In this lesson, we will learn two things: what is Hooke’s law?
and how is it useful in practical engineering applications.
Let’s get started!
If you have seen the bare bones structure of a building or a bridge,
you’ll notice that there are several steel beams, that are called as truss elements,
that make up the whole structure. The purpose of these truss elements is to
support the weight and other loads that are acting on the structure.
So, they are subjected to tremendous amount of stresses. But how do we know how big
are these stresses? And more importantly, why do we need this information?
Since they’re load bearing elements of the structure, the engineer who is designing them
must be aware of this knowledge to make sure that no element is stressed beyond its limit.
Otherwise, that element can fail and lead to a catastrophe.
So, how does this concern Hooke’s law? Well, it turns out that Hooke’s law can helps us find the
value of the stresses in such structures. Hooke’s law provides a way of calculating the
stresses developed in a part when it is subjected to certain amount of strain.
This law is applicable only when the deformation is elastic.
Let’s take a step back and learn what is elastic deformation.
Elastic deformation is nothing but when an object is deformed to an extent
such that it does not sustain any permanent deformations.
So, upon unloading, it should return to its original state.
Speaking in terms of energy, since there is no permanent change in its structure,
all the work that is done in deforming it is stored as internal energy and it is dissipated
as heat energy upon unloading. So, coming back to Hooke’s law, it states that
the stresses developed in a part are linearly proportional to the
strains that it is subjected to, if it is undergoing elastic deformation.
In other words, it states that there’s a linear relationship between the stresses and strain and
the constant of proportionality is called as Young’s modulus.
Most metals undergo elastic deformation up to a limit, called as elastic limit. And within
this limit, they follow a linear relation between the stresses and strains.
So, now you can see how the Hooke’s law is very helpful in calculating the stresses
developed in the truss elements of buildings or bridges.
Also, note that the Hooke’s law is not just limited to stress analysis of massive structures.
It can be used in many other applications such as studying the vibrational characteristics of
rotating equipment, or to see how various assemblies behave when they’re subjected to
harmonic or even random dynamic loads such as seismic activity.
Hooke’s law also provides a way of characterizing materials based on their proportionality constant
Youngs modulus. In this case, the linear relation ship between stress and strain
is the constitutive model and the proportionality constant Youngs modulus
is nothing but the material property. Materials with higher Young’s modulus offer more
resistance to deformation compared to those that have lower Young’s modulus.
If you inspect this table, you’ll notice that steel alloys have much higher Young’s modulus
compared to other metals such as Aluminum or some other class of materials such
as cherry hardwood. So, this clearly indicates why
steel is chosen for making the load bearing elements in massive structures.
The Hooke’s law is applicable to both the deviatoric and the volumetric behavior of
the material. It uses Poisson’s effect which states that when a material is stretched or
compressed in one direction, it deforms in the other perpendicular directions as well.
This is due to the incompressible behavior of materials.
And the ratio of strain in lateral direction
to the strain in direction of loading is nothing but the Poisson’s ratio.
The negative sign in this relation indicates
that the lateral strain is in the direction opposite to loading direction.
Typically, the value of Poisson’s ratio ranges between 0 and 0.5 with 0 indicating that the
material is compressible to 0.5 indicating that the material is incompressible.
Most materials fall somewhere between this range meaning they all resist change in volume to
some extent, but they may still experience some amount of change in volume.
Here’s a table of values of Poisson’s ratio for some common materials,
you’ll notice that the materials with smaller values are generally compressible in nature
and the values with higher Poisson’s ratio are usually incompressible.
For instance, cork is known to be compressible and has a Poisson’s ratio which is almost 0.
On the other hand, elastomers are nearly incompressible,
and their Poisson’s ratio is close to 0.5.
But why are we discussing this under Hooke’s law? We will learn in the following lessons how
this relationship helps us in formulating the stress tensors for more generic cases
using Hooke’s law. Most metals such as grades of steel
or aluminum fall somewhere between 0.27-0.3 and polymer plastic such as high-density
polyethylene is slightly over 0.4. So, in summary Hooke’s law provides a linear
relation between the stresses and strains for both the deviatoric and the volumetric behavior
and it is very useful in several industrial applications.
What we discussed in this lesson is only in one dimension,
but it can be extended to general 3-D cases.