Elastic properties

There is a paucity of reliable, consistent data for hydrate elastic properties. Since these properties depend on crystal structures, many of them can be estimated reliably. However, since 1998, there have been significant efforts to perform accurate measurements of these properties in order to help facilitate correct interpretation of sonic or seismic velocity field data obtained on hydrates in the natural environments. 

Whalley (1980) presented a theoretical argument to suggest that both the thermal expansivity and Poisson s ratio should be similar to that of ice. With the above two estimates, Whalley calculated the compressional velocity of sound in hydrates with a value of 3.8 km/s, a value later confirmed by Whiffen et al. (1982) via Brillouin spectroscopy. Kiefte et al. (1985) performed similar measurements on simple hydrates to obtain values for methane, propane, and hydrogen sulfide of 3.3, 3.7, and 3.35 km/s, respectively, in substantial agreement with calculations by Pearson et al. (1984). 

Pandit and King (1982) and Bathe et al. (1984) presented measurements using transducer techniques, which are somewhat different from the accepted values of Kiefte et al. (1985). The reason for the discrepancy of the sonic velocity values from those in Table 2.8 and above is not fully understood. It should be noted that compressional velocity values can vary significantly depending on the hydrate composition and occupancy. This has been demonstrated by lattice-dynamics calculations, which showed that the adiabatic elastic moduli of methane hydrate is larger than that of a hypothetical empty hydrate lattice (Shpakov et al., 1998). 

Shimizu et al. (2002) extended the previous Brillouin spectroscopy measurements by performing in situ measurements on a single crystal methane hydrate. They examined the effect of pressure on shear (TA) and compressional (LA) velocities, and compared these results to that for ice. The shear velocities of methane hydrate and ice were very similar, showing a slight decrease (about 2 to 1.85 km/s) with increasing pressure (0.02-0.6 GPa). Conversely, the compressional velocities of ice and methane hydrate were different. The 

Lee and Collett (2001) measured the compressional (P-wave) and shear (S-wave) velocities of natural hydrates in sediments (33% average total porosity) at the Mallik 2L-38 well. The P-wave velocity of nongas-hydrate-bearing sediment with 33% porosity was found to be about 2.2 km/s. The compressional velocity of gas-hydrate-bearing sediments with 30% gas hydrate concentration (water-filled porosity of 23%) was found to be about 2.7 km/s, and 3.3 km/s at 60% concentration (water-filled porosity of 13%), that is, about a 20% or 50% increase to nongas-hydrate-bearing sediment. The shear velocity was found to increase from 0.81 to 1.23 km/s. 

As previously mentioned, MTMS-dtxwtd silica aerogels exhibit unprecedented flexibility when compared with standard silica analogs. Let us discuss the properties of these outstanding materials in some detail and underline them by measurements  

Solids can be deformed by tension, torsion, shear, bending, and compression. The cause of deformation is an external load or stress which induces strain inside the body. According to Hooke s law, the stress is directly proportional to the strain [26]. In tension mode (4.5) holds, 

Generally, any deformation is directly proportional to applied force, provided the forces do not exceed a certain limit called elastic limit. Within this limit, a specimen returns to its original shape and size, due to elastic (nonplastic) deformations. In other words. 

Under quasi-static loading, or in conditions under which short-term loading responses are expected to occur [27], both meniscal and discal tissues may be modeled as linear elastic and orthotropic. Under a constant load rate, the non-linear behavior may be described by an exponential stress-strain relationship given by 

The macroscopic tensile strength of the entire meniscus was studied by Mathur et al [4] by gripping the horns of the meniscus, and stretching it to failure. The results suggested that the medial meniscus was significantly 

Strength and modulus of the meniscus vary with different locations and with different orientations of the specimen due to structural and compositional changes (Table B2.4). For loading parallel to the fibers, it appears that the meniscus may be stronger in the anterior location, and that the lateral meniscus may be stronger than the medial meniscus. This may be explained in part by the fact that the fiber orientation is more random in the posterior part of the medial meniscus. 

A similar trend is seen in the tensile modulus of meniscal specimens oriented parallel to the circumferential direction (Table B.2.5), and if a power law is used as the constitutive equation (equation B2.3), the coefficients A and C show an identical pattern [7, 10, 28]. 

