Transversal isotropy

For an oriented polymer, the magnitude of the observed second moment static magnetic field H0, which can be conveniently defined by the polar and azimuthal angles A, transverse isotropy, to which the following discussion is limited, the observed second moment will depend only on the angle A, there being no preferred orientation in the plane normal to the 3 direction. The treatment follows that originally presented by McBrierty and Ward 9>. 

Each unit of structure in the oriented polymer will also be considered to possess transverse isotropy. Its orientation can therefore be defined by polar and azimuthal angles (0, tp), but the condition of transverse isotropy for the whole sample means that the observed second moment will depend only on functions of 0 (in fact, P200 and P400) the functions involving (p taking fixed average values. 

Finally, a given internuclear vector rjk takes a direction with respect to the 3 axis of the unit of structure defined by polar and azimuthal angles ( , r[). Because each unit possesses transverse isotropy, the second moment will involve functions of h, only, the functions of r being replaced by their average values. 

We may then write the arrays for the elastic constants for various symmetries, the two most useful being hexagonal (also transverse isotropy as in fibre symmetry) and isotropic. Hexagonal gives ... 

Because of the asumed transverse isotropy it follows that = 1/2 (Cu - Cn). The terms Qj are the elastic stiffnesses expressed in the contracted (Voigt) notation. 

The liner is an elastic plastic isotropic material. Besides, the laminate behaviour is different from a layer to another and each layer behaves according to the fibre direction. The fibre is assumed to have a transverse isotropy and equivalent properties in the (2-3) plane which normal axis (1) refers to the fibre longitudinal direction, as shown in Figure 1. 

Assuming (1) axis is the longitudinal direction of the fibre, the compliance tensor 5c takes the same form than Si, whereas, taking into account the transversal isotropy, the compliance constants have the following expressions ... 

If one assumes fibre symmetry of the sample (like in uniaxial deformation), transverse isotropy for the molecular units (for instance ai U2 = 03), and additivity of polarizabilities, it is easily shown [9] that the difference in macroscopic polarizabilities along and perpendicular to the fibre axis of the sample is simply ... 

For isotropic media, there is a transverse isotropy in D, such that the transverse dispersion for n perpendicular to uD is = n D n. 

Since the Qj are simply related to the technical elastic moduli, such as Young s modulus (T), shear modulus (G), bulk modulus (iC), and others, it is possible to describe the moduli along any given direction. The full equations for the most general anisotropy are too long to present here. However, they can be found in Yoon and Katz [1976a]. Presented below are the simplified equations for the case of transverse isotropy. Young s modulus is... 

Transverse isotropy The symmetry arrangement of structure in which there is a unique axis perpendicular to a plane in which the other two axes are equivalent. The long bone direction is chosen as the unique axis. In crystals this symmetry is called hexagonal. 

If this process is repeated, one finds only three values of Poisson s ratio are needed, not six. For fiber-reinforced materials, the number of elastic constants may be further reduced if other symmetries appear. For example, in some materials short fibers are randomly oriented in a plane and this gives transverse isotropy. That is, there is an elastically isotropic plane but the stiffness and compliance constants will be different normal to this plane (five elastic constants are needed). 

Stated in these very general terms, it is by no means obvious that the problem of adequately describing orientation in a polymer can ever be completely solved. Indeed, there are certainly situations where it can be shown to be insoluble in terms of the available information. However, in many if not most of the oriented polymer situations which have been studied in detail, simplifications are possible. The most sweeping simplification is to assume transverse isotropy, i.e. that there is no preferential... 

Measurements of optical anisotropy, i.e. birefringence give the same type of orientation function. In this case, again assuming fibre symmetry, the polymer is considered to be an aggregate of polarisable units with transverse isotropy. Each unit is defined by the second rank polaris-ability tensor... 

In this chapter we will present the approach adopted by Ward and co-workers. Similar treatments have also been given independently by Yamagata and Hirota and by Slonim and Urman. Due to their appearance in the Japanese and Russian literature only, these latter previous treatments did not achieve prominence in the western literature. Furthermore, although it is perfectly possible to develop the theory in a very elementary manner, using Euler angle transformations, and this was the method of the earlier work, we choose to work here in terms of spherical harmonic analysis. The compactness of this representation has many advantages, particularly if the treatment is to be extended beyond transverse isotropy. 

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 ... 

For transverse isotropy of the sample, we may average over the range 0 27t with a uniform probability function ... 

Transverse isotropy Transverse isotropy Transverse isotropy Transverse isotropy... 

Studies of mechanical anisotropy in polymers have been made on specimens of two distinct types. Uniaxially drawn filaments or films have fibre symmetry, with isotropy in the plane perpendicular to the draw direction. Films drawn at constant width or films drawn uniaxially and subsequently rolled and annealed under closely controlled conditions, show orthorhombic symmetry. For fibre symmetry (also called transverse isotropy) the number of independent elastic constants reduces to five and the compliance matrix is... 

Studies of possible deformation mechanisms in drawn, rolled and annealed sheets have been made by Owen and Ward, as an extension of the model of Davies et a. for specimens having transverse isotropy. Because of the anneal involved in the production process c/c shear is a less important deformation mechanism in this form of material. If the lamellae exist as platelets of very large extent compared with the thickness of inter-lamellar material this latter could undergo only simple shear, with the shear direction parallel with the surface of the lamellae but if the platelets are effectively of infinite extent only in one direction then inter-lamellar material could deform when the platelets are subjected to a normal stress. The inter-lamellar material then undergoes pure shear. (Some consequences of this model have been discussed in Chapter 4.)... 

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