Frank elastic stresses

In a flowing liquid crystal, both the viscous stresses and Frank elastic stresses are normally important. Thus, the Ericksen theory for the viscous stresses, must somehow be combined with the Frank theory for the elastic stresses. This was accomplished by Leslie, who... 

Problem 10.3(b) (Worked Example) Derive Eq. (10-31), the time- or strain-dependent shear viscosity in the absence of Frank elastic stresses. 

To evaluate the second term of Eq. (10-10), we need to obtain N. Using the definition of N in Eq. (10-12) along with Eq. (10-3), which is valid in the absence of Frank elastic stresses, we get... 

The radius a of the onions in the intermediate shear-rate regime of lyotropic smectics depends on shear rate, scaling roughly as a A similar texture size scaling rule is found in nematics (see Section 10.2.7) there it reflects a balance of shear stress r y against Frank elastic stress. In smectics, the two important elastic constants B and Ki have differing... 

The LE theory is rather complex since it contains both viscous and elastic stresses. It can best be understood by considering viscous and elastic effects separately. If elastic effects are neglected, the LE equations reduce to Ericksens transversely isotropic fluidy while in the absence of flow the elastic stresses are just those of the Frank-Oseen theory (discussed below in Section 10.2.2). ... 

On the other hand materials deform plastically only when subjected to shear stress. According to Frenkel analysis, strength (yield stress) of an ideal crystalline solid is proportional to its elastic shear modulus [28,29]. The strength of a real crystal is controlled by lattice defects, such as dislocations or point defects, and is significantly smaller then that of an ideal crystal. Nevertheless, the shear stress needed for dislocation motion (Peierls stress) or multiplication (Frank-Read source) and thus for plastic deformation is also proportional to the elastic shear modulus of a deformed material. Recently Teter argued that in many hardness tests one measures plastic deformation which is closely linked to deformation of a shear character [17]. He compared Vickers hardness data to the bulk and shear... 

The static continuum theory of elasticity for nematic liquid crystals has been developed by Oseen, Ericksen, Frank and others [4]. It was Oseen who introduced the concept of the vector field of the director into the physics of liquid crystals and found that a nematic is completely described by four moduli of elasticity Kn, K22, K33, and K24 [4,5] that will be discussed below. Ericksen was the first who understood the importance of asymmetry of the stress tensor for the hydrostatics of nematic liquid crystals [6] and developed the theoretical basis for the general continuum theory of liquid crystals based on conservation equations for mass, linear and angular momentum. Later the dynamic approach was further developed by Leslie (Chapter 9) and nowadays the continuum theory of liquid crystal is called Ericksen-Leslie theory. As to Frank, he presented a very clear description of the hydrostatic part of the problem and made a great contribution to the theory of defects. In this Chapter we shall discuss elastic properties of nematics based on the most popular version of Frank [7]. 

The discussion so far considers the theoretical shear stress of a crystal in the absence of thermal fluctuations. Frank [81] considered that there are always local thermal fluctuations, which must be taken into account. At any temperature T, there is a significant chance of thermal fluctuations in the timescale of the experiments supplying an energy up to 50 kT. Furthermore, this discussion only relates the yield stress to the elastic energy whereas U is strictly the activation enthalpy. Analogous to the site model theory (see Section 7.3), we should discuss the Gibbs free energy AG, where AG — TAS and the shear strain rate is... 

An important aspect of the macroscopic structure of liquid crystals is their mechanical stability, which is described in terms of elastic properties. In the absence of flow, ordinary liquids cannot support a shear stress, while solids will support compressional, shear and torsional stresses. As might be expected the elastic properties of liquid crystals are intermediate between those of liquids and solids, and depend on the symmetry and phase type. Thus smectic phases with translational order in one direction will have elastic properties similar to those of a solid along that direction, and as the translational order of mesophases increases, so their mechanical properties become more solid-like. The development of the so-called continuum theory for nematic liquid crystals is recorded in a number of publications by Oseen [ 1 ], Frank [2], de Gennes and Frost [3] and Vertogen and de Jeu [4] extensions of the theory to smectic [5] and columnar phases [6] have also been developed. In this section it is intended to give an introduction to elasticity that we hope will make more detailed accounts accessible the importance of elastic properties in determining the... 

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