Young's Modulus of Elasticity Problems -

[joe465]

Young's Modulus of Elasticity Problems - URGENT

Homework Statement

Calculate Young's Modulus of Elasticity for the three materials using the graphs only.

Stress (N/m2) Strain
70 0.06
100 0.11
150 0.22
200 0.35
220 0.48

Homework Equations

ε = Stress / Strain

The Attempt at a Solution

I believed that to calculate Young's Modulus was a case of stress over strain but surely it should give the same value throughout the elastic region to be a constant. The stress and strain results provided must be correct as they were already provided.

70 / 0.06 = 1166.66
100 / 0.11 = 909.09
150 / 0.22 = 681.82
200 / 0.35 = 571.43
220 / 0.48 = 458.33

So my main question is why are these so different, am i applying an incorrect formulae. Please note i was only using the formulae because i wanted to check i was getting it right and wasn't sure how to use the graphs to obtain the value.

How do a derive the youngs modulus from a Stress-Strain graph?

Please help as i only have 1 more day before this is due in.

Many thanks, Joe

 

[CWatters]

Not my field but I found this which suggests the curve isn't allways straight...

Presumably the Youngs modulus normally quoted is the slope at the bottom left in the elastic region.

Perhaps wait for other suggestions.

 

[CWatters]

Do you know what the material is? What units for strain?..

I ask because a youngs modulus of 1000 N/m² seems quite low..

For example this page quotes that of rubber as 0.01 * 10⁹ to 0.1 * 10⁹ N/m²

http://www.engineeringtoolbox.com/young-modulus-d_417.html

 

[joe465]

That is the lowest values that were pre-given, the other 2 materials are

Stress Strain E
50000000 0.00128 39062500000
100000000 0.0016 62500000000
150000000 0.00187 80213903743
200000000 0.00213 93896713615
250000000 0.00227 1.10132E+11
300000000 0.00253 1.18577E+11
350000000 0.00339 1.03245E+11
375000000 0.00427 87822014052
400000000 0.00707 56577086280Stress Strain E
100000000 0.00109 91743119266
200000000 0.00145 1.37931E+11
300000000 0.00182 1.64835E+11
400000000 0.00215 1.86047E+11
470000000 0.00236 1.99153E+11
460000000 0.00273 1.68498E+11
460000000 0.00291 1.58076E+11
500000000 0.00318 1.57233E+11
550000000 0.00382 1.43979E+11
600000000 0.00436 1.37615E+11
The idea of the assignment is to calculate the modulus of E and then research into what materials they could be. I'm going to presume the one i have already given would be something such as glass, which would explain why its so different.

I'm meant to calculate the E from the graphs that i have anyway, how do i go about this?

I thought modulus was meant to be a consistant number across the whole elastic region so why does it change??

Thanks

 

[SteamKing]

Not every material has a perfectly elastic region in its stress-strain curve.

 

[joe465]

So what value would i use for the elasticity since there so different, would i have to take an average?

 

[Chestermiller]

Show us what the stress-strain graphs look like. (Strain = abscissa, Strain = ordinate). Seeing the graphs by eyeball should tell a lot.

Chet

 

[SteamKing]

strain = abscissa, stress = ordinate

 

[joe465]

http://imageshack.us/f/194/materialx.jpg/
http://imageshack.us/f/38/materialy.jpg/
http://imageshack.us/f/209/materialz.jpg/

http://imageshack.us/f/194/materialx.jpg/
http://imageshack.us/f/38/materialy.jpg
http://imageshack.us/f/209/materialz.jpg

There are the three stress-strain graphs

 

[Chestermiller]

strain = abscissa, stress = ordinate

Hi. Thanks for the correction.

Chet

 

[Chestermiller]

http://imageshack.us/f/194/materialx.jpg/
http://imageshack.us/f/38/materialy.jpg/
http://imageshack.us/f/209/materialz.jpg/

http://imageshack.us/f/194/materialx.jpg/
http://imageshack.us/f/38/materialy.jpg
http://imageshack.us/f/209/materialz.jpg

There are the three stress-strain graphs

Please include the strain and stress 0,0 on these arithmetic scale plots. Also, maybe you can provide plots of log(stress) vs. log(strain) so we can see the behavior expanded at small strains (of course without the point 0,0 on the log-log plot). Please also show the data points on the plots.

These plots that you have shown so far are already very revealing.

 

[joe465]

Thankyou, my initial task was to calculate the CSA which i did at 0.00002m2. I was pre-given Load and Extension data for 2 materials and given Stress-Strain data for the third. I was tasked to calculate the stress-strain data for all three and then plot the relevant graphs.

