Principal stretch direction

The uniaxial extension is illustrated in Fig. 8. If the Xi axis is aligned with the principal stretching direction, it can be noted that two fluid particles located on this axis will get further apart as the deformation proceeds. It is convenient to use a coordinate system in which the axes are oriented in the directions of the principal strain axes. For this choice, the strain rate components always have the general form... 

Expressions for timescales associated with stretching and molecular diffusion for nonreacting systems are available in the literature [112]. A fluid element undergoes shear in two-dimensional chaotic flow with the principal stretching direction Xi and compression direction X2- Molecular diffusion becomes important after a time... 

In principle, W can be determined from Eq. (2) if principal stresses at are measured as functions of applied principal stretch ratios X,-. However, since bW/bJ/ rather than W itself are more directly connected with the stress-strain relations [see Eq. (11)], their determination from the measurements of at and X,- is more feasible than that of W. 

The data in this work were obtained by quenching either uniaxially stretched samples for which the low thickness of the specimens allows rapid cooling, or samples deformed in simple shear where higher thickness of the specimens was required in order to perform the scattering experiments along the three principal shear directions. For both types of flow, special devices were developed to control the flow kinematics in the molten state as well as the quenching process which freezes the molecular orientation. These devices are briefly described in the next paragraphs. 

Figure 6.26 shows the chemical structure of PPV. Block letters from A to F show the eight carbon sites in one PPV monomer unit. The principal axes of proton hyperfine coupling of an unpaired tt-electron, defined in Figure 6.18, arc also shown for two proton sites of inequivalent bond orientations. ESR spectra of stretch-oriented undoped PPV films have shown anisotropic lineshapes with an average g value of 2.003 and a spin concentration of about 1 spin/10 PPV monomer units. The observed anisotropic behaviour of the. g value and the linewidth, both larger for the stretch direction, are qualitatively similar to those found in stretch-oriented polyacetylene films mentioned in Section 3.2 and... 

Ayy is the principal value for the C—proton in a Jt-electron radical with spin density p = 1, and vo is the free proton frequency. When the external field is applied parallel fo the stretching direction (y) the maximum ENDOR frequency v+ is estimated to be c.a. 21 MHz at which point the signal changes its slope. By substituting the experimental values of Ayy and (v -vo) in Eq. (7.33), p(0) is evaluated. Due to the uncertainty of the exact value for Ayy the value of p(0) ranges between 0.11 and... 

Fig. 7.51 Geometrical relationship for the main chain axis (OC) in a laboratory axis system (OX, OY, OZ) (a) and in the principal axis system (Ol, 02, 03) of the g tensor (b). The external field (OZ) and the stretching direction are located in the plane, YZ. The figure is adapted from [36] with permission from the American Chemical Society...

The stabilisation of one-layer ETFE-foil constructions takes place by mechanical pre-stressing.The shortened foil has to be mechanically stretched in order to be mounted on the boundary. In the pre-stressing state, these foil geometries display anticlastic curved surfaces, i.e. surfaces curved in the opposite direction in both principal stress directions. The Gaussian curvature as the product of both main curvatures ki and of the foil geometry is less than zero. 

In plane-strain tension the principal stretch is in the 2-direction. Thus,... 

An unswollen sample is deformed uniaxially. The principal extension ratio along the stretching direction is denoted X.. The sample is then immersed in solvent, and swelling occurs in directions normal to the stretching axis this is accompanied by a lateral change in dimensions. The quantities are the two principal extension ratios in the lateral directions with respect to the unswollen sample of volume Vq. The product of the three principal extension ratios is the inverse of the volume fraction of polymer, v, in the swollen network... 

The PCA can be interpreted geometrically by rotation of the m-dimensional coordinate system of the original variables into a new coordinate system of principal components. The new axes are stretched in such a way that the first principal component pi is extended in direction of the maximum variance of the data, p2 orthogonal to pi in direction of the remaining maximum variance etc. In Fig. 8.15 a schematic example is presented that shows the reduction of the three dimensions of the original data into two principal components. 

In anisotropic samples, it is convenient to concentrate on the principal directions of deformation. In uniaxial stretch, the parallel and perpendicular radii are given by... 

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