Elasticity
Elasticity is the ability of a material to return to its original shape and size after being stretched, compressed, twisted or bent. Elastic deformation (change of shape or size) lasts only as long as a deforming force is applied to the object, and disappears once the force is removed. Greater forces may cause permanent changes of shape or size, called plastic deformation.
In ordinary language, a substance is said to be "elastic" if it stretches easily. Therefore, rubber is considered a very elastic substance, and rubber bands are even called "elastics" by some people. Actually, however, most substances are somewhat elastic, including steel, glass, and other familiar materials.
The simplest description of elasticity is Hooke's Law, which states, "The stress is proportional to the strain." This relation was first expressed by the British scientist, Robert Hooke. He arrived at it through studies in which he placed weights on metal springs and measured how far the springs stretched in response. Hooke noted that the added length was always proportional to the weight; that is, doubling the weight doubled the added length.
In the modern statement of Hooke's law, the terms "stress" and "strain" have precise mathematical definitions. Stress is the applied force divided by the area the force acts on. Strain is the added length divided by the original length.
To understand why these special definitions are needed, first consider two bars of the same length, made of the same material. One bar is twice as thick as the other. Experiments have shown that both bars can be stretched to the same additional length only if twice as much weight is placed on the bar that is twice as thick. Thus, they both carry the same stress, as defined above.
The special definition of strain is required because, when an object is stretched, the stretch occurs along its entire length, not just at the end to which the weight is applied. The same stress applied to a long rod and a short rod will cause a greater extension of the long rod. The strain, however, will be the same on both rods.
The amount of stress required to produce a given amount of strain also depends on the material being stretched. Therefore, the ratio of stress to strain is a unique property of materials, different for each substance. It is called the elastic modulus (plural: moduli). It is also known as Young's modulus, after Thomas Young who first described it. It has been measured for thousands of materials. The greater the elastic modulus, the stiffer the material is. For example, the elastic modulus of rubber is about 600 psi (pounds per square inch). That of steel is about 30 million psi.
All deformations, no matter how complicated, can be described as the result of combinations of three basic types of stress. One is tension, which stretches an object along one direction only. Thus far, our discussion of elasticity has been entirely in terms of tension. Compression is the same type of stress, but acting in the opposite direction.
The second basic type of stress is shear stress. This results when two forces push on opposite ends of an object in opposite directions. Shear stress changes the object's shape. The shear modulus is the amount of shear stress divided by the angle through which the shape is strained.
Hydrostatic stress, the third basic stress, squeezes an object with equal force from all directions. A familiar example is the pressure on objects under water due to the weight of the water above them. Pure hydrostatic stress changes the volume only, not the shape of the object. Its modulus is called the bulk modulus.
The greatest stress a material can undergo and still return to its original dimensions is called the elastic limit. When stressed beyond the elastic limit, some materials fracture, or break. Others undergo plastic deformation, taking on a new permanent shape. An example is a nail bent by excessive shear stress of a hammer blow.
The elastic modulus and elastic limit reveal much about the strength of the bonds between the smallest particles of a substance, the atoms or molecules it is composed of. However, to understand elastic behavior on the level of atoms requires first distinguishing between materials that are crystalline and those that are not.
Metals are examples of crystalline materials. Solid pieces of metal contain millions of microscopically small crystals stuck together, often in random orientations. Within a single crystal, atoms are arranged in orderly rows. They are held by attractive forces on all sides. Scientists model the attractive force as a sort of a spring. When a spring is stretched, a restoring force tries to return it to its original length. When a metal rod is stretched in tension, its atoms are pulled apart slightly. The attractive force between the atoms tries to restore the original distance. The stronger the attraction, the more force must be applied to pull the atoms apart. Thus, stronger atomic forces result in larger elastic modulus.
Stresses greater than the elastic limit overcome the forces holding atoms in place. The atoms move to new positions. If they can form new bonds there, the material deforms plastically; that is, it remains in one piece but assumes a new shape. If new bonds cannot form, the material fractures.
The ball and spring model also explains why metals and other crystalline materials soften at higher temperatures. Heat energy causes atoms to vibrate. Their vibrations move them back and forth, stretching and compressing the spring. The higher the temperature, the larger the vibrations, and the greater the average distance between atoms. Less applied force is needed to separate the atoms because some of the stretching energy has been provided by the heat. The result is that the elastic modulus of metals decreases as temperature increases.
To explain the elastic behavior of materials like rubber requires a different model. Rubber consists of molecules, which are clusters of atoms joined by chemical bonds. Rubber molecules are very long and thin. They are polymers, long chain-like molecules built up by repeating small units. Rubber polymers consist of hundreds or thousands of atoms joined in a line. Many of the bonds are flexible, and can rotate. The result is a fine structure of kinks along the length of the molecule. The molecule itself is so long that it tends to bend and coil randomly, like a rope dropped on the ground. A piece of rubber, such as a rubber band, is made of vast numbers of such kinked, twisting, rope-like molecules.
When rubber is pulled, the first thing that happens is that the loops and coils of the "ropes" straighten out. The rubber extends as its molecules are pulled out to their full length. Still more stress causes the kinks to straighten out. Releasing the stress allows the kinks, coils, and loops to form again, and the rubber returns to its original dimensions. Materials made of long, tangled molecules stretch very easily. Their elastic modulus is very small. They are called elastomers because they are very "elastic" polymers.
The "kink" model explains a very unusual property of rubber. A stretched rubber band, when heated, will suddenly contract. It is thought that the added heat provides enough energy for the bonds to start rotating again. The kinks that had been stretched out of the material return to it, causing the length to contract.
Elasticity is involved whenever atoms vibrate. An example is the movement of sound waves. A sound wave consists of energy that pushes atoms closer together momentarily. The energy moves through the atoms, causing the region of compression to move forward. Behind it, the atoms spring further apart, as a result of the restoring force.
The speed with which sound travels through a substance depends in part on the strength of the forces between atoms of the substance. Strongly bound atoms readily affect one another, transferring the "push" due to the sound wave from each atom to its neighbor. Therefore, the stronger the bonding force, the faster sound travels through an object. This explains why it is possible to hear an approaching railroad train by putting one's ear to the track, long before it can be heard through the air. The sound wave travels more rapidly through the steel of the track than through the air, because the elastic modulus of steel is a million times greater than the bulk modulus of air.
The most direct way to determine the elastic modulus of a material is by placing a sample under increasing stresses, and measuring the resulting strains. The results are plotted as a graph, with strain along the horizontal axis and stress along the vertical axis. As long as the strain is small, the data form a straight line for most materials. This straight line is the "elastic region." The slope of the straight line equals the elastic modulus of the material. Alternatively, the elastic modulus can be calculated from measurements of the speed of sound through a sample of the material.
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