On Ductility

The benefit of elastoplastic materials

One of the most useful structural properties a material can have is ductility. In order to understand why this is so it is first necessary to explain some related concepts. We will start with stress and strain.

Stress is a measure of load intensity; more precisely it is the ratio of force divided by area. This concept can be visualised by considering how it is that someone can lie down on a bed of nails without being harmed, yet if the same person were to tread on a single nail it would pierce their foot. The reason a bed of nails causes no harm is because the person’s weight is spread over the cumulative area of many nails whereas a single nail concentrates a person’s weight over a very small area i.e. the bed of nails applies low stress, while the single nail applies high stress.

When a structural member is exposed to stress it will change in length. If the stress is tensile it will become longer and if it is compressive the reverse is true. If the material is uniform then the elongation or shortening is spread evenly along the length of the member. For example, a member that is half as long will change in length by half as much. The ratio of elongation or shortening per unit length is defined as strain.

Stress and strain are of course related; higher stress will cause higher strain. A useful way of expressing this relationship is to plot a stress-strain graph. The convention is to measure stress on the vertical axis and strain on the horizontal axis.

If a material is ductile, like steel, the relation of stress to strain will be directly proportional resulting in a straight line on our graph [1]. The inclination of the straight line is know as the elastic modulus, for reasons we will see shortly.

There will, however, come a point where strain will increase more quickly than stress and the graph is no longer a straight line. The proportional limit of the material has thus been breached. Soon after this point the graph will become horizontal, which means that strain will increase without a corresponding increase in stress. This threshold is known as the yield stress.

Beyond yield the internal structure of the material will begin to change at an atomic level. This process is known as strain hardening and it results in additional strength, and therefore higher stress, in return for further strains.

Eventually the stress-strain curve will again flatten leading to increased strains, but this time with decreasing stress. The point where this occurs is know as the ultimate stress.

Increased strain accompanied by decreasing stress is a rather curious effect. Why would increasing strain result from a reduced stress? There is of course a rational answer to this question. 

When any material is stretched there is a corresponding narrowing to facilitate the stretch. Generally this is too small to notice, however beyond the point at which the stress-strain curve flattens the effect becomes more pronounced and ‘necking’ occurs.

If we were to calculate true stress, based on a necked [reduced] cross-section, rather than continuing with a nominal stress, based on the original cross-section, then the stress-strain plot would in fact continue to grow. After a period of necking fracture will eventually occur.

Returning to the beginning of our stress-strain plot we can modify our approach. This time instead of continually increasing the stress applied to our test member we can load it to a known stress and then unload it again. We can then increase the stress to a slightly higher value and then unload again. This sequence of loading and unloading may be repeated.

When we do this we find that as long as we remain below the proportional limit the unloaded member will fully recover its original shape and will follow the straight line portion of the stress-strain graph. Beyond the proportional limit a full recovery will not be made; there will be some residual strain left behind.

When the member fully recovers its shape it is said to be elastic. The point at which it loses its elasticity is known as the elastic limit. Beyond the elastic limit materials are considered to be plastic. For many materials, like steel. The proportional limit, yield stress and elastic limit are relatively closer together and are in practise treated as if they were the same. For some materials, such as rubber, the elastic limit lies well beyond the proportional limit.

When large permanent strains start to occur within the plastic zone the term plastic flow is adopted. 

There are several reasons why ductility, incorporating both elastic and plastic behaviour, is important. We shall discuss one of them below and save another for the next post.

Supposing we wished to design a series of floor beams for a new building. The owners would not be terribly happy if they were to permanently deform and sag while the building was in use. For this reason we would want them to remain within the elastic range. We would therefore perform our design based on limiting stresses in the beams to the yield stress of the material.

This would satisfy the requirements for every day use of the building, however supposing an unforeseen or extreme event occurred, which caused the floor to become overloaded. In such circumstances you would not wish the floor beams to fail suddenly and without warning, for this would inevitably lead to a loss of life. 

Self-evidently it would be preferable for there to be a visible warning that something was wrong so that the occupants could vacate the floor safely and remedial action could be taken. This opportunity is provided by plastic behaviour in the floor beams. Although permanent deformation will occur within the plastic range the floor will at least remain stable and safe.

It is fairly obvious how this principle will work if the floor beams are made of steel, but perhaps less so for concrete. After all concrete is not a ductile material. It is weak in tension and fails explosively in compression.

The first problem is solved by casting steel reinforcement into the concrete in zones where the concrete is in tension. The reinforcing bars, rather than the concrete, resist the applied tensile stresses.

The addition of steel reinforcing bars also provides the means to deal with concrete’s lack of ductility. The reinforcement is deliberately designed to have a smaller capacity in tension than the concrete has in compression. This way as the reinforced concrete bends the reinforcing bars in the tension zones will reach their elastic limit before the concrete reaches its brittle limit in the compression zones. This principle is known as under-reinforcement and it is key to all reinforced concrete design. 

Thus reinforced concrete becomes a ductile material.

[1] In this example we are of course assuming the application of tensile stress