Stress-Strain Curves

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We had earlier introduced the idea of Stresses and Strains. Previously, we were working with a force-displacement curve. But remember, we ditched the load-displacement because it was not really useful for comparing two or more materials. ¹

Concept of elasticity

Let’s come back to our rod pulling experiment. We take a metal rod and start pulling it from one end while clamping the other end. After pulling for some distance (1mm maybe) we release the pulling force. This is when we notice that, in some time after we release the load and stop “pulling”, the rod returns to its original length.

We repeat this exercise a couple of times and each time we pull the rod a little more (say in increments of 1mm i.e 1mm, 2mm, 3mm etc.). Now what happens is, as we keep increasing the distance of the pull (1mm, 2mm, 3mm etc.) beyond a certain distance, the rod stops returning to its original position.

The more we pull, the higher is the load required because of stresses that develop in the rod. So we could also say that beyond a certain load, the rod stops returning to its original position.

So at the values less than this critical value, we say the rod deforms elastically (i.e it is NOT permanently deformed) while beyond this limit we say the rod deforms plastically (i.e it IS permanently deformed).

Hooke’s Law

It was observed from experiments that, within this elastic limit, the average stress is proportional to the average strain. In such a case the material is said to be following Hooke’s Law. ²

Thus, according to Hooke’s Law we can say

$\sigma = e \times E$

The constant E is the modulus of elasticity, or Young’s modulus, $\sigma$ is the stress and e is the strain. If we plot the strain on x axis with the corresponding stress on y axis with we get a stress-strain curve.

Interpreting a typical stress-strain curve

This is what a stress-strain curve looks like.

The initial linear portion of the curve is the elastic region within which Hooke’s law is obeyed (the region from origin to point 2).

Point 2 is the elastic limit, defined as the greatest stress that the metal can withstand without experiencing a permanent strain when the load is removed. The slope of the stress-strain curve in this region is the modulus of elasticity.

If you look at the point 2 in the diagram, you can see the change of nature from elastic to plastic deformation. Before point 2 the curve is linear (indicating elasticity) and beyond point 2 it is non-linear (indicating plastic behaviour). This point is called the yield point.³

Therefore, prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible.

The determination of the yield point is quite tedious, and dependent on the sensitivity of the strain-measuring instrument. For engineering purposes ⁴ the limit of usable elastic behaviour is described by the yield strength.

The yield strength is defined as the stress which will produce a small amount of permanent deformation, generally a strain equal to 0.2 per cent or 0.002 inches per inch.

So now if you want to compare two materials to see which will easily deform non-linearly, just compare their yield strength values. The lower the value, the easier it will get deformed plastically.

Plastic deformation begins when the elastic limit is exceeded. As the plastic deformation of the specimen increases, the load required to extend the specimen increases with further straining. Eventually the load reaches a maximum value (point 1). If you divide this maximum load by the original area of the specimen you get something that is called the ultimate tensile strength.

The diameter of the specimen begins to decrease rapidly beyond maximum load, so that the load required to continue deformation drops off until the specimen fractures. Since the average stress is based on the original area of the specimen, it also decreases from maximum load to fracture.

Ok, so now you know about these different points on the typical stress-strain curve. But why are we even interested in them? Simply because they are a means for us to compare the behaviour of different materials.

If we plot the stress-strain curves for different materials, we begin to see a pattern. Some materials have a curve like we just saw. Others may just have a linear section and then the sample breaks. Or they may have a small curve at the highest point and then break.

The general behaviour of materials under load can be classified as ductile or brittle depending upon whether or not the material exhibits the ability to undergo plastic deformation.

A completely brittle material would fracture almost at the elastic limit while a brittle metal, such as white cast iron, shows some slight measure of plasticity before fracture.

It is important to note here that brittleness is not an absolute property of a metal.

A metal such as tungsten, which is brittle at room temperature, is ductile at an elevated temperature. A metal which is brittle in tension may be ductile under hydrostatic compression.

A metal which is ductile in tension at room temperature can become brittle in the presence of notches, low temperature, high rates of loading, or embrittling agents such as hydrogen. We will look at this aspect in detail later, but the thing to remember here is that brittleness or ductility is not an absolute property of a metal

Another area of interest is what the area under these curves represent.

If we look at the area from 0 to point 2 limit in the graph we can call it the area of the elastic region. This area represents the work done on a unit volume of material, as a simple tensile force is gradually increased from zero to such a value that the elastic limit (i.e. yield point) of the material is reached.

This area is defined as the modulus of resilience.

Next if we look at the area from 0 to fracture limit it represents the work done on a unit volume of material as a simple tensile force is gradually increased from zero to the value causing rupture.

This area is defined as the modulus of toughness.

Footnotes

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