MECHANICAL PROPERTIES OF MATERIALS

Contents for this page	Related topics
Introduction Types of forces acting on materials Hooke's law Plasticity, creep and fracture Microscopic view of material strength Additional questions		Data Glossary
Learning Outcomes
After studying this section, you will (a) be familiar with the terms "elasticity" and "plasticity", (b) understand and be able to apply Hooke's Law, and (c) know what is meant by the term "strength of materials"

Introduction:

Above left: Parthenon, Athens (credit: Kallistos), Center: Burj Khalifa, Dubai (credit: Poco a poco), Right: Akashi-Kaikyo Bridge (credit: Hamilton)

A major question that confronts engineers and architects is whether the materials that they are considering using are fit for the purpose to which they are intended. No material lasts for ever or is always completely dependable when subjected to various forces. The problem is to choose a material whose strength is such that it will withstand such failures as creep, fracture and metal fatigue indefinitely, and not deform within certain permissible limits, when subjected to the sort of forces that are likely to be applied to the final product. This, in short, is the area of materials science known as STRENGTH OF MATERIALS.

In antiquity, large structures were built out of wood or solid stone, an example being the Parthenon, built some 2500 years ago. Nowaday, we have at our disposal all sorts of materials, each with their own physical characteristics, that can be used for specific purposes. This enables engineers to construct such technological marvels as the Burj Khalifa in Dubai (in 2010, the world's tallest building, 828 metres) and the Akashi-Kaikyo Bridge in Japan (in 2008, the world's longest suspension bridge, with a span of 1991 metres). A knowledge of the PERFORMANCE of materials is crucial if structures from which they are constructed are to avoid FAILURE.

Types of forces acting on materials:

When forces act on materials, DEFORMATION tends to occur. By this we mean that the material tends to change its shape under the influence of the forces that are applied. This change may be large or small, reversible or permanent, depending on the type of force and the material involved. Ultimately, the material will undergo FRACTURE, which is a nice way of saying that it will break!

Stress is measured in units of force per unit area of cross-section (N.m^-2), and is commonly given the symbol σ (greek "sigma"). Since the dimensions of stress are the same as those for pressure, stress is frequently measured in pascals. Strain is a pure number, and is given the symbol ε (greek "epsilon").

A material may also be subjected to bending and twisting forces. The diagram above shows a beam, mounted in two different ways. The beam will tend to bend under the force of gravity, and undergo both compression and tension. Some materials are very strong under compression but relatively weak under tension (stone, concrete). Others, such as wood or steel are strong under tension.

Hooke's law:

A material is said to be ELASTIC if, when deformed by an applied force, it returns to its original shape when the force is removed. There are many familiar examples of this, such as a steel springs, rubber bands, rubber balls, bows, and so on. Permanent deformation may occur if the stress is too large. However, many structural materials also present the property of RIGIDITY, in that they will offer resistance to stress.

The relationship between stress and strain in elastic materials was investigated by Robert Hooke, and led to what is known as Hooke's law.

In mathematical terms, a stress F = kX, where F is the stress, and X is the strain. k is a constant called the ELASTIC MODULUS or YOUNG'S MODULUS of the material.

This is easily demonstrated using a spring. When no weight is applied to the spring, the strain is zero, and we can measure its length, L. We can now attach various known masses to the spring, and determine the extensions, x, that is, the strains, caused by the stresses mg. A graph of strain against stress will be a straight line for a certain range of stresses. Beyond a certain limit, called the LINEAR LIMIT, Hooke's law is no longer obeyed. Within the linear region, the stress F = kx, where k is the SPRING CONSTANT (). The spring constant has units of N.m^-1 if the extension is given in metres, or simply N if the extension is expressed as a fraction or percentage of the original length.

Hooke's law is the principle underlying the spring balance. It consists of a spring fitted in ahousing, which bears a scale calibrated in mass units. A pointer attached to the spring indicates zero when nothing is attached to the balance. The "mass" of an object is detrmined from the pointer. The following points are worth remembering about a spring balance: It does not measure mass, but the force applied to it. Consequently, the extension of the spring will depend not only on the mass, but also on the value of g, the acceleration due to gravity at the point of use. On the moon, for example, the reading on the balance would be about one-sixth the reading on the earth for any given mass. If the spring has been stretched at any time beyond it elastic limit, the balance will be faulty.

