Modulus, elastic

The elastic modulus of a polymer crystal provides us with important information on the molecular conformation in the crystal lattice [24]. The elastic modulus (crystal modulus) of the crystalline regions in the direction parallel to the chain axis has been measured for a variety of polymers by X-ray diffraction [25]. Examination of the data so far accumulated enables us to relate the crystal modulus, namely, the extensivity of a polymer molecule, both to the molecular conformation and the mechanism of deformation in the crystal lattice. Furthermore, knowledge of the crystal modulus is of interest in connection with the mechanical properties of the polymer, because the crystal modulus gives the maximum attainable modulus for the specimen modulus of a polymer. 

The initial slope of the stress-strain curve of the crystalline regions gives the crystal modulus, when the changes in the crystal lattice spacing under a constant stress are monitored by X-ray diffraction. 

The elastic modulus E is the slope of stress-strain curve which is a constant for a linear elastic material for stresses less than the proportional limit. If E is the slope of a as a function of e curve, then 

We can see the effect of the DOP and Ti02 on the elastic modulus of the proposed material in Fig. 6.14. The elastic modulus of the synthetic elastomer decreases when the content of the DOP increases, as is shown in Fig. 6.14(a). The reduction in the elastic modulus can clearly be seen when the DOP content increases over 40 PHR. The reason for this decrease in the modulus is accounted for by the fact that the plasticizer makes the carbon chain s connection in the elastomer weaker. On the other hand, the impact of the Ti02 on the proposed elastomer is more complex. As is illustrated in Fig. 6.14(b), the elastic modulus increases when the content of the Ti02 is changed from 0 PHR to 10 PHR and from 10 PHR to 

Other mechanical properties of glasses are inherent to the material. The elastic modulus, E, is determined by the individual bonds in the material and by the structure of the network. The hardness of glasses is a function of the strength of individual bonds and the density of packing of the atoms in the structure. 

As classic brittle materials, glasses exhibit nearly perfect Hookian behavior on application of a stress. The ratio of the strain, e, resulting from application of a stress, a, is a constant which is known as the elastic modulus, or Young s modulus, E, which is defined by the expression  

If a tensile stress is applied to a specimen in the direction of the x-axis, the specimen will elongate in that direction. This elongation will be accompanied by contraction in the y and z directions. The ratio of the transverse strain to the axial strain is called Poisson s ratio. Poisson s ratio for oxide glasses generally lies between 0.2 and 0.3, although the value for vitreous silica is only 0.17. The shear modulus, G, which relates shear strain, y, to shear stress, t, is given by the expression  

Young s modulus, the shear modulus, and Poisson s ratio are related by the expression  

The elastic modulus of a material arises from the relation between an applied force and the resultant change in the average separation distance of the atoms which form the structure of that material. If we consider the Condon-Morse curve for force, F, as a function of atomic separation distance, r, we can write an expression of the form  

F is the force applied to the dragline, S is the cross sectional area of the double filament, L is the initial length of the dragline, and AL is the elongation of the dragline when the F is applied. The F and AL are determined from the force-elongation curve within the elastic limit point and the S from the electron scanning microscopy. 

Again it must be emphasized that this discussion is strictly limited to time-independent phenomena. 

Note that the choice of the coordinate symbols (x, y, z) for the directions parallel to the X, Y and Z sides, respectively, is arbitrary. In this regard, we should again remind ourselves that real pieces of material are three-dimensional. The three-dimensional aspect of a material means that both the location and direction of the forces on the sample s surfaces must be taken into account. As stated in the Introduction we must balance the forces in all directions on the sample, and similarly balance all the moments. Otherwise, the sample will start to accelerate, which would make the measurements difficult indeed  

Note that the sample has elongated in the stretch direction but, to preserve volume, has had to thin. The clamps at the ends also had to shrink to accommodate this motion and avoid distortion of the sample. In practice, of course, this can t be done, so an end correction must be applied unless the sample is very long and thin. Alternatively, one can visualize the virtual clamps as a transverse line of material points well removed from the physical clamps. 

We can see most easily how pressure works by introducing the concept of material stress, as opposed to the total or applied stress to the sample. Material stress can be regarded as the stress produced by the deformation of the material it always opposes the deformation. The force applied to the clamps must be 

Considering the tension experiment, we designate the direction of stretch as the x direction and the force causing this as Fx. This is also the direction of the normal to the plane to which the force is applied (the virtual clamp) its area will be designated Ax. The stress on this face is given by 

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The importance of polymer composites arises largely from the fact that such low density materials can have unusually high elastic modulus and tensile strength. Polymers have extensive applications in various fields of industry and agriculture. They are used as constructional materials or protective coatings. Exploitation of polymers is of special importance for products that may be exposed to the radiation or temperature, since the use of polymers make it possible to decrease the consumption of expensive (and, sometimes, deficient) metals and alloys, and to extent the lifetime of the whole product. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

In AFM, the relative approach of sample and tip is nonnally stopped after contact is reached. Flowever, the instrument may also be used as a nanoindenter, measuring the penetration deptli of the tip as it is pressed into the surface of the material under test. Infomiation such as the elastic modulus at a given point on the surface may be obtained in tliis way [114], altliough producing enough points to synthesize an elastic modulus image is very time consuming. 

