Problem 2781. Rule of mixtures (composites) - reverse engineering

Created by goc3

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The <a href = "http://en.wikipedia.org/wiki/Rule_of_mixtures">rule of mixtures</a> is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:<ul><li>Ef: elastic modulus of the fiber material</li><li>Em: elastic modulus of the matrix material</li><li>ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)</li></ul>Suppose that you are reverse engineering a composite article. You've tested coupons from the article to determine Ec and taken sections to determine the material properties and volume fraction for the fiber material. Using the known values, calculate modulus estimates for the matrix material based on both weightings of upper- and lower-bounds. Provide values for (upper-bound) weightings of 0, 1/4, 1/2, 3/4, and 1 (weighting is not related to fiber fraction or material volume).The lower-bound estimate of elastic modulus is calculated by:Ec = 1 / (ff / Ef + (1 – ff) / Em) [eq.1]The upper-bound (linear) estimate of elastic modulus is calculated by:Ec = ff * Ef + (1 – ff) * Em [eq.2]For example, for each test case you will calculate Em_l, the lower-bound matrix elastic modulus extrapolated by eq.1, and Em_u, the upper-bound matrix elastic modulus extrapolated by eq.2. The weighted matrix elastic modulus is given by Em = wt * Em_u + (1-wt) * Em_l. For a weighting of 1/4, the weighted modulus is therefore given by Em = 0.25 * Em_u + 0.75 * Em_l.

The <http://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures> is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:

* Ef: elastic modulus of the fiber material
* Em: elastic modulus of the matrix material
* ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)

Suppose that you are reverse engineering a composite article. You've tested coupons from the article to determine Ec and taken sections to determine the material properties and volume fraction for the fiber material. Using the known values, calculate modulus estimates for the matrix material based on both weightings of upper- and lower-bounds. Provide values for (upper-bound) weightings of 0, 1/4, 1/2, 3/4, and 1 (weighting is not related to fiber fraction or material volume).

The lower-bound estimate of elastic modulus is calculated by:

Ec = 1 / (ff / Ef + (1 – ff) / Em)   [eq.1]

The upper-bound (linear) estimate of elastic modulus is calculated by:

Ec = ff * Ef + (1 – ff) * Em   [eq.2]

For example, for each test case you will calculate Em_l, the lower-bound matrix elastic modulus extrapolated by eq.1, and Em_u, the upper-bound matrix elastic modulus extrapolated by eq.2. The weighted matrix elastic modulus is given by Em = wt * Em_u + (1-wt) * Em_l. For a weighting of 1/4, the weighted modulus is therefore given by Em = 0.25 * Em_u + 0.75 * Em_l.

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Last Solution submitted on Feb 05, 2023

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Jan Orwat on 16 Dec 2014

Grant, in the test suite you expect the answer (Em) to be linearly spaced while it is not. for example, in first test wt=0.5 you check for Em=17.7193 which gives Ec=32.9652. Correct answer is closer to Em=20.25 (when wt=0.5) based on equations you provide. Also, do you really want so small precision? 10e-1 equals 1.

Jan Orwat on 16 Dec 2014

I would also like to comment on the tag "THESE_ARE_JUST_PATHETIC". I don't know who has written this, I don't care. I just feel it inappropriate.

goc3 on 16 Dec 2014

Jan, I've revised the explanation for this problem to be clearer. I understand the confusion, as this is one of many different ways to slice such problems. This problem is specifically asking for extrapolated values of Em using both bound equations and then using the (provided) equally spaced weightings to determine equally spaced modulus estimates. Realistically, such a problem would likely implement statistical distributions, rather than evenly spaced weightings, but I didn't want to make the problem too difficult. Also, I've revised the precision for this and the other problems to be much tighter.

goc3 on 16 Dec 2014

I don't quite understand your second comment regarding the tag. Could you elucidate what you intended?

Jan Orwat on 17 Dec 2014

I meant, that writing such tags like the one mentioned above doesn't explain anything to anyone, but may be offending. I think it is a good practice to say what we enjoy or dislike via comments and let others to discuss it. (I assume it was not your tag, Grant). Anyway, it disappeared

goc3 on 18 Dec 2014

It was definitely not my tag; fortunately it disappeared. Thanks for commenting on it though, as you are right; apparently you influenced the offender to take it down.

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Problem 2781. Rule of mixtures (composites) - reverse engineering

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