Deformation, elastic

Let us now turn to the relationship between stresses and strains. We have already addressed this in Chapter 3 where we discussed a very broad spectrum of rheological properties found in various systems, namely, elasticity, plasticity, viscosity, and their numerous combinations. Some of the significant limitations that we adapted include a consideration of a single stressed state of a uniform shear and of near steady-state processes. Here, we will limit ourselves to a discussion of a single rheological behavior, that is, elasticity, and will focus on the particular peculiarities and generalizations pertinent to this field. 

The described initial linear (Hookean) behavior is typical for a very large number of solidlike objects. Conversely, objects like soap thin films or modeling clays do not exhibit this type of behavior. A single parameter characterizing the elastic properties of an object is Young s modulus, that is, the dimensional proportionality constant, E = a/e = Ff/AlS. In Chapter 4, we listed some typical values of E. In continuum mechanics, the elasticity moduli are typically described by the letter c and the inverse quantity, compliance, by the letter s, that is, 5 oc 1/c. 

In the most general case, the elasticity (or compliance) moduli relate every component of the stress tensor to every component of the strain tensor there are four components in each tensor in the 2D case and nine components in the 3D case. This corresponds to a total of 81 moduli c and compliances s. However, let us restrict ourselves to a discussion of the principal strains in uniform 

Physical-Chemical Mechanics of Disperse Systems and Materials 

In the simplest case of a uniaxial extension along the Ox axis, the uniaxial stress 

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It is rare to be able to observe elastic deformations (which occur for instance during earthquakes) since by definition an elastic deformation does not leave any record. However, many subsurface or surface features are related to the other two modes of deformation. The composition of the material, confining pressure, rate of deformation and temperature determine which type of deformation will be initiated. 

In effect of loads acting in the scope of elastic deformations (proportional) in polycristalline bodies, changes in distances between atomic surfaces of the net are observed, which change the initial magnetic permeability. 

Substances in this category include Krypton, sodium chloride, and diamond, as examples, and it is not surprising that differences in detail as to frictional behavior do occur. The softer solids tend to obey Amontons law with /i values in the normal range of 0.5-1.0, provided they are not too near their melting points. Ionic crystals, such as sodium chloride, tend to show irreversible surface damage, in the form of cracks, owing to their brittleness, but still tend to obey Amontons law. This suggests that the area of contact is mainly determined by plastic flow rather than by elastic deformation. 

An unsteady ain-flow unbalance that alternates between inlets can set up an alternating thrust pattern which can be very damagiag to beariags designed for low thrust load. Mechanical vibration and elastic deformation problems and diagnostic techniques for stmctural iaadequacies ia fan design are discussed ia Reference 16. 

Two approaches have been taken to produce metal-matrix composites (qv) incorporation of fibers into a matrix by mechanical means and in situ preparation of a two-phase fibrous or lamellar material by controlled solidification or heat treatment. The principles of strengthening for alloys prepared by the former technique are well estabUshed (24), primarily because yielding and even fracture of these materials occurs while the reinforcing phase is elastically deformed. Under these conditions both strength and modulus increase linearly with volume fraction of reinforcement. However, the deformation of in situ, ie, eutectic, eutectoid, peritectic, or peritectoid, composites usually involves some plastic deformation of the reinforcing phase, and this presents many complexities in analysis and prediction of properties. 

Elastohydrodynamic Lubrication (EHL). Lubrication needs in many machines ate minimized by carrying the load on concentrated contacts in ball and toUet beatings, gear teeth, cams, and some friction drives. With the load concentrated on a small elastically deformed area, these EHL contacts ate commonly characterized by a very thin separating hydrodynamic oil film which supports local stresses that tax the fatigue strength of the strongest steels. 

Rheology is the science of the deformation and flow of matter. It is concerned with the response of materials to appHed stress. That response may be irreversible viscous flow, reversible elastic deformation, or a combination of the two. Control of rheology is essential for the manufacture and handling of numerous materials and products, eg, foods, cosmetics, mbber, plastics, paints, inks, and drilling muds. Before control can be achieved, there must be an understanding of rheology and an ability to measure rheological properties. 

