Principal shear directions

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. 

From specimens cut in the x-z and y-z shearing planes, mean square chain dimensions similar to Rg j and Rg q can also be determined in the principal shear directions x, y and z. Typical isointensity patterns in the three shearing planes are shown in Fig. 28 for sample B. It can be seen that the shape of these curves in the x-z and y-z planes is nearly elliptical like in the x-y plane. However, due to the symmetry of the shear flow, the z direction is a principal direction for molecular orientation at all scales and the major and minor axes of the isointensity curves are indeed found to be parallel to the x,z and y,z directions. Therefore, the analysis of the scattering data in the x-z and y-z planes has been carried out by fixing the direction of the axes of the ellipse (to 0° and 90°) in the least-squares fit program. The major and minor axes of these ellipses then define the scattering vectors q, q and q in the principal shear directions and allow us to calculate the mean square chain dimensions Rg, Rg y and Rg 2-... 

The reliability of the experimental procedure (deformation in the melt, quenching, characterization of orientation at room temperature) could be verified in elongation (specimens quenched for different macroscopic extensions in steady elongational flow) as well as in simple shear (good correlation of the chain dimensions measured in all principal shear directions). 

Compare the transformed orthotropic compliances in Equation (2.88) with the anisotropic compliances in terms of engineering constants in Equation (2.91). Obviously an apparenf shear-extension coupling coefficient results when an orthotropic lamina is stressed in non-principal material coordinates. Redesignate the coordinates 1 and 2 in Equation (2.90) as X and y because, by definition, an anisotropic material has no principal material directions. Then, substitute the redesignated Sy from Equation (2.91) in Equation (2.88) along with the orthotropic compliances in Equation (2.62). Finally, the apparent engineering constants for an orthotropic iamina that is stressed in non-principal x-y coordinates are... 

An important implication of the presence of the shear-extension coupling coefficient is that off-axis (non-principal material direction) tensile loadings for composite materials result in shear deformation in addition to the usual axial extension. This subject is investigated further in Section 2.8. At this point, recognize that Equation (2.97) is a quantification of the foregoing implication for tensile tests and of the qualitative observations made in Section 1.2. 

Find the Tsai-Hill failure criterion for pure shear loading at various angles B to the principal material directions, i.e., the shear analog of Equation (2.134). 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

The data obtained with the scattering vector in the x-y principal shearing plane showed that the principal directions of orientation depend on the magnitude of the scattering vector, and therefore on the considered length scale on the chain. On the other hand, the position correlations within the chain in the neutral direction of the flow have been found to remain unaffected by the shear deformation. 

If we consider a two-dimensional Poiseuille flow (x flow direction, y shear direction), the principal stresses may be expressed as ... 

Shears of this nature, in which principal axes of the deformation undergo rotations equal to the irrotational shearing strains of eq. (7.7), are referred to as simple shear in distinction to the irrotational tensor shear strains that preserve principal axis directions of the deformation, which are referred to as pure shear. 

Especially in thick laminates with low fibre volume content, the absolute distance of a specific layer from the laminate surface may vary significantly. Therefore, the recommendation that the fibre orientation on the bond surface is to coincide with the principal loading direction is sometimes hard to fulfil. Furthermore, it is to be noted that machining may damage the fibres on the step shear surface. 

When ai = an = am, no principal shear stresses exist on any inclined plane according to Eq. (1.29) and stress state is hydrostatic . Table 1.1 provides information on the directional cosine values giving maximum shear stresses. 

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... 

The properties away from the principal fibre direction(s) are much lower than along the fibre direction. Moreover, the measurements can be made in various ways as there are different types of shear loading. 

The normal and shear components (T and x of traction T (also called the normal and shear stresses) read in the system of principal stress directions... 

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