Discover what tensile and compressive stress and strain are, and how they relate to each other. Then find out how we can use stress and strain to learn more about a material's elastic properties.
Stress and Strain
When looking at a force problem, you're probably used to being concerned only about how the object moves after being affected by the forces acting on it. What you probably haven't thought too much about is how the structure of the object might be affected by that same force. A common example of this happens with bridges. When people or vehicles move over a bridge, their weight creates a downwards force. The bridge doesn't change position due to this weight, but it can bend.
To help ensure that a bridge can hold the amount of weight that it was designed to withstand without breaking under heavy traffic, it undergoes something called a stress test. Stress is defined as internal force per unit area. Mathematically, we can write this as:
The equation for stress gives it standard units of Pascals (Pa). The Pascal is a common unit used for pressure. Both pressure and stress are forces over an area. The difference between the two is that pressure is external, whereas stress is internal to the object.
The amount of stress applied to an object will determine the level of strain the object experiences, where strain is the amount of deformation an object undergoes due to stress. In essence, stress is the internal force, and strain is the physical effect of that force on the object.
Tension and Compression
Depending on how force is applied to the object, it can undergo different types of stress and strain. Two of the most common types are tensile and compressive stress and strain. When an object is under tension it is experiencing an increase in length. A rubber band being stretched out is a common example of an object experiencing tensile stress and strain. The opposite of tension is compression, where an object is undergoing a decrease in length. If you've ever squeezed a rubber ball or a pet's squeak toy in your hands, you were creating a compressive stress and strain in the object.
In the case of tensile and compressive strain, the mathematical definition is the change in length divided by the original length of the object. The difference between the two is how that length changes. For tensile it's an increase in length, and for compressive it's a decrease in length. If we want to separate them into two different equations, we can write them out as follows.
Elastic Modulus and Hooke's Law
Any object is capable of experiencing tensile and compressive stress and strain, but not all react to that stress to the same degree. Let's imagine you have two small blocks in your hands. One is made of steel and the other is rubber. If the same amount of stress is applied to both of them, which will experience more strain? Which one will stretch or compress further? You know it's going to be the rubber block. The steel block is much stiffer than the rubber block, and won't stretch or compress as much.
In physics, we can use something called the elastic modulus to measure the stiffness of a material. It turns out all we need to find the elastic modulus is the stress applied to the object and the strain it experiences. We can write elastic modulus out as the ratio of the two. So, elastic modulus equals stress divided by strain.
The elastic modulus is actually a constant. This means any given material will always have the same elastic modulus. If we rearrange the equation to get stress by itself, we can see something interesting. We can get the equation in the form of
It's not immediately obvious, but you may have seen an equation like this before. You have stress, a force per unit area, equal to a constant multiplied by strain, a displacement. This is very similar to Hooke's law, which you might have looked at for springs, where force equals the spring constant times a displacement.
It turns out that our rearranged elastic modulus equation is also Hooke's law. Our rearranged elastic modulus equation is a more general form of Hooke's law that applies to all materials, and not just springs. Hooke's law shows us that the stress needed to stretch or compress a material is directly proportional to the distance it is stretched or compressed.
Another interesting way to compare stress and strain is to look at a graph of the two.
To create this graph, stress is applied to a material until it breaks. We can learn quite a bit about the material from looking at the curve of the graph. At the beginning of our graph, there is a steep diagonal line upwards. After the steep diagonal line you see the curve start to bend. At the beginning of this bend is the limit of proportionality. Beyond this point, Hooke's law no longer applies to our material. After the limit of proportionality, at the peak of the bend in our curve is the elastic limit. The range from the beginning of the curve to the elastic limit is known as the elastic region. In this region, if you stopped applying stress to our material it would return to its original shape.
After the elastic limit, the material loses its elasticity. We call the range from this point on to the end of our graph the plastic region because the material becomes malleable like plastic. After stress is applied, the material will not fully return to its original length. At the end of our bend, where the curve starts to straighten out again, is the yield point. From here on, it only takes a small amount of stress to change the length of our material considerably. We can see that because the curve is much less steep than before. Finally, at the end of our curve, where it suddenly stops, is the fracture point. Here, the material has ruptured. For example, if the material was experiencing tensional stress, it may have snapped into two pieces at this point.
When a force acts on an object, it can create stress and strain within it. Stress is an internal force per unit area, and the deformation the object undergoes due to the stress is the strain. Two of the most common types of stress and strain are tensile and compressive stress and strain. Tensile means there is an increase in length of the object, and compressive is a decrease in length.
From stress and strain we can find a material's elastic modulus, which is the measure of the stiffness of a material. We can then use elastic modulus to find a formula for Hooke's law that works for all materials instead of just springs. This general form of Hooke's law shows us that the stress needed to stretch or compress a material is directly proportional to the distance it is stretched or compressed.
Finally, we can learn a lot from looking at a graph of stress vs. strain for a material. We can see the range of stresses and strains for which the material has elastic properties, and for which it is malleable like plastic. We call these two ranges the elastic region and the plastic region. At the transition between these two regions, we can also see the point where Hooke's law stops applying to the material, and the point where it starts taking only a small amount of stress to cause a large strain. Finally the curve ends abruptly at the point where the material ruptures due to stress.
Achieve the following goals after reviewing the topics within this lesson:
- Distinguish between stress and strain, and list two common types
- Contrast tension and compression
- Write elastic modulus and calculate a formula for Hooke's law
- Illustrate a stress-strain curve graph and understand what it shows