Shear direction

Flow behaviour of polymer melts is still difficult to predict in detail. Here, we only mention two aspects. The viscosity of a polymer melt decreases with increasing shear rate. This phenomenon is called shear thinning [48]. Another particularity of the flow of non-Newtonian liquids is the appearance of stress nonnal to the shear direction [48]. This type of stress is responsible for the expansion of a polymer melt at the exit of a tube that it was forced tlirough. Shear thinning and nonnal stress are both due to the change of the chain confonnation under large shear. On the one hand, the compressed coil cross section leads to a smaller viscosity. On the other hand, when the stress is released, as for example at the exit of a tube, the coils fold back to their isotropic confonnation and, thus, give rise to the lateral expansion of the melt. 

Once again failure is just possible in the shear direction... 

There have been a number of past attempts to unify hardness measurements but they have not succeeded. In several cases, hardness numbers have been compared with scalar properties that is, with cohesive energies (Plendl and Gielisse, 1962) or bulk moduli (Cohen, 1988). But hardness is not based on scalar behavior, since it involves a change of shape and is anisotropic. Shape changes (shears) are vector quantities requiring a shear plane, and a shear direction for their definition. In this book, the fact that plastic... 

For an estimate of the ultimate shear strength, r0, of a single domain based on the lattice parameters we use a simple shear plane system proposed by Frenkel [19]. As shown in Fig. 19 it consists of a linear array of periodic force centres resembling the polymer chain. According to this model the relation between the relative displacement x along the shear direction and the shear stress is given by... 

Fig. 19 Shear plane system with periodic force centres spaced at a distance p along the shear direction x and with an interplanar spacing dc according to the model of Frenkel [19]...

In cellulose II with a chain modulus of 88 GPa the likely shear planes are the 110 and 020 lattice planes, both with a spacing of dc=0.41 nm [26]. The periodic spacing of the force centres in the shear direction along the chain axis is the distance between the interchain hydrogen bonds p=c/2=0.51 nm (c chain axis). There are four monomers in the unit cell with a volume Vcen=68-10-30 m3. The activation energy for creep of rayon yarns has been determined by Halsey et al. [37]. They found at a relative humidity (RH) of 57% that Wa=86.6 kj mole-1, at an RH of 4% Wa =97.5 kj mole 1 and at an RH of <0.5% Wa= 102.5 kj mole-1. Extrapolation to an RH of 65% gives Wa=86 kj mole-1 (the molar volume of cellulose taken by Halsey in his model for creep is equal to the volume of the unit cell instead of one fourth thereof). 

In the preceding section, it was demonstrated that the walls of the system can be neglected when the system is in the hydrodynamic regime, i.e., when D is sufficiently large. In this situation, it is often desirable to treat the system as a bulk fluid and apply shear directly without any boundary effects. In this section, we describe two different methods with which to shear bulk systems. 

When integrating the equations of motion, it is important to not impose the shear only at the boundaries because this would break translational invariance. Instead, we need to correct the position in the shear direction at each MD step of size At. This correction is done, for instance, in the following fashion ... 

When studying systems with mixed fluid and solid directions, it is important to keep in mind that each solid direction should be allowed to breathe and fluid directions need to be scaled isotropically or constrained to a constant value. Allowing two fluid directions to fluctuate independently from one another allows the simulation cell to become flat like a pancake, which we certainly would like to avoid. As an example, consider Figure 15, in which a lamellar block copolymer phase is sheared. The convention would be to have the shear direction parallel to x and the shear gradient direction parallel to y. No reason exists for the simulation cell to distort such that Lxz = Lyz = 0 would not be satisfied on average, so one may fix the values of Lxz and Lyz from the beginning. As a result, one solid direction exists plus two fluid directions. We can also constrain Lxx to a constant value, because the shear direction will always be fluid and another fluid direction can fluctuate. This result means that we should allow the simulation cell to fluctuate independently in only the directions of... 

Slip is not always a purely dissipative process, and some energy can be stored at the solid-liquid interface. In the case that storage and dissipation at the interface are independent processes, a two-parameter slip model can be used. This can occur for a surface oscillating in the shear direction. Such a situation involves bulk-mode acoustic wave devices operating in liquid, which is where our interest in hydrodynamic couphng effects stems from. This type of sensor, an example of which is the transverse-shear mode acoustic wave device, the oft-quoted quartz crystal microbalance (QCM), measures changes in acoustic properties, such as resonant frequency and dissipation, in response to perturbations at the surface-liquid interface of the device. 

This type of boundary condition has been explored extensively in the acoustics literature. Ferrante et al. [41] modeled the interface between a surface oscillating in the shear direction and a semi-infinite liquid as a spring connecting two masses (Fig. 2). The no-slip boundary condition for the displacements at the interface is replaced by a complex-valued ratio of the upper and lower displacements,... 

Note 1 The bands always lie perpendicular to the prior shear direction. 

Forced sinusoidal nonresonant shear directly applied by pole pieces of electromagnet to sample disk Forced sinusoidal shear strain imposed by mechanical drive of clamped annular plate of propellant... 

The transverse orientation, where the lamellar normal is parallel to the shear direction, that is not found for low molecular weight diblocks was observed for... 

Fig. 5.11 Scattering patterns obtained for a PE-PVCH diblock (M = 15 kgmor1, /PE = 0.52) at room temperature (Hamley et al. 19966) (a) SAXS pattern (b) WAXS pattern. The X-ray beam was incident perpendicular to the shear direction and to the lamellar normal (perpendicular orientation in Fig. 5.12).

Fig. 5.12 Model for the lamellar organization in semicrystalline PE-PVCH diblocks crystallized from the ordered melt (Hamley et al. 1996b). The PE chains are folded with stems parallel to the lamellar interface. The convention for labelling of the axis system with respect to the shear direction is also indicated.

At 25 °C, a diffuse shear morphology is observed, without any craze. In aged samples (30 h at 130 °C), fine bands (ca. 100 A thick) that grow in both the maximum shear directions have a tendency to collect and localise the shear deformation into ca. 3000 A-wide diffuse shear bands, as indicated by the arrow D in Fig. 76a. In the case of un-aged sample, the fine bands are less distinct and more delocalised, as shown in Fig. 76b. 

Figure 21.9 2D SAXS (logarithmic scale) from a shear-aligned specimen of the parent ABC block copolymer (a, c). The shear direction is horizontal in (a) and along the X-ray beam direction in (c), while the surface normal of the sample is vertical in both images. Radial averages (b, d) in both cases show a main peak at s = (2.63 + 0.05) x 10 2 nm4 (repeat spacing of 38.0 0.7 nm) and the dotted vertical lines indicate the allowed reflections for a hexagonal lattice.37 (Reprinted with permission from G. E. S. Toombes et al., Chem. Mater. 2008, 20, 3278-3287. Copyright 2008 American Chemical Society.)...

No shish are present in Fig. 51a. Indeed there is some question as to whether the oriented rows of bands are shish-kebabs many of the kebab bands fie at an oblique angle to the apparent shear direction. Possibly they were nucleated along fines of smeared particles. On the other hand, Fig. 51b shows unmistakable shish kebabs (350 °C, 30 min, followed by air-quenching). In the shish on the left all of the kebabs would be nucleated at the sides of the apparent fiber, a split being present between the kebabs growing in opposite directions. On the right shish, however, a number of the kebabs extend across the cen-... 

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