Hello everyone, and welcome to another video on the Efficient Engineer channel!
In this video we're going to take a look at a really interesting material
property, Poisson's ratio.
Poisson's ratio is such an important material property
because it gives us key information about how different materials deform
under loading. Let's get right into it!
We can start with a quick example,
the rubber band.
If you take a rubber band and stretch it along its length, its
length will obviously increase. But it is also quite intuitive that at the same time
the band will get thinner.
Poisson's ratio can tell us how much
thinner the band is going to get.
What about for a generic lump of material?
Take this little orange cuboid - if we apply a compressive force to it, it will
expand in the directions which are perpendicular to the direction in which
we are applying the load.
We call the direction along which the load is
applied the longitudinal direction, and we call the perpendicular directions the
lateral directions.
If we reverse the direction of the force to apply a tensile load, the cuboid will
obviously extend in the longitudinal direction, and it will contract in the
lateral directions.
The key concept here is that when you apply a load to a
material in one direction, the material will also deform in the lateral directions.
Poisson's ratio is the parameter that
tells us how much the material will deform in the lateral directions.
To define Poisson's ratio more precisely, let's take a closer look at the case
where a tensile force is applied.
First let's give names to the various parameters involved.
Lx, Ly and Lz are their original dimensions of our cuboid
along the X, Y and A axes.
Delta Lx, Delta Ly and Delta Lz are the changes in these
dimensions after the load has been applied.
They are shown as being divided by two
because the change in dimension occurs on both sides of the object.
Next we can write out the strains in all three directions.
As a quick reminder strain is
a measure of deformation, and it is defined as the change in length
divided by the original length.
It is denoted using the Greek letter epsilon and is usually
expressed as a percentage.
When we apply our longitudinal load, it turns out that
the resulting strains in the lateral directions are equal. And here's the
interesting part - they are proportional to the strain in the longitudinal direction.
The ratio between the longitudinal strain and the lateral
strain is actually a material constant.
You might have guessed it by now, but we
denote this material constant with the Greek letter nu, and we call it Poisson's ratio.
By the way, Poisson's ratio is named after this guy,
Siméon Denis Poisson,
the famous French mathematician who formally defined the
ratio in a note published in 1827.
You might be wondering why a minus sign
slipped into the equation.
Remember that by convention tensile strains are positive
and compressive strains are negative.
The minus sign is just there so that
for the typical case where the lateral strains are opposite in sign to
the longitudinal strains, we get a positive Poisson's ratio.
Now is probably a good time to mention
that this equation and the concepts described in this video
only apply for isotropic materials, which are materials which have the same
properties in all directions.
We are also assuming that the materials are
deforming within the elastic region. Things get a bit more complicated when
plastic deformation is involved.
Anyway to summarise, Poisson's ratio is a
dimensionless material property which tells us how much a given material will
contract in the lateral directions when we pull on it in a longitudinal direction.
Now that we know what Poisson's ratio is, we can look at some typical values for
different materials.
Without getting too deep into the math behind it, it's useful
to know that the theoretical range of possible values for Poisson's ratio
goes from -1 to 0.5.
In practice, most real materials have a Poisson's ratio of
somewhere between 0 and 0.5.
Most metals have a Poisson ratio of around 0.3.
Here are some typical values for a few selected materials.
Now let's look at how different values of Poisson's ratio affect how a
material deforms under loading.
We'll keep things simple and consider a
two-dimensional case.
Most materials have Poisson's ratios between zero and 0.5.
When a tensile force is applied, these materials contract in the
lateral directions as we have already seen with our orange cuboid.
Materials with a Poisson's ratio of zero are interesting.
When a longitudinal tensile force is applied
there is no deformation in the lateral directions.
One material that behaves in this way is cork,
which has Poisson's ratio close to zero.
This property makes it a very useful material for certain applications.
A great example is the cork in a bottle of wine.
Because it doesn't expand laterally when compressed,
as you can see here, it can easily be inserted into the neck of a bottle.
A material with a larger Poisson's ratio would be much more difficult to
insert, as it would expand in the lateral directions when compressed.
Finally we have materials with negative Poisson's ratios.
These are known as auxetic materials.
They expand laterally when pulled,
and contract laterally when compressed, which seems counter-intuitive.
These are mostly engineered materials, like special foams, rather than materials
occurring naturally in nature.
This animation should give you an idea of how
it is possible to get negative Poisson's ratios.
You can see that as the material
is compressed, both its longitudinal and lateral dimensions are reduced.
For a typical material with a Poisson's ratio larger than zero, you would expect the
lateral dimensions to increase when the material is compressed.
Because it tells us how a material deforms, Poisson's ratio is a very
important parameter in continuum mechanics for determining how a body
responds to applied stresses.
Let's look at a simple case of uniaxial stress,
the tensile test.
It is easy enough to determine the strains acting on a small
piece of the specimen under test.
The applied stress is a normal stress in the
X direction, sigma X. The strain in the longitudinal X direction is simply given
by Hooke's law, as sigma X divided by the Young's modulus E.
Even though there are no stresses acting
in the lateral Y and Z directions, there will be strains in these directions,
as the material is contracting laterally.
The strains in the lateral directions are obtained
by multiplying the longitudinal strain by
Poisson's ratio.
This is just using the definition of Poisson's ratio which we
covered earlier. But what if we look at a more complex case where we have
tri-axial stress, with different stresses in all three directions?
In this case we can't just use Hooke's law
to determine the strain in the X direction, because it
will also be affected by the strains in the Y and Z directions.
The simple version of Hookes law for uniaxial stress
no longer applies because the
strains in one direction will depend on the stresses applied in all three directions.
We can use Hooke's law in combination with the equation for
Poisson's ratio, and the principle of superposition, to obtain the equation for
strain in the X direction.
We can re-arrange this equation into a more practical form,
like so,
and then apply the same process to obtain equations for strains in the
Y and Z directions.
These equations form what is known as the Generalized Hooke's Law,
and can be used to determine deformations for a case of tri-axial stress.
There is one last case I would like to mention, which is materials with a Poisson ratio
of 0.5, the maximum value in the theoretical range we discussed earlier in the video.
Let's return to our cuboid to introduce the concept of volumetric strain.
Volumetric strain is a measure of the change in volume of an object under load.
I'm sure it will come as no surprise that it is closely linked to
Poisso'ns ratio.
We can calculate the volumetric strain by summing the strains
in all three directions.
Let's use the equations from the Generalized Hooke's Law
we just covered to expand the equation for volumetric strain.
We can then re-arrange the terms to end up with the following equation.
You might notice that something interesting
happens to the volumetric strain when Poisson's ratio
is equal to 0.5.
We end up with a volumetric strain equal to zero.
This means that for materials with a Poisson's ratio of 0.5 the volume of the material
remains constant as it deforms.
These are known as incompressible materials.
Rubber is an example of an incompressible material.
And that's it!
We've reached the end of our review of Poisson's ratio.
If you enjoyed watching, please remember to like
and subscribe for more engineering videos!