Symmetry unidirectional

Figure 2-8 Physical Symmetry of a Unidirectionally Reinforced Lamina...

For plane stress in the 1-2 plane of a unidirectional lamina with fibers in the 1-direction, < = T. 3 = r23 = 0- However, from the cross section of such a lamina in Figure 2-39, Y = Z from the obvious geometrical symmetry of the material construction. Thus, Equation (2.126) leads to... 

Kwon, Ki-Young Wong, Kin L. Pawin, Greg et. al. Unidirectional Adsorbate Motion on a High-Symmetry Surface Walking Molecules Can Stay the Course, Phys. Rev. Lett. 95, 166101 (2005) [Issue 16-14 October 2005]... 

For the mass balance, the following assumptions and operating conditions are considered (i) steady state, (ii) unidirectional, incompressible, continuous flow of a Newtonian fluid under laminar flow regime, (iii) only ordinary diffusion is significant for a mixture where the main component is water, (iv) azimuthal symmetry, (v) axial diffusion neglected as... 

The Bchl-b molecules of the special pair are arranged with a nearly perfect two-fold symmetry. The Bchl-b rings of the special pair are nearly parallel to this symmetry axis. A more subtle deviation from symmetry is the different degree of nonplanarity of the two Bchl-b ring systems of the special pair. The tetrapyrrole ring of BCmp (M = branch, P = special pair) is considerably more deformed than that of BCLP. This can cause an unequal charge distribution between the two Bchl-b systems of the special pair, which in turn can be part of the reason for unidirectional electron transfer, essential for the photosynthesis process [38]. 

The emergence of target waves is connected to a breaking of the coupling symmetry. To explore this we study a unidirectional coupling with respect to 0... 

Channel-type structure of (CH3 )4 N BO(OH)f 2 (NH2 )2 CO H2 0(12). In compound 12 [5c], both independent urea molecules and the BO(OH)2 ion in the asymmetric unit occupy special positions of symmetry m such that only the C-O or B-O bond lies on the mirror plane. A projection of the crystal structure along the [010] direction is presented in Figure 8.22. The host lattice consists of a parallel arrangement of unidirectional channels whose cross-section has the shape of a peanut. The diameter of each spheroidal half is about 7.04 A, and the separation between two opposite walls at the waist of the channel is about 5.85 A. The well-ordered tetramethylammonium cations are accommodated in double columns within each channel. 

An orthotropic material is called transversely isotropic when one of its principal planes is a plane of isotropy, i.e. at every point there is a plane on which the mechanical properties are the same in all directions [2]. Unidirectional carbon fibers packed in a hexagonal array with a relatively high volume fraction can be considered transversely isotropic, with the 2-3 plane normal to the fibers as the plane of isotropy (Figure 22.2). For a transversely isotropic material, it should be noted that the subscripts 2 and 3 (for a 2-3 plane of symmetry) in the material constants are interchangeable. Hence... 

Orthotropic n. Having three mutually perpendicular planes of elastic symmetry, as in composites having fibers miming in two perpendicular directions, or biaxially oriented sheet. If the fibers or orientation are unidirectional, the material is still orthotropic but also isotropic in the two directions perpendicular to the fibers or oriented polymer chains. Complete textile glossary. Celanese Corporation, New York. Brandrup J, Immergut EH (eds) (1989) Polymer handbook, 3rd edn. Wiley-Interscience, New York. 

Most unidirectional composites are manufactured from prepreged layers of yarns. Each layer is called a lamina and a group of lamina is called a laminate. Each lamina in the laminate can have a specific direction based on the design requirements starting from an arbitrary direction. The only important matter to remember is that these laminates must have a mirror symmetry around their neutral axis otherwise they will warp immediately after manufacture and cannot be straightened. This happens due to the difference in the thermal expansion coefficients between the fibers and the matrix and between different lamina causing one side to permanently expand more than... 

As pointed out by Meyer [14], the reflection symmetry of smectic-C liquid crystals can be removed if the constiment molecules are chiral, and thus it becomes possible to have spontaneous polarization. This phase is called the chiral smectic-C or smectic-C, and its stmcture is shown in Figure 4.7. Within a layer, the structure is the same as in smectic-C. The liquid crystal director n is, however, no longer oriented unidirectionally in space but twists from layer to layer as in the cholesteric phase [15]. The symmetry group is C2. The two-fold rotational symmetry axis is perpendicular to both the layer normal a and the director n. Now it is possible to have spontaneous polarization along the two-fold rotational symmetry axis. 

This chapter began by describing briehy the elasticity of anisotropic materials, providing the fundamental relationships and the allowed simplihcations by the existence of material planes of symmetry. The current unidirectional composites are usually classihed as transversely isotropic materials. In this case, only hve independent elastic constants are necessary to fully characterize unidirectional composites. The micromechanics provides the analytical and numerical approaches to predict the elastic constants based on the elastic properties of the composite constituents. Several classical closed formulas are revisited and compared with experimental data. Finally, stiffness and compliance transformations are given in the context of unidirectional composites. Experimental data are used to assess theoretical predictions and illustrate the off-axis in-plane elastic properties. 

Considering transverse isotropy, as applies to unidirectionally fiber reinforced composites, it is necessary to express (5.43) in terms of the individual and joint invariants of s,y and d[, for that particular symmetry. Such an expansion yields an expression for that involves 17 material parameters, all of which are functimis of m, T, d j2 and +dpj. Obviously, such a general formulation is practically intractable. 

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