Elastic constants 348 experimental determination

As a result of the above difficulties, alternative techniques were sought and have become established. One approach (sound velocity) involves passing ultrasonic waves through the material and determining their velocities. The other approach (dynamic resonance) uses the natural vibration of the material. The elastic constants are determined from specimen geometry and the resonant frequency. For these last two techniques, experimental accuracies of <0.1% are not uncommon. Both of these latter techniques can also be used on anisotropic materials but the current discussion will emphasize isotropic materials. 

The biaxially oriented PET sheets have been extensively studied with regard to their mechanical anisotropy and all nine independent elastic constants have been determined by a variety of experimental techniques 38,39). The complete set of compliances for a one-way drawn sheet of draw ratio 5 1 is shown in Table 7. It is interesting to note that these compliances clearly reflect the two major structural features, the high chain axis orientation and the preferential orientation of the benzene rings... 

The simple theory of electronegativity fails in this discussion because it is based merely on electron transfer energies and determines only the approximate number of electrons transferred, and it does not consider the interactions which take place after transfer. Several calculations in the alkali halides of the cohesive energy (24), the elastic constants (24), the equilibrium spacing (24), and the NMR chemical shift 17, 18, 22) and its pressure dependence (15) have assumed complete ionicity. Because these calculations based on complete ionicity agree remarkably well with the experimental data, we are not surprised that the electronegativity concept of covalency fails completely for the alkali iodide isomer shifts. 

Experimental determination of the components of the elastic force thus requires measurements of the changes in force with temperature at constant volnme and length. The constant volume requires the application of hydrostatic pressure during measurement of the force-temperature coefficient. This experiment is extremely difficult to perform 22-i3). 

The statistical theory of rubber elasticity predicts that isothermal simple elongation and compression at constant pressure must be accompanied by interchain effects resulting from the volume change on deformation. The correct experimental determination of these effects is difficult because of very small absolute values of the volume changes. These studies are, however, important for understanding the molecular mechanisms of rubber elasticity and checking the validity of the postulates of statistical theory. 

The coefficients Cn are called elasticity constants and the coefficients Su elastic compliance constants (Azaroff, 1960). Generally, they are described jointly as elasticity constants and constitute a set of strictly defined, in the physical sense, quantities relating to crystal structure. Their experimental determination is impossible in principle, since Cu = (doildefei, where / i, and hence it would be necessary to keep all e constant, except et. It is easier to satisfy the necessary conditions for determining Young s modulus E, when all but one normal stresses are constant, since... 

For anisotropic materials torsion is discussed in the books by Love, Lekhnitskii175 and Hearmon185. The torque M now depends not upon one elastic constant only, as in the isotropic case, but upon two. This makes the determination of shear modulus by a torsion test a difficult task and requires careful experimentation. Early work on this for polymers was done by Raumann195, by Ladizesky and Ward205 and by Arridge and Folkes165. 

When the full expression (21) will be determined, say, from the experimental second virial coefficient it will be possible to calculate all constants (compressibility, elastic constants, etc.) of these molecular lattices. 

Fits to the data on this basis are shown in Figure 8e. Values for the coupling coefficients can be extracted from experimentally determined relationships of this type if the bare elastic constants are known. 

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. 

Ro is determined by fitting to the experimental lattice parameter. Shell charges and spring constants were derived for the binary oxides by fitting to dielectric and elastic constants. All the binary potentials and shell parameters were transferred mchanged for the ternary and quaternary cuprates. 

Fig. 3.4.1. The experimental Fteedericksz geometries for the determination of the (a) splay, (b) twist and (c) bend elastic constants of a nematic liquid crystal.

Fig. 3.9.3 compares this relation with the experimental data of Chatelain and as can be seen the agreement is good. In principle a measurement of the intensity of the scattering and its angular variation offers a method of determining the elastic constants. 

There are two possible sources for the discrepancies between measured and predicted elastic constants when the orientation averages used are determined from models for molecular orientation the models may be incorrect and neither Voigt nor Reuss averaging may be appropriate. In order to examine the second of these possibilities more directly it is necessary to determine the orientation averages experimentally. 

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. 

Finally, we consider theoretical or computational means of determining the ideal elastic constants of some polymers together with their temperatme dependences and compare these with experimental values, determined, as much as possible, under conditions that are free of viscoelastic relaxations. We then provide small-strain energy-elastic constants of a variety of both glassy and semi-crystalline polymers. 

For TmCd and TmZn a variety of techniques have been applied to determine the important coupling constants and gj. they are listed in table 3. In addition to the temperature dependence of the symmetry elastic constant Cj (T), the parastriction method, the third-order susceptibility and the magnetic field dependence of the structural phase transition temperature Tq B) have been used. The different experimental methods have been described in sect. 2.4.1. It is seen from table 3 that the coupling constants determined with these different methods are in good agreement with each other. 

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