Fibre symmetry

This procedure may seem rather formidable but it readily will be illustrated by a simple example which forms the basis of much of the studies in this area. Consider an aggregate of units with fibre symmetry (i.e. defined by the orientation of a single axis), the aggregate also possessing fibre symmetry. The spectroscopic measurements will then yield values for (cos2 P>, (cos4 P>, etc. The distribution function will be a function of 0 only, and can be expanded in terms of P2(cos 0), P4(cos 0), etc. 

It was therefore with some confidence in the infra-red and Raman spectroscopic methods that a much more complex investigation was carried out on the molecular orientation in one-way drawn PET films which show uniplanar axial orientation 5). In such films the condition of fibre symmetry is removed in two ways (1). There is no longer uniaxial symmetry of the distribution of chain axes. 

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 ... 

A further complication arises with this material in that the diffraction pattern is not a true fibre pattern. The molecular chain axis in the crystallites is tilted with respect to the fibre axis (2,3) and this causes reflections to be displaced above and below the mean layer line position. Thus it is possible that some reflectionswhich would overlap in a true fibre pattern to such an extent that their intensities could not be determined separately, might be resolved with the present material. When this happens for a reflection of observable intensity, then it will be measured as a discrete reflection and the resolution of the final structure will be higher than if the material had true fibre symmetry. However, Stambaugh et al (4)... 

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 ... 

The cell constants for stage-1 compounds have been more precisely determined over the years. It must be emphasized here, as it has been before and certainly will be again later, that the availability of oriented polyacetylene has been crucial for the progress made in the interpretation of the diffraction data. For the quantitative comparison of intensities, Shirakawa polyacetylene is still used because its isotropic diffraction can be compared with powder calculations and does not need the tricky Lorentz corrections of fibre symmetries. Moreover, Shirakawa polyacetylene is known for its relatively high crystallographic coherence and the absence of impurity reflections. 

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... 

Other advantages of working in terms of spherical harmonic functions are that for cases with fibre symmetry, the Legendre addition theorem can be used, and affords considerable algebraic simplifications (see for example Ref. 25), and that for lower symmetries, the treatment can readily be generalised. It should be mentioned that the exact definitions of P2 cos9), etc., and p 9), can differ in different treatments due to the adoption of different normalisation procedures (see, for example. Chapter 5, Section 5.2).)... 

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... 

It should be noted that in the case of fibre symmetry (in which there are only 5 independent creep functions) tensile creep measurements on specimens cut at only three angles to the fibre axis (say 0°, 45° and 90°) will give three independent combinations of the five, say S22it [2S23(t)+S44(t)] and SssU). Whilst if lateral contraction measurements are also carried out during creep then all 5 functions, including the... 

A high degree of anisotropy of both isochronous compliance and nonlinearity of behaviour is found in highly drawn LDPE having fibre symmetry, as shown by Raumann and Saunders and Darlington and Saunders and thus provides a suitable system in which to examine the utility of the theory. 

The variation of the isochronous modulus at 100 s with the magnitude of the creep strain at 100 s for strains in the region OT-10% in samples cut at various angles to the fibre axis is shown in Fig. 3. The data were obtained using the isochronous stress-strain procedure, previously referred to, on LDPE drawn at 20 C so as to produce fibre symmetry with a draw ratio of 4-2. In this figure horizontal straight lines would indicate linear viscoelastic behaviour. The strain at which significant deviation firom... 

In Section 10.2 the effect of materials symmetry on the number of independent compliance constants Sij for linear elastic behaviour was presented. For the case of fibre symmetry, eqn. (3), we have in particular, Si3 = Sai = S23 = S32. For the linear viscoelastic case Rogers and Pipkin were able to show theoretically that without recourse to the arguments of irreversible thermodynamics it was not possible to show that Si3 = S31 and S23 = S32. Further the validity of all these equalities must be in doubt in non-linear behaviour at finite strains. 

