web.archive.org

elasticity: Definition, Synonyms and Much More from Answers.com

️Wed Jul 01 2015

Concept

Unlike fluids, solids do not respond to outside force by flowing or easily compressing. The term elasticity refers to the manner in which solids respond to stress, or the application of force over a given unit area. An understanding of elasticity—a concept that carries with it a rather extensive vocabulary of key terms—helps to illuminate the properties of objects from steel bars to rubber bands to human bones.

How It Works

Characteristics of a Solid

A number of parameters distinguish solids from fluids, a term that in physics includes both gases and liquids. Solids possess a definite volume and a definite shape, whereas gases have neither; liquids have no definite shape.

At the molecular level, particles of solids tend to be precise in their arrangement and close to one another. Liquid molecules are close in proximity (though not as much so as solid molecules), and their arrangement is random, while gas molecules are both random in arrangement and far removed in proximity. Gas molecules are extremely fast-moving, and exert little or no attraction toward one another. Liquid molecules move at moderate speeds and exert a moderate attraction, but solid particles are slow-moving, and have a strong attraction to one another.

One of several factors that distinguishes solids from fluids is their relative response to pressure. Gases tend to be highly compressible, meaning that they respond well to pressure. Liquids tend to be noncompressible, yet because of their fluid characteristics, they experience external pressure uniformly. If one applies pressure to a quantity of water in a closed container, the pressure is equal everywhere in the water. By contrast, if one places a champagne glass upright in a vise and applies pressure until it breaks, chances are that the stem or the base of the glass will be unaffected, because the pressure is not distributed equally throughout the glass.

If the surface of a solid is disturbed, it will resist, and if the force of the disturbance is sufficiently strong, it will deform—for instance, when a steel plate begins to bend under pressure. This deformation will be permanent if the force is powerful enough, as in the above example of the glass in a vise. By contrast, when the surface of a fluid is disturbed, it tends to flow.

Types of Stress

Deformation occurs as a result of stress, whether that stress be in the form of tension, compression, or shear. Tension occurs when equal and opposite forces are exerted along the ends of an object. These operate on the same line of action, but away from each other, thus stretching the object. A perfect example of an object under tension is a rope in the middle of a tug-of-war competition. The adjectival form of "tension" is "tensile": hence the term "tensile stress," which will be discussed later.

Earlier, stress was defined as the application of force over a given unit area, and in fact, the formula for stress can be written as F/A, where F is force and A area. This is also the formula for pressure, though in order for an object to be under pressure, the force must be applied in a direction perpendicular to—and in the same direction as—its surface. The one form of stress that clearly matches these parameters is compression, produced by the action of equal and opposite forces, whose effect is to reduce the length of a material. Thus compression (for example, crushing an aluminum can in one's hand) is both a form of stress and a form of pressure.

Note that compression was defined as reducing length, yet the example given involved a reduction in what most people would call the "width" or diameter of the aluminum can. In fact, width and height are the same as length, for the purposes of most discussions in physics. Length is, along with time, mass, and electric current, one of the fundamental units of measure used to express virtually all other physical quantities. Width and height are simply length expressed in terms of other planes, and within the subject of elasticity, it is not important to distinguish between these varieties of length. (By contrast, when discussing gravitational attraction—which is always vertical—it is obviously necessary to distinguish between "vertical length," or height, and horizontal length.)

The third variety of stress is shear, which occurs when a solid is subjected to equal and opposite forces that do not act along the same line, and which are parallel to the surface area of the object. If a thick hardbound book is lying flat, and a person places a finger on the spine and pushes the front cover away from the spine so that the covers and pages no longer constitute parallel planes, this is an example of shear. Stress resulting from shear is called shearing stress.

Hooke's Law and Elastic Limit

To sum up the three varieties of stress, tension stretches an object, compression shrinks it, and shear twists it. In each case, the object is deformed to some degree. This deformation is expressed in terms of strain, or the ratio between change in dimension and the original dimensions of the object. The formula for strain is δL/L_o, where δL is the change in length (δ, the Greek letter delta, means "change" in scientific notation) and L_o the original length.

