Changes in Length—Tension and Compression: Elastic Modulus

Changes in Length—Tension and Compression: Elastic Modulus

A change in length \(\Delta L\) is produced when a force is applied to a wire or rod parallel to its length \({L}_{0}\), either stretching it (a tension) or compressing it. (See the figure below.)

(a) Tension. The rod is stretched a length \(\Delta L\) when a force is applied parallel to its length. (b) Compression. The same rod is compressed by forces with the same magnitude in the opposite direction. For very small deformations and uniform materials, \(\Delta L\) is approximately the same for the same magnitude of tension or compression. For larger deformations, the cross-sectional area changes as the rod is compressed or stretched.

Experiments have shown that the change in length (\(\Delta L\)) depends on only a few variables. As already noted, \(\Delta L\) is proportional to the force \(F\) and depends on the substance from which the object is made. Additionally, the change in length is proportional to the original length \({L}_{0}\) and inversely proportional to the cross-sectional area of the wire or rod. For example, a long guitar string will stretch more than a short one, and a thick string will stretch less than a thin one. We can combine all these factors into one equation for \(\Delta L\):

where \(\Delta L\) is the change in length, \(F\) the applied force, \(Y\) is a factor, called the elastic modulus or Young’s modulus, that depends on the substance, \(A\) is the cross-sectional area, and \({L}_{0}\) is the original length. The table below lists values of \(Y\) for several materials—those with a large \(Y\) are said to have a large tensile stifness because they deform less for a given tension or compression.

Elastic Moduli

Young’s moduli are not listed for liquids and gases in the table above because they cannot be stretched or compressed in only one direction. Note that there is an assumption that the object does not accelerate, so that there are actually two applied forces of magnitude \(F\) acting in opposite directions. For example, the strings in the figure above are being pulled down by a force of magnitude \(w\) and held up by the ceiling, which also exerts a force of magnitude \(w\).

Example: The Stretch of a Long Cable

Suspension cables are used to carry gondolas at ski resorts. (See the figure below) Consider a suspension cable that includes an unsupported span of 3020 m. Calculate the amount of stretch in the steel cable. Assume that the cable has a diameter of 5.6 cm and the maximum tension it can withstand is \(3\text{.}0×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{N}\).

Gondolas travel along suspension cables at the Gala Yuzawa ski resort in Japan. (credit: Rudy Herman, Flickr)

Strategy

The force is equal to the maximum tension, or \(F=3\text{.}0×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{N}\). The cross-sectional area is \({\mathrm{\pi r}}^{2}=2\text{.}\text{46}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}\). The equation \(\Delta L=\frac{1}{Y}\frac{F}{A}{L}_{0}\) can be used to find the change in length.

Solution

All quantities are known. Thus,

\(\begin{array}{lll}\Delta L& =& \left(\frac{1}{\text{210}×{\text{10}}^{9}\phantom{\rule{0.25em}{0ex}}{\text{N/m}}^{2}}\right)\left(\frac{3\text{.}0×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{N}}{2.46×{10}^{–3}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}}\right)\left(\text{3020 m}\right)\\ & =& \text{18 m}.\end{array}\)

Discussion

This is quite a stretch, but only about 0.6% of the unsupported length. Effects of temperature upon length might be important in these environments.

Bones, on the whole, do not fracture due to tension or compression. Rather they generally fracture due to sideways impact or bending, resulting in the bone shearing or snapping. The behavior of bones under tension and compression is important because it determines the load the bones can carry. Bones are classified as weight-bearing structures such as columns in buildings and trees.

Weight-bearing structures have special features; columns in building have steel-reinforcing rods while trees and bones are fibrous. The bones in different parts of the body serve different structural functions and are prone to different stresses. Thus the bone in the top of the femur is arranged in thin sheets separated by marrow while in other places the bones can be cylindrical and filled with marrow or just solid. Overweight people have a tendency toward bone damage due to sustained compressions in bone joints and tendons.

Another biological example of Hooke’s law occurs in tendons. Functionally, the tendon (the tissue connecting muscle to bone) must stretch easily at first when a force is applied, but offer a much greater restoring force for a greater strain. The figure below shows a stress-strain relationship for a human tendon. Some tendons have a high collagen content so there is relatively little strain, or length change; others, like support tendons (as in the leg) can change length up to 10%.

Note that this stress-strain curve is nonlinear, since the slope of the line changes in different regions. In the first part of the stretch called the toe region, the fibers in the tendon begin to align in the direction of the stress—this is called uncrimping. In the linear region, the fibrils will be stretched, and in the failure region individual fibers begin to break.

A simple model of this relationship can be illustrated by springs in parallel: different springs are activated at different lengths of stretch. Examples of this are given in the problems at end of this tutorial. Ligaments (tissue connecting bone to bone) behave in a similar way.

Typical stress-strain curve for mammalian tendon. Three regions are shown: (1) toe region (2) linear region, and (3) failure region.

Unlike bones and tendons, which need to be strong as well as elastic, the arteries and lungs need to be very stretchable. The elastic properties of the arteries are essential for blood flow. The pressure in the arteries increases and arterial walls stretch when the blood is pumped out of the heart. When the aortic valve shuts, the pressure in the arteries drops and the arterial walls relax to maintain the blood flow.

When you feel your pulse, you are feeling exactly this—the elastic behavior of the arteries as the blood gushes through with each pump of the heart. If the arteries were rigid, you would not feel a pulse. The heart is also an organ with special elastic properties. The lungs expand with muscular effort when we breathe in but relax freely and elastically when we breathe out. Our skins are particularly elastic, especially for the young. A young person can go from 100 kg to 60 kg with no visible sag in their skins. The elasticity of all organs reduces with age. Gradual physiological aging through reduction in elasticity starts in the early 20s.

The equation for change in length is traditionally rearranged and written in the following form:

The ratio of force to area, \(\frac{F}{A}\), is defined as stress (measured in\({\text{N/m}}^{\text{2}}\)), and the ratio of the change in length to length, \(\frac{\Delta L}{{L}_{0}}\), is defined as strain (a unitless quantity). In other words,

In this form, the equation is analogous to Hooke’s law, with stress analogous to force and strain analogous to deformation. If we again rearrange this equation to the form

we see that it is the same as Hooke’s law with a proportionality constant

This general idea—that force and the deformation it causes are proportional for small deformations—applies to changes in length, sideways bending, and changes in volume.