classical mechanics - Clarifying the actual definition of elasticity. Is steel really more elastic than rubber?

Yes, I know it's steel. It's everywhere on the web and I did google. But I seek enlightenment.

My physics textbook defines elasticity as:

Property by virtue of which a material regains its shape.
Or, the ability of material to resist change in its shape or size.

While I get what my textbook intends to say, I strongly think that there is a subtle difference between the 2 definitions. I mean according to the first definition, certainly rubber is more elastic than steel as rubber has tendency to regain its shape even when stretched several times its natural length. On the other hand, a steel bar would become permanently set and even fracture if the strain increases ever so slightly (let's keep "but it requires tremendous force" out of the way here, that's not the main point here) . In this sense, obviously rubber is more elastic.

But the second definition makes clear that steel is the winner. Steel has greater tendency to resist its shape change and hence it should be more elastic.

So, it is very clear that we can define elasticity 2 ways, either by a picture of strain tolerance (winner = rubber) or by stress tolerance (winner = steel). Most of the physicists (but definitely not all) seem to prefer the stress tolerance definition (mostly without clarification). What I seek here is a logical(and maybe philosophical) answer to why? Why prefer one definition over other, especially the one which defies common sense of general public? When everyone seems to agree with rubber as winner, why change the rules?

Answer

There are two separate concepts here:

1. 
   

   the Young's modulus, which determines the force needed to stretch the material

   
   


2. 
   

   the elastic limit, aka yield strain, which determines how far the material can be stretched

   
   

As you say, the term elastic tends to be used in a vague way that conflates these two properties. Generally a high Young's modulus means the material is stiff so I would say steel is stiffer than rubber not more elastic than rubber. Steel also has a much smaller yield strain that rubber because you can't stretch steel far before it starts to deform while rubber can be stretched a long distance.

So if you're going to use the vaguely defined term elastic then steel is certainly less elastic than rubber in both meanings. However in a physics or engineering context you would use the precisely defined terms Young's modulus and yield strain instead.

Finally:

There is another meaning for elastic, which is what Rod has covered in his answer. I'm going to summarise it here for completeness but please upvote Rod's answer as he thought of it first!

If we say a collision is elastic it means no energy is lost in the collision. In this sense the collision between steel balls is highly elastic. That's why a Newton's cradle with steel balls will swing for ages once you set it going. By contrast collisions between rubber balls tend to be squidgier and lose more energy so in this sense they are less elastic than steel. It might be that this is why you have seen steel described as more elastic than rubber. The term elastic applies to the collision rather than the material.