In iso-strain (Voigt) and iso-stress (Reuss) models, what are the standard expressions for the longitudinal modulus E1 and the transverse modulus E2 using Vf and Vm?

• A. E1 ≈ Vf Ef + Vm Em; 1/E2 ≈ Vf/Em + Vm/Ef
• B. E1 ≈ Vf Ef + Vm Em; 1/E2 ≈ Vf/Ef + Vm/Em
• C. E1 ≈ Ef^Vf + Em^Vm; 1/E2 ≈ Ef/Vf + Em/Vm
• D. E1 ≈ Vf Ef - Vm Em; 1/E2 ≈ Vf/Ef - Vm/Em

Answer: (B)

This question tests how the iso-strain and iso-stress assumptions lead to the two classic bounds for a unidirectional composite’s stiffness. When loading along the fiber direction, the strain is the same in both fiber and matrix (iso-strain). That means the stiffness just adds in a weighted way, giving the longitudinal modulus as E1 = Vf Ef + Vm Em.

When loading perpendicular to the fibers, the stress is the same in both constituents (iso-stress). The overall response comes from the sum of the individual compliances, so the reciprocal of the transverse modulus is 1/E2 = Vf/ Ef + Vm/ Em, i.e., E2 = 1 / (Vf/ Ef + Vm/ Em).

These expressions use the fiber and matrix moduli (Ef and Em) and their volume fractions (Vf and Vm). The other forms would not follow from the equal-strain or equal-stress assumptions, so they don’t represent the standard iso-strain or iso-stress predictions.