Twist torsional elasticity

The strain tensor must conform to the symmetry of the liquid crystal phase, and as a result, for nonpolar, nonchiral uniaxial phases there are ten nonzero components of kij, of which four are independent ( i i, 22> A 33 and 24)- These material constants are known as torsional elastic constants for splay (k, 1), twist ( 22) bend ( 33) and saddle-splay ( 24) terms in 24 do not contribute to the free energy for configurations in which the director is constant within a plane, or parallel to a plane. The simplest torsional strains considered for liquid crystals are one dimensional, and so neglect of 24 is reasonable, but for more complex director configurations and at surfaces, k24 can contribute to the free energy [7]. In particular 24 is important for curved interfaces of liquid crystals, and so must be included in the description of lyotropic and membrane liquid crystals [8]. Evaluation of Eq. (16) making the stated assumptions, leads to [9] ... 

Torsional distortions can now be written in terms of derivatives of a and c, and it is found [10] that nine torsional elastic constants are required for the smectic C phase. Mention should be made of the biaxial smectic C phase, which has a twist axis along the normal to the smectic layers. This helix is associated with a twist in the c-di-rector, and so elastic strain energy associated with this can be described by terms similar to those evaluated for the chiral nematic phase. 

Another basic physical property has to due with the concept of stiffness. In engineering terms, stiffness refers to the ability of object to withstand an applied force without deformation. Stiffness is dependent on both the structure of the object, and the elastic modulus of the material(s) being used. Stiffness can be measured in bending, in twisting (torsion), or in a single direction (tension or compression in one axis) (Figure 3.4). 

The rotational relaxation of DNA from 1 to 150 ns is due mainly to Brownian torsional (twisting) deformations of the elastic filament. Partial relaxation of the FPA on a 30-ns time scale was observed and qualitatively attributed to torsional deformations already in 1970.(15) However, our quantitative understanding of DNA motions in the 0- to 150-ns time range has come from more accurate time-resolved measurements of the FPA in conjunction with new theory and has developed entirely since 1979. In that year, the first theoretical treatments of FPA relaxation by spontaneous torsional deformations appeared. 16 171 and the first commercial synch-pump dye laser systems were delivered. Experimental confirmation of the predicted FPA decay function and determination of the torsional rigidity of DNA were first reported in 1980.(18) Other labs 19 21" subsequently reported similar results, although their anisotropy formulas were not entirely correct, and they did not so rigorously test the predicted decay function or attempt to fit likely alternatives. The development of new instrumentation, new data analysis techniques, and new theory and their application to different DNAs in various circumstances have continued to advance this field up to the present time. 

Besides the oblique contact, tangential displacements may also be produced in the contact of two elastic spheres under the actions of a compressional twist, as shown in Fig. 2.13. Since the torsional couple does not give rise to a displacement in the z-direction, the pressure distribution is not influenced by the twist and is thus given by the Hertzian contact theory. 

In an earlier section, we discussed molecular chains with continuous curvature. These models enable one to st udy polymer conformational structure by means of notions of elasticity theory. As seen, geometrical properties (e.g., curvature, torsion, twist, writhe) and topological properties (e.g., linking) can be used to characterize the molecular shapes of ID models. This approach can be generalized to the study of 2D elastic surfaces. 

Another important aspect of finite elasticity theory is the ability to measure the strain energy fimction derivatives Wi = dW/dli and W2 = BW/9I2. Penn and Kearsley (94) showed how this is done nsing data from torsional experiments. An interesting aspect about torsion in finite deformations is that in order to maintain the cylinder at a constant length, it is necessary to apply normal forces at the ends of the cylinder. If the cylinder is left imrestrained, it will lengthen in an effect referred to as the Poynting (95) effect, first observed early in the last century in experiments with metal wires. When a cylinder of length L is twisted by an amount... 

Another mathematical model that describes the formation of perversions in helices can be found in literature [97-99] and models the transition from a twisted rod to a helix by making some approximations the rod must be elastic, slender and have intrinsic curvature and torsion. The use of this model allowed the calculation [103] of the applied tension on a APC suspended electrospun fiber and the critical tension, below which the fiber starts to twist, by counting the number of turns present in a suspended fiber from pictures obtained by SEM observation. 

D.H. Hodges, E.H. Dowell, Nonlinear equations of motion for the elastic bending and torsion of twisted nonuniform rotor blades. NASA Technical Note NASA TN D-7818, National Aeronautics and Space Administration, Washington DC (1974)... 

Here the unit vector n is the director, the elastic moduli Ku, K22, and K33 describe, respectively, transverse bending (splay), torsion (twist), and longitudinal bending (bend)... 

There are three independent shear moduli Gi = 1/ 44, G2 = I/S55 and G3 = l/see corresponding to shear in the 23, 13 and 12 planes respectively. For a sheet of general dimensions, torsion experiments where the sheet is twisted about the 1, 2 or 3 axis will involve a combination of shear compliances. This will be discussed in greater detail later, when methods of obtaining the elastic constants are described. 

The way in which 6 R) varies with position depends on the form of the torsional deformation applied to the liquid crystal, and in order to calculate the principal elastic constants, it makes sense to calculate the free energy density for normal mode deformations, i. e. those which correspond to splay, twist and bend. These can be achieved easily by confining the director to a plane, and assuming the undisturbed director at the origin to be along the z-axis. g is the wave-vector of the deformation, and for q constrained to the X, z plane, the components of the director as a function of position become ... 

Finally, there are some unique measurements of the mechanical torque connected with an elastic deformation of the nematic. Faetti et al. [63] determined the splay and bend elastic constants by means of such torsion measurements, and Grupp [ 139] made measurements of the twist elastic constant 22-... 

Elastic torsion Stress function 1/Shear modulus Twice the rate of twist... 

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