A student asks: I had a question about how stiffness relates to elasticity. Does a high stiffness mean high or low elasticity?

About elasticity:

Saying a material has a high stiffness doesn't say anything about its elasticity. But it does say something about its elastic modulus. Don't worry - many people get confused at first on this topic. I will attempt to explain why.

You may recall that when a stress is applied to a material, the material deforms. This deformation is called strain. If the material returns to its original shape when the stress is removed, then we say that:

a) the material behaved in an elastic manner,

b) the response was elastic, and /or

c) the strains were elastic.

If, when the stress is removed, the material does not return to its original shape, we say that it accumulated permanent deformation or plastic strain.

For Isotropic Materials, like metal, plastic, or rubber:

A stiff material does not deform much when a stress is applied. A stiff material has a high elastic modulus. If someone asks "what is the stiffness of this material?", the answer is usually "the value of the elastic modulus is X MPa".

A compliant material does deform a lot when a stress is applied. A compliant material has a low elastic modulus. If someone asks "what is the compliance of this material?" the person will usually scratch their head and finally say "Um, well, um...the elastic modulus is X MPa". But the correct answer would be "the inverse of the elastic modulus is Y m^2/N."

What confuses a lot of people is that difference between elasticity and elastic modulus. Elasticity is technically just the theory of materials that are assumed not to deform permanently. This is true when stresses are less than the materials yield strength.

But in general, people say a material has a high elasticity, or is highly elastic, if it can be stretched to large strains without failure. Rubber is an example of a highly elastic material. The problem is, that RUBBER IS HIGHLY ELASTIC (can be stretched a lot without being deformed permanently or breaking) BUT IT HAS A LOW ELASTIC MODULUS (large strains are caused by relatively small stresses). There are not very many materials with both a high elastic modulus and high elasticity.

For Anisotropic Materials, like composites or wood:

Stiffness and compliance have the same meanings. But, we have to quantify them with matrices since the composite has different stiffness properties in different directions.

To be precise, we should refer to the terms of the [Q] matrix as 'stiffness components' and the terms of the [S] matrix as 'compliance components'. However, since [Q] and [S] are inversely related, and are both the result of the theory of elasticity, we often call the terms of both of them 'elastic terms'. Just to be lazy, I guess. No term in the [Q] or [S] matrices should ever be called 'elastic modulus', however. This is becuase Elastic Modulus has a mathematical definition:

(elastic modulus) = (stress)/(strain) when there is only one stress applied to the material.

More confused now? Just remember:

A material with a high elastic modulus, or large [Q] terms, is stiff (or has high stiffness).

A material with high elasticity can be stretched a lot without failing.