STRESS AND ELASTIC CONSTANTS

John W. Dooley, Physics Department, Millersville University

DEFINITION OF STRESS

We begin by thinking of stress as the cause of strain; stress is the perturbation and strain is the response. In practice, we strain an object by applying a pair (or multiple pairs) of forces to it. A pair of forces is arranged so that the net force, by the pair on the object, is zero. This means that, once the object has distorted into its new (strained) shape, the acceleration of the object is zero.

Just as the displacement of the edges of a sample was not the preferred way to describe strain, the values of the forces applied to the sample are not preferred as a measure of stress. We defined strain using a fractional change in length of the sample. We will define stress associated with a force on the sample as the force, divided by the area to which the force is applied.

The virtue of this choice is that it gives a measure of stress that is independent of the boundaries of our sample. The force per unit area is intensive; characterizing the internal state of the material in the sample.

INTENSIVE AND EXTENSIVE PARAMETERS

An extensive parameter depends on the size of the sample that the parameter describes. Mass is an extensive parameter. An intensive parameter describes the material character of the sample without giving information about the size of the sample. Density is an intensive parameter.

The displacement of the edges of a strained sample is an extensive parameter. Strain of a sample is an intensive parameter. Force is an extensive parameter, while stress is an intensive parameter.

Here is a test that helps to discover if a particular parameter describing a sample object is extensive: Make an identical second object. Join it to the original to make an object that is twice the size of the original. (We call this "doubling" the object.) If the parameter has doubled as a result, it is an extensive parameter.

Before comparing force and stress, we return to the familiar parameters of length and strain, in a sample undergoing tensile strain.

Doubling a strained system.

The figure demonstrates the extensive character of the length of a strained sample. When the sample is doubled as shown, the length of the sample doubles.

On the other hand, the fractional change in length, The right half of the doubled sample is displaced to the right in two ways:

First, it has a displacement that is largest on its right face, because it is stretched. Second, it is shifted to the right by the stretch of the original block on its left.

Doubling a strained system.

A second way of doubling the system is sketched in the figure. In this case the length of the system does not change. As before, the fractional change in length and the strain remain the same.

An advantage of the intensive parameter (the strain) is that it does not change when the system is doubled. The extensive parameter (length) not only is sensitive to doubling the system; it is also sensitive to how the system is doubled.

We will often use extensive parameters as a starting point, since they are typically the most easy to measure. For understanding the distortions of the system, and comparing them to other systems, we will typically convert to the intensive parameters.

We are now prepared to consider the forces required to make the strain. We have an intensive parameter, the strain, to represent the response of a system to a perturbation. We now seek an intensive representation of the perturbation. The force is an obvious first candidate, but we shall see that the force is an extensive parameter:

Forces to double a strained system.

The figure shows the pair of forces used to cause a tensile strain. (We neglect the side effect.) The system is the same as the first one used to discuss tensile strain, except that some notations have been removed for clarity.

The two forces have equal magnitude and opposite directions; not because they are an "action-reaction" pair, but because they must add to zero: We wish the block to be strained, but not to accelerate.

We double the strained system, as shown. Block A is the original block, and B is the block added on to double the system. The strain in each block is the same. The force required to produce that strain in block A is the same as before.

The force on the right face of A is now provided by B. Newton's third law tells us that there is an equal and opposite force by A on the left face of B. The second half of the force pair on B is provided by us, on the right face of B.

Since B is identical to A, and all forces here have the same magnitude, the strain in B is the same as the strain in A.

The two forces we exert; on the left face of A, and on the right face of B are the same as the forces we exerted to strain block A alone. The force required to produce the change has not doubled, for this method of doubling the system.

Forces to double a strained system.

Now we double the system in a second way. In this case we need a force pair to stretch A, and another force pair to stretch B. To achieve the same strain, the net force on the right of the doubled system is twice the force for A alone.

For this method of doubling the system, we were required to double the force. The force required to produce a given distortion is not only extensive, it depends on the method of doubling the system. Force is unsatisfactory as a measure of the perturbation.

The intensive parameter we need will remain the same no matter how we choose to double the system. It turns out that we can achieve this if we divide the force applied to a block face, by the area of the face. In the first method of doubling, both the area and the force remain the same, as does their ratio. In the second method, both the area and the force are doubled, so their ratio remains the same.

Result: For a given strain, the ratio of force to area is constant, independent of the method of doubling the sample.

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