But this curve is just an approximation.

It doesn't represent the actual stress or strain in the test piece during the tensile test.

In this video, we're going to talk about true stress and true strain.

In the typical stress-strain curve, stress is defined as the applied force divided by the initial cross-sectional area of the test specimen, and strain is defined as the change in specimen length divided by the initial length.

These stress and strain values are actually just approximations of the true stress and strain in the specimen.

We call them engineering stress and engineering strain, and denote them using the subscript

E. This curve is known as the engineering stress-strain curve.

To determine the true stress and strain values, we would need to consider the fact that the dimensions of the specimen change throughout the duration of the test.

If we were to measure the true stress and strain, our stress-strain curve would look something like this.

You can differentiate a true curve from an engineering curve by noting that the engineering curve drops after necking, whereas the true curve is always increasing.

Remember that necking is the rapid reduction in the cross-sectional area of the test piece, which begins when the engineering curve reaches its maximum value, which is the ultimate tensile strength of the material.

So if they do not accurately reflect the true stress and strain in the test specimen, why do engineers commonly use engineering curves instead of true curves?

Well, there are two main reasons.

Firstly, it's quite difficult to measure the instantaneous cross-sectional area during a tensile test, and so most of the time we just don't have the true stress-strain curves.

And secondly, most of the time we are only analyzing or designing things which deform within the elastic region.

As you can see here, the engineering and true curves are very similar for small strain values.

But for cases where we have large plastic deformation, the difference between the two curves becomes significant.

This is in large part due to the sudden reduction in the cross-sectional area of the test piece when necking occurs.

When assessing cases where we have significant plastic deformation, it becomes important to use the true stress-strain curves.

Examples of this might be the analysis of manufacturing processes, or performing finite element analysis which models large strains.

By making a few assumptions, we can calculate the true stress-strain curve based on the engineering curve, and so we can avoid having to measure the instantaneous cross-sectional area during the tensile test.

Unlike engineering stress, which is calculated by dividing the applied force by the initial cross-sectional area of the test piece, true stress is calculated by dividing by the instantaneous cross-sectional area at each instant throughout the test.

It accounts for the fact that the cross-sectional area of the test piece is changing as the test is performed.

We can adjust this equation for true stress by assuming that the volume of the test piece remains constant.

This assumption is valid in the elastic region of the stress-strain curve, because any volume changes in the elastic region will be small, and it is valid in the plastic region of the curve because materials are considered to be incompressible during plastic deformation.

I talk more about material incompressibility in my video on Poisson's ratio.

If volume remains constant, the product of the instantaneous area and the instantaneous length is equal to the product of the original area and the original length.

But it is important to know that this assumption is not valid after necking has occurred, because of the associated change in the cross-sectional area.

Beyond necking, we would need to base the true stress on actual measurements of the cross-sectional area.

But anyway, we can rearrange this equation so that instantaneous area is on the left hand side, and then substitute it into our equation for true stress.

The definition of engineering strain is that it is the change in length divided by the original length.

We can rearrange this equation to the form L divided by L0 minus 1, and we can use this to obtain an equation for true stress which is a function of the engineering stress and engineering strain, both of which can easily be obtained from a tensile test.

Now let's derive an equation for true strain.

True strain needs to consider the fact that the original length of the specimen is continuously changing at each instant throughout the duration of the tensile test.

We could calculate it by splitting the tensile test into increments, and calculating the change in strain at each increment based on the length at the start and end of the increment.

In this example, I will consider 3 increments.

The increments can then be summed up to calculate the true strain at the end of increment 3, for example.

Instead of doing this manually with large increment sizes, this approach can be defined mathematically using integration, like this.

By remembering that the integral of 1 over x is ln of x plus c, we end up with this equation, which can be rearranged to be a function of the engineering strain.

Because of the form of this equation, true strain is also known as logarithmic strain, or natural strain.

I hope this has helped explain the differences between engineering and true stress-strain curves.

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