Material Science, Material Selection Question

[mathwurkz]

I have a problem with this homework. Please have a look at my work and see if it checks out.
ROCKET FUEL TANK
A company has asked you to help their business in space tourism. They have designed a rocket that will be powered by nitrous oxide (reacted with rubber), and you are to select materials for the fuel tank (pressure vessel).
a) Considering the flight will be at high altitude (the edge of space), suggest materials that will have a good fracture toughness low temperature. THe M1 index should consider brittle failure under a displacement-limited design. Find materials with M1 > 10.
b) To increase passenger safety and spacecraft performance, consider M2 a second material index for energy-limited failure under fracture at minimum mass. Hint: Recall that toughness is the integral of the stress-strain curve (energy), and find an index that will maximize fracture toughness.
M2 > 0.25 required.
We are given 4 formulas.
$$\sigma = \frac{S_f pR}{2t}$$, $$\epsilon= \frac{\sigma}{E}$$, $$K_{IC}= C\sigma _f \sqrt{\pi a}$$, $$U= \frac{1 \sigma}{2 E}$$
For part a), this is how I solved it.
$$\sigma = \frac{K_{IC}}{C \sqrt{\pi a)}}$$, divide both sides by E., we know, $$\frac{\sigma}{E} = \frac{\Delta l}{l}$$, so substituting gives me, $$\Delta l = \frac{l K_{IC}}{CE \sqrt{\pi a}}$$
all in all...
$$\Delta l = \left[ \frac{1}{C \sqrt{\pi a}}\right] \left[l\right]\left[ \frac{K_{IC}}{E}\right]$$ so...
$$M_1 = \frac{K_{IC}}{E}$$ to maximize the fracture toughness, under displacement-limited design.
b) $$U = \frac{1\sigma ^2}{2E} = \frac{1}{2E} \frac{K_{IC}^2}{C \sqrt{\pi a}}$$
so i get...
$$M_2 = \frac{K_{IC}^2}{E}$$
The problem is when I go to the CES materials selection program. The index I have for part a ends up dominating and gives me a small list of materials. The problem is the next part of the problem tells us that parts a and b should have made our selections come down to 2 composites and 2 polymers. But what I end up getting from the program are just polymers. All the composites are below the slope line on the materials selection chart. So at this point, I am thinking that I must have misinterpreted the question wrong, got a wrong material index, or maybe the program is all buggy. If anyone can check what I am doing or even can tell me if my indicies do yield 2 composites and 2 polymers i'd appreciate it. THanks.

 

[PerennialII]

Don't have CES available on this machine, but somehow for the 2nd index I would've used (intuitively before reading it completely) the integral of the stress-strain curve itself, but your question does state to go with fracture toughness on this as well (and if you've linear-elasticity going there would suggest even more so ... although fracture toughness will be maximized also if you just maximize the area under the stress-strain curve). So would've thought it reasonable to maximize fracture toughness and energy the structure can take without connecting the two.
Probably doesn't change anything but thought I'd mention if you want to try it as a criterion, but don't see anything 'wrong' in the way you've interpreted the question itself. Puzzling

 

[piercas]

it is important to carefully analyze and interpret data to ensure accurate results. In this case, it seems that the material index you have calculated for part a (M1 = K_IC/E) may not be the most appropriate for the given problem. It is possible that using this index alone may not accurately reflect the fracture toughness at low temperatures. It may be helpful to also consider other factors such as the material's ductility and ability to withstand thermal stresses at high altitudes.

Additionally, for part b, the material index you have calculated (M2 = K_IC^2/E) may not accurately reflect the energy-limited failure under fracture at minimum mass. It may be beneficial to consider other factors such as the material's strength and weight to find a more suitable index for this situation.

In order to ensure the best material selection for the rocket fuel tank, it may be helpful to consult with other experts in the field and use multiple material selection tools to compare and validate results. It is also important to carefully review the input data and assumptions used in the material selection program to ensure accurate and reliable results.