# Metal Stress, Strain, and Fatigue This titanium rod has been stretched to twice its original length, an engineering strain of 100%. Photo dunand.northwestern.edu

All metals deform (stretch or compress) when they are stressed, to a greater or lesser degree. This deformation is the visible sign of metal stress called metal strain and is possible because of a characteristic of these metals called ductility—their ability to be elongated or reduced in length without breaking.

## Calculating Stress

Stress is defined as force per unit area as shown in the equation σ = F / A.

Stress is often represented by the Greek letter sigma (σ) and expressed in newtons per square meter, or pascals (Pa). For greater stresses, it is expressed in megapascals (106 or 1 million Pa) or gigapascals (109 or 1 billion Pa).

Force (F) is mass x acceleration, and so 1 newton is the mass required to accelerate a 1-kilogram object at a rate of 1 meter per second squared. And the area (A) in the equation is specifically the cross-sectional area of the metal that undergoes stress.

Let's say a force of 6 newtons is applied to a bar with a diameter of 6 centimeters. The area of the cross section of the bar is calculated by using the formula A = π r2. The radius is half of the diameter, so the radius is 3 cm or 0.03 m and the area is 2.2826 x 10-3 m2.

A = 3.14 x (0.03 m)2 = 3.14 x 0.0009 m2 = 0.002826 m2 or 2.2826 x 10-3 m2

Now we use the area and the known force in the equation for calculating stress:

σ = 6 newtons / 2.2826 x 10-3 m2 = 2,123 newtons / m2 or 2,123 Pa

## Calculating Strain

Strain is the amount of deformation (either stretch or compression) caused by the stress divided by the initial length of the metal as shown in the equation ε = dl / l0. If there is an increase in the length of a piece of metal due to stress, it is referred to as tensile strain. If there's a reduction in length, it's called compressive strain.

Strain is often represented by the Greek letter epsilon (ε), and in the equation, dl is the change in length and l0 is the initial length.

Strain has no unit of measurement because it's a length divided by a length and so is expressed only as a number. For example, a wire that's initially 10 centimeters long is stretched to 11.5 centimeters; its strain is 0.15.

ε = 1.5 cm (the change in length or amount of stretch) / 10 cm (initial length) = 0.15

## Ductile Materials

Some metals, such as stainless steel and many other alloys, are ductile and yield under stress. Other metals, such as cast iron, fracture and break quickly under stress. Of course, even stainless steel finally weakens and breaks if it is put under enough stress.

Metals such as low-carbon steel bend rather than breaking under stress. At a certain level of stress, however, they reach a well-understood yield point. Once they reach that yield point, the metal becomes strain hardened. The metal becomes less ductile and, in one sense, becomes harder. But while strain hardening makes it less easy for the metal to deform, it also makes the metal more brittle. Brittle metal can break, or fail, quite easily.

## Brittle Materials

Some metals are intrinsically brittle, which means they are particularly liable to fracture. Brittle metals include high-carbon steels. Unlike ductile materials, these metals do not have a well-defined yield point. Instead, when they reach a certain stress level, they break.

Brittle metals behave very much like other brittle materials such as glass and concrete. Like these materials, they are strong in certain ways—but because they cannot bend or stretch, they are not appropriate for certain uses.

## Metal Fatigue

When ductile metals are stressed, they deform. If the stress is removed before the metal reaches its yield point, the metal returns to its former shape. While the metal appears to have returned to its original state, however, tiny faults have appeared at the molecular level.

Each time the metal deforms and then returns to its original shape, more molecular faults occur. After many deformations, there are so many molecular faults that the metal cracks. When enough cracks form for them to merge, irreversible metal fatigue occurs.

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