Stretching definition

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

We begin by remembering the mechanical definition of work and apply that definition to the stretching process of Fig. 3.1. Using the notation of Fig. 3.1, we can write the increment of elastic work we associated with an increment in elongation dL as... 

By combining random flight statistics from Chap. 1 with the statistical definition of entropy from the last section, we shall be able to develop a molecular model for the stress-strain relationship in a cross-linked network. It turns out to be more convenient to work with the ratio of stretched to unstretched lengths L/Lq than with y itself. Note the relationship between these variables ... 

Although it is clear that a grows with rt.o the molecule can be unstable at any definite temperature at definite length. If the chain is stretched sufficiently by the surface (intermolecular bonds are sufficiently removed from each other), then we can accept for the first approximation a 0, and a 1. Then ... 

Contrary to widespread opinion, the value of Ea is not a constant quantity. As was proved previously [52], the value of E is variable, since it depends on the ordering of macromolecules in the amorphous material of the fiber. At the same time, one can suppose that this ordering will be affected by the specificity of the fine structure of the fiber, and particularly by the type of substructure of the fiber. The relationship determining the modulus Ea appropriate for a definite type of fiber substructure can be derived from Eq. (11) when appropriate values of A are assumed. In the case of the microfibrillar substructure, i.e., for A < I, typical of PET fibers stretched, but not subjected to annealing, this equation has the form [52] ... 

The result is that while there is, in DM, something that might be called an information cone centered at each site, it is not really what we usually think of as a relativistic, light cone, for wliidi we can point to interior points and definitely say they arc causally related and know for sure that points outside of each other s light cones are completely independent. In DM it is simply false to say that only those events inside the information cone of the past can influence a present event the information cone can well consist of lights cones stretching into all directions, forward and back in time. 

More definite evidence comes from an MO study of the S—O stretching in dimethyl sulphoxide9, where three basis sets were employed a STO-3G one (I), a 4-31G one (double-zeta, II) and a 3G + d one (III). Table 6 reports the main results the small effect of the double-zeta, and the dramatic effect of the 3d functions, are clearly visible. Notice also how the C—S bond length and the bond angles are by far less sensitive to basis set changes. 

The presence of methylenic bands shifted at higher frequency in the very early stages of the polymerization reaction has also been reported by Nishimura and Thomas [114]. A few years later, Spoto et al. [30,77] reported an ethylene polymerization study on a Cr/silicalite, the aluminum-free ZSM-5 molecular sieve. This system is characterized by localized nests of hydroxyls [26,27,115], which can act as grafting centers for chromium ions, thus showing a definite propensity for the formation of mononuclear chromium species. In these samples two types of chromium are present those located in the internal nests and those located on the external surface. Besides the doublet at 2920-2850 cm two additional broad bands at 2931 and 2860 cm are observed. Even in this favorable case no evidence of CH3 groups was obtained [30,77]. The first doublet is assigned to the CH2 stretching mode of the chains formed on the external surface of the zeolite. The bands at 2931 and... 

The original length 6f the specimen Is LO and its stretched length is L. At very small deformations, all the strain definitions of Table 2 are equivalent, For shear tests (see Figure 2)... 

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