Hooke's law, formulated by English physicist Robert Hooke (1635-1703), relates strain to stress. Hooke's law can be stated in simple terms as "the strain is proportional to the stress," and can also be expressed in a formula, F = ks, where F is the applied force, s, the resulting change in dimension, and k, a constant whose value is related to the nature and size of the object under stress. The harder the material, the higher the value of k ; furthermore, the value of k is directly proportional to the object's cross-sectional area or thickness.

The elastic limit of a given solid is the maximum stress to which it can be subjected without experiencing permanent deformation. Elastic limit will be discussed in the context of several examples below; for now, it is important merely to know that Hooke's law is applicable only as long as the material in question has not reached its elastic limit. The same is true for any modulus of elasticity, or the ratio between a particular type of applied stress and the strain that results. (The term "modulus," whose plural is "modu li," is Latin for "small measure.")

Moduli of Elasticity

In cases of tension or compression, the modulus of elasticity is Young's modulus. Named after English physicist Thomas Young (1773-1829), Young's modulus is simply the ratio between F/A and δL/L_o—in other words, stress divided by strain. There are also modu li describing the behavior of objects exposed to shearing stress (shear modulus), and of objects exposed to compressive stress from all sides (bulk modulus).

Shear modulus is the relationship of shearing stress to shearing strain. This can be expressed as the ratio between F/A and φ. The latter symbol, the Greek letter phi, stands for the angle of shear—that is, the angle of deformation along the sides of an object exposed to shearing stress. The greater the amount of surface area A, the less that surface will be displaced by the force F. On the other hand, the greater the amount of force in proportion to A, the greater the value of φ, which measures the strain of an object exposed to shearing stress. (The value of φ, however, will usually be well below 90°, and certainly cannot exceed that magnitude.)

With tensile and compressive stress, A is a surface perpendicular to the direction of applied force, but with shearing stress, A is parallel to F. Consider again the illustration used above, of a thick hardbound book lying flat. As noted, when one pushes the front cover from the side so that the covers and pages no longer constitute parallel planes, this is an example of shear. If one pulled the spine and the long end of the pages away from one another, that would be tensile stress, whereas if one pushed in on the sides of the pages and spine, that would be compressive stress. Shearing stress, by contrast, would stress only the front cover, which is analogous to A for any object under shearing stress.

The third type of elastic modulus is bulk modulus, which occurs when an object is subjected to compression from all sides—that is, volume stress. Bulk modulus is the relationship of volume stress to volume strain, expressed as the ratio between F/A and δV/V_o, where δV is the change in volume and V_o is the original volume.

Real-Life Applications

Elastic and Plastic Deformation

As noted earlier, the elastic limit is the maximum stress to which a given solid can be subjected without experiencing permanent deformation, referred to as plastic deformation. Plastic deformation describes a permanent change in shape or size as a result of stress; by contrast, elastic deformation is only a temporary change in dimension.

A classic example of elastic deformation, and indeed, of highly elastic behavior, is a rubber band: it can be deformed to a length many times its original size, but upon release, it returns to its original shape. Examples of plastic deformation, on the other hand, include the bending of a steel rod under tension or the breaking of a glass under compression. Note that in the case of the steel rod, the object is deformed without rupturing—that is, without breaking or reducing to pieces. The breaking of the glass, however, is obviously an instance of rupturing.

Metals and Elasticity

Metals, in fact, exhibit a number of interesting characteristics with regard to elasticity. With the notable exception of cast iron, metals tend to possess a high degree of ductility, or the ability to be deformed beyond their elastic limits without experiencing rupture. Up to a certain point, the ratio of tension to elongation for metals is high: in other words, a high amount of tension produces only a small amount of elongation. Beyond the elastic limit, however, the ratio is much lower: that is, a relatively small amount of tension produces a high degree of elongation.

Because of their ductility, metals are highly malleable, and, therefore, capable of experiencing mechanical deformation through metallurgical processes, such as forging, rolling, and extrusion. Cold extrusion involves the application of high pressure—that is, a high bulk modulus—to a metal without heating it, and is used on materials such as tin, zinc, and copper to change their shape. Hot extrusion, on the other hand, involves heating a metal to a point of extremely high malleability, and then reshaping it. Metals may also be melted for the purposes of casting, or pouring the molten material into a mold.

