension curve. For metals, a yield stress can be defined as a 0.2% offset from the initial straight elastic response. However, the onset of non-linearity in polymers indicates a viscoelastic rather than a plastic response. The formation of a neck is the first sign of permanent deformation. 

---