The typical stress-strain curve for a ductile material, which is obtained by performing
a tensile test, looks something like this.
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 analysing 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 re-arrange 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 re-arrange this equation to the form, L divided by L0, minus one.
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 three 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/x is ln(x)+C, we end up with this
equation, which can be re-arranged 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.
Thanks for watching, and remember to hit the subscribe button and the bell to be notified
about future videos!