Ultimate Strength

The tension that a material can with stand is called its ultimate strength, and due to their ductile properties, most metals possess a high value of ultimate strength. It is possible, however, for a metal to break down due to repeated cycles of stress that are well below the level necessary to rupture it. This occurs, for instance, in metal machines such as automobile engines that experience a high frequency of stress cycles during operation.

The high ultimate strength of metals, both in tension and compression, makes them useful in a number of structural capacities. Steel has an ultimate compressive strength 25 times as great as concrete, and an ultimate tensile strength 250 times as great. For this reason, when concrete is poured for building bridges or other large structures, steel rods are inserted in the concrete. Called "rebar" (for "reinforced bars"), the steel rods have ridges along them in order to bond more firmly with the concrete as it dries. As a result, reinforced concrete has a much greater ability than plain concrete to with stand tension and compression.

Steel Bars and Rubber Bands Under Stress

Crystalline Materials

Metals are crystalline materials, meaning that they are composed of solids called crystals. Particles of crystals are highly ordered, with a definite geometric arrangement repeated in all directions, rather like a honeycomb. (It should be noted, however, that the crystals are not necessarily as uniform in size as the "cells" of the honeycomb.) The atoms of a crystal are arranged in orderly rows, bound to one another by strongly attractive forces that act like microscopic springs.

Just as a spring tends to return to its original length, the highly attractive atoms in a steel bar, when it is stretched, tend to restore it to its original dimensions. Likewise, it takes a great deal of force to pull apart the atoms. When the metal is subjected to plastic deformation, the atoms move to new positions and form new bonds. The atoms are incapable of forming bonds; however, when the metal has been subjected to stress exceeding its ultimate strength, at that point, the metal breaks.

The crystalline structure of metal influences its behavior under high temperatures. Heat causes atoms to vibrate, and in the case of metals, this means that the "springs" are stretching and compressing. As temperature increases, so do the vibrations, thus increasing the average distance between atoms. For this reason, under extremely high temperature, the elastic modulus of the metal decreases, and the metal becomes less resistant to stress.

Polymers and Elastomers

Rubber is so elastic in behavior that in everyday life, the term "elastic" is most often used for objects containing rubber: the waistband on a pair of underwear, for instance. The long, thin molecules of rubber, which are arranged side-by-side, are called "polymers," and the super-elastic polymers in rubber are called "elastomers." The chemical bonds between the atoms in a polymer are flexible, and tend to rotate, producing kinks along the length of the molecule.

When a piece of rubber is subjected to tension, as, for instance, if one pulls a rubber band by the ends, the kinks and loops in the elastomers straighten. Once the stress is released, however, the elastomers immediately return to their original shape. The more "kinky" the polymers, the higher the elastic modulus, and hence, the more capable the item is of stretching and rebounding.

It is interesting to note that steel and rubber, materials that are obviously quite different, are both useful in part for the same reason: their high elastic modulus when subjected to tension, and their strength under stress. But a rubber band exhibits behaviors under high temperatures that are quite different from that of a metal: when heated, rubber contracts. It does so quite suddenly, in fact, suggesting that the added energy of the heat allows the bonds in the elastomers to begin rotating again, thus restoring the kinked shape of the molecules.

Bones

The tensile strength in bone fibers comes from the protein collagen, while the compressive strength is largely due to the presence of inorganic (non-living) salt crystals. It may be hard to believe, but bone actually has an ultimate strength—both in tension and compression—greater than that of concrete!

The ultimate strength of most materials is rendered in factors of 10⁸ N/m₂—that is, 100,000,000 newtons (the metric unit of force) per square meter. For concrete under tensile stress, the ultimate strength is 0.02, whereas for bone, it is 1.3. Under compressive stress, the values are 0.2 and 1.7, respectively. In fact, the ultimate tensile strength of bone is close to that of cast iron (1.7), though the ultimate compressive strength of cast iron (5.5) is much higher than for bone.

