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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My kind request to post a video for every 2 or 3 week. Thank you for posting those videos.

nice video I really liked it

Very nice explanation!

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