Mathematical treatment of elasticity:

Consider a metal bar of cross-sectional area, A, and length, L, subjected to tension forces that tend to cause an increase in length. We define the TENSILE STRESS, σ as the force applied per unit area of the bar, that is,

σ = F/A

Note that the unit of tensile stress is the pascal, the same unit as pressure. The TENSILE STRAIN, ε, is the ratio of the increase in length, ΔL to the original length, that is,

ε = ΔL/L

ε is dimensionless. For relatively low values of ΔL, Hooke's law is obeyed, and we have

where Y is knows as YOUNG'S MODULUS, with units of pascals (see some values). It is a measure of the STIFFNESS of a material, namely, that material's resistance to deformation under stress. A material is stiff if a large applied force is required to produce a small strain. This is another statement of Hooke's law, as applied to rods, cylinders and wires of material under tension (or compression). It is instructive to examine in detail what happens when a wide range of stresses is applied to an elastic material:

In the stress range up to A, the LINEAR LIMIT, Hooke's law is obeyed. From A to B, Hooke's law is no longer obeyed, but removal of the stress cause the material to return to its original length. In other words, the deformation is reversible. B is known as the ELASTIC LIMIT, and the range up to B is called the ELASTIC RANGE. The stress corresponding to the point B is called the YIELD STRENGTH or YIELD POINT of the material. In the range B to D, called the PLASTIC REGION, the stress causes permanent deformation. We call this PLASTIC DEFORMATION. In this range, relatively large deformations occur for small additional stresses. In this range, the cross-sectional area of the material may decrease, a phenomenon called "necking-down", until the material breaks at D, the FRACTURE POINT. The stress corresponding to point C is called the ULTIMATE TENSILE STRENTH of the material. When we talk about the "strength" of a material, we refer to its ability to resist fracture under high stresses. The TOUGHNESS of a material refers to that material's ability to absorb large amounts of stress, deforming plastically before breaking. Polythene and steel are examples of tough materials. On the other hand, a BRITTLE material, such as glass, is elastic over a very limited range of stresses, and undergoes fracture without plastic deformation. RESILIENT materials are those that can absorb large stresses before they deform plastically. Coil springs are examples of this.

Plasticity, creep and fracture:

Plasticity:

Plasticity is the property of a material to undergo significant permanent deformation without undergoing fracture, when subjected to a stress. This happens to elastic materials when the stress exceeds a certain limit, known as the YIELD STRENGTH or YIELD POINT for that material. This would be the stress at the point labelled B in the graph shown above on the left. Normally, plastic materials undergo large deformation for a relatively small stress. A perfectly plastic material shows no tendencyto revert to its original shape once it has been subjected to a stress.

Creep:

Creep is the permanent deformation that a material experiences when it is subjected to stresses below the yield strength for extended periods of time. For example, if a spring is kept in a stretched or compressed state within the elastic region for long periods of time, the spring may not revert to its original, unstressed length. Creep is more likely to occur at elevated temperatures, and is a problem for materials subjected to high stresses at high temperatures, such as, for examples, the turbine blades of jet engines. Zinc, tin and lead, their alloys, as well as most polymers, can show creep at room temperature.

Fracture:

A fracture occurs when a material separates into two or more fragments when subjected to a stress. Materials that are particularly subject to fracture are called BRITTLE MATERIALS. Glass is an example of this. Materials that can be subjected to significant stresses without fracture are called DUCTILE MATERIALS. Most metals are ductile.

However, a ductile material, when subjected to repeated stresses that may well be below the ultimate tension strength, may develop microscopic cracks that lead eventually to fracture. This is known as METAL FATIGUE, a phenomenon that came into prominence in the 1950's, when the world's first jet airliner, the Comet, experienced several disastrous crashes with considerable losses of life.

Metals are malleable () and ductile (). Part of the reason for this is that the metal crystal consists of planes of atoms surrounded by delocalised electrons. It is therefore relatively easy to slip one plane over another when subjected to forces (shown as red arrows in the diagram below):

It turns out that metals (and other crystalline materials) are more easily deformed than expected in the light of the above "slip" model. The reason for this is that crystals are very seldom perfect, and contain what is known as DISLOCATIONS. One such dislocation is a VACANCY, where a row of atoms is incomplete:

What happens in effect is that the dislocation moves one atom at a time, a process that requires much less energy than if a whole plane of atoms were to be moved, as occurs in the "slip".

Grains and grain boundaries:

Crystals, unless specially "grown", do not consist of one large crystal, but rather of microcrystalline grains separated by GRAIN BOUNDARIES, as ilustrated in the diagram on the right. The smaller these grains, the more difficult it is for dislocations to move across the whole of the sample. The dislocations become tangled, and layers are not able to slip very far in any one direction. Metals can be made stronger, by a method called WORK HARDENING, in which the metal is repeatedly deformed plastically. Another way in which metals can be made stronger, is to introduce "foreign" atoms into their crystal structures. These atoms then cause dislocations. For example, adding small quantities of carbon to iron produces steel, a much stronger material. Alloys made of two or more different metals can also display more strength than the individual metals.

Additional questions

Picture credits:

Parthenon: Author: Kallistos (Creative Commons Attribution 3.0).
Burj Dubai: Author: Poco a poco (Creative Commons Attribution ShareAlike 3.0).
Akashi-Kaito Bridge: Author: Hamilton (Creative Commons Attribution ShareAlike 3.0).

Some values of Young's modulus for various materials:

Note that for homogeneous materials, such as metals, Young's modulus for tension and compression are usually the same. This is not true for inhomogeneous materials such as stone or concrete.