HOOKE'S LAW AND ELASTIC LIMIT
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/Lo, where δL is the change in length (δ, the Greek letter delta, means "change" in scientific notation) and Lo 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/Lo—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
THE MACHINE PICTURED HERE ROLLS OVER STEEL IN ORDER TO BEND IT INTO PIPES. BECAUSE OF ITS ELASTIC NATURE, STEEL CAN BE BENT WITHOUT BREAKING. (Photograph by Vince Streano/Corbis. Reproduced by permission.)
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/Vo, where δV is the change in volume and Vo 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
RUBBER BANDS, LIKE THE ONES SHOWN HERE FORMED INTO A BALL, ARE A CLASSIC EXAMPLE OF ELASTIC DEFORMATION. (Photograph by Klein/Corbis. Reproduced by permission.)
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
A HUMAN BONE HAS A GREATER "ULTIMATE STRENGTH" THAN THAT OF CONCRETE. (Ecoscene/Corbis. Reproduced by permission.)
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.
KEY TERMS
ANGLE OF SHEAR:
The angle of deformation on the sides of an object exposed to shearing stress. Its symbol is φ (the Greek letter phi), and its value will usually be well below 90°.
BULK MODULUS:
The modulus of elasticity for a material subjected to compression on all surfaces—that is, volume stress. Bulk modulus is the relationship of volume stress to volume strain, expressed as the ratio between F/A and dV/Vo, wheredV is the change in volume and Vo is the original volume.
COMPRESSION:
A form of stress produced by the action of equal and opposite forces, whose effect is to reduce the length of a material. Compression is a form of pressure. When compressive stress is applied to all surfaces of a material, this is known as volume stress.
DUCTILITY:
A property whereby a material is capable of being deformed far beyond its elastic limit without experiencing rupture—that is, without breaking. Most metals other than cast iron are highly ductile.
ELASTIC DEFORMATION:
A temporary change in shape or size experienced by a solid subjected to stress. Elastic deformation is thus less severe than plastic deformation.
ELASTIC LIMIT:
The maximum stress to which a given solid can be subjected without experiencing plastic deformation—that is, without being permanently deformed.
ELASTICITY:
The response of solids to stress.
HOOKE'S LAW:
A principle of elasticity formulated by English physicist Robert Hooke (1635-1703), who discovered that strain is proportional to stress. Hooke's lawcan be written as 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 being subjected to stress. Hooke's law applies only when the elastic limit has not been exceeded.
LENGTH:
In discussions of elasticity, "length" refers to an object's dimensions on any given plane, thus, it can be used not only to refer to what is called length in everyday language, but also to width or height.
MODULUS OF ELASTICITY:
The ratio between a type of applied stress (that is, tension, compression, and shear) and the strain that results in the object to which stress has been applied. Elastic modu li—including Young's modulus, shearing modulus, and bulk modulus—are applicable only as long as the object's elastic limit has not been reached.
PLASTIC DEFORMATION:
A permanent change in shape or size experienced by a solid subjected to stress. Plastic deformation is thus more severe than elasticdeformation.
PRESSURE:
The ratio of force to surface area, when force is applied in a direction perpendicular to, and in the same direction as, that surface.
SHEAR:
A form of stress resulting from equal and opposite forces that do not act along the same line. If a thick hard-bound book is lying flat, and 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.
SHEAR MODULUS:
The modulus of elasticity for an object exposed to shearing stress. It is expressed as the ratio between F/A and φ, where φ (the Greek letter phi) stands for the angle of shear.
STRAIN:
The ratio between the change in dimension experienced by an object that has been subjected to stress, and the original dimensions of the object. The formula for strain is dL/Lo, where dL is the change in length and Lo the original length. Hooke's law, as well as the various modu li of elasticity, relates strain to stress.
STRESS:
In general terms, stress is any attempt to deform a solid. Types of stress include tension, compression, and shear. More specifically, stress is the ratio of force to unit area, F/A, where F is force and A area. Thus, it is similar to pressure, and indeed, compression is a form of pressure.
TENSION:
A form of stress produced by a force which acts to stretch a material. The adjectival form of "tension" is "tensile": hence the terms "tensile stress" and "tensile strain."
ULTIMATE STRENGTH:
The tension that a material can with stand without rupturing. Due to their high levels of ductility, most metals have a high value of ultimate strength.
VOLUME STRESS:
The stress that occurs in a material when it is subjected to compression from all sides. The modus of elasticity for volume stress is the bulk modulus.
YOUNG'S MODULUS:
A modulus of elasticity describing the relationship between stress to strain for objects under either tension or compression. Named after English physicist Thomas Young (1773-1829), Young's modulus is simply the ratio between F/A and δL/Lo—in other words, stress divided by strain.
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