Definition of strain is not unique. Let’s start with 1-dimensional strain.
Why start with 1-D strain instead of directly talking about 3D strain?
Because in mechanical analysis and simulation, 1D strain is often used in material definition
and in determining material status. Here, we will discuss two widely used
strain definitions in engineering field: the engineering strain and the true strain. We
are going to explain the mathematical formulation of the two strains and the
essential differences between them. ​ Let’s first have a look of the engineering
strain. The definition of engineering strain is straightforward and intuitive:
it’s the ratio between the change in length and the original length.
How does engineering strain get this form. Let’s decompose the
total change of length to multiple steps and have a deeper look of this expression.
For engineering strain, there’s an assumption that for each step during deformation,
the change of deformation is uniform, which forms a step-incremental deformation.
Let’s see here, for first small step, change of length is delta_L1 for the entire step. It
jumps from 0 to delta_L1 sharply and remain as a straight line for first step. The strain value
for now is delta_L1/L0. For the second step, another increase in length happens, the value
is delta_L2. Accordingly, we draw another step increment on this time displacement graph. Now,
the strain value is the summation of the first step and second step value, as shown in here.
Same thing happens for the third step and all the following steps, till the final step. And the
strain at the last step is the summation of all these terms. Since the denominator is the same,
we can directly add up the numerator, which leads to the expression delta_L/L0.
So this zig-zag kind of presentation of deformation process, is it natural?
No. In real life, the deformation process is almost always continuous
instead of step-jump like this. Here the assumption is made in order to have
such a simplified strain definition that engineers are easy to measure or calculate the value.
And that’s basically why this definition is named as engineering strain.
Next, let’s have a look of another widely used strain measurement: True strain.
Here, instead of making any assumption of the deformation process, we use the reality,
the deformation is continuous. Just as implied by the
name “true” strain, there’s more truth here. For a linear problem, the time-displacement curve
is a straight line, as shown here. Compared to engineering strain, the definition of
true strain is a little more complicated, requiring the concept of integration.
So for this continuous deformation, let’s take an extremely small piece of length change dl, we call
such dl infinitesimal, which means it’s infinitely small. This dl can represent any infinitesimal
deformation during the entire deformation process. In another world, the total deformation
is formed by infinite number of such dl. For this infinitesimal change of length,
we define an infinitesimal strain, which is dl/L0. To have the strain for the entire deformation,
we need to do integration for the infinitesimal strain from 0 to total strain, which leads to
an integration from the original length to the deformed length. This integral form is basically
the full definition of true strain measurement. With some mathematical derivation, the expression
can be more concise in the log form, which is the form you find in most cases for true strain.
A direct comparison between engineering strain and true strain. Engineering strain
uses step-incremental deformation, which forms a linear strain. True strain uses continuous
deformation, which is a nonlinear expression. There’s a relationship between these two strains.
For true strain, this part in bracket is just the definition of engineering strain, so if we replace
it by engineering strain, we have the relationship between these two strain definitions.
Let’s verify that if these two strains meet the requirements of a stain definition,
that is when there’s no deformation, they should be absolute 0, and when the deformation is small,
they should yield to very similar values. So when the change of length Delta_L = 0, we can
see that both of the expression becomes 0. When there’s small deformation, say change
of length is 1/20 of the original length, the engineering strain is 0.05 and the true
strain is calculated to be 0.049. So the values are pretty close to each other.
However, when the deformation is comparatively large, the difference
between the two strains will show up. When the change of length is 30% of the
original length, the engineering strain is 0.3, but the true strain is calculated
to be 0.26. The difference is more than 10%. Let’s make the length change as horizontal axis
and plot the two strain values. Engineering strain is a straight line, while true strain
is a nonlinear curve, and they deviate from each other with increasing deformation.