Material X

TEST DATA FOR X
LOAD..EXT...CSA...Stress (F/A)..Strain...Orig Length E (Stress / strain)
1000..0.48...0.00002...50000000...0.00128...375...39062500000
2000..0.6...0.00002...100000000...0.0016.....375...62500000000
3000..0.7...0.00002...150000000...0.001866667...375...80357142857
4000..0.8...0.00002...200000000...0.002133333...375...93750000000
5000..0.85..0.00002...250000000..0.002266667...375...1.10294E+11
6000..0.95..0.00002...300000000..0.002533333...375...1.18421E+11
7000..1.27..0.00002...350000000..0.003386667...375...1.03346E+11
7500..1.6...0.00002...375000000..0.004266667...375...87890625000
8000..2.65..0.00002...400000000..0.007066667...375...56603773585
8500..5.5...0.00002...425000000..0.014666667...375...28977272727
8300..7...0.00002...415000000..0.018666667...375...22232142857
8000..7.9...0.00002...400000000..0.021066667...375...18987341772
7000..8.7...0.00002...350000000..0.0232...375...15086206897
6100..9.1...0.00002...305000000..0.024266667...375...12568681319

Material Y

TEST DATA FOR Y
LOAD...EXT...CSA... Stress...Strain...Orig Length...E (Stress / strain
2000...0.6...0.00002...100000000..0.001090909...550...91666666667
4000...0.8...0.00002...200000000..0.001454545...550...1.375E+11
6000...1...0.00002...300000000..0.001818182...550...1.65E+11
8000...1.18..0.00002...400000000..0.002145455...550...1.86441E+11
9400...1.3...0.00002...470000000..0.002363636...550...1.98846E+11
9200...1.5...0.00002...460000000..0.002727273...550...1.68667E+11
9200...1.6...0.00002...460000000..0.002909091...550...1.58125E+11
10000.1.75..0.00002...500000000..0.003181818...550...1.57143E+11
11000.2.1...0.00002...550000000..0.003818182...550...1.44048E+11
12000.2.4...0.00002...600000000..0.004363636...550...1.375E+11
14000.5...0.00002...700000000..0.009090909...550...77000000000
12000.7.2..0.00002...600000000..0.013090909...550...45833333333
10000.7.25.0.00002...500000000..0.013181818...550...37931034483
8000...7.27..0.00002...400000000..0.013218182...550...30261348006

For Material Z only the Stress-Strain data was given already

TEST DATA FOR Z
Stress...Strain...CSA......E (Stress / strain)
70... 0.06...0.00002...... 1166.666667
100... 0.11 ...0.00002......909.0909091
150... 0.22 ...0.00002......681.8181818
200... 0.35 ...0.00002......571.4285714
220... 0.48...0.00002......458.3333333

You have asked for 0,0 data but i was not given any data close to that amount. Should i still include this on the graphs and just start the line from thin air essentially.

Is my use of formulae correct. Should i change the scale data on my graphs?

 

[Chestermiller]

The point 0,0 is known to be an exact point on your graph. If there is no applied stress, then there is no strain, and vice versa. I asked for the results on a log-log plot because it will increase the resolution in the region of small strains where the stress-strain behavior is expected to be linear. On a log-log plot, the linear stress-strain region should have a slope of 45 degrees (assuming equal spacing of the decades in the horizontal and vertical axes). If you are using a graphics package, just change the scales from arithmetic to logarithmic. And if you are switching to log scales, make sure you omit the point 0,0.

Chet

 

[joe465]

So your suggesting the graph starts in mid air? I'm not sure what you mean but I'm confident i have to use arithmetic graphs and not log. Do i just use the gradient to determine Young's Modulus? So Change in Y axis / Change in X axis?

I have very little time left to get this done, i need to get a value and find the material.

Thanks, Joe

 

[Chestermiller]

So your suggesting the graph starts in mid air? I'm not sure what you mean but I'm confident i have to use arithmetic graphs and not log. Do i just use the gradient to determine Young's Modulus? So Change in Y axis / Change in X axis?

I have very little time left to get this done, i need to get a value and find the material.

Thanks, Joe

I'm not suggesting that the graph starts in mid air. I'm suggesting that there is an exact point on the graph at 0,0. How much was the strain before any stress was applied? Zero, right?

If you don't want to use log scales, then you need to zoom in on the data at small strains and stresses. To do this, omit the data points that are well beyond the linear region, and, by doing so, expand the scales on both axes. Then you will get a better eyeball view of what is happening at small strains.

Chet

 

[Chestermiller]

I plotted up your data for two of the cases, and, in both cases, the data is offset along the strain axis. Otherwise, the data is pretty linear in the region of small strains, and you should be able to get a pretty accurate determination of the Young's modulus. I don't know why your data is offset along the strain axis.

Chet

 

[joe465]

Thankyou, all the Load and extension data was provided. The area i believe to be correct and I'm confident my calculations are correct so any idea what to do now?

 

[Chestermiller]

Thankyou, all the Load and extension data was provided. The area i believe to be correct and I'm confident my calculations are correct so any idea what to do now?

The graphs are "screaming" that something is wrong with the strains. This must mean that something was wrong with the extension measurements. This must somehow be related to the zero point of the extension measurements.