The properties of intervertebral discs are more complex than those of the menisci, since properties vary with disc level, and discs must withstand loads and moments in three orthogonal directions. (Table B2.6). 

Until the discovery of the MAX phases, the price paid for high specific stiffness values has been a lack, or at least a difficulty, of machinability. It is important, therefore, to note that one of the most characteristic properties of the MAX phases is the ease with which they can be machined, with nothing more sophisticated than 

Also listed are the bulk moduli values (6 ), measured directly in an anvil cell, and Bf, calculated from the shear and longitudinal sound velocities. The references and values are color-coded the 312s are highlighted yellow, and 413s gray. 

Poisson s ratios for most of the MAX phases hover around 0.2, which is lower than the 0.3 of Ti, and closer to the 0.19 of near-stoichiometric TiC. 

Several theoretical reports have shown a one-to-one correspondence between the bulk moduli of the binary MX and ternary MAX compounds - a not too-surprising result given that the latter are comprised of blocks of the former (40, 70, 71]. 

An important, but subtle, factor influencing the B-values of the MAX phases is their stoichiometry, and more specifically their vacancy concentrations. This effect is best seen in the B-values of Ti2AlN, where theory and experiment show a decrease in lattice parameters as C is substituted for by N. Given that the lattice parameters shrink, it is not surprising that theory predicts that this substitution should increase the B-value, when, in fact, experimentally it decreases with increasing N-content [42]. This paradox is resolved when it is appreciated that B is a strong function of vacancies, and that the addition of N results in the formation of vacancies on the A1 and/or N sites. As discussed below, the presence of these defects also influence other properties [43, 44]. 

It has not proved possible to develop general analytical hard core models for liquid crystals, just as for normal liquids. Instead, computer simulations have played an important role in extending our understanding of the phase behaviour of hard particles. It has been found that a system of hard ellipsoids can form a nematic phase for ratios L/D 2.5 (rods) or L/D 0.4 (discs). However, such a system cannot form a smectic phase, as can be shown by a scaling argument in statistical mechanical theory. However, simulations show that a smectic phase can be formed by a system of hard spherocylinders. The critical volume fractions for stability of a smectic A phase depend on whether the model is that of parallel spherocylinders or, more realistically, freely rotating spherocyHnders. 

The elastic properties of liquid crystal phases can be modelled using continuum theory. As its name su ests, this involves treating the medium as a continuum at the level of the director, neglecting the structure at the molecular scale. 

An aligned monodomain of a nematic liquid crystal is characterized by a single director h. However, in imperfectly aligned or unaligned samples the director varies through space. The appropriate tensor order parameter to describe the director field is then 

Here fCi, K2 and K3 are elastic constants. The first, fCi, is associated with a splay deformation of the director field, K2 is associated with a twist deformation and K3 with bend (Fig. 5.20). These three elastic constants 

Continuum theory has also been applied to analyse the dynamics of flow of nematics. The equations provide the time-dependent velocity, director and pressure fields. These can be determined from equations for the fluid acceleration, the rate of change of director orientation in terms of the velocity gradients and the molecular field, and the incompressibility condition. Further details can be found in de Gennes and Frost (1993). Various combinations of elements of the viscosity tensor of a nematic define the so-called Leslie coefficients. 

---

Stannarius R 1998 Elastic properties of nematic liquid crystals 1998 Handbook of Liquid Crystals Vol 2A. Low Molecular Weight Liquid Crystals led D Demus, J Goodby, G W Gray, Fl-W Speiss and V Vill (New York Wiley-VCH)... 

Other elastomeric-type fibers iaclude the biconstituents, which usually combine a polyamide or polyester with a segmented polyurethane-based fiber. These two constituents ate melt-extmded simultaneously through the same spinneret hole and may be arranged either side by side or ia an eccentric sheath—cote configuration. As these fibers ate drawn, a differential shrinkage of the two components develops to produce a hehcal fiber configuration with elastic properties. An appHed tensile force pulls out the helix and is resisted by the elastomeric component. Kanebo Ltd. has iatroduced a nylon—spandex sheath—cote biconstituent fiber for hosiery with the trade name Sidetia (6). 

