Welcome to our second lesson in our introduction to material elasticity course. Today we'll be
focusing on how atomic bonding impacts elastic behavior. Let's focus on linear elastic behavior
for our discussion. Remember when we were looking at our tensile stress strain curves that different
material families had linear relationships between stress and strain in the elastic region.
The equations for shear and bulk modulus suggest linear relationships as well. Now elasticity
is actually a really complex topic. We'll highlight some of the complexities later in this lesson
but first let's start with a simple description. Linear elastic behavior can be described as a
macroscopic manifestation of atomic bonding. Let's start with a simple example- two atoms
bonded together. We can visualize this as our atoms being held together by a spring
with some length Ao. Now if I apply some small tensile force we can see that my atoms
are displaced by some distance delta. We can plot this on a force versus displacement curve
and see that for some displacement delta caused by some force F my spring will have a stiffness of S.
Now when I'm imparting a force on a material I'm imparting energy; so we can look at an energy
versus displacement curve and see that for a given force we are increasing the energy of our system
because we are moving our atoms out of their equilibrium position. The same occurs for
compressive forces; we are once again moving our atoms out of their equilibrium spacing.
Now this is great and all but what impact does this have on my linear elastic behavior?
Well if I'm assuming that my force is being applied to my atom over some
area Ao squared (provided that my atom is roughly a cube with length Ao)
and i have a resulting strain which is given by my displacement delta over Ao, I'm able to
calculate a stress equation shown here. Combine this all together we get that our Young's modulus
is roughly proportional to my bond stiffness S over Ao. Now this occurs when I'm applying
small forces and i'm staying within a region of my force displacement curve where there is a linear
relationship between force and displacement, exactly like how when I'm applying or have
macroscopic strain in my system that i'm having a linear relationship between stress and strain.
So we see that our atomic behavior is mirroring what we saw in our macroscopic system behavior.
But how does this explain why my ceramics and my metals have different Young's modulus?
For that we have to look at bond stiffness. Different material families
are made of different bond types: covalent, metallic, ionic, Van der Waals, hydrogen.
Describing the details of these bonds is for a different course but we do know that different
bond types have different bond stiffnesses shown here. Covalent bonds are very very stiff-
20 to 200 newtons per meter. An example of a covalent bond is a carbon-carbon bond like
those found in diamonds. Van der Waals bonds on the other hand are what hold
the long covalently bound polymeric chains together within a polymer material.
These have a stiffness of 0.5 to 5 newtons per meter; this is significantly different and it shows
in the slopes of our curves in our elastic region of our stress-strain curve. Even metallic and
covalent bonding are significantly different and again this shows when I'm looking at my ceramic
and metallic curves. So we've seen that bond stiffness has an impact on the modulus
of different atomic bonds and therefore are different material families but the example
system we used was just two atoms held together by one bond and a force only going in one direction.
Materials generally speaking (excluding things like graphene) are three-dimensional. This means that
the arrangement of the atoms within my system is going to have an impact on my mechanical behavior.
Now for simple bulk materials we often consider them to be isotropic. This means that
no matter what cube face I apply my force I will have the same stress strain response and therefore
the same modulus. Anisotropy occurs when there is some structural difference in my material along
different axes; this means that my mechanical behavior will be different along these axis.
We can visualize this by thinking about a composite. Let's say I have a composite
where the matrix is reinforced by fibers that are arranged along my vertical axis.
My stress and strain responses and therefore my Young's modulus will be very different if
I'm applying a tensile force along the axis of my fibers or perpendicular to the axis of my fibers.
This means that composites, depending on how the matrix and reinforcements are arranged, are often
anisotropic. Now anisotropy can be a great asset during design. Composites are often
designed this way on purpose because they're adding stiffness within a specific direction of
the material that they know is an important mode of loading within their design.
However this also adds complexity to the elastic response. This involves things like stiffness and
compliance tensors which is well beyond the scope of this course, but it's important to realize that
orientation-dependent properties do exist and they can have a profound impact on your design of your
product. Is your material roughly isotropic? Is it slightly anisotropic? Or is it highly anisotropic.
In this lesson we've covered the basics of how atomic bonding impacts linear elastic behavior.
As I said, elasticity is a complex topic we could be talking about for hours, but understanding that
the bonding and structure of my material has a direct impact on its macroscopic properties
is critical. It makes us better engineers and designers because we look at our materials
holistically from the atoms up when doing material selection for product design.
But what impact does elasticity actually have on my product design?
Our last lesson in this course will focus on that. We'll go through an example of how
stiffness as well as other design constraints can help us narrow down material selection.
So join us as we take a look at how elasticity impacts real world design