# Estimating Young's Modulus

In order to measure the Young's modulus of a rectangular sample of elastic material I subject it to a vertical load along its major axis using weights of gradually increasing importance and measure the variation in length. But, I face the following dilemma:

1. When estimating the Young's modulus using the ratio of engineering stress over engineering strain, $E_{eng} := \frac{PL_0}{A_0\delta{L}}$ I get a trajectory for Young's modulus that's constantly decreasing in relation to increasing force.
2. When estimating the Young's modulus using the ratio of true stress to true strain, $E_{true} := \frac{P/A}{ln\frac{L}{L_0}}$ I get a trajectory for Young's modulus that's constantly increasing in relation to increasing force.

Basically, the estimates $E_{true}$ and $E_{eng}$ diverge as the load increases. In fact, at 10% of the *maximum load for this material I observe that $\frac{E_{true}}{E_{eng}} \approx 60$. Is this something I should expect?

I must note that for $E_{true}$ I estimate $A:=W*T$, the cross-sectional area(or product of actual width and actual thickness), by assuming that $A \approx \frac{W_0*L_0}{L}*\frac{T_0*L_0}{L}=\frac{W_0*T_0*L_0^2}{L^2}$. Below are the graphs for the relationship between the engineering stress and engineering strain and the true stress against true strain respectively. The Young's Modulus is given by the slope of each curve which would ideally be approximately constant:

Note: Upon closer inspection, the lack of a linear relationship for the 'True' Young's Modulus is to be expected if we consider how the ratio of true stress to true strain would vary over time:

Subject to a linearly increasing load, $E_{true}(k) = \frac{(k*F_0)/A}{log(l_k/l_0)} \approx \frac{(k*F_0)/(V/l_k)}{log(l_k/l_0)}=\frac{k*F_0}{V}*\frac{l_k}{log(l_k/l_0)}$ where $V$ and $l_0$ are constant. Clearly, this function of $k$ is strictly increasing so $E_{true}$ never stabilizes even if $C = \frac{k*F_0}{l_k}$ was a constant. On the other hand, $E_{eng}$ would be constant if $\frac{k*F_0}{l_k}$ was a constant.

Alternatively, if you analyze the ratio $\frac{E_{true}}{E_{eng}}$. You find that $\frac{E_{true}}{E_{eng}} \approx \frac{k*F_0}{V}*\frac{l_k}{log(l_k/l_0)}*\frac{A_0*l_k}{F_0*k*l_0}=\frac{l_k^2}{l_0^2*log(l_k/l_0)}$.

The above analysis would hold just as well for any other elastic material with a prismatic geometry.

*this is computed using the tensile strength of the material

• If your $E$ is dependent on the load $P$ then you've gone beyond the linear domain of the material (where Young's modulus is now undefined). Jul 5, 2016 at 16:03
• Unless the Poisson's ratio of your material is equal to zero the real area will be less than $A_0$ so the stress will be higher than you calculate. That's why your Young's modulus is decreasing with increasing strain. Speaking as a non-engineer I have absolutely no idea where the equation you cite in (2) comes from or why you'd expect it to give a sensible answer. Jul 5, 2016 at 16:04
• Or what John said... Jul 5, 2016 at 16:05
• @JohnRennie The equation in (2) is the true stress over the true strain: doitpoms.ac.uk/tlplib/thermal-expansion/young-mod-def.php
– user29305
Jul 5, 2016 at 16:15
• According to Wikipedia the Young's modulus is defined using the engineering strain in which case the deviation from a straight line is presumably because you haven't taken account of the area change due to a non-zero Poisson's ratio. Actually your graph of the modulus calculated using the engineering strain looks pretty darned straight to me. Jul 5, 2016 at 16:41

$A \approx \frac{W_0*L_0}{L}*\frac{T_0*L_0}{L}=\frac{W_0*T_0*L_0^2}{L^2}$

If my interpretation is correct, you are assuming that $W*L=W_0*L_0$ and $T*L=T_0*L_0$

That would make the volume: $W_0*T_0*L_0^2/L$, which decreases when you stretch the material. For small strain, the Poisson ratio would approach 1. Poisson ratio should be between -1 and 0.5 for a stable, isotropic, linear elastic material. I don't know what the material is you're testing, but..

$A \approx \frac{A_0*L_0}{L}$ seems more suitable imo.

• This is a better approximation but if you check the note I added at the bottom of the question details, it appears that the problem lies in the definition of the 'True' Young's Modulus.
– user29305
Jul 5, 2016 at 23:50

The results you get in the experiment depend on the dimensions of the sample. Are T and W much smaller than L? The shape of the sample is also important in another way. Is it a dog bone shape, with a short middle section length being used as the sample length?

If T and W are small compared to L, then the cross sectional area used in calculating the true stress has to take into account the Poisson contraction.