Theory of Elastic Constants

Madhusudana and Prathiba [238], later corrected to 10 cgs units (10 N)byBar-bero and Durand [224] and i3 = 0.2A ]i for 5-CB by Lavrentovich and Pergamenshchik [190]. However, in view of the mathematical difficulties discussed above, one has to treat these quantitative data with proper care. 

Finally, it should be noted that surface effects on nematic elasticity do not amount only to the appearance of the and K24. elastic terms. Changes in the order parameter in a surface layer (see Barbero and Durand [239]) and references 1-6 therein), as well as unbalanced molecular interaction forces due to the broken symmetry of the nematic interaction potential at the surface [240], may lead to a spatial variation in the bulk elastic constants and the emergence of new elastic terms near the surface. 

The first theoretical dicussion of the temperature dependenee of elastic constants within the framework of the molecular-statistical Maier-Saupe theory was given by Saupe [241]. He attributed their temperature dependence to changes in the order parameter S and the molar volume with temperature, and he introduced reduced elastic constants 

should be practically temperature independent according to this theory, and the Frank elastic constants scale with the square of the nematic order parameter, while their ratios should be constant material parameters. In his first qualitative estimation of their relative quantities Saupe [241] derived a A ii 22 33 of-7 11 17. This non- 

In view of the desirable molecular design of liquid crystalline phases, the final goal of microscopic theories is not only to predict the temperature dependence of elastic coefficients, but also to reveal relations between the molecular and elastic properties of the mesophases. Since the original work of Saupe in 1960 [241], a large number of microscopic theories have been developed [181-183, 242-270]. The first molecular statistical theories of the Frank elastic constants were formulated by Priest [242] and Straley [243], and a significant. step was made by Poniewierski and Stecki [244,245] who applied density-functional theory in order to relate in a direct quantitative way the elastic constants to characteristic properties of a statistical ensemble of molecules with 

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The principal elastic constants for a nematic liquid crystal have already been defined in Sec. 5.1 as splay (A , j), twist(/ 22) and bend(fc33). In this section we shall outline the statistical theory of elastic constants, and show how they depend on molecular properties. The approach follows that of the generalised van der Waals theory developed by Gelbart and Ben-Shaul [40], which itself embraces a number of earlier models for the elasticity of nematic liquid crystals. Corresponding theories for smectic, columnar and biaxial phases have yet to be developed. 

The first attempt at a molecular theory of elastic constants due to Saupe and Nehring [44] assumed an attractive pair potential of the form Ui2= i>r P2(cosdi2), and set... 

A number of publications deal explicitly with the theory of elastic constants at phase transitions. The behaviour of nematic elastic constants in the vicinity of the nematic-smectic transition has been analysed by means of the Ginzburg-Landau Hamiltonian [295]. At the transition to the smectic A phase, a critical divergence of K22 and A 33 is predicted. 

In the following discussion of design theory the values of a number of elastic constants... 

The value of Ci is obtained from the plot of o/2(X - A ) vs. 1A and extrapolating to 1A = 0. By comparison with the theory of elasticity, it has been proposed that Cl = 1/2 NRT, where N is cross-link density, R the gas constant, and T the absolute temperature (of the measurement). To assure near-equilibrium response, stress-strain measurements are carried out at low strain rate, elevated temperature, and sometimes in the swollen state. °... 

The molecular theory of elasticity of polymeric networks which leads to the equation of state, Eq. (28), rests on the following basic postulates Undeformed polymeric chains of elastic networks adopt random configurations or spatial arrangements in the bulk amorphous state. The stress resulting from the deformation of such networks originates within the elastically active chains and not from interactions between them. It means that the stress exhibited by a strained network is assumed to be entirely intramolecular in origin and intermolecular interactions play no role in deformations (at constant volume and composition). 

