Twist deformation, elastic stress

The strain increases the energy of the solid as a stress is applied. The distortion of the director in liquid crystals causes an additional energy in a similar way. The energy is proportional to the square of the deformations and the correspondent coefficients are defined as the splay elastic constant, K, twisted elastic constant K22 and bend elastic constant Kx, i.e., the respective energies are the half of... 

When a stress Le. a force per unit area) is applied to a solid, for example it is stretched, sheared, twisted or squashed, it deforms, i.e. changes its length, or shape. For small deformations, the amount of deformation is proportional to the applied stress. The material is said to behave elastically. Beyond a certain deformation (the elastic limit) the material ceases to be elastic, and the material no longer returns to its initial shape when the stress is removed. This is called plastic deformation. If the material is deformed further then it will eventually break. Some materials, such as rubber, are elastic for large deformations, while others, such as plasticine, have a relatively small elastic limit but can then undergo large plastic deformations. Brittle materials, such as china, can only withstand small deformations before they break. 

In designing RPs, as reviewed certain important assumptions are made so that two materials act together and the stretching, compression, twisting of fibers and of plastics under load is the same that is, the strains in fiber and plastic are equal. Another assumption is that the RP is elastic, that is, strains are directly proportional to the stress applied, and when a load is removed, the deformation disappears. In engineering terms, the material obeys Hooke s Law (Hooke s law states, it... 

It can be appreciated that the problem is much more complex if part of the cross-section (the outer part) deforms in the non-linear range. If this occurs, the strain in the outer part is still given by eqn 4.73. The complexify arises because the stress generated at each value of r is less than that given by eqn 4.74 (see Figure 4.6). The problem is in some respects analogous to the twisted elastic-plastic rod. 

As we have repeatedly stressed, flexoelec-tricity is a phenomenon that is a priori independent of chirality. But we have also seen that some flexoelectric deformations do have a tendency to occur spontaneously in a chiral medium. All except the helical C state are, however, suppressed, because they are not space-filling. A flexoelectric deformation may of course also occur spontaneously in the nonchiral case, namely, under exactly the same conditions where the deformation is space-filling and does not give rise to defect structures. In other words, in creating the twist-bend structure which is characteristic of a helielectric. Imagine, for instance, that we have mesogens which have a pronounced bow shape and, in addition, some lateral dipole. Sterically they would prefer a helicoidal structure, as depicted in Fig. 52, which would minimize the elastic... 

The director deformations described by that do not lead to layer compressions, in the continuum range where the wavelengths A of the deformation are much larger than the molecular dimensions (A 10 nm) can be induced by stress K 27t/pf <10T N/m. This is usually smaller than of the layer compression modulus B l(f N/m , For this reason, deformations that do not lead to layer compression (such as splay in SmA) are usually called soft deformations, whereas those that require layer compression (such as bend and twist in SmA) are the so-called hard deformations. In SmC there will be six soft and three hard deformations, so it is basically impossible to take into account all elastic terms while keeping the transparent physics. (In the chiral smectic C materials, additional three terms are needed, as shown by de Gennes. ) Fortunately, however, the larger number of soft deformations enable for the material to avoid the hard deformations, which makes it possible to understand most of the elastic effects, even in SmC materials. 

If the bar is simply twisted, there is no volume change ( = 0), only distortion. (Lines scratched along the sides of the bar would be converted to helices by the strain.) This type of deformation is called shear, and the constant of proportionality between the instantaneous elastic shear stress and strain is the shear modulus, G, The shear modulus is related to Young s modulus by... 

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