Transversely isotropic unit

For this situation of a transversely isotropic aggregate of transversely isotropic units, the Legendre addition theorem gives... 

It is also comparatively straightforward to-calculate P200, P220, P420 and P o for a biaxially oriented aggregate of transversely isotropic units in terms of the principal extension ratios Xx, X2 and (with X,X2 3 = 1). 

Pseudo-affine model, the deformation process of polymers in cold drawing is very different from that in the rubbery state. Elements of the structure, such as crystallites, may retain their identity during deformation. In this case, a rather simple deformation scheme [12] can be used to calculate the orientation distribution function. The material is assumed to consist of transversely isotropic units whose symmetry axes rotate on stretching in the same way as lines joining pairs of points in the bulk material. The model is similar to the affine model but ignores changes in length of the units that would be required. The second moment of the orientation function is simply shown to be ... 

As an example, it can be shown that the Reuss average compliance 53333 for a uniaxial sample consisting of an aggregate of transversely isotropic units is given in terms of the compliances Sy/y of the unit by... 

Fig. 10. Comparison of the measured refractive indices for PET film with values calculated from the orientation functions determined from n.m.r. assuming transversely isotropic structural units. I, Experimental points predicted values. Reproduced from Polymer by permission of the publishers, Butterworth Co (Publishers) Ltd. (C)...

Total Sn0 for transversely isotropic structural units in trans configuration + 0.00424 —0.00227 + 0.00276... 

This expression is identical in form to equation (5.20). In the case of Raman scattering, however, it is necessary to compute the average Raman tensor, (. For a transversely isotropic system, the segment is free to spin about the r. axis, and the vector ni is averaged over the unit circle normal to r . In addition to (n() = 0 and equation (5.22), we require the result,... 

A particular sample of a uniaxially oriented polymer is composed of structural units each of which is transversely isotropic with respect to an axis Oz within the unit. The value of (cos 0) is 0.65, where 6 is the angle between the Oz axis of a typical unit and the draw direction. There are 4.2 x 10 structural units per m in the polymer and the polarisabilities of an individual unit for light polarised parallel and perpendicular to Oz are 2.6 x 10 and 1.9 x 10 F m, respectively. Calculate the polarisabilities of the sample for light polarised parallel and perpendicular to the draw direction and hence the birefringence of the sample. 

We consider first the situation of transverse isotropy. The transverse isotropy is assumed to arise as follows. First the units of structure are transversely isotropic. Secondly, there is no preferential orientation of the units of structure in a plane perpendicular to the draw direction. We rewrite the Van Vleck equation in terms of spherical harmonics ... 

The equations which predict the elastic constants of a partially oriented polymer involve orientation functions to define the orientation of the aggregate units. For example, the average extensional compliance S33 for a transversely isotropic aggregate of transversely isotropic structural units is given by S 33 = Su sin 0+S33Cos" 0-l-(2Si3-l-544) sin 0cos 9. 

The shear stress growth on the inception of shear flow may reflect the orientation of the liquid crystalline domains. Orientation seems to occur within less than 2 strain units in shear flow. This primary normal stress difference can exhibit different phenomena from the shear stress response. In particular for the 60 mole % PHB/PET system, values of N are positive and rise gradually to the equilibrium values whereas the 80 mole % PHB/PET system can exhibit negative values of N. Ericksen s transversely isotropic fluid theory can qualitatively handle some of the observed phenomena. Further studies which couple the transient flow behavior to the orientation and morphology need to be carried out. 

It is interesting to apply the aggregate model to these data, calculating bounds for the elastic constants of an equivalent fibre by averaging the sheet constants in the plane normal to the sheet draw direction. This requires an extension of the mathematical treatment of Section 8.6.2 to deal with the case of a transversely isotropic aggregate of orthorhombic units. The basic equations have been given in detail elsewhere [99] so only the key results will be summarised here. If the orthorhombic unit constants are n, 13,..., 66 (Section... 

In the simplest liquid-crystalline phase, namely the uniaxial nematic, there is at rest a special direction designated by a unit vector n called the director (see Fig. 10-2). In the plane transverse to the director, the fluid is isotropic. The most common nematics are composed of oblong molecules that tend to point in a common direction, which defines the director orientation. Oblate, or disc-like, molecules can also form uniaxial nematics for these discotic nematics, the director is defined by the average orientation of the short axis of the molecule. Lath-like molecules or micelles (shaped like rectangular slabs), in which all three dimensions of the molecule are significantly different from each other, can form biaxial nematics (Praefcke et al. 1991 Chandrasekhar 1992 Fialtkowski 1997). A biaxial... 

In our case of two infinite surfaces which are separated by a medium that transmits stress (e.g., the charged fluid discussed earlier), the force per unit area in the z direction between two rigid surfaces of area A whose normals are in the z and —z directions, is therefore given by /7zz(z = D) = ITc(D) where the subscript i denotes that this is the longitudinal component of the stress tensor. The transverse component of the stress tensor, /7,(z) = TIxx(z) = Hyy(z) (for a system that is isotropic in the plane perpendicular to the z direction), is related to the force in the Jc, y direction for displacements that increase the cross-sectional area of the system. The relevant normal vectors are also in... 

In accordance with the experiment, the transverse compression strains have an inverse sign to the longitudinal extensions, and their contribution, v, is on the order of 0.2-0.3. These two parameters, that is, the modulus with the units of stress (Pa), [E] = [o], and dimensionless Poisson ratio v are sufficient for describing arbitrary deformations of an isotropic solid. It is worth pointing out here that two parameters (E and v, G and v, or G and E) are the smallest pieces of fourth order tensors that are sufficient for the description of strains. It is not possible for one to get away with using a single parameter. Eor an arbitrary principal stress tensor. 

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