Dataset: Stang Aluminum Alloys

### Overview

In 1946 Stang, Greenspan, and Newman published data on structural alloys under high-strain conditions. We'll analyze the aluminum data.

### Observations

1. ↑ Stang, Greenspan, and Newman, "Poisson's ratio of some structural alloys for large strains" (1946) National Bureau of Standards [1]

## Team Alpha

* While Young's modulus (E) and shear modulus (mu) are supposed to be intensive properties that don't vary with thickness, the thickest sample (thick = 0.081) does seem to have significantly lower values of E and mu compared to the samples with the other thicknesses.

1. ↑ q4 graph

## Team Beta

* As the material is rolled thinner, it seems to be more elastic and have a higher poisson's ratio. This could be because of the larger grains due to the thinner rolling.
	+ https://www.the-warren.org/ALevelRevision/engineering/grainstructure.htm
* Q4: Does the [provided] graph support or contradict the claim above?
	+ (Weakly) contradicts. Based on the data, `E` and `mu` vary with thickness.
	+ Hypothesis: These material properties are better described by probability distributions than point estimates, and the oft reported values are merely statistics summarizing the distribution (e.g., the population mean).

## Team Gamma

* 0.081 inch samples had different Poisson ratio and tensile modulus that the other samples. This may be to some processing technique that affects the surface of the samples or there may be another hidden variable with the 0.081" group.

* There does not seem to be a linear relation between thickness and these properties based on the four thicknesses tested

* 45 degree cut samples may have some difference in Poisson ratio but not in modulus

* There are very few samples, it is hard to draw reliable conclusions from the dataset, and using box plots and density plots can lead to overconfidence.

Presentation on this data given in class

## Team Delta

* Seems like there is a Non-linear impact of thickness on E and mu.
* Lower E and Mu at 0.081, that seems to be a distinct group from the other measurements.
* Angle doesn't seem to have an impact. Why not?
	+ The applied force direction not a big impact.
* Are E and Mu proportional or have a linear relationship? Based on the E vs. mu graph, it seems possible.

## Team Epsilon

* E varies with thickness. [2]
* mu varies with thickness. https://github.com/branish/data-science-work/blob/master/c03-stang-assignment\_files/figure-gfm/unnamed-chunk-2-1.png
* E and mu appear proportional to each other.
* E is not affected by angle. https://github.com/branish/data-science-work/blob/master/c03-stang-assignment\_files/figure-gfm/unnamed-chunk-1-1.png
* mu is not affected by angle. https://github.com/branish/data-science-work/blob/master/c03-stang-assignment\_files/figure-gfm/unnamed-chunk-3-1.png

[3]

## Team Zeta

### **Thickness affects Modulus of Elasticity and Poisson’s ratio**

**Simple definitions for complicated terms:**

*Young’s modulus/Elasticity:* How stiff is this material?

*Poisson’s ratio:* How much does this squeeze as it is stretching?

*Intensive property:* Do the properties of this material change if there is more or less of the material?

**What is rolling?**

Squeezing a metal sheet thru two rollers (usually with heat) so it becomes thinner and uniform.

**Why do we care?**

We use rolled aluminum for high risk stuff like planes and cars.

### **Thickness affects Modulus of Elasticity and Poisson’s ratio**

The thickest aluminum (0.081") has the lowest modulus of elasticity (10,100 ksi or less), whereas all of the thinner pieces of aluminum have elasticity values greater than 10,250 ksi, especially the thinnest aluminum (0.022"), which has the highest modulus of elasticity (10,500 ksi or greater). Similarly, the thickest aluminum (0.081”) has the lowest median value for Poisson’s ratio (less than 0.31625).

### **Angle does not affect Modulus of Elasticity but does affect Poisson’s Ratio**

When comparing the Modulus of Elasticity against thickness, breaking down the values by angle, there are no distinct patterns that exist within each angle “bucket” or across angles.

The shape of the plot is consistent with a material that is homogenous orthogonal anisotropic (meaning that the material has different properties according to orientation). The frequently quoted values for Young’s modulus and Poisson’s ratio assume an isotropic material.

### **Modulus of Elasticity and Poisson’s Ratio *should* be intensive material properties.**

Though the data would indicate the contrary, as explored in the previous two points, we believe that this data has been impacted by manufacturing variability. When look at the measurement data in terms of sample thickness, we can specifically see that the various test thicknesses remain relatively closely grouped among themselves, suggesting that the source material may have some affect on the results.

This concept is further supported by the work done by Dr. Zachary del Rosario  on the subject of rigorous design margins and his design allowables framework, to precisely understand the variability in material properties for even well defined alloys, and how to accommodate that variability in design.

1. ↑ 'Modulus of Elasticity and Thickness'
2. ↑ 'Poisson’s Ratio and Thickness'
3. ↑ 'Young’s modulus by Thickness (facet by angle)'
4. ↑ 'Angle vs Poisson’s Ratio'
5. ↑ 'Grouping as a Function of Sample Source'
6. ↑ 'Cutting the Double Loop'
7. ↑ 'Margin as Model'