Concept Clarification 1: Strength & Stiffness

I realized that in some of my hurry to get to the fun stuff, I may gloss over some things that are a bit confusing, or I plain forgot to include it. Or hell, I may even have just made a mistake.

These Concept Clarification posts will try to help clear some of those issues up in a short and sweet post. This one will focus on the concepts of Strength and Stiffness.



Let's start with some definitions:

• Strength: The amount of stress a body can support before failing. Examples: Yield strength is the stress a body can support before yielding. Fracture strength is the stress a body can support before fracturing. Strength is a point along the stress-strain curve for a material and applies to bulk deformation.
• Stiffness: The amount of change in load that results from a given deformation increment. Stiffness is the slope of the force-displacement or the stress-strain curve.  The stiffness of the linear elastic portion of a stress-strain curve is called the Young's modulus.

First an analogy and mental exercise:
Think about a plain old coil spring.  You stretch it a little bit, it requires a little force. You stretch it a bit more, and it requires a bit more force. The amount that force changes in relation to the change in the amount of stretching you do is the stiffness of the spring.

If you instead think about it in terms of the force you apply: you pull on it with a little force and it stretches a little bit. You pull with a bit more force and you get a bit more stretch. This change in the stretch relative to the change in the force you apply is called the compliance of the spring. It is the reciprocal (inverse) of the stiffness.

Now let's say you just think about how much force you can apply to the spring. You apply a force, and let it go. At some point, you realize the spring isn't going back to its original shape. If you keep applying a force, at some point the spring will break. This is representative of passing the yield strength and fracture strength for the spring, respectively.

Now just remember that stress is just force per unit area and strain is displacement per unit length. In other words, a stress-strain diagram is just a force-displacement diagram that's had its axes normalized based on specific values.

On to some pictures:

Simple illustration of strength (values of \(\sigma\) noted), and stiffness (slope)

In the above image, two particular values of strength are noted on the stress (vertical) axis: the yield strength, \(\sigma_y\), and the fracture strength, \(\sigma_{fr}\). These values correspond to specific changes in material behavior. For example, from a stress of 0 to \(\sigma_y\), the material behaves elastically. Once the applied stress passes the yield strength, the behavior becomes plastic (by definition).

The two triangles labeled A & B represent the slope at two different parts of the stress-strain curve. They graphically represent the stiffness we talked about with the spring, but in terms of stress and strain: we make a small change in strain, and get a change in stress.

The slope at any point of the above stress-strain curve can be defined from the change in stress (\(\Delta \sigma\)) and change in strain (\(\Delta \epsilon\)) around that point as follows:
\[\mathrm{slope} = \lim_{\Delta \epsilon \rightarrow 0}{\frac{\Delta \sigma}{\Delta \epsilon}} = \frac{d\sigma}{d\epsilon}\]
So in calculus-speak, the stiffness refers to the derivative of the stress-strain (or force-displacement) curve.

Hope that helps.