Even with these figures, it may be hard to understand how bone can be stronger than concrete, but that is largely because the volume of concrete used in most situations is much greater than the volume of any bone in the body of a human being. By way of explanation, consider a piece of concrete no bigger than a typical bone: under relatively small amounts of stress, it would crumble.

Where to Learn More

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

"Dictionary of Metallurgy" Steelmill.com: The Polish Steel Industry Directory (Web site). <http://www.steelmill.com/DICTIONARY/Diction-ary.htm> (April 9, 2001).

"Engineering Processes." eFunda.com (Web site). <http://www.efunda.com/processes/processes_home/process.cfm> (April 9, 2001).

Gibson, Gary. Making Shapes. Illustrated by Tony Kenyon. Brookfield, CT: Copper Beech Books, 1996.

"Glossary of Materials Testing Terms" (Web site). <http://www.instron.com/apps/glossary> (April 9, 2001).

Goodwin, Peter H. Engineering Projects for Young Scientists. New York: Franklin Watts, 1987.

Johnston, Tom. The Forces with You! Illustrated by Sarah Pooley. Milwaukee, WI: Gareth Stevens Publishing, 1988.

Elasticity is a measure of the responsiveness of one variable to changes in some other variable. For example, advertising elasticity is the relationship between a change in a firm's advertising budget and the resulting change in product sales. Economists are often interested in the price elasticity of demand, which measures the response of the quantity of an item purchased to a change in the item's price. Elasticity measures are reported as a proportional or percent change in the variable being studied. The general formula for elasticity, represented by the letter "E" in the equation below, is:

E percent change in x / percent change in y.

Elasticity can be zero, one, greater than one, less than one, or infinite. When elasticity is equal to one, there is unit elasticity. This means the proportional change in one variable is equal to the proportional change in another variable, or in other words, the two variables are directly related and move together. When elasticity is greater than one, the proportional change in x is greater than the proportional change in y and the situation is said to be elastic.

Inelastic situations result when the proportional change in x is less than the proportional change in y. Perfectly inelastic situations result when any change in y will have an infinite effect in x. Finally, perfectly elastic situations result when any change in y will result in no change in x. A special case known as unitary elasticity of demand occurs if total revenue stays the same when prices change.

Elasticity for Managerial Decision Making

Economists compute several different elasticity measures, including the price elasticity of demand, the price elasticity of supply, and the income elasticity of demand. Elasticity is typically defined in terms of changes in total revenue since that is of primary importance to managers, CEOs, and marketers. For managers, a key point in the discussions of demand is what happens when they raise prices for their products and services. It is important to know the extent to which a percentage increase in unit price will affect the demand for a product. With elastic demand, total revenue will decrease if the price is raised. With inelastic demand, however, total revenue will increase if the price is raised.

The possibility of raising prices and increasing dollar sales (total revenue) at the same time is very attractive to managers. This occurs only if the demand curve is inelastic. Here total revenue will increase if the price is raised, but total costs probably will not increase and, in fact, could go down. Since profit is equal to total revenue minus total costs, profit will increase as price is increased when demand for a product is inelastic. It is important to note that an entire demand cure is neither elastic or inelastic; it only has the particular condition for a change in total revenue between two points on the curve (and not along the whole curve).

Demand elasticity is affected by the availability of substitutes, the urgency of need, and the importance of the item in the customer's budget. Substitutes are products that offer the buyer a choice. For example, many consumers see corn chips as a good or homogeneous substitute for potato chips, or see sliced ham as a substitute for sliced turkey. The more substitutes available, the greater will be the elasticity of demand. If consumers see products as extremely different or heterogeneous, however, then a particular need cannot easily be satisfied by substitutes. In contrast to a product with many substitutes, a product with few or no substitutes—like gasoline—will have an inelastic demand curve. Similarly, demand for products that are urgently needed or are very important to a person's budget will tend to be inelastic. It is important for managers to understand the price elasticity of their products and services in order to set prices appropriately to maximize firm profits and revenues.