The design of load-bearing structures for service at room temperature is generally based on the yield strength or for some appHcations on the tensile strength. The metal is expected to behave essentially in an elastic manner, that is, the stmcture undergoes an elastic deformation immediately upon load apphcation and no further deformation occurs with time. When the load is removed, the stmcture returns to its original dimensions. 

When a fiber is stressed, the instantaneous elongation that occurs is defined as instantaneous elastic deformation. The subsequent delayed additional elongation that occurs with increasing time is creep deformation. Upon stress removal, the instantaneous recovery that occurs is called instantaneous elastic recovery and is approximately equal to the instantaneous elastic deformation. If the subsequent creep recovery is 100%, ie, equal to the creep deformation, the specimen exhibits primary creep only and is thus completely elastic. In such a case, the specimen has probably not been extended beyond its yield point. If after loading and load removal, the specimen fails to recover to its original length, the portion of creep deformation that is recoverable is still called primary creep the portion that is nonrecoverable is called secondary creep. This nonrecoverable elongation is typically called permanent set. 

The more quickly and completely a fiber recovers from an imposed strain, the more nearly perfectly elastic it is. The ratio of the instantaneous elastic deformation to the total deformation may be used as a criterion of elasticity (62). The integrated divergences from a theoretical graph of perfect elasticity versus elongation is also used as a criterion for determination of the elasticity index. 

Grease Retention, Wrinkle Resistance, and Durable Press. On bending or creasing of a textile material, the external portion of each filament in the yam is placed under tension, and the internal portion is placed in compression. Thus, the wrinMe-recovery properties must be governed in part by the inherent, tensional elastic deformation and recovery properties of the fibers. In addition to the inherent fiber properties, the yam and fabric geometry must be considered. 

The melt temperature of a polyurethane is important for processibiUty. Melting should occur well below the decomposition temperature. Below the glass-transition temperature the molecular motion is frozen, and the material is only able to undergo small-scale elastic deformations. For amorphous polyurethane elastomers, the T of the soft segment is ca —50 to —60 " C, whereas for the amorphous hard segment, T is in the 20—100°C range. The T and T of the mote common macrodiols used in the manufacture of TPU are Hsted in Table 2. 

Elastic Behavior. Elastic deformation is defined as the reversible deformation that occurs when a load is appHed. Most ceramics deform in a linear elastic fashion, ie, the amount of reversible deformation is a linear function of the appHed stress up to a certain stress level. If the appHed stress is increased any further the ceramic fractures catastrophically. This is in contrast to most metals which initially deform elastically and then begin to deform plastically. Plastic deformation allows stresses to be dissipated rather than building to the point where bonds break irreversibly. 

For stainless steel, the stress-strain curve (see Fig. 26-37) has no sharp yield point at the upper stress limit of elastic deformation. Yield strength is generally defined as the stress at 2 percent elongation. 

The first consequence of the work assumption may be established immediately for elastic deformations. Consider an arbitrary finite smooth closed cycle of homogeneous deformation %(ti, f ) which lies entirely in the elastic... 

Assuming constant volume (valid if v = 0.5 or, if not, plastic deformation elastic deformation) ... 

There is one obvious drawback of high-hysteresis rubber. In normal rolling operation, considerable elastic deformations still take place in the tyre wall, and high-loss tyres will consume fuel and generate considerable heat. The way out is to use a low-loss tyre covered with a high-loss tread - another example of design using composite materials (Fig. 26.9). 

In summary, then, design with polymers requires special attention to time-dependent effects, large elastic deformation and the effects of temperature, even close to room temperature. Room temperature data for the generic polymers are presented in Table 21.5. As emphasised already, they are approximate, suitable only for the first step of the design project. For the next step you should consult books (see Further reading), and when the choice has narrowed to one or a few candidates, data for them should be sought from manufacturers data sheets, or from your own tests. Many polymers contain additives - plasticisers, fillers, colourants - which change the mechanical properties. Manufacturers will identify the polymers they sell, but will rarely disclose their... 

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