Understanding True Stress and True Strain

Understanding True Stress and True Strain

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
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10 Replies to “Understanding True Stress and True Strain”

  1. I'm from Brazil and I learn a lot watching your videos. I'm engineering student and I have no patience for waiting the class for learn. Because of people like you we can learn everywhere. Thanks a lot.

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