The fundamental quantities in elasticity are second-order tensors, or dyadicx the deformation is represented by the strain thudte. and the internal forces are represented by Ihe stress dyadic. The physical constitution of the defurmuble body determines ihe relation between the strain dyadic and the stress dyadic, which relation is. in the infinitesimal theory, assumed lo be linear and homogeneous. While for anisotropic bodies this relation may involve as much as 21 independent constants, in the euse of isotropic bodies, the number of elastic constants is reduced lo two. 

Oseen likewise proceeded by setting up an expression for energy density, in terms of chosen measures of curvature. However, he based his argument on the postulate that the energy is expressible as a sum of energies between molecules taken in pairs. This is analogous to the way in which Cauchy set up the theory of elasticity for solids, and in that case it is known that the theory predicted fewer independent elastic constants than actually exist, and we may anticipate a similar consequence with Oseen s theory. 

Priest (1973) and Straley (1973), in terms of the classical virial expansion, the Onsager theory (referred to in Section 2.1) and the curvature moduli theory, derived the elastic constants of rigid liquid crystalline polymers. The free energy varies according to the change of the excluded volume of the rods due to the deformation. The numerical calculation of elastic constants (Lee, 1987) are shown in Table 6.2. 

Several experiments were carried out to investigate the elastic constants of nematic polymers. They were essentially in agreement with the theory. But the available data are insufficient for checking theoretical predictions. Systematic and careful experiments are required to investigate the relationship of elastic constants to the flexibility and molecular length. We will introduce some measurement techniques and experimental data for elastic constants. 

Design theory shows that the values of a number of elastic constants must be known in addition to the strength properties of the fibers, resin, and their combination. Reasonable assumptions are made in carrying out designs. In the examples used, more or less arbitrary values of elastic constants and strength values have been chosen to illustrate the theory. Any other values could be used. 

In this analysis, it is assumed that all the glass fibers are straight however, it is unlikely that this is true, particularly with fabrics. In practice, the load is increased with fibers not necessarily failing at the same time. Values of a number of elastic constants must be known in addition to strength properties of the resins, fibers, and combinations. In this analysis, arbitrary values are used that are low for elastic constants and strength values. Any values can be used here the theory is illxtstrated. 

Reasonably accurate analytical methods have been established for predicting elastic constants of a unidirectional RP. Using laminated plate and shell theory (macromechanics), the elastic constants of multidirectional RPs are derived from the elastic constants of the unidirectional layer (Chapter 8). In addition to the work on prediction of elastic constants, work has been done in predicting strength of multidirectional RPs based on experimentally determined strengths of unidirectional RPs. 

Figure 3.16 Comparison of elastic constants of an open-cell alumina foam with the Gibson and Ashby (GA) theory.

According to Hooke s law, Young s modulus E, is the coefficient between a small deformation, Sp, and stress, cr a = ESp. However, the experimental measurements of Young s modulus show that it is not a constant parameter, as assumed in the classic theory of elasticity it depends on both temperature and time scale. It is known that an increase in the deformation rate (or deformation frequency) causes an increase in the value of the modulus [44-53]. Hence, Young s modulus is essentially a relaxational parameter. 

In the course of the past decade several explanations have been put forward to account for the anomalous diffusion in bcc metals. The different models based on the presence of extrinsic vacancies (Kidson, 1963), on the temperature dependence of elastic constants (Aaronson and Shewmon, 1967), on dislocation enhanced diffusion (Peart and Askill, 1967) or on vacancy anharmonicity (Gilder and Lazarus, 1975) have been thoroughly discussed in the original and in the various review papers. Two additional models, the activated interstitial and, more recently, the w-embryo model, have been proposed as alternative explanations for anomalous bcc diffusion. These two models and their possible common interpretation on the basis of the Engel-Brewer electron correlation theory will be discussed in the next sub-section. 

In the foregoing treatment, the influence of the elastic strain energy has deliberately been neglected. With the help of the theory of elastically deformed bodies, it is found that, in the case where there is a change in lattice constant with composition, the free energy has an additional term which is positive and reads ... 

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