Secondly, the ultimate properties of polymers are of continuous interest. Ultimate properties are the properties of ideal, defect free, structures. So far, for polymer crystals the ultimate elastic modulus and the ultimate tensile strength have not been calculated at an appropriate level. In particular, convergence as a function of basis set size has not been demonstrated, and most calculations have been applied to a single isolated chain rather than a three-dimensional polymer crystal. Using the Car-Parrinello method, we have been able to achieve basis set convergence for the elastic modulus of a three-dimensional infinite polyethylene crystal. These results will also be fliscussed. 

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

The glass-ceramic phase assemblage, ie, the types of crystals and the proportion of crystals to glass, is responsible for many of the physical and chemical properties, such as thermal and electrical characteristics, chemical durabiUty, elastic modulus, and hardness. In many cases these properties are additive for example, a phase assemblage comprising high and low expansion crystals has a bulk thermal expansion proportional to the amounts of each of these crystals. 

Nylon-6. Nylon-6—clay nanometer composites using montmorillonite clay intercalated with 12-aminolauric acid have been produced (37,38). When mixed with S-caprolactam and polymerized at 100°C for 30 min, a nylon clay—hybrid (NCH) was produced. Transmission electron microscopy (tern) and x-ray diffraction of the NCH confirm both the intercalation and molecular level of mixing between the two phases. The benefits of such materials over ordinary nylon-6 or nonmolecularly mixed, clay-reinforced nylon-6 include increased heat distortion temperature, elastic modulus, tensile strength, and dynamic elastic modulus throughout the —150 to 250°C temperature range. 

Hardness is a measure of a material s resistance to deformation. In this article hardness is taken to be the measure of a material s resistance to indentation by a tool or indenter harder than itself This seems a relatively simple concept until mathematical analysis is attempted the elastic, plastic, and elastic recovery properties of a material are involved, making the relationship quite complex. Further complications are introduced by variations in elastic modulus and frictional coefficients. 

For most hydrardic pressure-driven processes (eg, reverse osmosis), dense membranes in hoUow-fiber configuration can be employed only if the internal diameters of the fibers are kept within the order of magnitude of the fiber-wall thickness. The asymmetric hoUow fiber has to have a high elastic modulus to prevent catastrophic coUapse of the filament. The yield-stress CJy of the fiber material, operating under hydrardic pressure, can be related to the fiber coUapse pressure to yield a more reaUstic estimate of plastic coUapse ... 

Industrially, polyurethane flexible foam manufacturers combine a version of the carbamate-forming reaction and the amine—isocyanate reaction to provide both density reduction and elastic modulus increases. The overall scheme involves the reaction of one mole of water with one mole of isocyanate to produce a carbamic acid intermediate. The carbamic acid intermediate spontaneously loses carbon dioxide to yield a primary amine which reacts with a second mole of isocyanate to yield a substituted urea. 

Metal-Matrix Composites. A metal-matrix composite (MMC) is comprised of a metal ahoy, less than 50% by volume that is reinforced by one or more constituents with a significantly higher elastic modulus. Reinforcement materials include carbides, oxides, graphite, borides, intermetahics or even polymeric products. These materials can be used in the form of whiskers, continuous or discontinuous fibers, or particles. Matrices can be made from metal ahoys of Mg, Al, Ti, Cu, Ni or Fe. In addition, intermetahic compounds such as titanium and nickel aluminides, Ti Al and Ni Al, respectively, are also used as a matrix material (58,59). P/M MMC can be formed by a variety of full-density hot consolidation processes, including hot pressing, hot isostatic pressing, extmsion, or forging. 

Fig. 11. Modulus inciease as a function of fibei volume fraction alumina fiber-reinforced aluminum—lithium alloy matrix for (a) E (elastic modulus),...

Al—Li. Ahoys containing about two to three percent lithium [7439-93-2] Li, (Fig. 15) received much attention in the 1980s because of their low density and high elastic modulus. Each weight percent of lithium in aluminum ahoys decreases density by about three percent and increases elastic modulus by about six percent. The system is characteri2ed by a eutectic reaction at 8.1% Li at 579°C. The maximum soHd solubiHty is 4.7% Li. The strengthening precipitate in binary Al—Li ahoys is metastable Al Li [12359-85-2] having the cubic LI2 crystal stmcture, and the equhibrium precipitate is complex cubic... 

Nonoxide fibers, such as carbides, nitrides, and carbons, are produced by high temperature chemical processes that often result in fiber lengths shorter than those of oxide fibers. Mechanical properties such as high elastic modulus and tensile strength of these materials make them excellent as reinforcements for plastics, glass, metals, and ceramics. Because these products oxidize at high temperatures, they are primarily suited for use in vacuum or inert atmospheres, but may also be used for relatively short exposures in oxidizing atmospheres above 1000°C. 

The most important properties of refractory fibers are thermal conductivity, resistance to thermal and physical degradation at high temperatures, tensile strength, and elastic modulus. Thermal conductivity is affected by the material s bulk density, its fiber diameter, the amount of unfiberized material in the product, and the mean temperature of the insulation. Products fabricated from fine fibers with few unfiberized additions have the lowest thermal conductivities at high temperatures. A plot of thermal conductivity versus mean temperature for three oxide fibers having equal bulk densities is shown in Figure 2. 

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