Fig. 10. Creep data for cold-drawn LDPE with fibre symmetry. Draw ratio 1-4. (From Clayton. ) Note that the values of St 3 and Si 2 are negative.

The time dependence of the shear compliance 544(f) was studied over a range of draw ratios in cold-drawn LDPE by Qayton using torsional creep apparatus. In general, good agreement was obtained between this directly measured shear compliance and that calculated by inserting the tensile creep data above into eqn. (8) (see Qayton et al ). It should be mentioned that the measurement of S44. for a material with fibre symmetry is one of the cases where the difficulties mentioned in Section 10.3.2 are least severe, since torsion occurs about an axis which is perpendicular to a plane of isotropy in the specimen. 

A detailed examination of the problems of direct measurement of 544(f) and 566(f) in highly cold-drawn LDPE with fibre symmetry was carried out by Ladizesky and Ward using torsional creep apparatus. They found only a small dependence of 544(t) upon time at high draw ratio, in agreement with the above mentioned studies on tensile and torsional creep. The direct determination of 566(fX by experiments in torsion was, however, shown to require a complicated double extrapolation procedure reasonable agreement then being obtained between the measured values and those derived from the measurements of 5t 1 and 5i2 during tensile creep. 

Creep studies have also been carried out on LDPE drawn at 55 C, drawn at 95°C and drawn at room temperature followed in this case by various thermal treatments at 55°C all the sheets having fibre symmetry (see Darlington et The sheets drawn at 55°C and room temperature had high values of draw ratio whilst those drawn at 95°C were produced with a full range of draw ratios. 

Creep testing of oriented polymers, intended to fully characterise the anisotropy of stiffness behaviour, presents formidable difficulties. For materials with fibre symmetry, techniques are now available which allow complete characterisation. These techniques are considerably more sophisticated than simple creep testing in isotropic materials. For lower symmetries it is still not possible to achieve full characterisation. 

For the discussion all SAXS- and WAXS-2D-pattems are shown with vertical tensile direction. For specimens with fibre symmetry this means that the fibre axis and so the scattering vector ss is also vertical, the scattering vector si2 is always horizontal. 

The evaluation of SAXS images is strongly related to the approach for materials with fibre symmetry developed by Stribeck (Stribeck, 2007). Data processing was realised with the software package pv-wave from Visual Numerics. 

Complete information about the specimen would be available only by tomographic methods with a stepwise rotation of the sample (see e.g. Schroer, 2006) or using inherent symmetry properties of the sample. Under the assumption of fibre symmetry of the stretched specimen around the tensile axis, from the slices through the squared FT-structure the three-dimensional squared FT-structure in reciprocal space can be reconstructed and hence also the projection of the squared FT-structure in reciprocal space. The Fourier back-transformation of the latter delivers slices through the autocorrelation function of the initial structure. Stribeck pointed out that the chord distribution function (CDF) as Laplace transform of the autocorrelation function can be computed from the scattering intensity l(s) simply by multiplying I(s) by the factor L(s) = prior to the Fourier back-... 

Under the condition of fibre symmetry signals in the pattern vertical to the tensile direction report highly oriented lamellae or lamellar fragments in tensile direction. This final pattern remains mainly constant also during further heating to about 160 °C (below the melting point) under load as well as unloaded. 

This construction enables the components of a two-dimensional stress (or strain), expressed in terms of a given set of perpendicular axes, to be converted into the components relative to any other set of perpendicular axes. In particular it provides a simple method for determining the principal axes of stress and strain. For fibre symmetry all planes that contain the fibre axis as one of the perpendicular axes are identieal thus the eonstruction is again applicable. 

The Elastic Constants for Specimens Possessing Fibre Symmetry... 

Stribeck N (1997) Data analysis of 2D-SAXS patterns with fibre symmetry from some elastomers. Fibre Diffraction Rev 6 20-24. 

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