Elasticity - 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 δ L / L o , where δ L 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 δ L / 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

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.)
T HE MACHINE PICTURED HERE ROLLS OVER STEEL IN ORDER TO BEND IT INTO PIPES . B ECAUSE 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 δ V / V o , where δ V is the change in volume and V o is the original volume.

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