# Hooke's Law is valid upto what limit?

My textbook states:

" the extension produced in the wire is directly proportional to the load applied,within elastic limit."

But my Physics professor said that it is valid upto only proportionality limit.

My guess : It should be upto elastic limit as the body should not reform and behave as a plastic.

Your prof is correct: there are two distinct limits here dealing with two different concepts:

1. The proportionality limit, which defines the maximum stress / strain that an object can undergo before the relationship $\sigma \propto \epsilon$ ($\sigma$ is stress, $\epsilon$ is strain) becomes invalid. If you think about it, this notion needs to be further qualified to be fully meaningful: in general the relationship is nonlinear i.e. $\sigma = f(\epsilon)$ for some general nonlinear function $f$ that approximates to $\sigma=k\,\epsilon$ for small $\epsilon$. So, to be clear, one must define how much one deviates from linearity to define a limit: there is always some deviation. You might specify a "proportional to within 1%" limit, for example.

2. The elastic limit, which defines the maximum stress / strain that an object can withstand and respond reversibly to stress in the sense that the body returns exactly to the same, zero-stress shape when the stress is released as it had before the application of stress and the body's stress / strain relationship stays the same upon the release of stress and its re-application. Strain beyond the elastic limit leads to a change of the body's zero-stress state shape.

To my knowledge, the limiting stress /strain in 2. is always greater than that in 1., sometimes considerably so. I'm not aware of any materials where 1. is greater than 2.; such behavior would be weird and interesting and perhaps someone else can answer whether it is possible (but that's another question).

I believe the definition section of the following wikipedia page will be of use:

https://en.m.wikipedia.org/wiki/Yield_%28engineering%29

Your teacher is correct to say Hooke's law is only really valid up to the proportionality limit, and this is simply because Hooke's law is a proportionality law, i.e. Force (or stress) is proportional to extension (or strain). Beyond the proportionality limit, by definition the force and extension are no longer proportional and so Hooke's law is not entirely valid.

However, the elastic behaviour itself, where deformations are temporary, lasts up to the elastic limit. This does indeed mean there is a region where you obtain non-proportional elastic behaviour, between the proportionality and elastic limits. This non-linearity arises here because, in my best understanding, Hooke's law is simply an approximation that is accurate until you approach the limit of elasticity. For metals, like steel, this non-linear elastic region is usually small, and in some practical cases it is ignored altogether! However, materials like elastomers will have a much larger non-linear elastic limit.

In this context, elastic means that the force is proportional to the stretch.

Notice, let $\delta$ be the extension produced by applying a load say $F$ then Hooke's law in mathematical form is $$\delta\propto F\ \ \ \ or\ \ F=k\delta$$ where, $k$ is a constant

1. The linear relationship $F=k\delta$ (between the extension $\delta$ & the load applied $F$) holds good upto proportionality limit i.e. the extension is linearly proportional to the load applied

2. From the point of proportionality limit to the point of elastic limit, the linear relationship $F=k\delta$ doesn't hold good i.e. the graph between load vs extension is nonlinear the extension $\delta$ is not directly proportional to the load $F$ although the material can reform.