stress to strain. It differs by material, and depends upon what kind of stress is applied.

Values for modulus of elasticity apply only for a range of stress values. Here, we specifically use changes in length to illustrate this point. If you

stress an object beyond its material’s proportionality limit, the strain ceases to be linearly proportional to stress. A “mild steel” rod will exhibit a

linear relation between stress and strain for a stress on it up to about 230,000,000 N/m^2. This is its proportionality limit.

Beyond a further point called the elastic limit, the object becomes permanently deformed and will not return to its original shape when the

stress ceases to be applied. You have exceeded its yield strength. At some point, you pull so far it ruptures. At that rupture point, you have

exceeded the material’s ultimate strength.

Concept 2 shows a stress-strain graph typical of a material like soft steel (a type of steel that is easily cut or bent) or copper. Graphs like these

are commonly used by engineers. They are created with testing machines similar to the one above by carefully applying force to a material

over time. The graph plots the minimum stress required to achieve a certain amount of strain, which in this case is measured as the

lengthening of a rod of the material. (We say “minimum stress” because the strain may vary depending on whether a given amount of stress is

applied suddenly or is slowly increased to the same value.)

You may notice that the graph has an interesting property: Near the end, the curve flattens and its slope decreases. Less stress is required to

generate a certain strain. The material has become ductile, or stretchy like taffy, and it is easier to stretch than it was before. Soft steel and

copper are ductile. Other materials will reach the rupture point without becoming stretchy; they are called brittle. Concrete and glass are two

brittle materials, and hardened steel is more brittle than soft steel.

11.7 - Tensile stress

Tensile stress: A stress that stretches.

On the right, you see a rod being stretched. Tensile forces cause materials to lengthen.

Tensile stress is the force per cross-sectional unit area of the object. Here, the cross-

sectional area equals the surface area of the end of the rod. The strain is measured as

the rod’s change in length divided by its initial length.

Young’s modulus equals tensile stress divided by strain. The letter Y denotes Young’s

modulus, which is measured in units of newtons per square meter. The value for

Young’s modulus for various materials is shown in Concept 2. Note the scale used in

the table: billions of newtons per square meter.

To correctly apply the equation in Equation 1 on the right, you must be careful with the

definitions of stress and strain. First, stress is force per unit area. Second, strain is the

fractional change in size. The relevant area for a rod in calculating tensile stress is the

area of its end, as the diagrams on the right reflect.

The equation on the right can be used for compression as well as expansion. When a

material is compressed, ǻL is the decrease in length. For some materials, Young’s

modulus is roughly the same for compressing (shortening) as for stretching, so you can

use the same modulus when a compressive force is applied.

In addition to supplying the values for Young’s modulus, we supply values for the yield

strength (elastic limit) of a few of the materials. The table can give you a sense of why

certain materials are used in certain settings. Steel, for example, has both a high

Young’s modulus and high yield strength. This means it requires a lot of stress to

stretch steel elastically, as well as a lot of stress to deform it permanently.

Some materials have different yield strengths for compression and tension. Bone, for

instance, resists compressive forces better than tensile forces. The value listed in the

table is for compressive forces.

Tensile stress

Causes stretching/compression along a

line

Stress: force per cross-section unit area

Strain: fractional change in length

Young's modulus

Relates stress, strain

(^210) Copyright 2000-2007 Kinetic Books Co. Chapter 11