Before you read this, I suggest you read post 20.9.
In everyday English “toughness” and “strength” are used to mean the same thing. But in science such words are often given very specific meanings to avoid confusion (post 17.14). Then “toughness” and “strength” have precise and different meanings.
It is sometimes helpful to have an intuitive idea of a scientific concept before you start to think about it in detail. Then you can think of a strong material as one that can withstand a high stress (post 20.9) and a tough material as one that can continue to withstand an increasing stress for a long time after it sustains damage. In the remainder of this post, I will explain this difference in more detail and introduce some other, related concepts.
To explain the difference between toughness and strength, in more detail, I have drawn the stress(σ)-strain(ε) curve (post 20.3) shown in the picture above. Stress-strain curves look different for different kinds of materials. The “typical” stress-strain curves shown in books are usually the stress-strain curves for metals that show elastic behaviour over a wide range of strain values. I have invented the example in the picture to illustrate the ideas I want to explain.
My fictitious material obeys Hooke’s law (post 20.3) until the strain reaches a value of σ2. Then the curve is no longer a straight line and I suppose that the material now deforms irreversibly. Many materials behave in this way because they can no longer store their deformation energy to use in recoil (post 20.2). This irreversible deformation is called plastic deformation and σ2 is called the yield stress.
In the picture above, the material can withstand a stress greater than a σ2 until σ equals of maximum value of σ1, the ultimate tensile stress. Now the stress decreases but the strain continues to increase until the material fractures (post 20.9). The value of σ at which the material fractures is called its fracture stress. When the material fractures, the value of σ falls to zero. In post 20.9 I used ultimate tensile stress as a measure of the strength of a material. I could equally well have chosen the fracture stress. My reason for using ultimate tensile stress is that reliable values are available for a wide range of materials.
In post 20.6, we saw that the area under a stress-strain curve was equal to the work done on a unit volume of material; the red area in the picture above, is the work done on and, therefore, the energy transferred to a unit volume of material when its stress increases from zero to σ’.
In the picture above, the red area represents to energy transmitted to a unit volume of material up to the point at which it fractures. This is defined to be the toughness of the material.
The picture above shows the behaviour of two different materials. The stronger can withstand the higher stress; the tougher can absorb the most deformation energy. In this picture, the stronger material does not undergo appreciable plastic deformation before it fractures – a material like this is called a brittle material. So, this material is strong but brittle. In general, it is a mistake to think that something that something brittle is not strong. The two ideas are completely different; unlike the material in our picture, our potato chip in post 20.9 was brittle but not strong. The tougher material, in the picture, undergoes appreciable plastic deformation before it fractures; it is called a ductile material.
In a later post, I hope to explore the implications of these ideas for fracture.
Related posts
20.9 Stiffness and strength
20.6 Elasticity
20.5 Poisson’s ratio
20.3 Hooke’s law
20.2 Deformation of objects
Follow-up posts
Thinking about Science with David Hukins 
Hi David
As usual, a very well explained post. One small point; if, as drawn in your example, the stress-strain curve ends suddenly in catastrophic failure, then the energy under the total curve can be an over-estimate of the toughness, as some of the “excess energy” may be used to produce noise or as kinetic energy to shoot bits of the test piece around. Think what happens when you pop a balloon with a pin; as well as rupturing the rubber, much of the stored energy is used as noise and propelling fragments around. It is easy to supply too much energy and simply smash a brittle material into multiple pieces, e.g. using a hammer to crack a nut. (I have been told that if you smash sugar cubes in the dark with a hammer, sometimes you can see some of the excess “fracture energy” being converted into flashes of light, but I have never tried this!). If the energy under the stress-strain curve is just enough to propagate the fracture, then the area under the curve is a good measure of toughness.
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Peter has made an important point. In post 20.10, some of the energy transferred to the material can be dissipated as kinetic energy of broken fragments or of molecular motion in the form of sound (post 18.13). Then the true toughness of the material will be the area under the graph, as described in my post, minus this kinetic energy.
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