Define the rule of mixtures for the axial Young's modulus E1 in a unidirectional composite.

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Multiple Choice

Define the rule of mixtures for the axial Young's modulus E1 in a unidirectional composite.

Explanation:
The axial stiffness of a unidirectional composite under a load along the fibers follows a parallel (iso-strain) rule of mixtures: the strains in fiber and matrix are the same, so their moduli contribute additively weighted by how much of each phase is present. This gives E1 = E_f V_f + E_m V_m, where E_f and E_m are the fiber and matrix moduli and V_f and V_m are their volume fractions (with V_f + V_m = 1). If the fiber modulus is much larger than the matrix modulus, the composite stiffness is driven by the fiber content, but the exact value is a weighted average as shown. The other forms fail because they either ignore volume fractions, swap the moduli between phases, or multiply sums in a way that doesn’t reflect the parallel loading scenario.

The axial stiffness of a unidirectional composite under a load along the fibers follows a parallel (iso-strain) rule of mixtures: the strains in fiber and matrix are the same, so their moduli contribute additively weighted by how much of each phase is present. This gives E1 = E_f V_f + E_m V_m, where E_f and E_m are the fiber and matrix moduli and V_f and V_m are their volume fractions (with V_f + V_m = 1). If the fiber modulus is much larger than the matrix modulus, the composite stiffness is driven by the fiber content, but the exact value is a weighted average as shown. The other forms fail because they either ignore volume fractions, swap the moduli between phases, or multiply sums in a way that doesn’t reflect the parallel loading scenario.

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