Simple elongation

In the simplest cases, the optical anisotropy of polymer systems is studied under the conditions of simple elongation, when the elongation velocity gradient i/ii is given. The system investigated then becomes, generally speaking, a triaxial dielectric crystal with components of the relative permittivity tensor 

For a system under elongational deformation along direction 1, for a beam of light propagating in direction 3, according to (10.17) one obtains different refractive indices for different polarisation of the beam, so that, for polarisation in directions 1 and 2, one has a difference of refractive indices 

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The various elastic and viscoelastic phenomena we discuss in this chapter will be developed in stages. We begin with the simplest the case of a sample that displays a purely elastic response when deformed by simple elongation. On the basis of Hooke s law, we expect that the force of deformation—the stress—and the distortion that results-the strain-will be directly proportional, at least for small deformations. In addition, the energy spent to produce the deformation is recoverable The material snaps back when the force is released. We are interested in the molecular origin of this property for polymeric materials but, before we can get to that, we need to define the variables more quantitatively. 

Starkweather H.W., Cocrystallization and polymer miscibility, J. Appl. Polym. Sci., 25, 139, 1980. Seguela R. and Pmd homme J., Affinity of grain deformation in mesomorphic block polymers submitted to simple elongation. Macromolecules, 21, 635, 1988. 

Fig. 96.—Theoretical and experimental stress-strain curves for simple elongation of gum-vulcanized rubber. (Treloar. )...

This is possible because the result must conform to the traditional theory of elasticity when crystallinity is zero (w = 0). This happens if r is interpreted as an average chain vector (made up of N links), the components x, y, z of which deform affinely, so that in simple elongation along z... 

Application of both approaches to describe simple elongation experiments yields that either theory predicts the so-called reduced stress a to be equal to the shear modulus G and to be independent of strain. 

The theory (9) predicts that in simple elongation, the ratio fc/f ph decreases with increasing strain and eventually goes to zero (phantom network). Furthermore, at a =1, the theory holds that... 

The classical statistical theory of rubber elasticity1) for a Gaussian polymer network which took into account not only the change of conformational entropy of elastically active chains in the network but also the change of the conformation energy, led to the following equation of state for simple elongation or compression 19-2,1... 

This expression is fully identical with the expression for t for simple elongation of solids (see 2.2.2). 

Priss 60> proposed another tube model and obtained the following expression for the elastic force at simple elongation... 

The free energy and the elastic force for simple elongation or compression in the primitive path model is62)... 

The measurements of the temperature dependence of the restoring force are usually carried out in the temperature range 350 100 K. Determination of f requires a rather far extrapolation of experimental results and it restricts the accuracy of this method. This type of thermoelastic measurements requires equilibrium conditions. Most widely, this method is used for simple elongation and seldom for compression 67) and torsion 68,69). 

The statistical theory of rubber elasticity predicts that isothermal simple elongation and compression at constant pressure must be accompanied by interchain effects resulting from the volume change on deformation. The correct experimental determination of these effects is difficult because of very small absolute values of the volume changes. These studies are, however, important for understanding the molecular mechanisms of rubber elasticity and checking the validity of the postulates of statistical theory. 

Kilian 91 has also used calorimetric determination of mechanical and thermal energy exchange in isothermal simple elongation for various polymer networks 24) and demonstrated that it can be described by relations which define thermomechanical properties of van der Waals networks (Fig. 4). 

In spite of the gross conformational differences between the Ned dimers, there are only minor differences between the individual motor domains. The overall fold of the motor domain is very similar to that of kinesin-1 and other N-type motors. Major differences are (1) The N-terminal lobe of Ned is enlarged (+9 amino acids) compared with rat kinesin-1. The additional residues are located between /11b and /11c (the L2 finger ). This, however, does not result in a simple elongation of the /1-hairpin as in HsKSP and in M-type motors (see below). In fact, the tip of the L2 finger is rather broadened and forms a short a-helix. (2) Loop L5, the insert in the P-loop helix z 2. is quite short (approximately eight residues compared with 12 in rat kinesin-1), due to three residues that are missing in the primary structure of DmNcd. (3) Switch-1 helix z.3 is short and loop L9, the linker between z.3 and / 6 that includes the switch-1... 

A simple elongational flow is developed as the filament is stretched with the following components of the rate of deformation ... 

For a Newtonian fluid in a simple elongational flow, the constitutive equation becomes... 

Kobayashi and Nagasawa (1970) calculated the acceleration of nucleation by means of simple elongation in the ideal case where there is no relaxation after elongation. They found that the nucleation and hence the crystallisation may be orders of magnitude higher than in spherulitic crystallisation. 

There are a couple of things about this relationship. First of all it is only an approximation. We ll get back to that in a while. Second, we have only considered simple elongation so far. There is a modulus associated with shear and also a bulk modulus. The most important point, however, is that the modulus determined this way, dividing stress by strain, is a material property and independent of the shape of an object. It is what we mean when we talk about the stiffness of a material. Stiffness is crucial in many engineering applications. If a strain of just 1.6% were allowed in an aircraft s wing spar booms, for example, it would look something like Figure 13-8. 

MX. Derive an equation relating the shear stress, t, to the shear strain, y. Assume 7 = 1 - MX. Comment on the difference between this stress/strain relationship and the one for simple elongation. 

An upper-convected Maxwell model has been used with the full relaxation spectrum for the calculation of the stress, but for calculating the birefringence this spectrum has been restricted to long relaxation times as shown in Fig. 12. The model predictions for the data of the Fig. 9 are shown in Fig. 13. The deviations from the linear stress-optical nole are well accounted for by this very simple model. However, the model does not describe the stress data in simple elongation, and in particular the initial stress values at temperatures close to the Tg. 

Simple elongate symmetrical basin Concentration of groundwater flow towards the Icng flanks A and B of the basin... 

Simple elongate symmetrical curved basin Preferred hydrocarbon migration towards concave long flank A migration preference A over E over C... 

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