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I do not understand something about the "rule of mixtures". Specifically, between upper bound elastic modulus and lower bound elastic modulus calculations, I seriously do not understand what value to choose based on transverse or longitudinal loading and on the basis of iso-stress and iso-strain conditions. Some insight into making these decisions will certainly help me out! enter image description here

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The derivation is given in the same place that you obtained the graph (Wikipedia).

The simplest case is a parallel fiber composite.

Iso-strain conditions are assumed when you pull parallel to the fibers. The fibers and the matrix feel the same applied force, must have the same relative change in length but will carry different stress amounts because of their relative area distributions. If this condition is not met, you are proposing to have the fibers pull away (stretch a different relative length) compared to the matrix (or the matrix from the fibers).

Iso-stress conditions are assumed when you pull perpendicular to the fibers. The fibers and the matrix distribute the same stresses but feel different degrees of force because of the different relative area exposed and will therefore stretch differently from each other. If this condition is not met, you are proposing to a case with non-uniform stress distributions on the surface area perpendicular to the applied force.

The complex rules of mixtures takes an approach that true composites may be somewhere between these two rules.

$$E^n = \sum E_j^n $$

For the simple rule of mixtures, $n = 1$. For the inverse rule of mixtures, $n = -1$.

We do not make fiber composites with just any amount of fiber. At some point, a limit is reached where the matrix cannot "glue together" the given amount of the fiber. This is an upper bound of fiber volume fraction. If we have too little fiber, the system is not providing any significant benefit for its composition. This is a lower bounds. The lower bounds can be set by what your ultimate operation mode will be. Do you want the composite to be improved for loads parallel to the fibers or do you want it to not fail in a certain way for loads perpendicular to the fibers. Sometimes, the decision involves a mixture of both conditions, and both the simple rule of mixtures (upper bounds) and inverse rule of mixtures (lower bounds) on elastic modulus are important in the design.

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I think below linked Wiki page can help you to get start. You should also search the web for iso-stress, and iso-strain conditions of a composite material to gain better understanding.

https://en.wikipedia.org/wiki/Rule_of_mixtures

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