Is There A Relationship Between Young's Modulus And Spring Constant?

FredericChopin

We know that the Young's modulus of an object is defined as the ratio between its stress and strain:

Y = σ/ε

, or:

Y = F*L/A*ΔL

We also know that Hooke's law, which can be applied to any linear elastic object, can find spring constant:

F = k*x

, rearranged:

k = F/x

But is there a relationship between the Young's modulus and the spring constant of an object? Is there a formula that shows this relationship?

Thank you.

vanhees71

I think, the following website illustrates the close relationship between both quantities well:

https://notendur.hi.is/eme1/skoli/ed...ngsModulus.htm

ZealScience

Sure, we can make a closer comparision
Y = F*L/A*ΔL

Considering the ideal springs, when you put two in parallel the strength would double, whereas series would half. This means that k is proportional to A/L. Put it like k=cA/L where c is another constant, then F=c*A*ΔL/L, rearrange into c=F*L/A*ΔL which is analogous to the previous one.

However Young's modulus is applied to string or rod like material, which means the c would equal to the Young's modulus when k is for a string or rod. Otherwise, you need to consider the shape of the spring.

FredericChopin

Just to clarify, from reading vanhees71's weblink and your post, the relationship between the Young's modulus and the spring constant of a string or rod-like object is:

k = Y*A/L ?

Also, would this formula only hold true up to the proportional limit of an object (since Hooke's law also states this limit)?

Studiot

Yes that's good.

Youngs modulus is simply for stress (loading) in a single dimension, such as stretching a string or compressing a column. So a single dimension (length) is involved.

Vanhees reference refers to a more general elastic system that might have loading in two or three dimensions and so area or volume might be involved. In this case the response is still proportional to the load, by spring constant but this does not relate so simply to youngs modulus.

This is what zealscience also meant by considering the shape.

FredericChopin

Excellent! Thank you.