Elastic Properties. The abiUty of a fiber to deform under below-mpture loads and to return to its original configuration or dimension upon load removal is an important performance criterion. Permanent deformation may be as detrimental as actual breakage, rendering a product inadequate for further use. Thus, the repeated stress or strain characteristics are of significance in predicting or evaluating functional properties. 

Plasticizers. It was found in 1926 that solutions of PVC, prepared at elevated temperatures with high boiling solvents, possessed unusual elastic properties when cooled to room temperature (137). Such solutions are flexible, elastic, and exhibit a high degree of chemical inertness and solvent resistance. 

The role of yeast in fermenting dough maturation is even less clear. The alcohol and carbon dioxide developed during fermentation must influence the elastic properties of the protein matrix. However, experimental procedures that would permit this to be checked in the absence of yeast have not been developed. 

Mechanical Properties. The hexagonal symmetry of a graphite crystal causes the elastic properties to be transversely isotropic ia the layer plane only five independent constants are necessary to define the complete set. The self-consistent set of elastic constants given ia Table 2 has been measured ia air at room temperature for highly ordered pyrolytic graphite (20). With the exception of these values are expected to be representative of... 

Many curing systems bring about aHquid—to—soHd conversion of the polysulfide polymers. Most curing agents produce an initial soHd mass characterized by a high degree of plasticity and poor elasticity. The development of elastic properties is so slow that the materials are not suitable for the accurate reproduction of undercut areas found ia oral stmctures. 

Selection of the right elastomer has to take iato account not only the low temperature resistance, the elastic properties, and mechanical properties, but also price, which can vary widely ia specialty elastomers. 

Mechanical properties depend considerably on the stmctural characteristics of the EPM/EPDM and the type and amount of fillers in the compound. A wide range of hardnesses can be obtained with EPM/EPDM vulcanisates. The elastic properties are by far superior to those of many other synthetic mbber vulcanizates, particularly of butyl mbber, but they do not reach the level obtained with NR or SBR vulcanizates. The resistance to compression set is surprisingly good, in particular for EPDM with a high ENB content. 

Fluids without any sohdlike elastic behavior do not undergo any reverse deformation when shear stress is removed, and are called purely viscous fluids. The shear stress depends only on the rate of deformation, and not on the extent of derormation (strain). Those which exhibit both viscous and elastic properties are called viscoelastic fluids. 

Optimum combination of elastic properties for the mirror support... 

To obtain high molecular weight polymers the tetramer is equilibrated with a trace of alkaline catalyst for several hours at 150-200°C. The product is a viscous gum with no elastic properties. The molecular weight is controlled by CcU eful addition of monofunctional material. 

Heating of the cyclic polymer at 250°C will also lead to the production of the linear polymer, which is rubbery and stable to 350°C. On standing, however, the material hydrolyses and after a few day loses its elastic properties and becomes hard and covered with drops of hydrochloric acid solution. 

Recovery from deformation and general high-elasticity properties. 

Most materials scientists at an early stage in their university courses learn some elementary aspects of what is still miscalled strength of materials . This field incorporates elementary treatments of problems such as the elastic response of beams to continuous or localised loading, the distribution of torque across a shaft under torsion, or the elastic stresses in the components of a simple girder. Materials come into it only insofar as the specific elastic properties of a particular metal or timber determine the numerical values for some of the symbols in the algebraic treatment. This kind of simple theory is an example of continuum mechanics, and its derivation does not require any knowledge of the crystal structure or crystal properties of simple materials or of the microstructure of more complex materials. The specific aim is to design simple structures that will not exceed their elastic limit under load. 

In a semicrystalline polymer, the crystals are embedded in a matrix of amorphous polymer whose properties depend on the ambient temperature relative to its glass transition temperature. Thus, the overall elastic properties of the semicrystalline polymer can be predicted by treating the polymer as a composite material... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

Elasticity Property of materials whereby they tend to retain or recover... 

Some typical elastic properties for unidirectional fibre composites are given in Table 3.4. 

Typical elastic properties of unidirectional fibre reinforced plastics... 

The previous section illustrated how to obtain the elastic properties of a unidirectional lamina. In practice considerably more information may be required about the behavioural characteristics of a single lamina. To obtain details of the stresses and strains at various orientations in a single ply the following type of analysis is required. 

---