Deformation plane

If A < 1, Eq. (10-4) has no solutions in the shearing plane. This means that the director rotates endlessly in the deformation plane. The period of this rotation—that is, the time it takes to rotate through an angle of tt, is... 

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 < 9 — nn < it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

Local-scaling transformations made their appearance in density functional theory (although in a disguised manner) in the works of Macke [58, 59]. Because the Thomas-Fermi theory corresponded to a free-electron gas model, and as such it was cast in terms of plane waves, any improvement on this theory required the introduction of deformed plane waves. Thus, initially local-scaling transformations were implicitly used when plane waves (defined in the volume V in ft3 and having uniform density p0 = N/V) ... 

Let us notice that local-scaling transformations generalize Eq. (27) the most general deformed plane wave becomes... 

Good measurements of contact are difficult to carry out and even more difficult to interpret because of the individualistic character of a given surface. Therefore the tendency has been to fit experimental results to behavior inferred from models. In one of the early simple models, a surface is viewed as an assembly of spherical asperities, and one of the basic schemes of contact is the mechanical interaction between a deformable plane surface and a spherical asperity. If the deformation of the sphere is elastic the deformed area on the sphere is a circle, and the relation between the load pressing the flat against the sphere and the radius of the circle is given by the familiar formula of elastic theory... 

In non-Newtonian liquids there appears another factor, the existence of "normal stresses." In contrast to shear stresses, these represent stresses in the same direction as the deformation plane, resulting in "stretching" of the liquid and swelling of the melt extruded from a tube or die. This is actually an elastic contribution to the deformation of the body, which does not exist in simple (Newtonian) liquids. The normal stresses also cause "climbing" of the polymer melt on a stirrer when polymer solutions or melts are mixed. 

Armero, F. 2000. On the locking and stability of finite elements in finite deformation plane strain problems , Computers Structures 75 261-290. 

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