Measuring elastic constants

Here we describe only the principal features of methods of measuring the elastic constants, and refer the reader either to the references cited or to Ward s more advanced text [3] for details of the experimental arrangements. 

The experimental methods employed for filaments are different from those used for flat sheets, and the two cases will therefore be discussed separately. 

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In this chapter we will be concerned with the line shift due to uniform strain. From this shift the strain may be calculated and, knowing the strain, we can determine the stress present, either by a calculation involving the mechanically measured elastic constants of the material, or by a calibration procedure involving measurement of the strains produced by known stresses. X-ray diffraction can therefore be used as a method of stress measurement. Note, however, that stress is not measured directly by the x-ray method or, for that matter, by any other method of stress measurement. It is always strain that is measured the stress is determined indirectly, by calculation or calibration. 

These considerations are amply supported by experiment. By making x-ray measurements on materials subjected to known stresses, we can determine the stress constant K experimentally. The values of K so obtained can differ substantially from the values calculated from the mechanically measured elastic constants. Moreover, for the same material the measured values of K usually vary with the indices hkl) of the reflecting planes. 

The most accurate method of measuring elastic constants is based on the use of ultrasonic sound velocity measurements. This is a dynamical... 

The Raleigh light scattering, another important approach to measure elastic constants and viscosities simultaneously, is applicable to liquid crystalline polymers as well. 

Experimental shock-wave methods, or Brillouin scattering techniques used to measure elastic constants at either ambient, or non-ambient conditions. 

These standing waves are fundamental in vibration problems and are known as natural vibrations, or vibrational modes. Other types of progressive waves could be analyzed in a similar fashion but this is left to other texts. Resonant vibrations can be used as a technique to measure elastic constants and this will be discussed in the next section. 

Strain dependence should also be considered when comparing acoustically measured elastic constants with statically measured values. As an example, for polyethylene at room temperature, the modulus is independent of strain up to a strain of about 10 (7). Beyond this point, the modulus decreases as the strain increases. Typically, acoustic measurements are made in the strain range 10 where the moduli are strain-independent, but static measin-ements... 

MEASURING ELASTIC CONSTANTS and one independent shear modulus ... 

Freedericksz Transitions as a Method for Measuring Elastic Constants... 

This section contains an outline of the theory and results of experimental studies of the elastic properties of nematics. First, a short introduction of the standard theories is given and the characteristic quantities, used to describe nematic phase elasticity are introduced. After an overview of the standard methods of measuring elastic constants, a summary of the experimental results is given. In particular, we list a collection of papers dealing with the extensively explored cyanobiphenyls and the standard substance 4-methyloxy-4 -butylbenzylidene-aniline (MBBA). The next part is devoted to the less-common surface-like elastic constants, and this is followed by a sketch of the theoretical approaches to the microscopic interpretation of elastic constants and the Landau-de Gennes expansion. The section is concluded by a brief discussion of elastic theory for biaxial nematic phases. 

The standard method for measuring elastic constants in nematics is the analysis of onedimensional director deformations in thin planar cells with defined anchoring conditions. This approach allows separate study of splay, twist or bend deformations in a pure form at the onset of the transition. 

Employing quasielastic light scattering, Dierker and Pindak [94] have measured elastic constants and viscosity of two-layer SmI films of 2M4P8BC in the vicinity of the weakly first order SmC -SmI transition. With the help of an applied magnetic field (along the x-axis), monodomain SmC and SmI samples were prepared. Similar to the case for the SmC phase, the intensity of depolarized light scattered by thermal molecular orientation fluctuations can be written as. 

It is common practice to measure elastic constants and their temperature dependence experimentally, either via static (e.g. three- or four-point bending, XRD or neutron scattering in connection with strain gauges) or via dynamic tests (e.g. sound velocity methods or resonant frequency techniques). It is well known from solid state physics, cf e.g. [Ashcroft Mermin 1976, Kittel 1988], that the elastic constants determine the velocity of sound waves in solids. For example, the velocity of transversal waves (shear waves) Vj, is given by... 

Concluding this section we would like to add some praetieal remarks. It seems that no complete theory is available to predict the temperature (or entropy) dependence of the elastic constants and thus usually they have to be determined from experimental measurements. Elastic constants determined by static measurements, e.g. from load-deflection curves in three-point bending via the formula [Mencik 1992]... 

The last fact was not taken into account when deriving Equation 4.55. Typical room-temperature values of D+, which are in the range 1-2 x 10 m s", are in agreement with Equation 4.55 if we take into account measured elastic constants, the ratio w+/We ( T5), and the deformation-potential constant ej. From a comparison of Equations 4.55 and 4.58, it follows that the phonon scattering is higher than the electron scattering at all temperatures, with the possible exception of very low temperatures, where T/To 1. However, at these temperatures, is limited by residual impurities, even in the purest metals available. 

The measured elastic constants for 4,4 octylcyanobiphenyl (8CB) are also plotted as a fimction of temperature in FIGURE 2, where it can be seen that the elastic constants for twist and bend increase as the underlying smectic A phase is approached on cooling in the nematic phase. This is a general result, since the smectic A phase cannot support either a twist or bend torsional distortion. 

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