Elasticity defined, figure

Alternatively, a yield stress can be defined as that at which the ratio of total stress to total strain is some selected amount, say 50% or 70% of the elastic modulus (Figure 3.3b). In the first case the yield stress is conveniently located graphically by offsetting to the right the stated amount of 0.2% (or 1%) and drawing a line paralleling that drawn for the elastic modulus. The point at which this line intersects the observed stress-strain line defined the yield stress. In the second case also the point of intersection of the line drawn with a slope of 0.7E, for instance, with the observed stress-strain line determines the yield stress. 

In 1885 Joseph Boussinesq (6), trying to extend the validity of these results to the case of axi-symetrical rigid convex punches indenting a flat semi-infinite elastic medium, demonstrates that, without an adequate boundary comlition, the size of the contact area is generally unknown. In ordo to overcome this difficulty, he imposes that normal stresses vanish on the border of the contact area. In other words, the profile of the distorted medium must be tangent to the surfiice of the punch on the border of the contact area. Note that this condition is the same as the condition presupposed by the Hertz s theory. With this assumption, the size oh of the contact area and the penetration depth 5h are completely defined (Figure 1). 

For materials having a nonlinear elastic region (Figure 6.6), use of the strain offset method is not possible, and the usual practice is to define the yield strength as the stress required to produce some amount of strain (e.g., e = 0.005). 

Figure 3.6 Definition of variables to define the shear deformation of an elastic body.

The elasticity of a fiber describes its abiUty to return to original dimensions upon release of a deforming stress, and is quantitatively described by the stress or tenacity at the yield point. The final fiber quaUty factor is its toughness, which describes its abiUty to absorb work. Toughness may be quantitatively designated by the work required to mpture the fiber, which may be evaluated from the area under the total stress-strain curve. The usual textile unit for this property is mass pet unit linear density. The toughness index, defined as one-half the product of the stress and strain at break also in units of mass pet unit linear density, is frequentiy used as an approximation of the work required to mpture a fiber. The stress-strain curves of some typical textile fibers ate shown in Figure 5. 

Force curve gives the relationship between the z-piezo displacement and the cantilever deflection as shown in Figure 21.10b. When a cantilever approaches to a stiff sample surface, cantilever deflection. A, is equal to the z-piezo displacement, z — Zo- The value of zo is defined as the position where the tip-sample contact is realized. On the other hand, z-piezo displacement becomes larger to achieve the preset trigger value (set point) of the cantilever deflection in the case of an elastic sample due to the deformation of the sample itself. In other words, we can obtain information about a sample deformation, 8, from the force-distance curve of the elastic surface by the following relationship ... 

Viscoelasticity illustrates materials that exhibit both viscous and elastic characteristics. Viscous materials tike honey resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Viscoelasticity is the result of the diffusion of atoms or molecules inside an amorphous material. Rubber is highly elastic, but yet a viscous material. This property can be defined by the term viscoelasticity. Viscoelasticity is a combination of two separate mechanisms occurring at the same time in mbber. A spring represents the elastic portion, and a dashpot represents the viscous component (Figure 28.7). 

A tube-plate is essentially a perforated plate with an unperforated rim, supported at its periphery. The tube holes weaken the plate and reduce its flexual rigidity. The equations developed for the stress analysis of unperforated plates (Section 13.3.5) can be used for perforated plates by substituting virtual (effective) values for the elastic constants E and v, in place of the normal values for the plate material. The virtual elastic constants E and v are functions of the plate ligament efficiency, Figure 13.38 see O Donnell and Langer (1962). The ligament efficiency of a perforated plate is defined as ... 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

These are obviously extreme examples, most products behave much more closely to what is termed unit elasticity— which is defined as being when a 1% reduction in price will cause a 1 % increase in demand (sales). The equihbrium price can then be defined as the price at which supply=demand (Figure 13.9 b and c). Of course the actual price will vary with the grade of product made and thus its end use, examples include pharmaceutical or technical grade chemicals, or potable or fuel alcohols. Obviously the price-demand behaviour of inelastic products demonstrates their vital importance to some consumers, especially when unchallenged by other producer companies or competing products. This class of products is obviously attractive, not least because of the relative ease with which high R D costs can be recovered. The demand, or lack of demand for products and also other considerations, such as brand loyalty on the part of the consumer and restrictions in supply due to companies exploitation of their patents tend to distort precise descriptions of market behaviour. 

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