# 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?

• Steel has the larger elastic modulus. Maybe I sometimes say that it has a high elasticity. But I would never say that it is very elastic. This is about language usage, not so much about physics. – Pieter Apr 4 '18 at 8:25
• Have you ever dropped a steel ball bearing onto a hard floor?It bounces rather well. Yes, rubber is easier to stretch or compress than steel, OTOH, if you get a rubber band and stretch it and let it relax quickly several times it soon warms up, which indicates that the process is not so elastic. – PM 2Ring Apr 4 '18 at 9:53
• Steel will permanently deform if is stressed beyond the elastic limit. As long as the stresses don't go to that point (plastic deformation) it is quite elastic. The rubber ball will not go back to exactly the same conformation. In any event, I agree with others: those statements are not definitions of elasticity. – garyp Apr 4 '18 at 13:49
• Be careful with terminology here. The quotes talk about "elasticity"; which may be colloquial shorthand for "modulus of elasticity" which doesn't account for how much strain it can take while remaining elastic. Having a higher modulus of elasticity doesn't make it more "elastic" by all definitions though. I think your sources were a bit careless with their wording is all. – JMac Apr 4 '18 at 14:35
• Does your textbook really use it's when they mean its? – TKK Apr 4 '18 at 16:00

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.

• "Yield point" refers to two things: the yield stress and the yield strain. – Pieter Apr 4 '18 at 8:48
• I agree. But what about the genral consensus that steel is more elastic? Should I just assume that all those people are referring to young's modulus and not the elasticity in real sense? – Sarthak123 Apr 4 '18 at 11:33
• @Sarthak123 I have to say that I have never heard anyone say that steel is more elastic than rubber. – John Rennie Apr 4 '18 at 11:42
• Well that's surprising because this was the first question that introduced me to the topic of elasticity. In fact, it's all over the web, a quick google search "steel vs rubber elasticity" brings up a list of countless articles claiming steel to be more elastic. – Sarthak123 Apr 4 '18 at 12:15
• Also worth noting that in informal context, "elasticity" is often shorthand for "modulus of elasticity" which is probably where "steel has greater elasticity" comes from. You're less likely to see someone say "steel is more elastic than rubber"; because that has less implication that you're only talking about "modulus of elasticity"; and are more likely also considering the elastic limit. – JMac Apr 4 '18 at 14:32

Both the OP and John Rennie have well illustrated the imperfections in the usage of the word "elastic" in physics and how the word can create confusion between "stiffness" and a material's ability to brook strain.

But an important point to be made is that the one important field where one hears the vague statement that "steel is more elastic than rubber" is in the context of Newtonian collision problems. So what's meant here is that steel objects typically undergo more elastic, i.e. kinetic energy conserving, collisions than rubber ones.

Newtonian collision problems come up very early and prominently in an undergraduate physics course, so this may well create the (probably mistaken) impression that physicists tend to mean stiffness rather than ability to brook strain by the word "elastic". Indeed, the fields wherein physicists, as opposed to specialist material scientists, mostly use the word "elastic" are those where the word refers to collisions and interactions, and in these contexts the word means "conserving of total kinetic energy of all colliding bodies" or "not energy converting". Elastic optical interactions such as Rayleigh scattering or Fresnel reflexions are those where the incident and scattered light have the same wavelength, thus photon energy, and no energy is dissipated in or transferred to the scatterer. Likewise with all particle physics specialities.

An interesting comment from user Jasper:

In other words, rubber's stress-strain curve has more hysteresis (as a fraction of the maximum strain energy in the loop) when the strain goes from negative to positive and back.

Intuitively, it's probably part of the cause, maybe the main cause in some materials but there are rubbers where other mechanisms account for the loss, according to some cursory research I've been doing into rubbers in recent weeks for bearings in an adaptive optics system I've been working on. I'm certainly no expert, but common models used are all linear differential equation models wherein the loss comes from damping terms. Look up the Kelvin Voigt Model and Maxwell-Wiechert Model and Standardized Linear Model. Synthetic rubber manufacturers often specify the loss properties of their wares by loss tangents and complex-valued Young's modulusses (which show a phase delay for sinusoidal force excitation). Mechanisms other than hysteresis that can give rise to loss tangents are viscous drag between neighboring molecules; this can be simply linear damping of the form $-\mu\,\dot{x}$ where $x$ measures the strain and $\mu$ a viscous drag term. To be clear: by "hysteresis" I mean a nonlinear, instantaneous two-valuedness of a strain-stress response curve where which of the two function branches is traversed is set by the direction of the variation. Each cycle around a $B\, vs.\,H$ loop in a ferromagnetic material or around a $\sigma\,vs.\,\epsilon$ loop in a deformable material transfers energy proportional to the loop area to the material. This is different from viscous drag.

• Ah, good point. I'd overlooked the issue of energy dissipation in collisions. – John Rennie Apr 4 '18 at 14:17
• In other words, rubber's stress-strain curve has more hysteresis (as a fraction of the maximum strain energy in the loop) when the strain goes from negative to positive and back. – Jasper Apr 4 '18 at 17:53
• @Jasper See my update. – WetSavannaAnimal Apr 5 '18 at 0:45
• @david Indeed hysteresis gives rise to loss. By hysteresis I (and I think Jason) mean a nonlinear, instantaneous two-valuedness of a response curve where which of the two function branches is traversed is set by the direction of the variation. Each cycle around a B-H loop in a ferromagnetic material or around a $\sigma-\epsilon$ loop in a deformable material transfers energy proportional to the loop area to the material. – WetSavannaAnimal Apr 5 '18 at 9:00
• @Jasper Actually, BTW you are right that hysteresis can be modelled - to an approximation - by a loss term similar to a viscous drag. I thinking of the resistive component in the transformer hysteresis model and getting confused with the electrical/ mechanical state variable analogy: it is charge that is the analogue of strain / displacement rather than current as I was assuming. Therefore, the viscous loss term is indeed analogous to $-R\,\dot{Q} = R I$, and I know empirically that $R\,I^2$ losses are found to be a good approximate model for hysteresis loss, at least where the amplitude .... – WetSavannaAnimal Apr 5 '18 at 9:12

My interpretation of the statement "steel is more elastic than rubber" is very different from yours.

I would say that it means rubber is viscoelastic and that there is a time dependence to the stress-strain relationship; it flows when you shear it. Steel will be very nearly perfectly elastic until reaching yield.

Understood this way, we can say that for a given stress OR strain applied, rubber will never be perfectly elastic. This is, by the way, basically equivalent to saying that no energy is lost in elastic collisions, as that energy is going into rearranging long chains of hydrocarbons in rubber instead of just vibrating an iron-carbon lattice and slightly heating it up.

• If rubber means for example a styrene butadiene copolymer then it does not show viscous flow. It's a rather lossy material, but does not permanently deform under a sustained stress. Or at least not unless we're considering very long timescales like days or weeks. – John Rennie Apr 4 '18 at 17:15
• What makes a timescale long or number of cycles large? My suspicion is that these effects are what is meant by such a general claim that the material is "less elastic". – William Grobman Apr 4 '18 at 17:29

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