In both the isostrain condition and the isostress condition the modulus of elasticity is determined differently. First we will take a look at the isostrain condition where the composite is loaded parallel to the direction of the fibres. In this case the fibres and matrix are strained the same amount.
The isostrain condition can be represented by an equation where ε = strain. Subscripts: c = composite, f = fibres, and m = matrix.
Using the definition of Young's modulus, the equation can be rewritten to take into account stress, σ, and modulus, E, terms.
In the isostrain condition the stress is distributed in the composite according to the rule of mixtures where v = volume fraction.
We then substitute for stress using the definition of modulus.
In the isostrain condition the strains are equal.
We can then cancel the strain terms.
This ultimately leaves us with an equation for modulus.
This shows that when a composite is loaded parallel to the direction of the fibres then the modulus of elasticity follows the rule of mixtures and is simply a weighted average of the modulus of the fibres and matrix.