Y= K*(L+ΔL)/A

The Young Modulus Y in Mpa = N/mm2

The spring constant K in N/mm

L+ ΔL in mm

the Area A in mm2

k*(L+ΔL)/A= [ (N/mm)*(mm)]/mm2

= N/mm2 = Y in Mpa

Your formula is correct

Dear Raj Zh,

I believe that there isn't contrary issue because in the expression Y = F*L/A*ΔL the correct is use the true strain, ε = ΔL/(L+ΔL), where L is the initial length of the bar. With this, the expression become the same.

Geraldo Belmonte

Geraldo J B Dos Santos

Dear Geraldo J B Dos Santos

Thank you for your reply. I also considered as you said, but found it usually use the relative strain or Engineering strain, i.e. ΔL/L to calculated the Young's modulus in the text book. So I feel a little confused.

Thank you

Dear Raj Zh

In the elastic region, Young's modulus and spring constant remain constant. In the equation, Y = σ/ε, i.e. Y = F*L/A*ΔL, ε is engineering strain, and L is initial length, not total length after the deformation.

So what you are changing in the equation Y= k*L/A is L to L+ΔL after the small deformation, but the initial length is clearly constant. So there is no contrary.

Ved

Dear Raj Zh,

The way you calculate the stress and strain in the deformed configuration is incorrect.

Simple answer first: Y= k*L/A is still correct for the deformed configuration (L+ΔL) if we do not take any Poisson's ratio effect into account at this moment.

Let's consider two states:

a) Stress-free state (σ0=0, ε0=0): Length: L, Area: A, Elastic stiffness k,

then any Force F1 that causes small Deformation ΔL1 may lead to

stress σ = σ1 - σ0= F/A = k*ΔL1/A; strain ε = ε1 - ε0= ΔL1/L (Engineering strain here).

We derive Elastic Modulus Y= σ/ε = k*L/A.

b) Deformed state (with a small deformation ΔL1 at beginning):

Assume the object has the stress σ1(≠0), and, the strain ε1(≠0) under an initial force F1 (with a corresponding length deformation ΔL1);

then any additional Force F2 that is applied to this deformed configuration may lead to:

corresponding deformation ΔL2

stress σ = σ2 - σ1 = (F2+F1)/A - F1/A = F2/A = k*ΔL2/A;

strain ε = ε2 - ε1 = (ΔL2+ΔL1)/L - ΔL1/L = ΔL2/L

Therefore Y= σ/ε = (k*ΔL2/A)/(ΔL2/L) = k*L/A.

See, they are still the same (Keep in mind: stress σ, strain ε we used here for calculating the elastic modulus Y are relative quantities to the ones from initial configuration (stress-free in case a, or deformed state in case b).

Hope it helps.

Best wishes,

Xiao