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

The 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.