 

[joe465]

I've double checked them and i can't see where its wrong. The extension was pre-given in mm and the Original length of sample was also pre-given in mm.

Strain = Change of Length / Original Length

For Material X

Original Sample Length = 375mm

Ext (mm) / Original Sample Length (mm)
0.48 / 375 = 0.00128Is this correct?

 

[Chestermiller]

I've double checked them and i can't see where its wrong. The extension was pre-given in mm and the Original length of sample was also pre-given in mm.

Strain = Change of Length / Original Length

For Material X

Original Sample Length = 375mm

Ext (mm) / Original Sample Length (mm)
0.48 / 375 = 0.00128Is this correct?

Yes, it looks correct. But, as best I can tell, the extensions look off (too large) by about 0.38mm (and the strains look off by about 0.001, at least for this case) . The only thing I can recommend is that you take the slope of the stress vs strain graph in the region of small strains. You will find that the data are fit very nicely by a straight line. The slope is what I would report as the Young's modulus. For this case, I would estimate the slope to be about 2 x 10¹¹.

 

[joe465]

Thanks, so to find the Young's Modulus i need to find the gradient of the slope being:

Gradient = Change in Y axis / Change in X or is it

Young's Modulus = Original length/(gradient * CSA)

Would you recommend me re-drawing the graph with using only a few pieces of data allowing for more accurate axis data?

 

[Chestermiller]

yes.

 

[joe465]

Which formula is right? Or are both used? There seems to be a few online suggesting different things.

Joe

 

[Chestermiller]

Which formula is right? Or are both used? There seems to be a few online suggesting different things.

Joe

Well, ordinarily, stress would be directly proportional to strain in the linear region, and you would put your best line through the origin to get the Young's modulus. Then, stress/strain would be the same as the slope. However, considering the problems with your data, your best shot would be to use the slope, assuming that the problems with the offset of the strains can somehow be explained at a later time.

Chet

 

[joe465]

Thanks, Does this mean that the gradient i.e Difference in Y / Difference in X would give me the correct answer and the second formula is not required?

Gradient = Difference in Y axis / Difference in X axis

Gradient = 150000000 – 100000000 / 0.00187 – 0.0016

Gradient = 185185185185.185 (3 dp)

Converting to GPa would be 185 being around the level for stainless steel or Tantalum which is at 186GPa??

 

[Chestermiller]

It would be better if you used all the data points, by drawing your best straight line through the data on a graph, and calculating the slope of the straight line. That way, no single data point would determine the entire outcome. After all, there is going to be experimental uncertainty in the individual data points.

 

[joe465]

I've used the gradiant that was already drawn and i got

Material X = 185.185 GPa
Material Y = 270.270 GPa
Material Z = 3.63636 × 10-7 GPa

Material X = Possibly Tantalum at 186
Material Y = Maybe Chromium at 279
Material Z = ?

Not sure about Material Z, doesn't seem like anything is even close.

How close would you say the Young's for that material would it have to be?

 

[Chestermiller]

I've used the gradiant that was already drawn and i got

Material X = 185.185 GPa
Material Y = 270.270 GPa
Material Z = 3.63636 × 10-7 GPa

Material X = Possibly Tantalum at 186
Material Y = Maybe Chromium at 279
Material Z = ?

Not sure about Material Z, doesn't seem like anything is even close.

How close would you say the Young's for that material would it have to be?

I don't know what to say about material Z. Those stresses seem awfully low, and the strains seem awfully high. Maybe some kind of very low modulus rubber.

 

[joe465]

Any ideas how i could prove which material they are from the Young's Modulus? Calculate anything else to back up the argument?

 

[Chestermiller]

Any ideas how i could prove which material they are from the Young's Modulus? Calculate anything else to back up the argument?

If you don't think your E values are accurate enough to identify the exact materials, why don't you just report the few materials you consider the most likely candidates.

 

[joe465]

Still having problems finding which materials they are. Material Y young's modulus is 270 GPa and looking at the stress strain graph it has a UTS of 700 MPa. I can't find a material which matches this, please help. Chromium has a Young's Modulus of 279 GPa but i can't find the UTS anywhere.

 

[Chestermiller]

Still having problems finding which materials they are. Material Y young's modulus is 270 GPa and looking at the stress strain graph it has a UTS of 700 MPa. I can't find a material which matches this, please help. Chromium has a Young's Modulus of 279 GPa but i can't find the UTS anywhere.

Steel looks like a possibility.

 

[joe465]

Yeah i went for steel because the stress strain shows two yield points which is a common steel reflection. Now just material Z. My tutor said it should be between 500-600 Pascals but i can't get this value at all no matter which units i use?

Stress Strain
70 0.06
100 0.11
150 0.22
200 0.35
220 0.48

Thats material Z's Stress-Strain Data.

 

[Chestermiller]

I fit a quadratic to these points, and got ~700 Pa at ~0 Pa stress. That should be good enough. If the units are really Pascals, I don't know of any materials with that low a Young's modulus.